Excitation dependent photoluminescence measurements of the nonradiative lifetime and quantum efficiency in GaAs S. R. Johnson, D. Ding, J.-B. Wang, S.-Q. Yu, and Y.-H. Zhang Citation: Journal of Vacuum Science & Technology B 25, 1077 (2007); doi: 10.1116/1.2720864 View online: http://dx.doi.org/10.1116/1.2720864 View Table of Contents: http://scitation.aip.org/content/avs/journal/jvstb/25/3?ver=pdfcov Published by the AVS: Science & Technology of Materials, Interfaces, and Processing Articles you may be interested in Strong excitation intensity dependence of the photoluminescence line shape in GaAs1−xBix single quantum well samples J. Appl. Phys. 113, 144308 (2013); 10.1063/1.4801429 Optically pumped lasing from a single pillar microcavity with InGaAs/GaAs quantum well potential fluctuation quantum dots J. Appl. Phys. 105, 053513 (2009); 10.1063/1.3074364 Time-resolved probing of the Purcell effect for InAs quantum boxes in GaAs microdisks Appl. Phys. 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Zhang Center for Solid State Electronics Research, Arizona State University, Tempe, Arizona 85287-6206 and Department of Electrical Engineering, Arizona State University, Tempe, Arizona 85287-6206 共Received 6 November 2006; accepted 5 March 2007; published 31 May 2007兲 The nonradiative lifetime and spontaneous emission quantum efficiency in molecular-beam epitaxy grown bulk GaAs is determined using injection level dependent photoluminescence 共PL兲 measurements. These measurements were performed at temperatures of 300, 230, 100, and 50 K using a HeNe pump laser with powers ranging from 0.3 to 40 mW. The quantum efficiency and lifetime is inferred from the power law relation linking pump power and integrated PL signal that is predicted by the rate equations. The nonradiative lifetime for bulk GaAs is determined to be 0.3 ␮s, with an additional temperature dependent component attributed to the AlGaAs barriers that rapidly reduces the nonradiative lifetime at temperatures above 230 K. The peak quantum efficiency is ⬎0.96 at 300 K and ⬎0.99 at temperatures below 230 K. © 2007 American Vacuum Society. 关DOI: 10.1116/1.2720864兴 I. INTRODUCTION Near unity quantum efficiency is critical to the performance of many luminescence based III-V devices. In order to grow these high quality materials by molecular-beam epitaxy 共MBE兲 it is necessary to establish routine, straightforward, and accurate measurement techniques for the material quantum efficiency to feedback to the growth. Although substantial efforts have been devoted to such measurements,1–4 there are few straightforward methods that offer reliable data. In this work, an unambiguous experimental approach is presented for the determination of the nonradiative lifetime and the injection dependent spontaneous emission quantum efficiency that is based on the power law relationship between the various types of recombination found in III-V semiconductors. In typical photoluminescence 共PL兲 measurements the pump power PPL 共mW兲 that is absorbed in the active region, is proportional to the electron-hole pair photoexcitation density 共cm−3 s−1兲, which is equal to the total electron-hole pair recombination rate within the active region. Furthermore, the PL signal integrated over energy, LPL 共photons/s兲, at the photodetector is proportional to the spontaneous emission rate per unit area per unit length 共cm−3 s−1兲 from the active region. These relationships are described by PPL = ca关An + 共1 − ␥r兲Bn + Cn 兴, 2 3 2 LPL = cbBn , 共1兲 where ␥r is the fraction of the spontaneous emission that is reabsorbed by the active region,5 and ca and cb are constants of proportionality that are determined by sample and measurement geometry. Here the recombination rates are written in terms of powers of the electron-hole concentration n, that physically reflect the various recombination processes that the electron-hole population undergoes. These processes are designated as Shockley-Read-Hall 共SRH兲 recombination for a兲 Author to whom correspondence should be addressed; electronic mail: shane.johnson@asu.edu 1077 J. Vac. Sci. Technol. B 25„3…, May/Jun 2007 n, radiative recombination for n2, and Auger recombination for n3, with leading coefficients A, B, and C, respectively,6 which are valid under low injection, n2 ⬍ NcNv, where Nc and Nv are the effective density of states for the electrons and holes. The assumptions implied in Eq. 共1兲 are that the fraction of the spontaneous emission recycled and the fraction of the pump power that photoexcites electron-hole pairs are independent of injection level, which are, respectively, valid under low injection and when the pump photon energy is substantially larger than the active material bandgap. It is important to include the photon recycling factor ␥r because it substantially affects the electron-hole density and the quantum efficiency. Each time a spontaneously emitted photon is recycled through band to band absorption, an electron-hole pair is generated which can be captured by either SRH or Auger recombination rather than emitting a photon, effectively increasing the nonradiative recombination by a factor of 共1 − ␥r兲−1. From Eq. 共1兲 the pump power can be written as powers of the PL signal with a leading coefficient for each of the recombination processes denoted as APL, BPL, and CPL, for the SRH, radiative, and Auger, respectively, with PPL = APL共LPL兲1/2 + BPLLPL + CPL共LPL兲3/2 . 共2兲 The constants APL, BPL, and CPL are the best fit parameters determined by fitting Eq. 共2兲 to pump power versus PL intensity measurements, from which the SRH and Auger coefficients are obtained by noting that A = APL 1 冑c a 冉 B共1 − ␥r兲 BPL C = CPL冑ca 冉 冊 1/2 B共1 − ␥r兲 BPL , 冊 3/2 , cb = ca共1 − ␥r兲/BPL , 共3兲 where B, ca, and ␥r are straightforward to calculate. B can be inferred from the thermal radiation 关see Eqs. 共6兲–共8兲兴; ca is a 1071-1023/2007/25„3…/1077/6/$23.00 ©2007 American Vacuum Society 1077 Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 209.147.144.20 On: Sat, 07 Feb 2015 23:28:34 1078 Johnson et al.: Excitation dependent photoluminescence measurements function of photoexcited active volume and the fraction of pump power actively absorbed 关see Eq. 共9兲兴; and ␥r is the fraction of spontaneous emission reabsorbed in the active region, which is determined via ray tracing over all solid angles. On the other hand, the luminescence constant of proportionality, cb, is extremely difficult to determine, as it depends on sample emittance, absolute throughput of the spectrometer, absolute responsivity of the detector, and the collection efficiency of the sample luminescence, each of which are generally not accurately known. Fortunately, cb can be written in terms of ca 关see Eq. 共3兲兴 and hence is eliminated from the calculation of the nonradiative coefficients. A good figure of merit for the assessment of material quality, that is independent of sample geometry and accessible via PL measurements, is the spontaneous emission quantum efficiency ␩q, which is defined as Bn2 ␩q ⬅ An + Bn2 + Cn3 冉 = 1+ APL CPL 共1 − ␥r兲共LPL兲−1/2 + 共1 − ␥r兲共LPL兲1/2 BPL BPL 冊 , and in the second equation is written in terms of the integrated PL intensity and other experimental parameters. From the Boltzmann approximation 共valid under low injection兲 the injection level 共i.e., bandgap energy Eg less quasi-Fermi level separation ⌬F兲 in terms of the electron-hole concentration is given by 冉 冊 冉 冊 共5兲 and is expressed in terms of pump power and ␩q on the right hand side, which is obtained by solving both the left hand equation in Eq. 共1兲 and the left hand equation in Eq. 共4兲 for An + Bn2 + Cn3, equating the two results, and solving the ensuing equation for n2. The radiative coefficient B is a measure of the efficacy of the spontaneous emission from a semiconductor material, and its underlying fundamental properties are governed by thermodynamics, which ultimately establishes the light emission properties through the density of photon states and the photon occupation of those states. Therefore, it is straightforward to calculate Bn2i 共where ni is the intrinsic, thermal equilibrium, electron-hole pair population兲, by noting that it is equivalent to the internal blackbody emission rate per unit area per unit length integrated over photon energies h␯ above the bandgap energy, with Bn2i = ⬇ 8␲c 共hc兲3 冕 ⬁ Eg ␣o共h␯兲共no共h␯兲兲2 共h␯兲2dh␯ eh␯/kT − 1 8␲c ␣gn2g共Eg + kT兲2kTe−Eg/kT , 共hc兲3 FIG. 1. Cross section 共with sample surface on right hand side兲 and growth temperature profile of bulk GaAs sample 共run B1657兲 used in photoluminescence measurements. The decimal numbers listed above each layer give the fraction of the pump radiation absorbed in that layer: upper for 300 K and lower for 77 K. ous emission energy Eg + kT. The intrinsic electron-hole concentration is accurately described by the Boltzmann approximation with −1 共4兲 ca共1 − ␩q␥r兲BNcNv Eg − ⌬F N cN v = ln = ln , kT n2 ␩q PPL 1078 共6兲 where ng = no共Eg + kT兲 and ␣g = ␣o共Eg + kT兲 are the index of refraction and absorption coefficient at the average spontane- n2i = NcNve−Eg/kT = 32␲3共mcmv兲3/2 冉 冊 m · kT h2 3 e−Eg/kT , 共7兲 where m is the electron mass, mc = m*e / m and mv = m*h / m are dimensionless, and m*e and m*h are the electron and hole effective masses, respectively. From Eqs. 共6兲 and 共7兲 the radiative coefficient is B⬇ 冉 冊冉 冊 1 2␲ 2 h mc 3 冉 冊 2 n2gc␣g Eg + 1 , 共mcmv兲3/2 kT 共8兲 where the term 共h / mc兲3 has units of volume and is the Compton wavelength 共2.43 pm兲 cubed. The term n2g共mcmv兲−3/2c␣g is a rate 共⬃1018 s−1兲 that is only weakly dependent on temperature and that is roughly constant across most direct band gap III-V semiconductors. Furthermore, the radiative coefficient goes by inverse temperature squared and band gap energy squared 共which is weakly temperature dependent兲. From Eq. 共8兲, the absorption coefficient and the band gap energy are key material figures of merit for spontaneous emission devices. II. EXPERIMENTAL RESULTS The GaAs/ AlGaAs sample used in the PL measurements in this work was grown by MBE according to the cross section shown in Fig. 1, which is designed so that 共i兲 the additional carriers generated in the graded AlGaAs layers are funneled into the active region increasing the injection level, 共ii兲 the fraction of the pump power that contributes to the photoexcited electron-hole population is independent of temperature, and 共iii兲 the pump radiation that passes through the active region is absorbed in a low temperature grown AlGaAs beam stop layer. All layers other than the active region are grown at low temperature to reduce their luminescence efficiency, minimizing pumping of the active region by spontaneous emission from other layers, particularly the 2000 nm thick beam stop layer; this was verified by confirming that no signal was present in the PL spectrum in the J. Vac. Sci. Technol. B, Vol. 25, No. 3, May/Jun 2007 Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 209.147.144.20 On: Sat, 07 Feb 2015 23:28:34 1079 Johnson et al.: Excitation dependent photoluminescence measurements 1079 TABLE I. Summary of parameter values used in calculations. Parameter Symbol Value Source GaAs absorption coefficient at Eg + kT GaAs index of refraction at Eg + kT Dimensionless effective electron mass Dimensionless effective hole mass Photon recycling factor GaAs combined density of states GaAs zero temperature band gap Einstein model coupling constant Einstein temperature Pump energy per excitation event Detector signal per generated photon GaAs Auger coefficient Radiation coefficient model constant Radiation coefficient model temperature ␣g ng mc mv ␥r N cN v Eg共0兲 Sg TE ca cb C Bo Tb 9050 cm−1 3.63 0.067 0.51 0.248 3.98⫻ 1036 共T / 300兲3 cm−6 1.519 eV 5.982 262.9 K 1.99⫻ 10−23 mW s cm3 9.5⫻ 10−23 V s cm3 7 ⫻ 10−30 cm6 s−1 7.076⫻ 10−11 cm3 s−1 951.2 K Refs. 7 and 10 Ref. 11 Ref. 6 Ref. 6 Calculation Ref. 6 Ref. 12 Ref. 8 Ref. 8 Eq. 共9兲 Eq. 共3兲 Ref. 9 Fit Fit Vah␯HeNe , THeNe共1 − RHeNe兲AHeNe 共9兲 vicinity of the Al0.2Ga0.8As band gap energy. Temperature independent photoexcitation for a given pump power is accomplished by utilizing a 130 nm thick AlGaAs absorbing layer above the active region. Since the absorption coefficient of all layers is reduced at low temperature 共the material band gap increases兲, the thickness of this layer is selected so that the additional pump radiation that reaches the active region at low temperature precisely compensates for the reduced absorption. A thicker 共thinner兲 absorbing layer would result in the active region absorbing the largest fraction of pump radiation at low 共room兲 temperature. The fraction of the pump radiation that enters the sample and that is absorbed in each layer is given by the decimal number listed above each layer in Fig. 1, the upper number is for 300 K and the lower number is for 77 K; although the fraction absorbed in each layer varies with temperature, the total fractional contribution to the GaAs active layer remains constant. The absorption lengths for the 633 nm HeNe pump radiation used in the design are 1 / ␣GaAs共300 K兲 = 260 nm, 1 / ␣GaAs共77 K兲 = 300 nm, 1 / ␣AlGaAs共300 K兲 = 490 nm, and 1 / ␣AlGaAs共77 K兲 = 730 nm. The room temperature values were taken from Ref. 7 and the low temperature values were estimated from the room temperature values using the shift in band gap energy 共and hence absorption spectrum兲 with temperature. The linearly graded AlGaAs layers are grown at a constant growth rate of 16.0 nm/ min, with the GaAs 共AlAs兲 growth rate linearly varied from 12.8 nm/ min 共3.2 nm/ min兲 at the 20% Al mole fraction to 15.2 nm/ min 共0.8 nm/ min兲 at the 5% mole fraction. The Al0.4Ga0.6As layers are transparent to the pump radiation and serve as barriers to prevent carrier spillage from one region of the sample to another. Although not absolutely necessary, it is convenient to design the sample so that the photoexcitation density is independent of sample temperature, thus making the constant ca 共pump power/photoexcitation density兲 independent of temperature, with ca = where THeNe = 0.739 is the measured fraction of the HeNe pump power that is transmitted through the optical system into the cryostat and onto the sample; 1 − RHeNe = 0.65 is the fraction of THeNe that enters the sample 共calculated using the optical constants in Ref. 7 and which changes ⬍1% over the temperatures in this work兲; and AHeNe = 0.63 is the calculated fraction of THeNe共1 − RHeNe兲 that is actively absorbed 共which as mentioned above is roughly independent of temperature by design兲. The active volume of the photoexcited electronhole population is Va = ␲共DHeNe / 2兲2da = 1.92⫻ 10−8 cm3, where DHeNe = 350 ␮m is the pump spot diameter and da = 200 nm is the bulk GaAs layer thickness. The pump photon energy is h␯in = 3.14⫻ 10−16 mJ 共or mW s兲. Resulting in ca = 1.99⫻ 10−23 cm3 mW s, this and other parameters used in the calculations in this paper are listed in Table I. Pump power dependent PL measurements of bulk GaAs are performed on the MBE grown 共run B1657兲 sample using a HeNe laser with pump powers PPL ranging from 0.3 to 40 mW at temperatures T = 300, 230, 100, and 50 K. These results are plotted in terms of pump power versus integrated PL signal in Fig. 2; the solid curves are fits of Eq. 共2兲 to the data. Note that the injection level is not large enough to observe the 3 / 2 Auger power law dependence in the data, as Cn / B共1 − ␥r兲 Ⰶ 1; therefore, the coefficient CPL is negligible and cannot be accurately determined from these measurements. The best fit parameters APL and BPL are given in Table II. As the injection increases in Fig. 2, the radiative lifetime decreases, resulting in the convergence of the curves toward the unity quantum efficiency line. The curves also move toward this line as temperature decreases due to an increasing radiative coefficient. Furthermore, if the injection level was increased further the experimental curves will eventually move away from the unity quantum efficiency line as Auger recombination becomes the dominant process. JVST B - Microelectronics and Nanometer Structures Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 209.147.144.20 On: Sat, 07 Feb 2015 23:28:34 1080 Johnson et al.: Excitation dependent photoluminescence measurements FIG. 2. Pump power vs PL signal from bulk GaAs at various temperatures. The solid curves are fits of a power law model to the data 关see Eq. 共2兲兴. The PL signal LPL is obtained by integrating the PL spectrum 共in units of detector output signal per unit wavelength兲 over wavelength, following the normalization of the spectrum for any wavelength dependent variations in the optical throughput of the monochromator-detector system, which is useful when comparing signals at different sample temperatures because the emission wavelength changes with temperature. This result is then normalized by multiplying it by the relative changes in the inverse of the monochromator slit width squared, i.e., ⫻1 / 4 for 共0.5 mm/ 1.0 mm兲2; this compensates for changes in the optical throughput of the monochromator due to changes in slit width 共spectral resolution兲, which is typically reduced when sample temperature is reduced to avoid detector saturation from the larger low temperature luminescence intensities. The spectral resolution is 1.0 nm when the input and output slit widths are 1.0 mm. The spectral resolution was more than adequate to resolve the PL line shape for all slit widths used. Attentive tracking and adjustment for optical throughput variations result in an approximately constant value of cb = ca共1 − ␥r兲 / BPL = 9.5 ⫻ 10−23 V s cm3. III. DISCUSSION Using a physical model, namely, the Einstein single oscillator photon occupation model, to describe the temperature TABLE II. Experimental fitting parameters, SRH coefficients ascertained using fitting parameters, and calculated radiative coefficients. Temperature 共K兲 APL 共mW/ V1/2兲 BPL 共mW/V兲 50 100 230 300 0.0381 0.0886 0.2007 1.3396 0.1543 0.1571 0.1606 0.1562 A 共s−1兲 B 共cm3 s−1兲 3.01⫻ 106 2.55⫻ 10−8 3.46⫻ 106 6.33⫻ 10−9 3.29⫻ 106 1.14⫻ 10−9 1.67⫻ 107 6.41⫻ 10−10 1080 FIG. 3. Radiative coefficient vs temperature for GaAs; the solid gray curve is a calculation using Eq. 共10兲 and the solid black curve is a fit of the model shown in the plot to the calculated curve over the 30 to 320 K temperature range. dependence of the band gap energy, the radiative coefficient given in Eq. 共8兲 is explicitly written in terms of temperature as B⬇ 冉 冊冉 冊 冉冉 冊 冊 1 2␲ ⬇ Bo 2 h mc Tb T 2 3 冉 n2gc␣g Eg共0兲 SgTE/T − T /T +1 共mcmv兲3/2 kT e E −1 −1 , 冊 2 共10兲 where Eg共0兲 is the zero temperature band gap, Sg is a dimensionless coupling constant, and TE is the Einstein temperature where kTE / 共eTE/T − 1兲 is the average thermal energy of the phonon population.8 The radiative coefficient calculated using Eq. 共10兲 is shown in Fig. 3 as a solid gray curve and the parameters used in the calculation are listed in Table I. The specific temperature dependence of the radiative coefficient over the temperature range of interest is determined by fitting an empirical two parameter model 关see right hand equation in Eq. 共10兲兴 to the calculated values, which is shown as the solid black curve that nicely overlaps the calculated curve in Fig. 3. The best fit parameters over the 30– 320 K temperature range are Bo = 7.076⫻ 10−11 cm3 s−1 and Tb = 951.2 K. The values of the radiative coefficient, which vary by a factor of 40 over the temperature range of the measurements, are given in Table II; this large variation is a major contributor to the improvement in quantum efficiency at low temperature. The SRH coefficients A are calculated using Eq. 共3兲 and are given in Table II. The spontaneous emission quantum efficiency ␩q is calculated using the fitting parameters APL and BPL, Eq. 共4兲, and the integrated PL signal LPL and is plotted against injection level, 共Eg − ⌬F兲 / kT, in Fig. 4; 共Eg − ⌬F兲 / kT is calculated using Eq. 共5兲 and the pump power PPL. The solid curves are fits of the model, J. Vac. Sci. Technol. B, Vol. 25, No. 3, May/Jun 2007 Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 209.147.144.20 On: Sat, 07 Feb 2015 23:28:34 1081 Johnson et al.: Excitation dependent photoluminescence measurements FIG. 4. Spontaneous emission quantum efficiency 共QE兲 vs injection level; the peak quantum efficiency region is shown in the inset. The solid curves are fits of Eq. 共11兲 to the data. 冉 冑 A ␩q = 1 + B e共Eg−⌬F兲/kT C + N cN v B 冑 N cN v e共Eg−⌬F兲/kT 冊 −1 共11兲 , to the data, where A is a fitting parameter and C = 7 ⫻ 10−30 cm6 s−1 is fixed at a recent value given in the literature.9 The values for A obtained from these fits are within 1% of those calculated using Eq. 共3兲. The peak quantum efficiency region is shown in the inset of Fig. 4. It should be noted that in this analysis 共Eg − ⌬F兲 / kT is calculated from both LPL and PPL since ␩q appears in Eq. 共5兲; to write 共Eg − ⌬F兲 / kT strictly in terms of PPL the cubic relation given in Eq. 共1兲 must be used, which in general results in a much more complex expression. However, since CPL does not appear in the present measurements, the injection level in terms of PPL alone can be written in relatively simple terms as 冉 冊 Eg − ⌬F N cN v = ln kT n2 冉 = ln 4BPL 2 APL ca共1 − ␥r兲BNcNv 2 共冑1 + 共4BPL/APL 兲PPL − 1兲2 冊 . 共12兲 The injection levels obtained from Eq. 共12兲 are essentially the same as those obtained from Eq. 共5兲. The experimentally determined SRH coefficient as a function of sample temperature is plotted in Fig. 5, where the solid blue curve is a fit of an Arrhenius model to the data; the model is given in the plot. This data indicates that there are two components to the SRH coefficient 共or nonradiative lifetime兲: 共i兲 a constant component A0 = 3.2⫻ 106 s−1 共1 / A0 = 310 ns兲 that most likely originates from deep levels 共traps兲 in the bulk GaAs active layer and 共ii兲 a temperature dependent component Aae−Ea/kT with frequency Aa = 4.9⫻ 1014 s−1 and characteristic activation energy Ea = 450 meV that rapidly increases near room temperature and that most likely originates from nonradiative recombination in the AlGaAs barriers. 1081 FIG. 5. SRH coefficient vs sample temperature; the solid curve is a fit to the data of the model shown in the plot. The sample design and measurement approach presented here makes the analysis straightforward. However, it is not absolutely necessary, since the above spontaneous emission quantum efficiency and lifetime measurements can be performed on most luminescence samples, where the constant of proportionality ca may have to be determined separately for each temperature and the fitting constant BPL may differ for each temperature if variations in the optical throughput are not monitored and corrected for. Furthermore, if sufficient pump power is available, higher injection can be achieved without funneling additional carriers into the active layer, eliminating the need for the graded AlGaAs layers. The beam stop layer can also be omitted provided a very thin GaAs buffer 共⬃20 nm兲 is grown on a semi-insulating substrate so that an insignificant amount of luminescence is generated in the buffer compared to the active region. Moreover, it is best to minimize photon recycling since its effect must be calculated and hence can add uncertainty to the determination of the nonradiative coefficients. The photon recycling factor ␥r can be as small as 0.1 for thin 共quantum well兲 active layers grown on semi-insulating substrates which parasitically absorb spontaneous emission without any further emission that can be reabsorbed by the active region. The definition and role of the photon recycling factor is further discussed in Ref. 5. IV. CONCLUSIONS The nonradiative lifetime and spontaneous emission quantum efficiency of bulk GaAs is determined using temperature and excitation density dependent photoluminescence. The quantum efficiency is inferred from the power law relation between integrated PL intensity and pump power, where the PL intensity increases as the square of the pump power when SRH recombination dominates, linearly when the quantum efficiency is unity, and sublinearly when Auger recombination dominates. At temperatures below 230 K the nonradiative lifetime of bulk GaAs is 310 ns, while at temperatures JVST B - Microelectronics and Nanometer Structures Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 209.147.144.20 On: Sat, 07 Feb 2015 23:28:34 1082 Johnson et al.: Excitation dependent photoluminescence measurements above 230 K the nonradiative lifetime rapidly decreases to 60 ns at 300 K. The temperature dependent part of the nonradiative lifetime is attributed to deep levels in the AlGaAs barriers. The peak spontaneous emission quantum efficiency is determined to be ⬎0.96 at 300 K and ⬎0.99 at temperatures below 230 K. ACKNOWLEDGMENT This work is supported by a MURI program from the Air Force Office of Scientific Research Grant No. FA9550-04-10374. 1 T. Fleck, M. Schmidt, and C. Klingshirn, Phys. Status Solidi A 198, 248 共2003兲. 1082 2 R. Westphaling, P. Ullrich, J. Hoffmann, H. Kalt, C. Klingshirn, K. Ohkawa, and D. 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