## Abstract

We present an efficient optical model to study spontaneous emission in a
cylindrically layered nanostructure. The total emission power of an emitter in the
nanostructure is efficiently calculated. A formula is derived to calculate the
lateral-surface emission power. As examples of practical interest, spontaneous
emission properties, including radiative transition rate of the emitter, the
assignment of the emission to lateralsurface emission and waveguided emission, are
comprehensively studied at the first time for an isolated ZnO nanowire and a
ZnO/SiO_{2} nanocable.

© 2007 Optical Society of America

## 1. Introduction

One-dimensional semiconductor nanostructures, such as nanowires (NWs) [1–2], core/shell nanocables (NCs) [3–5] and nanotubes (NTs) [6], have been under intensive research in recent years owing to their impressive functionalities in building up novel nanoscale photonic elements such as light-emitting diodes (LEDs), lasers and biosensors. These nanostructures can be approximately considered as cylindrically layered structures and their optical properties depend on the structural parameters. For example, the quantum efficiency of a NW LED should be different from that of a planar multilayered thin film LED since radiative transition rates of the emitter in these structures are distinct [7–9]. In addition, the assignment of the total emission power to the lateral-surface emission and the waveguided emission (propagating along longitudinal axis of the nanostructure) also depends on the size of the nanostructure. These properties are critical to NWs or NCs for LED and fluorescence labeling applications. In this paper these issues will be systematically studied.

The theoretical description of spontaneous emission (SE) in a cylindrically layered
nanostructure is based on the classical electromagnetics with the emitter modeled as an
incoherent electric dipole (point source) running at a constant current [8]. The radiative transition rate of the quantum
emitter and its lateral-surface emission efficiency can be characterized through the
total emission power and lateral-surface emission power radiated by the electric dipole
[8,9].
Spontaneous emission and the response of a classical dipole source in cylindrical
structures have been studied for decades from various angles by using different methods
[10–16], including cylindrical wave decomposition method [10,13,14], a combination of Fourier integral and multipole methods [15] and the three-dimensional finite difference time
domain (3D-FDTD) method [16]. However, the
methods in refs.[10,13–15] need special
treatments for the waveguide modes in the cylindrical structure and are inefficient for
the evaluation of the total emission power. The 3D-FDTD method is also inefficient. In
this paper, we use a simple technique without any special treatment for the waveguide
modes to calculate the total emission power and give an explicit formula to calculate
the lateral-surface emission power. The theory will be given in Section 2. SE
properties, including the radiative transition rate and the assignment of the emission
to lateral-surface emission, are studied in Section 3 for an isolated ZnO NW and a
ZnO/SiO_{2} NC. Conclusion is addressed in Section 4.

## 2. Theory

The geometry of a cylindrically multilayered nanostructure is shown in Fig. 1. An emitting layer is sandwiched between two
stacks of shells, i.e., *N* outer shells and *M* inner
shells. The emitting medium and the outer most medium (e.g. air) are assumed to be
non-absorbing at the emitting wavelength while the other shells can be either
transparent or absorptive. The quantum emitter is modeled as an incoherent classical
electric dipole (point source) with a constant current. The orientation of the electric
dipole is parallel to the electric field of the excitation in photoluminance (PL) [17, 18] and
is random in space in electroluminance. The SE properties of the emitter can be
characterized by the total emission power *F*, the lateralsurface
emission power *U* and waveguided emission power *W* (all
are normalized by the total emission power of the dipole in infinite medium) of the
dipole in the cylindrically layered nanostructure. If the materials are all lossless,
one has *F* =*U* + *W*. As a consequence of
Fermi’s golden rule, the radiative transition rate is optical-environment dependent as
[8,9]

where ^{0}
* _{r}* Γ and Γ

_{r}are the radiative transition rate in the infinite medium and the cylindrically layered media, respectively. Assuming that the nonradiative transition rate in the cylindrically layered media is Γ

_{nr}, the internal quantum efficiency

*η*, lateral-surface emission efficiency

_{F}*η*

_{U}and waveguided emission efficiency *η _{W}* are obtained as

where *η*
_{0}=Γ^{0}
* _{r}*/(Γ

^{0}

*+Γ*

_{r}*) is the initial internal quantum efficiency. In the cylindrically layered media, the electric field induced by an electric dipole can be written as*

_{nr}where *E*
^{0} and *E ^{r}* are the electric field of the dipole in the infinite medium and the reflected
electric field by the other shells, respectively. The total emission power can be
obtained as [8]

where *µ*, *ω*, *k*, Θ,
*α*⃑,and *r*⃑′ are the permeability, angular frequency,
wavenumber in the emitting medium, dipole moment, orientation and location of the
dipole, respectively. Here Im () stands for the imaginary part of () and the superscript
of *F* denotes the orientation. To calculate *F* and
*U*, we decompose the z-component of *E ^{0}* and

*H*(magnetic field of the dipole in the infinite medium) in terms of cylindrical waves in polar coordinates (

^{0}*ρ*,

*θ*,

*z*) [10,19]

Here $\epsilon ,{k}_{\rho}=\sqrt{{k}^{2}-{k}_{z}^{2}}$ and *k _{z}* are the permittivity, the radical and z components of the wavenumber,
respectively.

*J*and

_{v}*H*

^{(1)}

*denote the Bessel function and Hankel function of first kind at order*

_{v}*v*. The two stacks of shells are considered as two “black” shells characterized by the total downward reflection matrix

*N*

_{0v}, the total upward reflection matrix

*M*

_{0v}and the total transmission matrix

*T*

_{Nv}. Then the z component of the reflected field in the emitting layer and the transmitted field in the outermost medium can be obtained as

The field transverse to the z-direction can be readily obtained from the z component of
the field. For the detailed derivations of *N*
_{0v}, *M*
_{0v} and *T*
_{Nv}, we refer the readers to ref. [10].

#### 2.1 Calculation of the total emission power F

As seen from Eqs. (6) and (9), the calculation of
*F* involves the evaluation of an integral which has pole
singularities, physically corresponding to the guided modes in the cylindrically
layered structure. This is similar to the case of a planar multilayered structure
[20] and a direct evaluation of the
integral in Eq. (9) tends to fail.
These poles are located in the real axis of *k*
_{z} or in the first quadrant of the complex plane of *k*
_{z}. Here, we extend the integration domain in Eq. (9) to the complex plane of *k*
_{z} and use the Cauchy integral theorem to perform the integration. We
select an integration path in the fourth quadrant such that the domain enclosed by
the new path and the real axis of *k*
_{z} does not contain any pole. The new path shown as the dashed line in
Fig. 2 consists of a semicircle in the
fourth quadrant of the complex plane and a straight line along the real axis. The
integral in Eq. (9) can be written
as

where *k _{r}* is the radius of the semicircle

*C*.

_{R}*k*should be large enough so that the semicircle can bypass all the poles of the integrand. In this way, the integration is efficiently evaluated.

_{r}#### 2.2 Calculation of the lateral-surface radiation power

The normalized lateral-surface emission power *U* is the power leaving
the cylindrically layered structure from the lateral surface. Thus it can be
expressed as

where ${L}_{0}=\sqrt{\mu \u2044{\epsilon}_{\mathrm{out}}}{k}_{\mathrm{out}}^{2}\u204448{\pi}^{3}$is the normalization constant and *Re* () stands for
the real part of (). Here *ε _{out}* and

*k*are the permittivity and wavenumber of the outermost medium, respectively. After some algebra manipulation, Eq.(13) can be written as

_{out}If the outermost medium is lossless, *U* will not depend on
*ρ*.

## 3. Results and Discussion

In this section, the radiative transition rate, the lateral-surface emission efficiency
*η _{U}* and the waveguided emission efficiency

*η*are studied for an isolated ZnO NW and a ZnO/SiO

_{W}_{2}NC at a wavelength of 556 nm, i.e. the PL peak of a broadband ZnO NW LED [2]. The crosssection of the ZnO NW, which is actually hexagonal, is treated approximately as circular here. In our simulation, an initial internal quantum efficiency

*η*

_{0}=0.85 is used according to ref. [21]. The refractive indices of 2.45 and 1.46 are used for ZnO and silica, respectively. Below, the normalized total emission powers of the emitter orienting along the radical, azimuthal and z directions are denoted as

*F*,

^{ρ}*F*and

^{θ}*F*, respectively. Similar definitions also apply for

^{z}*U*.

#### 3.1 SE of an isolated ZnO NW

Figure 3(a) shows the dependences of
*F ^{x}* and

*U*on the radius of the ZnO NW for the emitter located at origin of the NW and with orientation

^{x}*x*(radical, azimuthal or z direction). Several remarkable features can be observed in Fig. 3(a). First of all,

*F*is smaller than

^{ρ}*F*by nearly an order of magnitude for the ZnO NW with small radius. As the radius of the NW further decreases,

^{z}*F*has a limiting value of 0.033. This limiting value depends on the refractive index of the NW and becomes smaller as the refractive index increases as shown in the inset of Fig. 3(a). Since the radiative transition rate is proportional to

^{ρ}*F*(c.f. Eq. (1)) and radiative transition and non-radiative transition are two competing processes, the internal quantum efficiency of the emitter orienting along the radical direction (below we will call it as

*ρ*-oriented emitter) is significantly low (c.f. Eq.(2)). Thus for the NWs with small radii the

*z*-oriented emitters are much more efficient. Secondly, there are two critical radii of the NW, i.e. 45 nm and 95 nm for the

*ρ*-oriented emitter and

*z*-oriented emitter, respectively. For the radius less than 45nm,

*η*for both kinds of emitter is less than 1%. But for a radius between 45nm and 95nm, only

_{W}*η*for z-oriented emitter is less than 1%. As the radius exceeds 45 nm,

_{W}*η*of the

_{W}*ρ*-oriented emitter first increases and then oscillates. Here a larger critical radius for the

*z*-oriented emitter is due to the fact that the emission by the emitter located at the origin of the NW can not couple to the fundamental HE

_{11}mode but have a good coupling to waveguide mode starting from TM

_{01}mode which is cutoff for the radius smaller than 95nm. Once the radius exceeds 95 nm, an abrupt change of

*U*occurs for the

*z*-oriented emitter.

Fig. 3(b) shows the variation of
*η _{U}* and

*η*as the change of the radius of the ZnO NW for the emitter located at the origin of the NW. As observed in Fig. 3(b),

_{W}*η*of the

_{U}*z*-oriented emitter is nearly four times larger than that of the

*ρ*-oriented emitter for the NWs with the radii smaller than 30nm. For the NW with a radius of 95 nm,

*η*of the

_{W}*ρ*-oriented emitter reaches the maximum value of 0.73, approximately 5.5 times larger than

*η*. We have also studied the SE properties of the emitter located near the boundary of the NW. Figure 3(c) shows the variation of

_{U}*η*and

_{U}*η*as the change of the radius for the emitters located near the boundary of the NW. Different from Fig. 3(b),

_{W}*η*of the emitters with different orientations have the same critical radius of 45 nm, below which

_{W}*η*is less than 1%. This is because the emission for the emitter near the boundary of the NW with any orientation (including along z direction) couples to the waveguided emission starting from the fundamental HE

_{W}_{11}mode.

From Fig. 3(a)–(c), one sees that for the ZnO NW with a radius less than 45 nm
almost no waveguided emission can be observed. For the NWs with such small radii,
*η _{U}* of the

*z*-oriented emitter is much larger than that of the

*ρ*-oriented or

*θ*-oriented emitter. Since in PL the orientation of the emitter is parallel to the electric field of the excitation [14, 15],

*η*by PL will be sensitive to the polarization of the excitation light.

_{U}#### 3.2 SE of a ZnO/SiO2 nanocable

Semiconductor NWs with coaxial shells of different kinds of materials or dopants are
interesting owing to their potential applications [3–5]. Semiconductor NWs with silica
shell can reduce non-radiative recombinations and protect them from aggregation,
leading to an improved chemical stability [4]. In this subsection, the SE properties
of a ZnO/SiO2 nanocable (NC) will be studied. As seen in Fig. 3(a) and 3(b), for
the emitter located at origin of the isolated ZnO NW with a radius between 45 nm and
95 nm, *η _{W}* of the

*ρ*-oriented emitter exists while that of the

*z*-oriented emitter is almost zero. Thus we expect that for a ZnO/SiO2 NC with a small core radius and a suitable shell,

*η*of the

_{W}*ρ*-oriented or

*θ*-oriented emitter will be much larger that of z-oriented emitter, meanwhile

*η*of z-oriented emitter is larger. Thus for such a coaxial ZnO/SiO

_{U}_{2}NC,

*η*and

_{W}*η*will show opposite polarization anisotropy in PL. To validate this, we study the SE properties of the ZnO/SiO

_{U}_{2}NCs with a core radius of 10 nm and silica shell of various thicknesses. Fig. 3(d) shows the variation of

*η*and

_{U}*η*as the change of the shell thickness for the emitters located near the boundary of the ZnO core with different orientations. Similar results are obtained for the emitter located at the origin of ZnO core. The curves for the

_{W}*ρ*-oriented emitter and

*θ*-oriented emitter are almost identical due to the small radius of the ZnO core. Two critical shell thicknesses, 50 nm and 180nm, are observed for the

*ρ*-oriented emitter and z-oriented emitter, respectively. Below the critical thickness,

*η*of the emitter with any orientation is less than 1%. While for the shell thickness between 50 nm and 180 nm, the waveguided emission exists for

_{W}*ρ*-oriented or

*θ*-oriented emitter but is nearly zero for the

*z*-oriented emitter. In the meantime,

*η*of the

_{U}*z*-oriented emitter is much larger than that of the ρ-oriented emitter.

For the ZnO NW or ZnO/SiO_{2} NC with a finite length, an optical cavity is
formed in the NW due to the facet reflection. In this case, the facet emission power
roughly equals the waveguided emission power due to the following two reasons (i) the
cavity effect is weak since the power reflection coefficient is less than 0.177 (as
roughly given by (*n*
_{ZnO}-*n*
_{air})^{2}/(*n*
_{ZnO}+*n*
_{air})^{2}) (ii) the waveguided emission for the infinitely long NW
equals the facet emission of the emitter located in the middle of the node and
antinode of the standing wave formed in the cavity. The facet emission of the NW is
the averaged emission by the emitters located within the node and antinode of the
standing wave.

## 4. Conclusions

In conclusion, we have presented an efficient and accurate optical modeling of SE for
cylindrically layered nanostructures. The total emission power is efficiently calculated
by selecting a new integration path in the complex plane. An explicit formula has been
derived to calculate the lateral-surface emission power. The SE properties of an
isolated ZnO NW and a ZnO/SiO_{2} NC are comprehensively studied. The radiative
transition rate of the emitter in the NW depends strongly on its orientation. The
*z*-oriented emitters are usually much more efficient than the
*ρ*-oriented or *θ*-oriented emitters for a NW with a
small radius since the transition rate of the latter is smaller than that of the former
by nearly one order of magnitude. There is a critical value of the radius of the NW
below which the waveguided emission efficiency is almost zero. For a ZnO/SiO_{2}
nanocable with a suitable silica shell thickness the lateral-surface emission and
waveguided emission show opposite polarization anisotropy.

## Acknowledgments

We would like to acknowledge the support of the university development fund, the small project fund (#10207444) of the University of Hong Kong, the grant (#14300.324.01) from the Research Grant Council of Hong Kong Special Administrative Region, and the grant (#60688401) from the Natural Science Foundation of China.

## References and links

**1. **C. J. Barrelet, A.
B. Greytak, and C.
M. Lieber, “Nanowire photonic circuit
elements,” Nano Lett. **4**, 1981–1985
(2004). [CrossRef]

**2. **J. M. Bao, M.
A. Zimmler, F. Capasso, X.
W. Wang, and Z.F. Ren,
“Broadband ZnO single-nanowire lightemitting
diode,” Nano Lett. **6**, 1719–1722
(2006). [CrossRef] [PubMed]

**3. **L Dai, X.
L. Chen, X. Zhang, T. Zhou, and H. Hu
“Coaxial ZnO/SiO_{2} nanocables fabricated by thermal
evaporation/oxidation,” Appl. Phys. A **78**, 557–559
(2004). [CrossRef]

**4. **Y. Wang, Z. Tang, X. Liang, L.
M. Liz-Marzan, and N.
A. Kotov, “SiO_{2}-Coated
CdTe Nanowires: Bristled Nano Centipedes, ”Nano
Lett. **4**, 225–231 (2004). [CrossRef]

**5. **O. Hayden, A.
B. Greytak, and D.
C. Bell, “Core-shell nanowire
light-emitting diodes,”Adv. Mater. **17**, 701–704
(2005). [CrossRef]

**6. **J. Goldberger, R.
R. He, Y. F. Zhang, S.
K. Lee, H. Q. Yan, H.
J. Choi, and P.
D. Yang, “Single-crystal gallium
nitride nanotubes,” Nature **422**, 599–602
(2003). [CrossRef] [PubMed]

**7. **E.
M. Purcell, “Spontaneous emission
probabilities at radio frequencies,” Phys. Rev. **69**, 681–681 (1946).

**8. **W. Lukosz,
“Theory of optical-environment-dependent spontaneous emission rates
for emitters in thin layers,”Phys. Rev. B **22**, 3030–3038
(1980). [CrossRef]

**9. **X. W. Chen, W. C.
H. Choy, S. L. He, and P.
C. Chui, “Comprehensive Analysis
and Optimal Design of Top-Emitting Organic Light Emitting
Devices,” J. Appl. Phys. **101**, 113107 (2007). [CrossRef]

**10. **J. R. Lovell and W.
C. Chew, “Response of a point
source in a multicylindrically layered medium,” IEEE
Trans. Geosci. Remote Sensing , **GE-25**,
850–858 (1987). [CrossRef]

**11. **I. Vurgaftman and J. Singh,
“Spatial and spectral characteristics of spontaneous emission from
semiconductor quantum wells in microscopic cylindrical cavities,”
Appl. Phys. Lett. **67**, 3865–3867
(1995). [CrossRef]

**12. **K. Oshiro
K and K. Kakazu,
“Spontaneous emission in coaxial cylindrical
cavities,” Prog. Theor. Phys **98**, 533–550
(1997). [CrossRef]

**13. **C. C. Wang and Z. Ye,
“Spontaneous emission in cylindrical periodically-layered
structures,” Phys. Stat. Solid I A- Applied
Research **174**, 527–540
(1999). [CrossRef]

**14. **W. Zakowicz and M. Janowicz,
“Spontaneous emission in the presence of a dielectric
cylinder,” Phys. Rev. A , **62**,
013820 (2000). [CrossRef]

**15. **D. P. Fussell, R.
C. McPhedran, and C. M. de
Sterke,
“Decay rate and level shift in a circular dielectric
waveguide,” Phys. Rev. A **71**, 013815 (2005). [CrossRef]

**16. **P. Bermel, J.
D. Joannopoulos, and Y. Fink,
“Properties of radiating pointlike sources in cylindrically
omnidirectionally reflecting waveguides,” Phys. Rev.
B **69**, 035316 (2004). [CrossRef]

**17. **S. Wakelin and C.
R. Bagshaw, “A prism combination for
near isotropic fluorescence excitation by total internal
reflection,” J. Microsc **209**, 143–148
(2003). [CrossRef] [PubMed]

**18. **J. Enderlein,
“Theoretical study of single molecule fluorescence in a metallic
nanocavity,” Appl. Phys. Lett. **80**, 315–317
(2002). [CrossRef]

**19. **W.
C. Chew, *Waves and Fields in Inhomogeneous
Media*, *Chap. 3* (IEEE Press, New
York, 1995).

**20. **P. Gay-Balmaz and J.
R. Mosig, “Three-Dimensional
planar radiating structures in stratified media,” Int. J.
Microwave Millimeter Wave Computer-Aided Eng. **37**, 330–343(1997). [CrossRef]

**21. **Y. Zhang and R.
E. Russo,“Quantum efficiency of
ZnO nanowire nanolasers,”Appl. Phys. Lett. **87**, 043106 (2005). [CrossRef]