224 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 49, NO. 2, FEBRUARY 2013 On the Efficiency Decrease of the GaN Light-Emitting Nanorod Arrays Liang-Yi Chen, Chi-Kang Li, Jin-Yi Tan, Li-Chuan Huang, Yuh-Renn Wu, Member, IEEE, and Jian Jang Huang, Senior Member, IEEE Abstract— Nanostructure light emitting arrays, with the mitigated quantum confined stark effect, provide a different angle to investigate the efficiency decrease in the GaN based LEDs. In this paper, the external quantum efficiency and the electroluminescent spectra of GaN based nanorod LEDs are characterized through experiments and simulations. The strains in the InGaN/GaNnanorods are varied with the choice of nanorod sidewall passivation materials. Our results indicate that Auger recombination dominates at low-level currents. However, even though the effect of Auger accounts fora higher percentage weighting, the increase number of leakage carriers out of quantum wells is responsible for the efficiency drop at high current levels. Index Terms— Light-emitting diodes, nanofabrication and quantum wells, nanostructures. I. I NTRODUCTION R ECENTLY, GaN based nanorod light emitting diodes (LEDs) have been widely explored [1]–[4] due to the unique properties of nanostructures, such as the relaxed strain in the InGaN/GaN multiple quantum wells(MQWs) and the improved light extraction as most of the generated photons fall within the escape cone along nanorods. The nanostructureLEDs have the potential for next-generation general lighting. In the past, we have demonstrated GaN nanorod LEDs with 6807mW/cm2 optical output at 32A/cm2 and the reverse current in the nA scale [5]. Furthermore, we discovered that strain in the nanorod structure is affected by the passivation materials, such as the spin-on-glass (SOG) or the PECVD (plasma enhanced chemical vapor deposition) grown SiO2 , for filling up the nanorod spacing [6]. The strain in the MQWs tilts the energy band and results in various amounts of peak wavelength shifts in the EL(electroluminescent) spectra with the injection currents. The k•p method and the piezoelectric field were both applied to identify the correlation between the strain, band tilt and light-emitting wavelength of the nanorod samples [6]. However, previous work at low injection currents has neglected both the quantum efficiency and the screening effect in the quantum well at high carrier injections. Manuscript received October 2, 2012; revised December 19, 2012; accepted December 28, 2012. Date of publication January 4, 2013; date of current version January 18, 2013. This work was supported in part by the National Science Council in Taiwan, under Grant 100-2628-E-002-030-MY3. The authors are with the Graduate Institute of Photonics and Optoelectronics, National Taiwan University, Taipei 106, Taiwan (e-mail: jjhuang@cc.ee.ntu.edu.tw). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JQE.2013.2237885 Up to now, the effect of efficiency decrease (or called droop effect) has not yet thoroughly investigated in GaN based nanorod LED arrays. In the present work, nanorod LED arrays at high injection currents are characterized. Nanostructure lighting emitting arrays, with the mitigated QCSE (quantum confined stark effect) [7], [8], provide a different angle to investigate the root cause of efficiency drop in the GaN based LEDs. As various effects that influence the decrease of efficiency droop, such as Auger [9], [10], carrier leakage (or overflow) [11], [12], and junction temperature [13] can’t be singled out in typical LED structure, we intend to investigate the EQE (external quantum efficiency) based on the correlation between the experiment and simulation. In this study, choice of passivation materials provides various degrees of strain relaxation (and thus the polarization and carrier leakage) in the MQWs. And effects of carrier screening, leakage and Auger recombination are considered in the simulation to investigate the efficiency decrease. II. E XPERIMENTS , M EASUREMENTS AND S IMULATIONS In contrast to the selective-area epitaxy for forming GaN-based nanorods [14], [15], in this study, dry-etching method for forming GaN-based nanorod was used. Before the dry-etching process, the two dimensional (2-D) nano-mask is used to define the pattern of nanorod structure. Recent works have reported the fabrication of highly-ordered 2-D patterning in large-area wafer-scale region by using rapid convective deposition method of silica micro-/nano-particle monolayer arrays [16], [17] or diblock-copolymer lithography methods [18]. The current works focused on the use of spin-coating method in deposition of the silica colloidal lithography step for patterning the nanorods LEDs [8].We prepare two types of InGaN/GaN-based nanorod LED arrays, one is passivated with PECVD grown SiO2 and the other is with spin-coated SOG. The PECVD process was performed using SiH4 /Ar (40sccm) and N2 O (160sccm) as source gases, at the radio frequency power of 140W under the substrate temperature of 300°. The Siloxanes family SOG was spin-coated, reflowed at 72° for 15 mins and then baked in the over at 400° for one hour. The device fabrication is very similar to our previous work [6], [8], [19] except using the epi-wafers from various MOCVD (metal-organic chemical vapor deposition) growth runs. The patterning steps of the nanorods were carried out using spincoated silica colloidal lithography process, as described in [8]. Fig. 1 shows the schematic diagram of the nanorod structure. 0018–9197/05$31.00 © 2013 IEEE CHEN et al.: EFFICIENCY DECREASE OF THE GaN LIGHT-EMITTING NANOROD ARRAYS 225 (SiO2 or SOG) Fig. 1. Schematic of the nanorod LED array. The passivation layer is either spin-coated SOG or PECVD-grown SiO2 . For convenience, nanorod LEDs passivated with SOG are denoted as the LED-SOG while nanorods passivated with SiO2 are LED-SiO2 . To extract the strain of the nanorod LED structure, the micro-Raman measurement was conducted using the laser at the wavelength of 532 nm as the excitation source. As for the electroluminescence (EL) and Luminescence-current (L-I) measurements, in order to minimize the effect of junction heating, pulsed currents were injected with a 1% duty cycle and a pulse period of 50 ms throughout the experiment. As for simulation, we employed one-dimension selfconsistent finite element method (FEM) solver [20], [21]. The method solves Poisson and drift–diffusion equations selfconsistently so that the internal quantum efficiency (IQE) and the optical transition energy can be obtained. (a) III. R ESULTS AND D ISCUSSION A. L-I Measurement and the Introduction of the Simulation Model Five devices of each type (LED-SOG and LED-SiO2 ) of nanorod arrays were employed for L-I measurement. The statistic data are shown in Fig. 2(a) (right axis) with the error bars labeled. For convenience, the following numerical values of L-I and EQE are represented by the mean value. At the injection current of 100mA, the optical power of LED-SOG is 14.2% higher than that of LED-SiO2 . The optical power of these two type of devices is influenced by the sidewall passivation based on the following reasons. First, part of the photons generated in MQWs are extracted from the sidewalls and are affected by the refractive index of SiO2 and SOG. Second, the IQE of the devices is affected by the nanorod sidewall defects induced during ICP etching. And third, as suggested by our previous work [6], the choice of passivation material results in various strains in InGaN/GaN MQWs. For the first case, the refractive index of SiO2 and SOG is 1.46 and 1.40. If the small difference of the refractive index is the root-cause, the number of photons that escapes from the active layers (n = 2.5) to the air through the passivated material of LED-SiO2 should be higher than that of LED-SOG due to a larger refractive index, which contradicts our experimental observations. And in the second case, since the nanorod formation of both devices were carried out simultaneously, the sidewall defects picked up during ICP etching should have the same amount. Also, the reverse current at –5V is lower than 18nA (see Fig. 2(b)), indicating that the sidewall defects are effectively suppressed. As a result, the strain may (b) Fig. 2. (a) L–I curves (right axis) and the corresponding EQE of nanorod LEDs (left axis). In this plot, five devices of each type were measured. The mean values along with the error bars are shown. (b) I–V curves of LED-SOG (black line) and LED-SiO2 (red line). play a key role in the light output behaviors and will be scrutinized latter in this work. Furthermore, we also plot the external quantum efficiency (EQE), defined as the measured output power divided by the injection current, in the left axis of Fig. 2(a) by normalizing the maximum efficiency of LED-SOG to 1. The EQE curves show the droop effect of both devices. And the difference of maximum EQE between two devices is 6.16%. As suggested by our previous work [6], the passivation layer of either SOG or SiO2 contributes to different degree of strain relaxation in the InGaN/GaN nanorod MQWs and results in various piezoelectric fields. The quantum well band profiles and the carrier distribution are thus affected. To further study the EQE behaviors with injection currents/carriers, numerical simulation was next conducted. The internal quantum efficiency (IQE) is defined by the following equation. (1) IQE = Irad Itotal (2) Itotal = ISRH + Irad + IAug + Ileak 226 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 49, NO. 2, FEBRUARY 2013 TABLE I PARAMETERS E MPLOYED IN THE S IMULATION Thickness (nm) [30] μe electron mobility (cm2 /Vs) [30] μh hole mobility (cm2 /Vs) Doping density (1/cm3 ) Activation energy (meV) LED-SOG carrier nonradiative lifetimesτ [23] (s) n-Gan i-InGaN i-GaN p-GaN 2000 3 14 160 200 300 200 200 10 10 10 5 5.00E+18 1.00E+15 1.00E+15 3.00E+19 25 25 25 170 2.172E-08 2.172E-08 2.172E-08 2.172E-08 (a) LED-SiO2 carrier nonradiative lifetimesτ [23] (s) 1.812E-08 1.812E-08 1.812E-08 1.812E-08 Radiative recombination coefficient B[22] 1.00E-11 1.00E-11 1.00E-11 1.00E-11 Auger recombination coefficient C[24] 2.00E-31 2.00E-31 2.00E-31 2.00E-31 Where Irad = Bnp is the radiative recombination current (n and p are the free electron and hole carrier density respectively, and B is the radiative recombination coefficient) [22], ISRH is the Shockley–Read–Hall recombination current and is defined in [23]. Furthermore, IAug = C(n2 p + p2 n) is the current contributed from the Auger recombination, in which C is the Auger coefficient [24]. And Ileak is the current contributed from the carrier leakage, of which is mainly attributed to the following two reasons. First of all, the polarization induced energy band tilt leads to the escape of electrons out of the quantum wells. And second, since the mobility and carrier density of holes are much lower than electrons, there is spatial imbalance of electron and hole current across the quantum well region. The above two phenomena become more obvious at high injection current density when electrons fill up the energy states of quantum wells. Thus, the leakage current leads to reduction in current injection efficiency [25], [26], which will be severely quenched at high current density. Recent works have shown the importance of considering the current injection efficiency in InGaNbased LEDs [25], which is consistent with the general physics of current injection efficiency in quantum well lasers/LEDs [26]. Moreover, the overflow current may recombine in the p-GaN or mostly surface recombine in the p-type contact. The simulated carrier leakage considers the band profile by first obtaining the piezoelectric field Epz in the quantum well and then by solving the 1-dimension (1-D) FEM Poisson and drift–diffusion equations. Note that the strain in the nanorods was assumed as uniform in biaxial directions, however there exists spatial 2-D/3-D strain distribution within the nanorods from the ICP etching and passivation methods. The current (b) Fig. 3. (a) Micro-Raman measurement of InGaN/GaN nanorod LEDs. (b) Close-up view of InGaN E2H phonon mode shoulder around the wavenumber of 560 cm−1 . distribution of the entire device can thus be obtained. We then sum the current which escapes from the quantum well and divided by the injection current to be the overflow ratio, Ileak /Itotal [22]. Ileak can thus be determined. At the steady state, carriers fallen out of the MQWs are considered as the leakage current while those in the quantum wells are applied to the model using A, B and C coefficients of which the IQE, ηIQE , is expressed as ηIQE = Irad Irad = . ISRH + Irad + IAug + Ileak Itotal (3) The detailed parameters of the simulation are listed in Table I. In order to calculate the IQE of the devices with straininduced piezoelectric field, micro-Raman spectroscopy was conducted on both LED-SOG and LED-SiO2 . The results are shown in Fig. 3(a). The main peak near 570cm−1 is the E2H mode of GaN while the shoulder near 560cm−1 is the InGaN E2H mode correlated to the biaxial strain of InGaN quantum CHEN et al.: EFFICIENCY DECREASE OF THE GaN LIGHT-EMITTING NANOROD ARRAYS (a) (b) Fig. 4. (a) Simulated IQEs by setting the Auger coefficient C = 0 and C = 2 × 10−31 cm6 s−1 on both devices. In the calculation, the strain obtained in Fig. 3 is considered. (b) Comparisons of the simulation with the experimental work. The results indicate that the effect of Auger has to be considered in order to correlate the simulation with the measurement. wells. The Lorentzian curves are applied for curve fitting and peaks of the InGaN E2H phonon mode are labeled by black arrows in close-up views of Fig. 3(b). The peak positions of LED-SiO2 and LED-SOG are 562.09 cm−1 and 561.23 cm−1 respectively. In our previous work [6], we developed a series of calculation that transforms the wavenumber shift of the Raman spectrum into the piezoelectric field within the InGaN layer. From Fig. 3(b), the strain is calculated to be −1.67% and −1.53% for LED-SiO2 and LED-SOG, respectively. As the strain in the MQWs of LED-SOG is more relaxed than that of LED-SiO2 , the LED-SOG possesses less piezoelectric field and flatter band profile, which increases the probability of electron-hole overlap in the quantum wells. Other methods for suppressing the charge separation issue in InGaN QW LEDs [27] had also been reported by using non-/semi-polar QW [28], polar QW with large overlap designs [27], and 227 Fig. 5. Percentage weightings of the carrier leakage out of MQWs (Ileak /Itotal , left axis) and of the Auger recombination (IAug /Itotal , right axis) of both LED-SOG and LED-SiO2 . ternary substrate/template for strain mismatch reduction [29]. The optical output power of LED-SOG is thus higher than that of LED-SiO2 . In the following calculation, the strain obtained from the Raman spectra will be employed. To reconciliate experimental observations with theoretical predictions, the effect of Auger is first neglected by setting C = 0. From Fig. 4(a), the simulated IQE (C = 0) finds its maximum at a larger current than the experimental ones and then gradually decreases. On the other hand, with the Auger coefficient considered, the IQE peaks up at the low injection current. And beyond the efficiency maximum, a more rapid decrease of IQE is observed due to the effect of the non-radiative term C(n2 p + p2 n). As the current is further increased, the slope of IQE decrement becomes saturated. For both devices, from Fig. 4(a), regardless of the consideration of Auger recombination, the slops of IQE curves approach to a nearly constant value at high current injections, indicating the saturation of the effect of Auger. The measured IQE follows the trend predicted by the simulation with the Auger coefficient considered in Fig. 4(a). As a result, the simulation indicates that in the GaN based nanorod LED array, the effect of Auger plays a key role in the efficiency decrease, especially at the low injection currents. B. Comparison Between Simulation and Experimental Results Here we assume the extraction efficiency, defined as the light output power divided by the internally generated light power, is the same in both devices. Also, the strain obtained from the Raman measurement was employed in the calculation. In Fig. 4(b), the measured and simulated IQEs of both devices are compared. The simulated IQEs are normalized by the peak IQE value of LED-SOG. The non-radiative carrier lifetimes of LED-SiO2 and LEDSOG are 1.81 × 108 s and 2.17 × 108 s, respectively. The difference of carrier lifetime is associated with the capability of sidewall passivation using either SiO2 or SOG. 228 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 49, NO. 2, FEBRUARY 2013 (a) (b) (c) (d) Fig. 6. Electron densities (#/cm2 ) at (a) 3 mA (near the maximum IQE) at 100 mA and (b) LED-SOG (black lines) and LED-SiO2 (red lines). (c) and (d) Corresponding band profiles at 3 and 100 mA, respectively, of both devices. The lower carrier lifetime of LED-SiO2 may also be attributed to the plasma damage during the PECVD deposition. In Figure 4(b), the experimental data are well-fitted to the simulated data. Under high injection currents, IQE gradually decreases and both devices show a similar slope. It implies that different degrees of strain relaxation can’t be the sole reason of the efficiency drop. The effect of strain is further divided to the carrier screening in the MQWs and the carrier leakage out the wells. The percentage weightings of Auger recombination (defined as IAug/Itotal ) and carrier leakage (defined as Ileak /Itotal ) are further extracted from the simulation data and are shown in Fig. 5. From the plot, IAug/Itotal quickly increases with the injection currents and saturates when the current is over 20mA for both devices. The rapid increase of Auger recombination below 20mA contributes to a steeper decline of the IQE at the onset of the efficiency maximum. And the saturation of Auger beyond 20mA explains the steady decrease of IQE in Fig. 4(b). On the other hand, Ileak /Itotal, even with a less percentage value, increases almost linearly over the entire current range except at the small current before the energy levels filled up. In Fig. 5, Ileak /Itotal of LED-SOG is smaller due to a more relaxed strain and less band tilt. Even though both Auger recombination and carrier overflow decreases the radiative recombination in the LED structure, they carry different physical meanings. Auger recombination occurs in the quantum well region while carrier leakage represents the carriers which do not stay in quantum wells. Despite a smaller percentage of Ileak /Itotal, the effect of carrier overflow can be understood by plotting carrier distribution in the quantum wells. Fig. 6(a) and (b) show the simulated electron density distributions at the peak IQE and at the injection current of 100mA, respectively, for both devices. The p-type is placed on the left side while the n-type is on the right. The electron density reaches its maximum within five quantum wells and a significant carrier leakage can be found at 100mA in the p-type GaN (about three order of magnitude higher than the current at peak IQE). Moreover, due to a more relaxed strain, carrier CHEN et al.: EFFICIENCY DECREASE OF THE GaN LIGHT-EMITTING NANOROD ARRAYS 229 (a) (b) (c) (d) Fig. 7. EL spectra of (a) LED-SOG and (b) LED-SiO2 at the injection current ranging from 1 to 100 mA. (c) and (d) Close-up views of EL spectra within the current range of 0–20 mA for LED-SOG and LED-SiO2 , respectively. leakage in the case of LED-SOG is smaller than that in LED-SiO2 . To further investigate the correlation between carrier screening and band structure, Fig 6(c) and (d) show the simulated band diagram of two types devises under different injection currents. For both devices, flatter band profiles are observed at 100mA, which indicates a more obvious screening effect, as compared with the case at a smaller injection current. The flatter band profile also indicates the optical transition energy is different from the case at low level currents. The band profile in Fig. 6 has enabled us to next conduct the electroluminescence(EL) measurement for investigating the optical transition energy within the quantum wells at various carrier injections. C. EL Measurement The EL spectra were extracted with the injection current ranging from 1mA to 100mA and are shown in Fig. 7. The current dependent EL spectra reveal some interesting correlations with the EQE decrease. The EL peak position of LED-SOG is 2.720eV, 2.758eV and 2.785eV at the injection current of 1mA, 20mA and 100mA, respectively. And for the LED-SiO2 , the energy peak is 2.706, 2.755eV and 2.784eV at injection current 1mA, 20mA and 100mA, respectively. The peak energy difference suggests that under low carrier injections (below 20mA), both devices demonstrate an obvious blue shift with the increase of current (see the close-up views in Fig. 7(c) and (d)). The larger blue shift of LED-SiO2 again indicates QCSE in the MQWs is correlated to the strain [8]. And the screening of piezoelectric field results in a flatter band profile which leads to higher EL peak energy. On the other hand, at the high carrier injections (20 ∼ 100mA), a relatively smaller energy peak shift is observed, which indicates that strain relaxation isn’t as much a dominant factor as that at lower currents. The phenomenon implies that since the increased carriers are not fully contributed to radiative recombination, part of them overflow out of the quantum wells and mitigate the effect of carrier screening in the quantum wells. The above phenomena are next verified by simulation. Since the self-consistent Poisson can also solve the Schrödinger equation, the EL energy peaks are calculated based on the optical transition between the confined energy states of the conduction band and valence bands [17]. Fig. 8 shows both the experimental and simulation results of the EL peak energy shifts. For LED-SOG, the calculated energy shift is 43.8meV (vs. 36.9meV in the measurement) at the injection currents 230 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 49, NO. 2, FEBRUARY 2013 R EFERENCES Fig. 8. Comparisons of the EL peak shifts between the simulation and measurement data. between 1mA and 20mA and 20.2meV (vs. 26.9meV in the measurement) at the current 20 ∼ 100mA. On the other hand, we obtained a 52.0meV (vs. 49.3meV in the measurement) shift at 1 ∼ 20mA and 21.6meV (vs. 28.2meV in the measurement) at 20 ∼ 100mA for LED-SiO2 . Although the simulation results do not fully coincide with the measurement, they reveal a certain consistency as both results have a larger peak shift at lower currents and a smaller shift at large injections. The phenomenon suggests that even the optical transition energy becomes higher due to the carrier screening (carrier pile-up) as the injection current is increased, a significant carrier overflow to the barrier starts to be observed at the current larger than 20mA. IV. C ONCLUSION For the nanorod LED structure with different strains, the dominant factors for efficiency decrease can be understood from the experiment and simulation at various injection current levels. Our results on nanorods show that: first, Auger recombination dominates the efficiency decrease near the EQE maximum, and then it saturates at the higher current levels, which contradicts general thought that the effect of Auger is more obvious at high currents. 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Nakamura, “444.9 nm semipolar (112 2) laser diode grown on an intentionally stress relaxed InGaN waveguiding layer,” Appl. Phys. Lett., vol. 100, no. 2, pp. 021104-1–021104-4, Jan. 2012. [30] T. T. Mnatsakanov, M. E. Levinshtein, L. I. Pomortseva, S. N. Yurkov, G. S. Simin, and M. A. Khan, “Carrier mobility model for GaN,” Solid State Electron., vol. 47, no. 1, pp. 111–115, Jan. 2003. Liang-Yi Chen received the B.S. degree in mechanical engineering and the M.S. degree from the Graduate Institute of Photonics and Optoelectronics, National Taiwan University, Taipei, Taiwan, in 2006 and 2008, respectively, where he is currently pursuing the Ph.D. degree. His current research interests include fabrication and theoretical calculation of the wideband-gap materials and nanostructures. Chi-Kang Li received the B.S. degree in mechanical engineering and M.S. degree from the Graduate Institute of Photonics and Optoelectronics, National Taiwan University, Taipei, Taiwan, in 2007 and 2009, respectively, where he is currently pursuing the Ph.D. degree. His current research interests include simulation of optoelectronic devices and high-power electronics. 231 Jin-Yi Tan received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, in 2011, where he is currently pursuing the M.S. degree. His current research interests include fabrication and modulation characteristics of wide band-gap materials and nanostructures. Li-Chuan Huang received the B.S. degree in electrical engineering from National Tsing Hua University, Hsinchu, Taiwan, in 2010, where she is currently pursuing the M.S. degree. Her current research interests include fabrication and analysis of light efficiency of InGaN/GaN nanorod light-emitting diode arrays. Yuh-Renn Wu (S’02–M’07) received the B.S. degree in physics and the M.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, in 1998 and 2000, respectively, and the Ph.D. degree in electrical engineering from the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, in 2006. He is currently an Associate Professor with the Institute of Photonics and Optoelectronic and the Department of Electrical Engineering, National Taiwan University, where he is involved in research on physics, design of optoelectronic devices, and high-power electronics. His current research interests include studies of nitride-based quantum wells, quantum wires, and quantum dot light-emitting diodes, high-power and high-speed electronics, ferroelectrics, and optoelectronic devices. Jian Jang Huang (M’98–SM’08) received the B.S. degree in electrical engineering and the M.S. degree from the Graduate Institute of Photonics and Optoelectronics (GIPO), National Taiwan University (NTU), Taipei, Taiwan, in 1994 and 1996, respectively, and the Ph.D. degree in electrical engineering from the University of Illinois, Urbana-Champaign, in 2002. He is currently a Professor at GIPO, NTU, where he is involved in research on applying nanostructures to optoelectronic devices. He developed a spincoating method for nanosphere lithography, which can be applied to nanomaterials or nanostructures for significant performance improvement of light emitting diodes, solar cells, and nanorod devices. Prof. Huang was a recipient of numerous awards for his research contributions, including the Wu Da-Yu Award, the most prestigious one for young researchers in Taiwan sponsored by National Science Council, in 2008 and the award for the most excellent young electrical engineer from the Chinese Institute of Electrical Engineering in 2008. He is a member of the Phi Tau Phi Scholastic Honor Society. He is the Chair of the SPIE (San Diego, CA), International Conference on Solid State Lighting, and the Board Director of Global Communication Semiconductor, Inc., CA. He is the Editor of the IEEE T RANSACTIONS ON E LECTRON D EVICES .