RECENTLY, nanowires have attracted great attention

Transcription

1 146 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 42, NO. 2, FEBRUARY 2006 Analysis of Mode Quality Factors and Mode Reflectivities for Nanowire Cavity by FDTD Technique Miao-Qing Wang, Yong-Zhen Huang, Senior Member, IEEE, Qin Chen, and Zhi-Ping Cai Abstract The mode frequency and the quality factor of nanowire cavities are calculated from the intensity spectrum obtained by the finite-difference time-domain (FDTD) technique and the Padé approximation. In a free-standing nanowire cavity with dielectric constant = 6 0 and a length of 5 m, quality factors of 130, 159, and 151 are obtained for the HE 11 modes with a wavelength around 375 nm, at cavity radius of 60, 75, and 90 nm, respectively. The corresponding quality factors reduce to 78, 94, and 86 for a nanowire cavity standing on a sapphire substrate with a refractive index of 1.8. The mode quality factors are also calculated for the TE 01 and TM 01 modes, and the mode reflectivities are calculated from the mode quality factors. Index Terms Cavity resonators, FDTD methods, laser modes, nanowire cavity, quality factor, reflection coefficient. I. INTRODUCTION RECENTLY, nanowires have attracted great attention because of their unique electrical and optical properties. The strong two-dimensional confinement of electrons, holes and photons makes them particularly attractive as potential building blocks for nanoscale electronics and optoelectronic devices [1], [2], including lasers [3] [5] and nonlinear optical frequency converters [6]. Nanowires have a very small lateral size with a diameter of about nm and a longitudinal size of only a few micrometers, making them one of the smallest lasers available at present. Semiconductor nanowire lasers based on GaN [3], CdS [4], and ZnO [5] have been realized experimentally, with a single nanowire functioning as a stand-alone optical cavity and gain medium. The mode quality factor, which reflects the confinement of guided modes in the nanowire cavity, is a key factor in designing nanowire lasers. The reflection coefficients of the guided modes have been calculated with the finite-difference time-domain (FDTD) Manuscript received July 12, 2005; revised October 25, This work was supported by the National Nature Science Foundation of China under Grant , and the 863 project plan under Grant 2003AA M.-Q. Wang is with the State Key Laboratory on Integrated Optoelectronics, Institute of Semiconductors, Chinese Academy of Sciences, Beijing , China, and also with the Department of Physics, Xiamen University, Xiamen , China. Y.-Z. Huang is with the State Key Laboratory on Integrated Optoelectronics, Institute of Semiconductors, Chinese Academy of Sciences, Beijing , China, and also with the College of Information Science and Engineering, Huaqiao University, Quanzhou , China. Q. Chen is with the State Key Laboratory on Integrated Optoelectronics, Institute of Semiconductors, Chinese Academy of Sciences, Beijing , China. Z.-P. Cai is with the Department of Electronic Engineering, Xiamen University, Xiamen , China. Digital Object Identifier /JQE technique [7], and the quality factor has been estimated from the reflection coefficients. In this paper, we directly calculate the mode frequency and the quality factor from the mode field intensity spectrum with the FDTD technique [8] and the Padé approximation [9].,, and modes in a free-standing nanowire cavity and the fundamental mode in a cavity grown on a sapphire substrate are considered for simulating the corresponding mode intensity spectrum. Furthermore, the mode reflectivities are calculated from the obtained mode quality factors and the mode group index for the semiconductor air and semiconductor sapphire interfaces of the nanowire cavity. II. NUMERICAL MODELS A free-standing nanowire cavity, which is totally surrounded by air, and a practical nanowire cavity grown on a sapphire substrate are considered in numerical simulation. The dielectric constant of the nanowire is taken to be, which is close to that of GaN, ZnO, and CdS at the lasing frequencies. To increase the computing efficiency of the FDTD simulation, we transform the three-dimensional (3-D) problem to a two-dimensional (2-D) one in cylindrical coordinates based on the azimuth symmetry of the nanowires. The components of the electromagnetic field in an axially symmetric cylindrical coordinate system can be expressed as follows [10]: where is the integer azimuthal mode number;,,, and,, are the components of the electric and magnetic fields, respectively, and,,, and,, are the corresponding reduced field components, which do not depend on. The electromagnetic fields have six nonzero components as. The corresponding modes are defined as hybrid electromagnetic modes (HE or EH). When, the modes can be expressed as transverse electric (TE) modes with nonzero field components of (,, ) or transverse magnetic (TM) modes with nonzero field components of (,, ). A schematic diagram of a free-standing cylindrical nanowire cavity and the corresponding FDTD computational space do- (1) /$ IEEE

2 WANG et al.: ANALYSIS OF MODE QUALITY FACTORS AND MODE REFLECTIVITIES FOR NANOWIRE CAVITY BY FDTD TECHNIQUE 147 Fig. 1. Schematic of the cross section of a free-standing nanocavity and the corresponding FDTD computational domain surrounded by A, B, C, and D. main are shown in Fig. 1, where,,, and are the boundaries that truncate the computational window. For HE or EH modes, Mur s first-order absorbing boundary condition (ABC) [8] is applied to the field components and at boundary and to and at boundaries and, respectively. In order to save computing time and load, the boundary is moved away from the z axis several meshes [10]. The analytical solution of the field components and for a circular dielectric rod can be expressed as the Bessel function inside the rod with azimuthal number [11]. We expect that the mode field components and of the nanowire cavity are still Bessel functions when and approaches zero. The asymptote of the Bessel function as approaches zero is We can apply (2) to the field components and at boundary to truncate the computing region when. When, we apply Mur s ABC to the field components or at boundary and or at boundaries and for the TE or TM modes, respectively. At boundary, the integral form of Maxwell s equation in the time domain as approaches zero can be expressed as where and are the finite difference area and the integral path. Thus, we can apply (3) to the field component for the TM mode and to for the TE mode at. It should be noted that the perfectly matched layer boundary condition is better than Mur s ABC in FDTD simulations. We use Mur s ABC because we have the code for it in hand. In the numerical simulation, an exciting source with a cosine impulse modulated by a Gaussian function is added to one component of the electromagnetic field at a point inside the nanowire cavity, where and are the times of the pulse center and the pulse half width, respectively, (2) (3) (4) and is the center frequency of the pulse. Then the time variation of a selected field component at some points inside the nanocavity is recorded as the FDTD output, and the Padé approximation based on Baker s algorithm [9] is used to calculate the field spectrum for obtaining the -factor and the mode frequencies. The code of the Padé approximation is presented in an appendix in [12]. In the calculation, the field spectrum obtained by the recursion relations of the Padé approximation approaches stability as the FDTD output length increases [13]. The convergence of the intensity spectrum can be proved since it does not vary with a longer FDTD output. The same mode frequency and quality factor can be obtained by using any component of the electric or magnetic field. The mode frequency and the -factor are calculated from the peak position of the spectrum and the ratio of the frequency to the full width at half maximum of the peak as. III. NUMERICAL RESULTS A. -Factor of a Free-Standing Nanowire Cavity For the free-standing nanowire cavity, Mur s absorbing boundary is set at six times the radius of the nanocavity, where the electric and magnetic fields can be considered to be approximately zero. Boundaries and are placed 2 m from the top and bottom facets, and boundary is moved away four meshes from the axis for the mode. In the FDTD simulation, the spatial step is chosen to be nm and nm, respectively, which is less than a fortieth of the wavelength in the media, thereby ensuring the accuracy. To save computing time, we usually choose the Courant limit as the time step. However, we cannot get stable results for the and modes with such a time step. To ensure the numerical stability, 1.0 and 0.91 times the Courant limit are used as the time step for the mode and for the and modes, respectively. The corresponding time step is for the mode and for the and modes. The fundamental mode in a free-standing nanowire cavity is first simulated for a nanowire cavity with cavity length m and radius nm. The total number of cells in the simulation is 900 in the z-direction and 120 in the -direction, respectively. The intensity spectrum obtained with an impulse (4) at and is plotted in the insert of Fig. 2, where the multiple peaks correspond to different longitudinal modes. We cannot get an accurate mode frequency or -factor by fitting the intensity spectra with multiple Lorentzian peaks because too many modes exist in the spectrum and some modes are not clear. So we choose an impulse (4) with and to excite only one single longitudinal mode. The mode intensity spectrum obtained by a single mode excitation and the Padé approximation method is plotted with squares in Fig. 2, and the fitting Lorentzian curve is also plotted with a solid line. About four hours are required for the step FDTD simulation on a Dell Workstation 670 with a 2.8-GHz CPU and 2 G memory. Because a hard source is used to excite the cavity, the FDTD output is greatly affected by

3 148 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 42, NO. 2, FEBRUARY 2006 Fig. 2. Intensity spectrum obtained by Padé approximation for HE in a free-standing nanowire of the length L =5m and the radius R =60nm. Fig. 4. Quality factor versus the wavelength for HE, TE, and TM in the free-standing nanowire cavity with cavity length of 5 m and the radius of 60, 75, and 90 nm. Fig. 5. Quality factor for HE in a free-standing nanocavity versus L at R = 60 nm and versus R at L = 5m, respectively, at the wavelength of about 375 nm. Fig. 3. Field distribution of the mode field H for HE in a quarter of the cross section of the nanowire cavity with L =5m and R =60nm at the wavelength nm. the exciting source. The mode intensity spectrum is usually obtained from the output of the last 5000 FDTD steps in the single mode simulation, and the obtained -factors are not influenced by the source. The field distribution of the mode field in a quarter of the cross section of the nanowire cavity is plotted in Fig. 3 for the fundamental mode at a wavelength of nm. The thick solid lines indicate the boundaries of the nanowire cavity. The results show that the nanowire cavity with a diameter of 120 nm, which is smaller than the lasing wavelength, still confines the guided modes very well. The -factors of the,, and modes versus the wavelength are plotted in Fig. 4 for free-standing nanowire cavities with cavity length m and radius 60, 75, and 90 nm, respectively. For the fundamental mode, the -factor increases 20 30% as the radius increases from 60 to 75 nm. For the and modes, the -factors increase quickly with the increase of the radius. Analytical solutions of a dielectric cylinder yield a cutoff wavelength of nm for and in the nanowire with radius nm. The -factors of the mode with a wavelength of about 380 nm are 22, 42, and 100 at the radii of 60, 75, and 90 nm, respectively. The corresponding -factors for the mode are 250 and 355 at radii of 75 and 90 nm, respectively, much larger than that of the and modes. The mode wavelength is determined by the size of the nanowire cavity and the mode numbers. So we can only get a mode -factor at discrete mode wavelengths. The quality factors versus cavity length at a radius of 60 nm and versus radius at a cavity length of 5 m are plotted in Fig. 5 for the mode with a wavelength of about 375 nm in the free-standing nanowire cavity. The -factor increases linearly with the increase of cavity length at different radii. For a cavity with a length of 5 m, the -factor increases linearly as the radius increases from 40 to 70 nm and saturates as the radius further increases. For a cavity length of 10 m, a 10 step

4 WANG et al.: ANALYSIS OF MODE QUALITY FACTORS AND MODE REFLECTIVITIES FOR NANOWIRE CAVITY BY FDTD TECHNIQUE 149 Fig. 6. Quality factor versus the wavelength for HE in the nanowire cavity grown on the sapphire substrate with the radius of 60, 75, and 90 nm. FDTD simulation is required because the mode interval is half of that of a cavity with m. B. -Factor of a Nanowire Cavity on a Substrate We consider a practical nanocavity standing on a sapphire substrate of refractive index. In the FDTD simulation, the boundary conditions are taken to be the same as those of the free-standing nanowire cavity except that boundary is put at a distance of 4 m from the bottom facet. The mode -factor of the fundamental mode versus wavelength is plotted in Fig. 6 for a nanowire cavity with length m and radius 60, 75 and 90 nm. The -factors at a mode wavelength of 375 nm are about 78, 94, and 86 for 60, 75, and 90 nm, respectively, values which are about 60% of those for the free-standing nanowire cavity. The interface between the nanowire and substrate has a much smaller reflectivity than the nanocavity/air interface. We think that the fluctuation of the -factor at wavelengths of 375 and 380 nm and nm is mainly an artifact of the simulation. In fact, the deviations of the -factor from the smoothed curve are only about 5%, which is within the error of the calculated -factors. C. Mode Reflection Coefficients In this section, we estimate the mode reflection coefficients from the obtained -factors. From the mirror loss, we obtain the relation between the -factor and the reflection coefficients and of the nanowire cavity where and are the mode wavenumber and speed of light in vacuum, respectively, and is the mode group index. Using the longitudinal mode wavelength interval obtained by FDTD simulation, we obtain the group indexes as 3.23, 2.97, and 2.88 at the wavelength 375 nm for 60, 75, and 90 nm, respectively. It should be noted that the obtained group index is only related to the dispersion of the nanowire waveguide. The material dispersion (5) Fig. 7. Mode reflection coefficients for HE at the interfaces of semiconductor air and semiconductor sapphire are plotted as the function of mode wavelength for the nanowire cavities with the radius of 60, 75, and 90 nm. is not included in our model. The results are well in agreement with the values 3.24, 3.03, and 2.86 obtained from the analytical dispersion curves of the nanowire rod waveguide [12]. From the -factors of the free-standing nanowire cavity, we can calculate the mode reflection coefficients at the semiconductor air interface of the nanowire cavity from (5). For the nanowire cavity on the sapphire, the mode reflection coefficients are and at the semiconductor sapphire interface of the nanowire cavity. The mode reflection coefficient can then be calculated from (5) using based on the -factor values and for the nanowire cavity on the sapphire substrate. We plot the mode reflection coefficients and for the mode in Fig. 7. For the nanowire cavity with radii of 75 and 90 nm, is about 0.45, which is a little larger than the reflection coefficient 0.42 of a normal incident wave on an infinite interface, and is about 0.14 due to the small refractive index difference. We also calculate for the and modes, and plot the results in Fig. 8. The reflection coefficient is about 0.7 for the mode at a radius of 90 nm, which is larger than that of the mode. A large mode reflection coefficient can be expected as the mode light ray strikes the interface at a large angle of incidence. In fact, the large confinement factor of the nanowire cavity [14] can also partly explained by the transmission direction of the mode light ray. As in a strong slab waveguide, the path of the zigzag propagating mode light ray in the center layer can be longer than the transmission length in the -direction. Furthermore, it was observed that the mode gain was larger than the material gain for the SiO Si SiO slab waveguide [15]. Finally, we compare the numerical results of the reflection coefficients with those of [7] in Table I at 0.99, 1.24, and 1.49, respectively, corresponding to 60, 75, and 90 nm and a wavelength of 380 nm. Reflection coefficients associated with our results are calculated from the -factors given in Figs. 7 and 8. We find of to be at from Fig. 7, which is underestimated due to the artifact fluctuation.

5 150 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 42, NO. 2, FEBRUARY 2006 ACKNOWLEDGMENT The authors would like to thank C. Z. Ning and A.V. Maslov of NASA Ames Research Center, CA, for providing the data of reflection coefficients. Fig. 8. Mode reflection coefficients for TE and TM at the interfaces of semiconductor air are plotted as the function of mode wavelength for the nanowire cavities with the radius of 60, 75, and 90 nm. TABLE I COMPARISON OF OUR RESULTS OF REFLECTION COEFFICIENTS AND THAT OF [7] A more reasonable value is about which we obtain by smoothing the corresponding curve. The differences between the calculated reflection coefficients and those of [7] are less than 10% except for the of at. REFERENCES [1] X. Duan, Y. Huang, Y. Cui, J. Wang, and C. M. Lieber, Indium phosphide nanowires as building blocks for nanoscale electronic and optoelctronic devices, Nature, vol. 409, pp , Jan [2] J. Wang, M. S. Gudiksen, X. Duan, Y. Cui, and C. M. Lieber, Highly polarized photoluminescence and photodetection from single Indium phosphide nanowires, Science, vol. 293, pp , [3] J. C. Johnson, H. J. Choi, K. P. Knutsen, R. D. Schaller, P. Yang, and R. J. Saykally, Single gallium nitride nanowire lasers, Nature Mater., vol. 1, pp , Oct [4] X. Duan, Y. Huang, R. Agarwal, and C. M. Lieber, Single-nanowire electrically driven lasers, Nature, vol. 421, pp , Jan [5] M. H. Huang, S. Mao, H. Feick, H. Yan, Y. Wu, H. Kind, E. Weber, R. Russo, and P. Yang, Room-temperature ultraviolet nanowire nanolasers, Science, vol. 292, pp , Jan [6] J. C. Johnson, H. Yan, R. D. Schaller, P. B. Petersen, P. Yang, and R. J. Saykally, Near-field imaging of nonlinear optical mixing in single zinc oxide nanowires, Nano Lett., vol. 2, pp , [7] A. V. Maslov and C. Z. Ning, Reflection of guided modes in a semiconductor nanowire laser, Appl. Phys. Lett., vol. 83, pp , [8] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method. London, U.K.: Artech House, [9] W. H. Guo, W. J. Li, and Y. Z. Huang, Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation, IEEE Microw. Wireless Compon. Lett., vol. 11, no. 5, pp , May [10] B. J. Li and P. L. Liu, Numerical analysis of the Whispering Gallery modes by the finite-difference time-domain method, IEEE J. Quantum Electron., vol. 32, no. 9, pp , Sep [11] M. J. Adams, An Introduction to Optical Waveguides. Chichester, U.K.: Wiley, 1981, p [12] Y. Z. Huang, Q. Chen, W. H. Guo, and L. J. Yu, Application of Padé approximation in simulating photonic crystals, Chin. J. Semicond., vol. 26, no. 7, pp , Jul [13] Q. Chen, Y. Z. Huang, W. H. Guo, and L. J. Yu, Calculation of propagation loss in photonic crystal waveguides by FDTD technique and Padé approximation, Opt. Commun., vol. 248, pp , [14] A. V. Maslov and C. Z. Ning, Modal gain in a semiconductor nanowire laser with anisotropic bandstructure, IEEE J. Quantum Electron., vol. 40, no. 10, pp , Oct [15] Y. Z. Huang, Comparison of modal gain and material gain for strong guiding slab waveguides, Proc. IEE Optoelectron., vol. 148, no. 3, pp , Jun IV. CONCLUSION We have calculated mode -factors and reflection coefficients for nanowire cavities by FDTD simulation and the Padé approximation. The results show that the mode has a much larger -factor than those of the and modes for a cavity radius that is one sixth of the wavelength, but the -factor of the mode increases quickly with the increase of the radius. The mode can have a much larger -factor than those of the and modes for a nanowire cavity with a radius of about a quarter of the mode wavelength. Considering that the whole nanowire is a gain medium and the confinement factor is very large for the nanowire cavity, we can expect that a nanowire cavity with a -factor of several hundred would have a threshold gain comparable to or even less than that of an edge-emitting heterostructure laser. Miao-Qing Wang was born in Zhejiang Province, China, in He received the B.Sc. degree in physics from Xiamen University, Xiamen, China, in He is currently working toward the M.S. degree at Xiamen University, Xiamen, China, and at the Institute of Semiconductors, Chinese Academy of Sciences, Beijing, China. Yong-Zhen Huang (M 95-SM 01) was born in Fujian Province, China, in He received the B.Sc., M.Sc., and Ph.D. degrees in physics from Peking University, Beijing, China, in 1983, 1986, and 1989, respectively. In 1989, he joined the Institute of Semiconductors, Chinese Academy of Sciences, Beijing, China, where he worked on the tunneling time for quantum barriers, asymmetric Fabry Perot cavity light modulators, and VCSELs. In 1994, he was a Visiting Scholar at BT Laboratories, Ipswich, U.K., where he was involved in the fabrication of the 1550-nm InGaAsP VCSEL. Since 1997, he has been a Professor with the Institute of Semiconductors, Chinese Academy of Sciences, and is the Director of the Optoelectronic Research and Development center. His current research interests involved microcavity lasers, semiconductor optical amplifiers, and photonic crystals.

6 WANG et al.: ANALYSIS OF MODE QUALITY FACTORS AND MODE REFLECTIVITIES FOR NANOWIRE CAVITY BY FDTD TECHNIQUE 151 Qin Chen was born in Hubei Province, China, in He received the B.Sc. degree in physics from Wuhan University, Wuhan, China, in He is currently working toward the Ph.D. degree at the Institute of Semiconductors, Chinese Academy of Sciences, Beijing, China, and studying the FDTD simulation and fabrication of microcavities and photonic crystals. Zhi-Ping Cai was born in Fujian Province, China, in He received the Ph.D. degree from University of Nice, Nice, France in He was a Senior Visiting Scholar from 1994 to 1996 and Guest Professor from 2000 to 2001 in ENSSAT, Lannion, France. Since 1998, he has been a Professor in the Department of Electronic Engineering, Xiamen University, Xiamen, China. His research interests cover laser physics, optoelectronics, meso-optics, and their applications. He is the author of over 80 publications in refereed journals and conference proceedings.