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Excitonic polariton structures in Wurtzite GaN
Physica B 302–303 (2001) 268–276
Excitonic polariton structures in Wurtzite GaN Kousuke Toriia,1, Shigefusa F. Chichibua,*, Takahiro Deguchib,2, Hisayuki Nakanishic, Takayuki Sotab, Shuji Nakamurad,3 a Institute of Applied Physics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan Department of Electrical, Electronics, and Computer Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku, Tokyo 169-8555, Japan c Department of Electrical Engineering, Science University of Tokyo, 2641 Yamazaki, Noda, Chiba 278-8510, Japan d Department of Research and Development, Nichia Chemical Industries Ltd., 491 Oka, Kaminaka, Anan, Tokushima 774-8601, Japan
b
Abstract Reflectance, photoreflectance and photoluminescence spectra of GaN substrate prepared by lateral epitaxial overgrowth technique were measured at low temperature. All spectra were discussed based on model exciton-polariton picture, and the spectra were successfully explained by the model. Temperature dependence of transition energies was well described using a model that assume Einstein phonons. # 2001 Elsevier Science B.V. All rights reserved. PACS: 78.20.e; 78.20.Jq; 78.66.Fd Keywords: GaN; Excitonic polariton; Lateral epitaxial overgrowth
1. Introduction The exciton polariton is an elementary excitation consisting of free excitons and electromagnetic waves, and was proposed by Hopfield [1]. It is known that the exciton-polariton dominates optical transitions near the band edge in direct bandgap semiconductors such as GaAs [2,3] (for a review see Excitons in Ref. [4]), CdS [4–6], and so on. Polariton effects in GaN have been studied by *Corresponding author. Tel.: +81-298-53-5289; fax: +81298-53-5205. E-mail address: [email protected] (S.F. Chichibu). 1 On leave from Waseda University. 2 Present address: Department of Electrical Engineering, Science University of Tokyo, 2641 Yamazaki, Noda, Chiba 278-8510, Japan. 3 Present address: Department of Materials Engineering, University of California, Santa Barbara, CA 93106, USA.
some groups [7–10]. Gil et al. [7] have discussed emission and reflection spectra at T ¼ 2 K from GaN epitaxial layer, which had been grown on sapphire substrates, within coupled A, B, and C exciton-polariton pictures assuming optical isotropy. The GaN epitaxial layer grown on sapphire substrates (GaN/sapphire) suffers residual biaxial compressive strain due to difference in thermal expansion coefficients between GaN and sapphire, though a part of strain is relaxed emitting highdensity misfit dislocations. Stepniewski et al. [8] have measured reflectance and emission spectra of high-quality, unstrained homoepitaxial layers on GaN bulk crystal, which was grown using very high pressure of nitrogen, at T ¼ 1:8 K, and have analyzed by exciton-polariton picture assuming optical isotropy, where the background dielectric function including contributions from the exciton excited states and from continuum states has been
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K. Torii et al. / Physica B 302–303 (2001) 268–276
fully taken into account, as well as the existence of surface exciton free layer. They have succeeded to reproduce reflectance spectrum by calculated one, but their discussions on the emission properties was not yet insufficient because of poor spectral resolution due to high concentration of residual impurities. Our group [9,10] has also measured low temperature reflectance [9], emission [9,10], and photoreflectance (PR) [10] spectra of high quality GaN substrates grown by lateral epitaxial overgrowth (LEO) technique, and succeeded to describe the emission spectra by a model excitonpolariton picture, where the optical anisotropy and existence of surface exciton free layer have been fully taken into account. Recently, high quality GaN substrates have been prepared by several modifications of LEO techniques [11–14] during metalorganic vapor phase epitaxy (MOVPE). In such GaN substrate, nearly threading dislocation (TD)-free region was grown laterally over patterned masks on GaN/ sapphire. One can research intrinsic material physic using such high quality GaN substrate with reduced TD and residual impurity density. Yamaguchi et al. [15,16] have studied reflectance and photoluminescence (PL) spectra of thick GaN and GaN substrate prepared by halide VPE-LEO to determine some valence band parameters. Mukai et al. [17] and Chichibu et al. [18] have shown that emission intensity and fundamental emission mechanisms in InGaN QWs having higher InN mole fraction were insensitive to the TD density by comparing QWs grown on LEO GaN base. In this paper, reflectance, PL, and PR spectra of high quality wurtzite GaN substrate, which was prepared by MOVPE-LEO with subsequent sapphire removal, are reviewed. The spectra were analyzed based on a model exciton-polariton picture in which A, B, and C excitons couple simultaneously to an electromagnetic wave, where effective-mass anisotropy and optical anisotropy were take into account. To determine unknown physical parameters, measured reflectance spectra was fitted by calculated one. The lifetime of exciton-polariton branch was calculated from the imaginary part of the wave vectors of corresponding polariton. Subsequently, PL and PR spectra were interpreted. It is found that the PL spectra
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are explained as the emission of excitonic polaritons accumulated at bottleneck portions of exciton-polariton branches, and PR signal is sensitive to bottleneck portions of exciton-polariton branches. Temperature dependence of the PL peak energies and transition energies estimated from the PR spectra are also analyzed by a model assuming Einstein phonons.
2. Experiments We measured reflectance, PR and PL spectra of GaN substrate. The GaN substrate was grown using the two-flow MOVPE method [19]. Selective growth of GaN was carried out on a 2 mm-thick GaN layer grown on a (0 0 0 1) C-face sapphire substrate under a low pressure of 100 Torr. The 0.1 mm-thick SiO2 mask was patterned to form 4 mm-wide stripe windows with a periodicity of 11 mm in GaN h1 1% 0 0i direction. Following the 10 mm-thick GaN growth on the SiO2 mask patterns, the coalescence of the selectively grown GaN made it possible to achieve a flat GaN surface over the entire substrate. After obtaining a 10 mm LEO GaN, the GaN growth was continued up to a 100 mm thickness. After that the sapphire substrate and the LEO GaN with SiO2 mask patterns were removed by polishing to obtain freestanding GaN with a thickness of approximately 70 mm. No film bending was observed. The lattice ( and c ¼ constants are a ¼3.1898 0.0002 A ( and crystal is nearly strain5.1855 0.0002 A, free. The residual electron density at 300 K was estimated to be smaller than 1014 cm3 by Hall effect measurement. This value is a limit of our equipment. Nonradiative defect density was characterized by time-resolved PL measurement. This sample exhibited biexponential-type decay. The decay constants are t1 ¼ 130 ps and t2 ¼ 860 ps at 300 K [18], and this value is larger than that of standard 1 mm-thick GaN/sapphire, t1 ¼ 80 ps and t2 ¼ 260 ps. This means that nonradiative defect density is quite small in our sample. Details of the preparation procedures and structural properties are given Refs. [9,14,20]. Reflectance spectra were measured using a 150 W Xe lamp as a light source at 10 K with
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incident angle smaller than 78. Polarized reflectance spectra with incident angle 508 were also measured using Glan-Thomas prism polaziser. Depolarized reflected light was dispersed by an f ¼ 133 cm focal length grating monochromator. For PL spectra, photons with energy larger than 3.2 eV from a Xe lamp, second-harmonic photons of 3.52 eV from a Ti : sapphire laser with a repetition rate of 80 MHz and a pulse duration of 100 fs, and photons of 3.81 eV from cw He–Cd laser were used as excitation scources. PL spectra, using Xe lamp and Ti : sapphire as an excitor, were measured at 10 K, and emission light were dispersed by an f ¼ 133 cm focal length grating monochromator. PL spectra using He–Cd laser were measured as a function of temperature, and emission light were dispersed by an f ¼ 67 cm focal length grating monochromator. PR spectra were measured as a function of temperature using the 325.0 nm line of a cw He–Cd laser as a modulation source. A Xe lamp was used as probe light, and the reflected light was dispersed by an f ¼ 67 cm focal length grating monochromator. All signals were detected by a photomultiplier using phase sensitive detection. The resolution was less than 0.1 meV for 133 cm monochromator, and 0.3 meV for 67 cm monochromator.
3. Formulation For the exciton, spatial dispersed oscilator, in the uniaxial crystal belonging to the space group C46v , the equation of motion for polarization is given by " # ! k2k q2 k2? q 2 þ o0 þ o0 h þ þ G P ¼ o20 aE; qt qt2 M? Mk ð1Þ where k, o0 , M, G, a, P, and E are wave vector, exciton resonance frequency, exciton mass, damping constant, polarizability tensor, polarization due to exciton, and electric field of light respectively, and ?(k) denote perpendicular (parallel) component to c-axis. From Eq. (1) and relation eE ¼ e0 E þ 4pP;
ð2Þ
dielectric tensor e is determined, where e0 is background dielectric tensor. For uniaxial crystal, e (e0 ) can be written by 0 1 e? B C e¼@ ð3Þ e? A: ek For GaN, taking into account the symmetry of valence bands [21], the dielectric function can be written by e? ðk; oÞ ¼ e0? þ
X n¼B; C
Sn? o20n o2
þ
o20n
k2k k2? þ o0n h þ Mn? Mnk
!
; iGn
ð4aÞ ek ðk; oÞ ¼ e0k þ
X n¼A; B; C
Snk o20n o2
þ
o20n
k2k k2? þ o0n h þ Mn? Mnk
!
; iGn
ð4bÞ where S ¼ 4pa is the oscillator strength, o is the angular frequency of polariton, and n denote A, B and C exciton. To reproduce the experimental results (experimental results exhibits first excited state of A exciton, An¼2 ), we took e0 including excited exciton states and continuum states [8,22]. The implicit dispersion relations of polariton are given by o2 e? ðk; oÞ k2 ¼ 0; c2 for the ordinary wave, and k2k o2 k2? ¼ 0; c2 ek ðk; oÞ e? ðk; oÞ
ð5aÞ
ð5bÞ
for the extraordinary wave, where c is the velocity of light in vacuum. Applying the electromagnetic boundary conditions and the additional boundary conditions (in the present case continuity of vector of excitonic polarization [23] were used), the dispersion relation of polariton can be calculated from Eqs. (4a), (4b), (5a) and (5b) numerically. Once the dispersion relation is obtained, the effective refractive index n% of GaN is determined.
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Taking into account an exciton-free surface layer, the so-called ‘‘dead layer’’, [1] the effective reflactive index are modified. From the effective refractive index, reflection coefficient R is calculated as 1 n% 2 : R¼ ð6Þ 1 þ n% The imaginary part of wave vector means attenuation of the wave. So we estimated the polariton lifetime t from t¼
1 ; k i vg
ð7Þ
where ki is the imaginary part of the wave vector and vg is the group velocity of the corresponding polaritons obtained as vg ¼
qo ; qkr
ð8Þ
with the real part of the wave vector, kr .
4. Results and discussions 4.1. Reflectance spectra In Fig. 1, reflectance spectrum with incident angle 78 (middle panel), S polarized one with incident angle 508 (top panel), and P polarized one with incident angle 508 (bottom panel) are shown by open circles. All spectra exhibit reflection anomalies corresponding to the ground state of A, B, and C free excitons and the first excited states of A exciton (An¼2 ). To obtain physical parameters, experimental spectra were fitted by theoretical spectra using Eq. (6). Fitting procedures were carried out to reproduce the near normal incidence reflectance and polarized reflectance spectra using the same parameters. The best-fit curves are shown in Fig. 1 by solid curves. Satisfactory agreement is achieved. Parameters obtained are summarized in Table 1. 4.2. PL spectra The PL spectra are shown in upper panel of Fig. 2. All spectra exhibit a neutral donor bound
Fig. 1. Reflectance spectra from high quality wurtzite GaN substrate. Middle panel shows near normal incidence (incident angle about 78) reflectance spectrum. Top and bottom panel shows S or P polarized reflectance spectra with incident angle 508.
exciton peak I2 and a small impurity-related peak ‘a’ [24] at the same energy, respectively. In He–Cd excited spectrum, peaks corresponding to A, B, and An¼2 excitons are observed. In Xe excited and Ti : sapphire excited spectra, the peak corresponding to A excitons split into two peaks. To explain these peaks, we estimate the lifetimes of excitonpolariton branches by Eq. (7) using physical parameters obtained by fitting procedures. Resultant polariton lifetimes are shown in lower panel of Fig. 2. It is known that exciton-polariton emission occurs at bottleneck energies where the lifetime of polariton approaches their maximum [25]. In Fig. 2, the peak energies related to A and B excitons in Xe excited spectrum are denoted by vertical dashed line. It is find that, lower and upper
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Table 1 Values of physical parameters obtained through the fitting procedure of the reflectance spectra. Eg is bandgap and m0 is electrom mass Parameters
ho0 (eV) S? Sk G (meV) M? =m0 Mk =m0 Eg (eV) Gðn > 1Þ e0? e0k
a
Exciton A
B
C
3.4791 0.0002 0.0066 0.0006 0 0.20 0.01 0.5a 1.1 0.1 3.5025 1.8 0.1
3.4844 0.0002 0.0058 0.0006 0.0056 0.0008 0.70 0.03 0.6a 1.0 0.1 3.5080 1.5a Background dielectric constants 8.6 10.1 Dead layer depth 6 (nm)
3.5027 0.0002 0.0013 0.0006 0.0014a 1.20 0.05 0.8a 0.4a } }
The reflectance spectrum is not sensitive to the parameters.
peaks related to A exciton are located at the energies, where the lifetime of lower branch of A excitonic polariton (LPBA ) and of a branch combined upper branch of A excitonic polariton and lower branch of B excitonic polariton (UPBA þLPBB ) approach their maximum, respectively. The peak related to B exciton is also located at where the lifetime of UPBB þLPBC approach its maximum. Thus, these peaks are due to LPBA , UPBA þLPBB , and UPBB þLPBC . The lower peak energy related to A exciton for Ti : sapphire excited PL spectrum agrees with that for Xe excited spectrum. But the upper peak related to A exciton and the peak related to B exciton for Ti : sapphire excited spectrum are located at slightly higher energy than those for Xe excited spectrum. This indicates that many more excitonic polaritons accumulate at the bottleneck portion under the high power excitation condition. The reason why lower peak related to A exciton do not shift to the higher energy is probably that the curvature of the corresponding dispersion relation are smaller than for remaining two. In He–Cd excited spectrum, there is only one peak related to A exciton, while in Xe and Ti : sapphire excited spectrum, there are two. But, the full width at half maximum (FWHM) of this
peak is about four times broader than that of I2 , and this peak can not be fitted by single Lorentzian line-shape function. So, this peak is assigned to convolution of upper and lower A excitonic polaritons. The reason why these peaks are not resolved may be due to the use of a He–Cd laser having high-energy photons that produces excitons having large momentum. From the viewpoint of excitonic polaritons, one more PL peak related to B exciton should exsist at lower energy side of observed peak related to B exciton. But in this study, such peak is not observed clearly for weak intensity. 4.3. PR spectrum The PR spectrum is shown in upper panel of Fig. 3. Reflection anomalies corresponding to A, B, C, and An¼2 excitons are found. PR is a photomodulated version of an electroreflectance, thus the PR spectrum was analyzed by the low-field ER lineshape function [26] ( ) p X
mj DR iyj ¼ Re ; ð9Þ Cj e E Eg; j þ iGj R j¼1 where p is the number of the spectral function to be fitted, E is the phonon energy, and Cj , yj , Eg;j
K. Torii et al. / Physica B 302–303 (2001) 268–276
273
Fig. 3. PR spectrum from high quarity wurtzite GaN substrate (upper panel), and the lifetime of exciton-polariton branches (lower panel). The PR spectrum exhibit reflectance anomalies corresponding to A, B, C, and An¼2 excitons. The lifetime is the same with those in Fig. 2.
Fig. 2. PL spectra from high quality wurtzite GaN substrate (upper panel), and the lifetime of exciton-polariton branches (lower panel). The PL spectra shows complex structure around 3.48 eV. In lower panel, LPB and UPB denote lower polariton branch and upper polariton branch, respectively, and subscripts denote A, B, or C exciton.
and Gj are amplitude, phase, energy and broadening parameter of the jth feature, respectively. We tentatively assumed that the transitions are due to free exciton resonance (mj ¼ 2). Obtained transition energies are summarized in Table 2 with exciton energies obtained from reflectance spectra, and shown by vertical dashed lines in Fig. 3. Exciton energies obtained from reflectance spectra
and transition energies obtained from PR spectrum are different from each other. Comparing the lifetime of polariton (shown in lower panel of Fig. 3) and the PR spectrum, it is found that the PR transition energies agree with the bottleneck energies of excitonic polaritons. This means PR monitors exciton-polaritons at low temperature. It is natural to admit that PR monitors excitonpolaritons rather than exciton or free carriers since the PR signal is deduced as [26] DR 2na ¼ Re ¼ Re½ða ibÞDe R nð e e a Þ ¼ aDe1 þ bDe2 ;
ð10Þ
where n2 ¼ e, n2a ¼ ea , na is the real reflactive index, De ¼ De1 þ iDe2 is the perturbation-induces change in the dielectric function e, and a and b are Seraphin coefficients. Eq. (10) means that the signal DR=R is purely sensitive to De.
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Table 2 Values of free exciton energies obtained from the fitting procedure of the OR spectra and transition energies obtained from the analysis of PR spectrum assuming the transitions as pure free excitons for the GaN substrate at 10 K Method
OR PR
Parameter
Exciton resonance energy (eV) Transition energy (eV)
Exciton/transition A
B
C
3.4791 0.0002 3.4803 0.0003
3.4844 0.0002 3.4852 0.0003
3.5027 0.0002 3.5033 0.0003
Fig. 5. Temperature dependence of PL peak energies and PR transition energies. Solid curve is theoretical curve calculated by assuming Einstein phonon. Good agreement between experimental data and theory is obtained.
Fig. 4. PR (upper panel) and PL (lower panel) spectra as a function of temperature. The PL spectra are dominated by exciton up to RT.
4.4. Temperature dependence Fig. 4 shows PR (upper panel) and PL (lower panel) spectra of GaN as a function of temperature. The A and B excitonic transitions are well
recognized up to 300 K. The PL spectra are dominated by A and B free exciton emissions above 75 K. Surprisingly, recombination of the first excited states of A excitons is recognized up to 200 K. In Fig. 5, transition energies obtained from the PR spectra and PL peak energies are plotted as a function of temperature. All plots show the same temperature dependence. It has been shown [27] that in polar semiconductors that have optical phonon branches energetically next to the acoustic ones, fitting of temperature variation of a bandgap using the empirical equation proposed by Cody [28], which assumes Einstein phonons, is better
K. Torii et al. / Physica B 302–303 (2001) 268–276
than that using the Varshni fitting [29], which assumes Debye phonons. Therefore, we fit A transition energy EA ðTÞ by the equation after Cody [28] as EA ðTÞ ¼ 3:480 0:162=½expð366=TÞ 1 ;
ð11Þ
where T is temperature, A transition energy at T ¼ 0 K is 3.480 eV and Einstein characteristic temperature YE is 366 K. Calculated curve is shown in Fig. 5 by dotted line. The obtained Einstein characteristic temperature corresponds to the energy maximum of the low-energy group of bulk phonon density of states in wurtzite GaN reported in Ref. [30] (260 cm1 or 370 K).
5. Conclusions OR, PR and PL spectra of high quality wurtzite GaN substrate prepared by MOCVD-LEO were measured. To understand the spectra, theoretical calculations based on exciton-polariton picture was carried out. The complex PL structure was explained by a model based on exciton-polariton picture. The PR method was shown to be sensitive to bottleneck of excitonic polariton branches rather than to free exciton resonance. The GaN substrate exhibited excitonic emissions up to RT. Temperature dependence of A transition energy was well discribed using a model, which assumes Eistein phonons.
Acknowledgements The authors are grateful to M. Sugiyama and R. Nakai for their help in experiment. One of us (S.F.C.) is thankful to Professor Fumio Hasedawa, Professor Steven P. DenBaars, Professor Umesh K. Mishra and Professor Jacques I. Pankove for continuous encouragements.
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Excitonic polariton structures in Wurtzite GaN Kousuke Toriia,1, Shigefusa F. Chichibua,*, Takahiro Deguchib,2, Hisayuki Nakanishic, Takayuki Sotab, Shuji Nakamurad,3 a Institute of Applied Physics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan Department of Electrical, Electronics, and Computer Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku, Tokyo 169-8555, Japan c Department of Electrical Engineering, Science University of Tokyo, 2641 Yamazaki, Noda, Chiba 278-8510, Japan d Department of Research and Development, Nichia Chemical Industries Ltd., 491 Oka, Kaminaka, Anan, Tokushima 774-8601, Japan
b
Abstract Reflectance, photoreflectance and photoluminescence spectra of GaN substrate prepared by lateral epitaxial overgrowth technique were measured at low temperature. All spectra were discussed based on model exciton-polariton picture, and the spectra were successfully explained by the model. Temperature dependence of transition energies was well described using a model that assume Einstein phonons. # 2001 Elsevier Science B.V. All rights reserved. PACS: 78.20.e; 78.20.Jq; 78.66.Fd Keywords: GaN; Excitonic polariton; Lateral epitaxial overgrowth
1. Introduction The exciton polariton is an elementary excitation consisting of free excitons and electromagnetic waves, and was proposed by Hopfield [1]. It is known that the exciton-polariton dominates optical transitions near the band edge in direct bandgap semiconductors such as GaAs [2,3] (for a review see Excitons in Ref. [4]), CdS [4–6], and so on. Polariton effects in GaN have been studied by *Corresponding author. Tel.: +81-298-53-5289; fax: +81298-53-5205. E-mail address: [email protected] (S.F. Chichibu). 1 On leave from Waseda University. 2 Present address: Department of Electrical Engineering, Science University of Tokyo, 2641 Yamazaki, Noda, Chiba 278-8510, Japan. 3 Present address: Department of Materials Engineering, University of California, Santa Barbara, CA 93106, USA.
some groups [7–10]. Gil et al. [7] have discussed emission and reflection spectra at T ¼ 2 K from GaN epitaxial layer, which had been grown on sapphire substrates, within coupled A, B, and C exciton-polariton pictures assuming optical isotropy. The GaN epitaxial layer grown on sapphire substrates (GaN/sapphire) suffers residual biaxial compressive strain due to difference in thermal expansion coefficients between GaN and sapphire, though a part of strain is relaxed emitting highdensity misfit dislocations. Stepniewski et al. [8] have measured reflectance and emission spectra of high-quality, unstrained homoepitaxial layers on GaN bulk crystal, which was grown using very high pressure of nitrogen, at T ¼ 1:8 K, and have analyzed by exciton-polariton picture assuming optical isotropy, where the background dielectric function including contributions from the exciton excited states and from continuum states has been
0921-4526/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 4 4 0 - 9
K. Torii et al. / Physica B 302–303 (2001) 268–276
fully taken into account, as well as the existence of surface exciton free layer. They have succeeded to reproduce reflectance spectrum by calculated one, but their discussions on the emission properties was not yet insufficient because of poor spectral resolution due to high concentration of residual impurities. Our group [9,10] has also measured low temperature reflectance [9], emission [9,10], and photoreflectance (PR) [10] spectra of high quality GaN substrates grown by lateral epitaxial overgrowth (LEO) technique, and succeeded to describe the emission spectra by a model excitonpolariton picture, where the optical anisotropy and existence of surface exciton free layer have been fully taken into account. Recently, high quality GaN substrates have been prepared by several modifications of LEO techniques [11–14] during metalorganic vapor phase epitaxy (MOVPE). In such GaN substrate, nearly threading dislocation (TD)-free region was grown laterally over patterned masks on GaN/ sapphire. One can research intrinsic material physic using such high quality GaN substrate with reduced TD and residual impurity density. Yamaguchi et al. [15,16] have studied reflectance and photoluminescence (PL) spectra of thick GaN and GaN substrate prepared by halide VPE-LEO to determine some valence band parameters. Mukai et al. [17] and Chichibu et al. [18] have shown that emission intensity and fundamental emission mechanisms in InGaN QWs having higher InN mole fraction were insensitive to the TD density by comparing QWs grown on LEO GaN base. In this paper, reflectance, PL, and PR spectra of high quality wurtzite GaN substrate, which was prepared by MOVPE-LEO with subsequent sapphire removal, are reviewed. The spectra were analyzed based on a model exciton-polariton picture in which A, B, and C excitons couple simultaneously to an electromagnetic wave, where effective-mass anisotropy and optical anisotropy were take into account. To determine unknown physical parameters, measured reflectance spectra was fitted by calculated one. The lifetime of exciton-polariton branch was calculated from the imaginary part of the wave vectors of corresponding polariton. Subsequently, PL and PR spectra were interpreted. It is found that the PL spectra
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are explained as the emission of excitonic polaritons accumulated at bottleneck portions of exciton-polariton branches, and PR signal is sensitive to bottleneck portions of exciton-polariton branches. Temperature dependence of the PL peak energies and transition energies estimated from the PR spectra are also analyzed by a model assuming Einstein phonons.
2. Experiments We measured reflectance, PR and PL spectra of GaN substrate. The GaN substrate was grown using the two-flow MOVPE method [19]. Selective growth of GaN was carried out on a 2 mm-thick GaN layer grown on a (0 0 0 1) C-face sapphire substrate under a low pressure of 100 Torr. The 0.1 mm-thick SiO2 mask was patterned to form 4 mm-wide stripe windows with a periodicity of 11 mm in GaN h1 1% 0 0i direction. Following the 10 mm-thick GaN growth on the SiO2 mask patterns, the coalescence of the selectively grown GaN made it possible to achieve a flat GaN surface over the entire substrate. After obtaining a 10 mm LEO GaN, the GaN growth was continued up to a 100 mm thickness. After that the sapphire substrate and the LEO GaN with SiO2 mask patterns were removed by polishing to obtain freestanding GaN with a thickness of approximately 70 mm. No film bending was observed. The lattice ( and c ¼ constants are a ¼3.1898 0.0002 A ( and crystal is nearly strain5.1855 0.0002 A, free. The residual electron density at 300 K was estimated to be smaller than 1014 cm3 by Hall effect measurement. This value is a limit of our equipment. Nonradiative defect density was characterized by time-resolved PL measurement. This sample exhibited biexponential-type decay. The decay constants are t1 ¼ 130 ps and t2 ¼ 860 ps at 300 K [18], and this value is larger than that of standard 1 mm-thick GaN/sapphire, t1 ¼ 80 ps and t2 ¼ 260 ps. This means that nonradiative defect density is quite small in our sample. Details of the preparation procedures and structural properties are given Refs. [9,14,20]. Reflectance spectra were measured using a 150 W Xe lamp as a light source at 10 K with
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incident angle smaller than 78. Polarized reflectance spectra with incident angle 508 were also measured using Glan-Thomas prism polaziser. Depolarized reflected light was dispersed by an f ¼ 133 cm focal length grating monochromator. For PL spectra, photons with energy larger than 3.2 eV from a Xe lamp, second-harmonic photons of 3.52 eV from a Ti : sapphire laser with a repetition rate of 80 MHz and a pulse duration of 100 fs, and photons of 3.81 eV from cw He–Cd laser were used as excitation scources. PL spectra, using Xe lamp and Ti : sapphire as an excitor, were measured at 10 K, and emission light were dispersed by an f ¼ 133 cm focal length grating monochromator. PL spectra using He–Cd laser were measured as a function of temperature, and emission light were dispersed by an f ¼ 67 cm focal length grating monochromator. PR spectra were measured as a function of temperature using the 325.0 nm line of a cw He–Cd laser as a modulation source. A Xe lamp was used as probe light, and the reflected light was dispersed by an f ¼ 67 cm focal length grating monochromator. All signals were detected by a photomultiplier using phase sensitive detection. The resolution was less than 0.1 meV for 133 cm monochromator, and 0.3 meV for 67 cm monochromator.
3. Formulation For the exciton, spatial dispersed oscilator, in the uniaxial crystal belonging to the space group C46v , the equation of motion for polarization is given by " # ! k2k q2 k2? q 2 þ o0 þ o0 h þ þ G P ¼ o20 aE; qt qt2 M? Mk ð1Þ where k, o0 , M, G, a, P, and E are wave vector, exciton resonance frequency, exciton mass, damping constant, polarizability tensor, polarization due to exciton, and electric field of light respectively, and ?(k) denote perpendicular (parallel) component to c-axis. From Eq. (1) and relation eE ¼ e0 E þ 4pP;
ð2Þ
dielectric tensor e is determined, where e0 is background dielectric tensor. For uniaxial crystal, e (e0 ) can be written by 0 1 e? B C e¼@ ð3Þ e? A: ek For GaN, taking into account the symmetry of valence bands [21], the dielectric function can be written by e? ðk; oÞ ¼ e0? þ
X n¼B; C
Sn? o20n o2
þ
o20n
k2k k2? þ o0n h þ Mn? Mnk
!
; iGn
ð4aÞ ek ðk; oÞ ¼ e0k þ
X n¼A; B; C
Snk o20n o2
þ
o20n
k2k k2? þ o0n h þ Mn? Mnk
!
; iGn
ð4bÞ where S ¼ 4pa is the oscillator strength, o is the angular frequency of polariton, and n denote A, B and C exciton. To reproduce the experimental results (experimental results exhibits first excited state of A exciton, An¼2 ), we took e0 including excited exciton states and continuum states [8,22]. The implicit dispersion relations of polariton are given by o2 e? ðk; oÞ k2 ¼ 0; c2 for the ordinary wave, and k2k o2 k2? ¼ 0; c2 ek ðk; oÞ e? ðk; oÞ
ð5aÞ
ð5bÞ
for the extraordinary wave, where c is the velocity of light in vacuum. Applying the electromagnetic boundary conditions and the additional boundary conditions (in the present case continuity of vector of excitonic polarization [23] were used), the dispersion relation of polariton can be calculated from Eqs. (4a), (4b), (5a) and (5b) numerically. Once the dispersion relation is obtained, the effective refractive index n% of GaN is determined.
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Taking into account an exciton-free surface layer, the so-called ‘‘dead layer’’, [1] the effective reflactive index are modified. From the effective refractive index, reflection coefficient R is calculated as 1 n% 2 : R¼ ð6Þ 1 þ n% The imaginary part of wave vector means attenuation of the wave. So we estimated the polariton lifetime t from t¼
1 ; k i vg
ð7Þ
where ki is the imaginary part of the wave vector and vg is the group velocity of the corresponding polaritons obtained as vg ¼
qo ; qkr
ð8Þ
with the real part of the wave vector, kr .
4. Results and discussions 4.1. Reflectance spectra In Fig. 1, reflectance spectrum with incident angle 78 (middle panel), S polarized one with incident angle 508 (top panel), and P polarized one with incident angle 508 (bottom panel) are shown by open circles. All spectra exhibit reflection anomalies corresponding to the ground state of A, B, and C free excitons and the first excited states of A exciton (An¼2 ). To obtain physical parameters, experimental spectra were fitted by theoretical spectra using Eq. (6). Fitting procedures were carried out to reproduce the near normal incidence reflectance and polarized reflectance spectra using the same parameters. The best-fit curves are shown in Fig. 1 by solid curves. Satisfactory agreement is achieved. Parameters obtained are summarized in Table 1. 4.2. PL spectra The PL spectra are shown in upper panel of Fig. 2. All spectra exhibit a neutral donor bound
Fig. 1. Reflectance spectra from high quality wurtzite GaN substrate. Middle panel shows near normal incidence (incident angle about 78) reflectance spectrum. Top and bottom panel shows S or P polarized reflectance spectra with incident angle 508.
exciton peak I2 and a small impurity-related peak ‘a’ [24] at the same energy, respectively. In He–Cd excited spectrum, peaks corresponding to A, B, and An¼2 excitons are observed. In Xe excited and Ti : sapphire excited spectra, the peak corresponding to A excitons split into two peaks. To explain these peaks, we estimate the lifetimes of excitonpolariton branches by Eq. (7) using physical parameters obtained by fitting procedures. Resultant polariton lifetimes are shown in lower panel of Fig. 2. It is known that exciton-polariton emission occurs at bottleneck energies where the lifetime of polariton approaches their maximum [25]. In Fig. 2, the peak energies related to A and B excitons in Xe excited spectrum are denoted by vertical dashed line. It is find that, lower and upper
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Table 1 Values of physical parameters obtained through the fitting procedure of the reflectance spectra. Eg is bandgap and m0 is electrom mass Parameters
ho0 (eV) S? Sk G (meV) M? =m0 Mk =m0 Eg (eV) Gðn > 1Þ e0? e0k
a
Exciton A
B
C
3.4791 0.0002 0.0066 0.0006 0 0.20 0.01 0.5a 1.1 0.1 3.5025 1.8 0.1
3.4844 0.0002 0.0058 0.0006 0.0056 0.0008 0.70 0.03 0.6a 1.0 0.1 3.5080 1.5a Background dielectric constants 8.6 10.1 Dead layer depth 6 (nm)
3.5027 0.0002 0.0013 0.0006 0.0014a 1.20 0.05 0.8a 0.4a } }
The reflectance spectrum is not sensitive to the parameters.
peaks related to A exciton are located at the energies, where the lifetime of lower branch of A excitonic polariton (LPBA ) and of a branch combined upper branch of A excitonic polariton and lower branch of B excitonic polariton (UPBA þLPBB ) approach their maximum, respectively. The peak related to B exciton is also located at where the lifetime of UPBB þLPBC approach its maximum. Thus, these peaks are due to LPBA , UPBA þLPBB , and UPBB þLPBC . The lower peak energy related to A exciton for Ti : sapphire excited PL spectrum agrees with that for Xe excited spectrum. But the upper peak related to A exciton and the peak related to B exciton for Ti : sapphire excited spectrum are located at slightly higher energy than those for Xe excited spectrum. This indicates that many more excitonic polaritons accumulate at the bottleneck portion under the high power excitation condition. The reason why lower peak related to A exciton do not shift to the higher energy is probably that the curvature of the corresponding dispersion relation are smaller than for remaining two. In He–Cd excited spectrum, there is only one peak related to A exciton, while in Xe and Ti : sapphire excited spectrum, there are two. But, the full width at half maximum (FWHM) of this
peak is about four times broader than that of I2 , and this peak can not be fitted by single Lorentzian line-shape function. So, this peak is assigned to convolution of upper and lower A excitonic polaritons. The reason why these peaks are not resolved may be due to the use of a He–Cd laser having high-energy photons that produces excitons having large momentum. From the viewpoint of excitonic polaritons, one more PL peak related to B exciton should exsist at lower energy side of observed peak related to B exciton. But in this study, such peak is not observed clearly for weak intensity. 4.3. PR spectrum The PR spectrum is shown in upper panel of Fig. 3. Reflection anomalies corresponding to A, B, C, and An¼2 excitons are found. PR is a photomodulated version of an electroreflectance, thus the PR spectrum was analyzed by the low-field ER lineshape function [26] ( ) p X
mj DR iyj ¼ Re ; ð9Þ Cj e E Eg; j þ iGj R j¼1 where p is the number of the spectral function to be fitted, E is the phonon energy, and Cj , yj , Eg;j
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273
Fig. 3. PR spectrum from high quarity wurtzite GaN substrate (upper panel), and the lifetime of exciton-polariton branches (lower panel). The PR spectrum exhibit reflectance anomalies corresponding to A, B, C, and An¼2 excitons. The lifetime is the same with those in Fig. 2.
Fig. 2. PL spectra from high quality wurtzite GaN substrate (upper panel), and the lifetime of exciton-polariton branches (lower panel). The PL spectra shows complex structure around 3.48 eV. In lower panel, LPB and UPB denote lower polariton branch and upper polariton branch, respectively, and subscripts denote A, B, or C exciton.
and Gj are amplitude, phase, energy and broadening parameter of the jth feature, respectively. We tentatively assumed that the transitions are due to free exciton resonance (mj ¼ 2). Obtained transition energies are summarized in Table 2 with exciton energies obtained from reflectance spectra, and shown by vertical dashed lines in Fig. 3. Exciton energies obtained from reflectance spectra
and transition energies obtained from PR spectrum are different from each other. Comparing the lifetime of polariton (shown in lower panel of Fig. 3) and the PR spectrum, it is found that the PR transition energies agree with the bottleneck energies of excitonic polaritons. This means PR monitors exciton-polaritons at low temperature. It is natural to admit that PR monitors excitonpolaritons rather than exciton or free carriers since the PR signal is deduced as [26] DR 2na ¼ Re ¼ Re½ða ibÞDe R nð e e a Þ ¼ aDe1 þ bDe2 ;
ð10Þ
where n2 ¼ e, n2a ¼ ea , na is the real reflactive index, De ¼ De1 þ iDe2 is the perturbation-induces change in the dielectric function e, and a and b are Seraphin coefficients. Eq. (10) means that the signal DR=R is purely sensitive to De.
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Table 2 Values of free exciton energies obtained from the fitting procedure of the OR spectra and transition energies obtained from the analysis of PR spectrum assuming the transitions as pure free excitons for the GaN substrate at 10 K Method
OR PR
Parameter
Exciton resonance energy (eV) Transition energy (eV)
Exciton/transition A
B
C
3.4791 0.0002 3.4803 0.0003
3.4844 0.0002 3.4852 0.0003
3.5027 0.0002 3.5033 0.0003
Fig. 5. Temperature dependence of PL peak energies and PR transition energies. Solid curve is theoretical curve calculated by assuming Einstein phonon. Good agreement between experimental data and theory is obtained.
Fig. 4. PR (upper panel) and PL (lower panel) spectra as a function of temperature. The PL spectra are dominated by exciton up to RT.
4.4. Temperature dependence Fig. 4 shows PR (upper panel) and PL (lower panel) spectra of GaN as a function of temperature. The A and B excitonic transitions are well
recognized up to 300 K. The PL spectra are dominated by A and B free exciton emissions above 75 K. Surprisingly, recombination of the first excited states of A excitons is recognized up to 200 K. In Fig. 5, transition energies obtained from the PR spectra and PL peak energies are plotted as a function of temperature. All plots show the same temperature dependence. It has been shown [27] that in polar semiconductors that have optical phonon branches energetically next to the acoustic ones, fitting of temperature variation of a bandgap using the empirical equation proposed by Cody [28], which assumes Einstein phonons, is better
K. Torii et al. / Physica B 302–303 (2001) 268–276
than that using the Varshni fitting [29], which assumes Debye phonons. Therefore, we fit A transition energy EA ðTÞ by the equation after Cody [28] as EA ðTÞ ¼ 3:480 0:162=½expð366=TÞ 1 ;
ð11Þ
where T is temperature, A transition energy at T ¼ 0 K is 3.480 eV and Einstein characteristic temperature YE is 366 K. Calculated curve is shown in Fig. 5 by dotted line. The obtained Einstein characteristic temperature corresponds to the energy maximum of the low-energy group of bulk phonon density of states in wurtzite GaN reported in Ref. [30] (260 cm1 or 370 K).
5. Conclusions OR, PR and PL spectra of high quality wurtzite GaN substrate prepared by MOCVD-LEO were measured. To understand the spectra, theoretical calculations based on exciton-polariton picture was carried out. The complex PL structure was explained by a model based on exciton-polariton picture. The PR method was shown to be sensitive to bottleneck of excitonic polariton branches rather than to free exciton resonance. The GaN substrate exhibited excitonic emissions up to RT. Temperature dependence of A transition energy was well discribed using a model, which assumes Eistein phonons.
Acknowledgements The authors are grateful to M. Sugiyama and R. Nakai for their help in experiment. One of us (S.F.C.) is thankful to Professor Fumio Hasedawa, Professor Steven P. DenBaars, Professor Umesh K. Mishra and Professor Jacques I. Pankove for continuous encouragements.
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