# Incorporating the Substrate Effect in the Polariton Dispersion of a Waveguided Nanocavity

Jung Hoon Song and Jang-Won Kang∗

## Abstract

The waveguide mode dispersion of hexagonal-cylinder-shaped 1D ZnO nanostructures has
been derived by assuming the shape to be a circular cylinder; this assumption does
not clearly reflect the strong exciton–photon coupling effect in the nanostructure.
We report a rational pathway to produce an exciton–polariton dispersion in the waveguide
of hexagonal-cylinder-shaped ZnO nanorods using the coupled oscillator model (COM).
The multiple peaks in the photoluminescence emissions from the nanorod ends were considered
as lower polariton (LP) eigenmodes with energy and momentum spacings following the
*E-k _{z}* relation. To calculate the dispersion curve that satisfies the LP eigenmodes, the
effective mode dispersion, as a photon contribution of COM, was calculated using the
finite-difference time-domain (FDTD) method. The calculation considered the actual
experimental circumstances, namely, a hexagonal-cylinder-shaped nanocavity lying on
a SiO

_{2}/Si substrate. The FDTD results show that a relative loss of the mode profile may occur due to the dielectric surroundings, such as the SiO

_{2}substrate. Calculating the effective mode dispersion is a more realistic approach to study the polariton feature in a hexagonal-cylinder-shaped waveguide than the ideal cylindrical waveguide theory, which excludes the substrate effect. This study presents a realistic route for calculating the

*E-k*dispersion in a ZnO nanocavity.

_{z}**Keywords:**Waveguide, ZnO nanorod, Exciton–polariton, Polariton dispersion

## 1. Introduction

One-dimensional (1D) semiconductor nanostructures, including nanowires and nanorods, are promising building blocks for the production of novel miniaturized optical devices, such as waveguides and lasers [13]. The end facets of an elongated 1D nanostructure can produce multiple mirror reflections owing to the difference in the refractive indices of the semiconductor and air, which leads to Fabry– Perot (FP) resonance along the long axis of the nanostructure [2,3]. In light–matter interactions, the optical resonance along the long axis can strongly interact with an exciton, corresponding to the matter component, and enables the observation of exciton–polariton phenomena in 1D nanostructures [4,5]. Because semiconductors with large exciton binding energies, such as ZnO, enable the realization of strong exciton–photon coupling at room temperature, ZnO 1D nanostructures have been widely used to study polariton features in 1D optical cavities [5,6]. The strong exciton–photon coupling can be characterized by the anti-crossing of the lower and upper branches on the energy-propagation vector dispersion. The coupling strength, known as the Rabi energy, is determined by the splitting energy between the lower and upper branches at zero detuning. The dispersion of the exciton–polariton in a planar microcavity can be readily observed using angle-resolved photoluminescence (PL) or reflectivity mapping. However, in a 1D optical nanocavity, owing to the measurement configuration of the 1D nanostructure lying on the substrate, the light diffracted from the two end facets is superimposed, resulting in interference. The interference of emissions from the two ends prevents the observation of the dispersive feature of polaritons, even if the anglemapping method is used for the 1D nanocavity [7]. Thus, instead of directly observing the dispersion by angle-resolved mapping, the polariton dispersion for a 1D nanocavity was analytically calculated by matching the dispersion with the multiple eigenmodes (spectral peaks) of near-band-edge emission coupled with the FP resonance [4,8], which is known as waveguide polariton dispersion.

Waveguided polariton dispersion has been widely investigated to explain the strong
exciton–photon coupling effect of a variety of semiconductor nanostructures, such
as CdS nanowires [4,8], ZnO nanowires/ nanorods [2,6], and perovskite nanoribbons [9,10]. Typically, as the photon part in the two coupled oscillators, the photon dispersion
for the waveguide mode is calculated based on the geometry of the nanocavity, which
could, for example, have a cylindrical or rectangular shape [4,810]. In particular, in ZnO nanostructures, even though the 1D ZnO nanostructures have
the shape of a hexagonal cylinder, the waveguide mode dispersion has been derived
by assuming the shape to be a circular cylinder [6]. However, this assumption does not reflect the experimental conditions under which
the hexagonal cylindrical nanocavity exists on the SiO_{2}/Si substrate. Accordingly, considering the experimental situation, a realistic approach
is required to produce polariton dispersion in the hexagonal ZnO nanocavity based
on a calculation of the appropriate photon mode dispersion.

In this work, we report a rational route to produce waveguided polariton dispersion
in a ZnO nanorod with a hexagonal cylindrical shape. Importantly, we show that the
dielectric surroundings, such as the SiO_{2}/Si substrate, of the ZnO nanorod can strongly affect the field profile and photon
(or waveguide mode) dispersion arising from the optical loss path into the SiO_{2} layer. In addition, we emphasize that the calculated photon dispersion, which considers
the dielectric contribution of the substrate, provides a more realistic pathway for
discussing the polariton dispersion and strong coupling effect in the waveguide nanostructure.

## 2. Experimental details

The ZnO nanorods were grown by immersing a photoresist-patterned substrate with a
ZnO (100 nm) thin film on a sapphire substrate in an aqueous solution of zinc nitrate
hydrate (Aldrich) and hexamethylenetetramine (Aldrich) at 95 °C [11]. The diameter and length of the nanorods could be controlled by adjusting the synthesis
time and molar concentration of the additives. Subsequently, a post-annealing process
was used to improve the crystal quality of ZnO in a nitrogen atmosphere at 500 °C
for 30 min. For optical measurements, the ZnO nanorods were directly dry-transferred
onto the SiO_{2}/Si substrate. The PL measurement was performed using a custom-built microscope equipped
with a CW 325-nm He-Cd laser as an excitation source. The emissions from the body
and ends of the nanorods were detected by controlling the position of the optical
fiber entrance on the image plane of the microscope. Numerical calculations were performed
using commercial calculation software (Lumerical FDTD and MATLAB).

## 3. Results and discussion

### 3.1. Optical properties

Figure 1(a) shows a schematic of the configuration used for PL measurements and an explanation
of the orientation of the hexagonal ZnO nanorods on the SiO_{2}/Si substrate. Although the excitation laser beam was focused on the body region of
the nanorod, the spatially resolved PL measurement technique allowed the detection
position to be selected. The inset of Fig. 1(a) shows a scanning electron microscopy image of a single ZnO nanorod with the shape
of a hexagonal cylinder on the SiO_{2}/Si substrate. The diameter and length of the nanorods lying on the SiO_{2} layer were 560 nm and 2.36 µm, respectively. The room-temperature PL spectrum of
the body region is shown in Fig. 1(b), indicating that emission from the body is dominated by the near-band edge transition
of ZnO at ~3.26 eV, which is consistent with the typical PL feature of ZnO. In contrast,
in Fig. 1(c), the emission from the end of the nanorod is excitonic with multiple peaks that result
from the longitudinal FP resonance along the length direction of the nanorod, which
is a typical feature of the polariton effect in waveguide-type nanocavities [4,6]. The multiple peaks from the end emission are typically considered as eigenmodes
of the lower polaritons (LP) with mode spacing, arising due to the dependence of the
coupling of the photons of FP modes on the length of the nanorod (i.e., *L _{z}*: the cavity length). In waveguide-type nanocavities, the mode spacing is inversely
proportional to the cavity length.

where ℏ is the Planck constant, *c* is the speed of light, and *n* is the refractive index. Accordingly, the energies of multiple eigenmodes from the
end emission, which can be defined as the corresponding energy of the peak position
by Lorentzian fitting, as shown in Fig. 1(c), can be distributed with equidistant π/*L _{z}* intervals on the propagation vector (

*k*) [4,8].

_{z}### 3.2. Numerical calculations

The features of the polariton in the optical cavity of the nanorod were studied by
calculating the field profile for the resonance mode using the finite-difference time-domain
(FDTD) method. The polarization of excitons in ZnO was reflected by setting the polarization
of the point dipole source with a wavelength range of 350–450 nm along the direction
perpendicular to the *c*-axis of the nanorod. In all calculations, perfectly matched layers were used as the
simulation boundary. In addition, the FP resonance behavior along the longitudinal
direction of the nanorods was investigated by recording the field intensity spectrum.
Figures 2(a) and 2(b) present the cross-sectional field profiles at the resonance wavelength of ~390 nm
to show the features of FP resonance attributable to the end reflection. In addition,
the mode profile on the hexagonal plane corresponds to the HE_{22} mode of a cylindrical waveguide [6]. Importantly, in Fig. 2(b), the field profile along the long axis of the nanorod shows the optical loss in the
SiO_{2} layer. These results suggest that the hexagonal nanorods lying on the SiO_{2} layer can undergo optical loss owing to their dielectric surroundings, resulting
in a change in the effective mode index and dispersion. To clarify this problem, we
used cylindrical waveguide theory to calculate the HE_{22} mode dispersion using the following equation for the HE_{vm} and EH_{vm} modes [6,8].

where the subscripts *v* and *m* denote the order and *m*-th root, respectively, *J* is the Bessel function of the first kind, and *K* is the modified Bessel function of the second kind.
*r* is the radius of the nanorod, *k _{0}* is the freespace wavevector,

*k*is the propagation wave vector along the long axis of the nanorod, and λ is the wavelength in free space. In this calculation, the refractive index of ZnO (

_{z}*n*

_{ZnO}) was obtained from the dielectric function in [12], including the longitudinal-transverse splitting energy from the two excitonic transitions (that is, the A and B excitons), and background dielectric function of bulk ZnO. To exclude the issue of large excitonic damping (or thermal broadening) at room temperature, dielectric constants below the ZnO bandgap were used in the calculation. Figure 2(c) shows the calculated HE

_{22}mode dispersion using Eq. (2) of the cylindrical waveguide theory, indicating that the calculated HE

_{22}mode based on the cylindrical waveguide theory is quite different from the effective mode dispersion obtained by the FDTD method. When considering the optical loss in the SiO

_{2}layer, the cylindrical waveguide theory is not sufficient to calculate the photon dispersion of the FP resonance mode in a hexagonal nanorod. Thus, rather than using the photon dispersion from the cylindrical waveguide theory, the effective mode dispersion calculated by FDTD is more realistic for studying the polariton features when considering the experimental situation.

### 3.3. Polariton dispersion curve

The strong coupling effect between the photon and exciton resonances can be characterized
by the anti-crossing behavior of the dispersion curve, which is split by the Rabi
energy (ℏΩ_{R}) and consists of lower and upper polariton (UP) branches [13,14]. Owing to the rapid relaxation of the UPs, the emission in the polariton system
mainly originates from the LPs. In particular, in waveguide-type nanocavities, it
is known that the polariton dispersion curve and strong coupling effect can be interpreted
by LP eigenmodes with a spacing of π/*L _{z}* on the

*k*[5,6], as shown in Fig. 1(c). The energies (

_{z}*E*) of the LP eigenmodes on

*k*are plotted on a dispersive curve along with the mode spacing, as shown in Fig. 3. To fit this behavior with the polariton theoretical model, the

_{z}*E*-

*k*dispersion can be analytically calculated using the coupled oscillator model (COM) as follows [14]:

_{z}where *E*_{ex} and *E*_{ph} are the exciton and photon energies, respectively, and γ_{ex} and γ_{ph} are the half-width at half-maximum values of the exciton and photon, respectively.
Further, α and β are eigenvectors, also known as Hopfield coefficients and satisfy
*g* is the coupling strength and is used as the fitting parameter for the analytical
calculation. The eigenvalues *E* represent the energies of the polariton modes, leading to the equation

where the Rabi energy
*E*_{ph} by directly substituting the values of *E*_{ph} in the above equation that expresses the eigenvalue *E*. Adjusting the coupling strength *g*, the calculated curve can fit the LP eigenmodes measured by the PL spectrum. Finally,
using ℏΩ_{R} ∼ 2*g* (*g*~88 meV), the Rabi energy can be estimated to be ~176 meV.

The Rabi energy and coupling strength are as large as the values reported for the ZnO microwire cavity and planar microcavity [15,16], indicating that our calculation is valid for the ZnO nanocavity. Because the photon dispersion from the effective mode calculation takes into consideration the actual experimental situation, we consider this polariton dispersion to be more suitable for understanding the strong coupling effect in the 1D nanocavity than the photon dispersion based on the ideal-cylinder-type waveguide that was previously used.

## 4. Conclusions

The LP eigenmodes of ZnO nanorods with the shape of a hexagonal cylinder were defined
by measuring the PL emission from the ends of the nanorods. To construct the exciton–polariton
dispersion, the effective mode dispersion was calculated as a photon contribution
of COM by using the FDTD method, taking into account the actual experimental environment,
that is, a nanocavity with the shape of a hexagonal cylinder lying on the SiO_{2}/Si substrate. The effective mode dispersion allowed us to calculate the *E*-*k _{z}* dispersion from the COM. The calculated

*E*-

*k*dispersion, which considers the dielectric contribution of the substrate, provides a more realistic and rational route for studying the strong exciton–photon coupling effect in the waveguide nanostructure.

_{z}## Article information

###### Articles from Applied Science and Convergence Technology are provided here courtesy of **Applied Science and Convergence Technology**

## References

- Das A., Heo J., Bayraktaroglu A., Guo W., Ng T.-K., Phillips J., Ooi B. S., and Bhattacharya P.. Array 2012;20.