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Dielectric functions and interband transitions of InxAl1 − xP alloys
Current Applied Physics 14 (2014) 1273e1276
Contents lists available at ScienceDirect
Current Applied Physics journal homepage: www.elsevier.com/locate/cap
Dielectric functions and interband transitions of InxAl1
xP
alloys
T.J. Kim a, *, S.Y. Hwang a, J.S. Byun a, D.E. Aspnes b, E.H. Lee c, J.D. Song c, C.-T. Liang d, Y.-C. Chang d, H.G. Park a, J. Choi a, J.Y. Kim a, Y.R. Kang a, J.C. Park a, Y.D. Kim a, * a
Nano-Optical Property Laboratory and Department of Physics, Kyung Hee University, Seoul 130-701, Republic of Korea Department of Physics, North Carolina State University, Raleigh, NC 27695-8202, United States c Center for Opto-Electronic Convergence Systems, Korea Institute of Science and Technology, Seoul 136-791, Republic of Korea d Research Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan, ROC b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 April 2014 Received in revised form 23 June 2014 Accepted 29 June 2014 Available online 8 July 2014
We report pseudodielectric functions <ε> from 1.5 to 6.0 eV of InxAl1 xP ternary alloy films. Data were obtained by spectroscopic ellipsometry on 1.2 mm thick films grown on (001) GaAs substrates by molecular beam epitaxy. Artifacts were minimized by real-time assessment of overlayer removal, leading to accurate representations of the bulk dielectric responses of these materials. Critical-point (CP) energies were obtained from numerically calculated second energy derivatives, and their Brillouin-zone origins identified by band-structure calculations using the linear augmented Slater-type orbital method. © 2014 Elsevier B.V. All rights reserved.
Keywords: Ellipsometry InAlP Dielectric function Critical point
1. Introduction Phosphorus-based wide-bandgap III-V semiconductors are increasingly important for many applications such as highelectron-mobility transistors and optoelectronic devices [1e3]. Among III-V compounds, the InxAl1xP alloys are interesting because they exhibit the smallest refractive index and largest band gap of any arsenic- or phosphorus-based alloy that is lattice matched to GaAs [4]. As a result, InxAl1xP is a useful material for high-performance laser diodes and light-emitting diodes in the visible spectral range [1]. To properly design and understand devices, the optical properties of InxAl1 xP are needed. Several studies of these properties, determined by spectroscopic ellipsometry (SE), have been reported [5e9]. However, data obtained to date have In compositions limited to x ~ 0.5. Here, we report pseudodielectric function <ε> ¼ <ε1> þ i<ε2> spectra from 1.5 to 6.0 eV of InxAl1 xP alloys over the entire composition range 0 x 1. Measurements were done with SE, with overlayer removal assessed in real time to ensure that the <ε> data are as close as possible to the intrinsic dielectric responses of bulk material. We determined critical point (CP)
* Corresponding authors. E-mail addresses: [email protected] (T.J. Kim), [email protected] (Y.D. Kim). http://dx.doi.org/10.1016/j.cap.2014.06.026 1567-1739/© 2014 Elsevier B.V. All rights reserved.
energies from numerically calculated second derivatives, and identified the origin of these features from electronic bandstructure calculations done using the linear augmented Slatertype orbital (LASTO) method [10]. 2. Experimental InxAl1 xP ternary alloy films for x ¼ 0.186, 0.310, 0.475, 0.715, and 0.831 were grown on (001) GaAs substrates by molecular beam epitaxy. The layers are about 1.2 mm thick, well beyond their critical thicknesses, so the associated optical properties can be regarded as characteristic of strain-relaxed materials. The layers exhibited streaky reflection-high-energy-electron-diffraction (RHEED) patterns during growth, indicating that the growth was laminar. We used high-resolution X-ray diffraction to determine In compositions. <ε> spectra were obtained from 1.5 to 6.0 eV using a spectroscopic ellipsometer that has been described previously [11], now converted to rotating-compensator operation. The samples were at room temperature, and the angle of incidence was 67.08 . To obtain the best possible approximation to intrinsic dielectric responses, overlayer removal was assessed in real time while the vertically mounted samples were treated with various etchants. The samples were maintained in high-purity N2 during processing and measurement to prevent reoxidation and the accumulation of physisorbed overlayers.
1274
T.J. Kim et al. / Current Applied Physics 14 (2014) 1273e1276
3. Results and discussion
120
. d2 ε du2 ¼ nðn 1ÞAeif ðu E þ iGÞn2 ; ¼ Aeif ðu E þ iGÞ2 ;
n ¼ 0;
ns0;
(1)
100
E0'
E0
AlP
80
< 1>
x = 0.186
60
x = 0.310 x = 0.475
40 E0
x = 0.715
E1
20
x = 0.831
E0' 0
InP
E2 2
3
4
5
6
E (eV) 120
E1
(b)
E0' AlP
100
x = 0.186
E0
80
< 2>
Fig. 1 shows <ε> data obtained during the removal of overlayers by chemical processing, using In0.475Al0.525P as an example. We first applied methanol to remove physisorbed contamination. The highest value of <ε2> at the E2 peak indicates the closest approximation to an abrupt interface, that is, the elimination of oxide and contamination overlayers and nanoscopic roughness [12]. We achieved this with a treatment of Br in methanol followed by a NH4OH rinse [13]. Any residual overlayers, which for example could be nonstoichiometric residual oxides or surface roughness, have a negligible effect on the values of CP energies determined in the region of the E1 peaks, although the situation for the E2 region may be different [14]. While we cannot ensure that oxides and roughness have been eliminated completely, the fact that the peak values of <ε2> reached hard limits unaffected by additional processing for the different alloys is a strong indication that the most abrupt interface for our procedures had been attained. Fig. 2(a) and (b) show the full set of <ε1> and <ε2> spectra, with each spectrum offset successively by 15. The spectra for the binary end points InP and AlP were taken from Refs. [12] and [15], respectively. We can easily observe the redshift of the E1 and E00 CPs with increasing composition x. The oscillations below the E0 feature are interference effects involving light back-reflected at the substrateefilm interface as a result of the films being transparent below their fundamental absorption edges. To determine the CP energies of the CP features, we calculated second-energy-derivatives d2<ε>/dE2 of these spectra numerically, using appropriate smoothing [13,15] to reduce noise levels. We then fit the results to the standard analytic CP expression [16],
E1
(a)
x = 0.310 x = 0.475
60
x = 0.715
where E is the CP energy, G is its broadening parameter, A is its amplitude, and f is its phase angle. The exponent n is 1, 1/2, 0, and 1/2 for excitonic, one-, two-, and three-dimensional CPs, respectively. Results for several InxAl1 xP alloys are shown in Fig. 3. The open circles are values calculated from the <ε1> data, while the solid and
40
20
E0
x = 0.831
E0'
E1
E2 InP
0
2
3
4
5
6
E (eV) Fig. 2. (a) Real and (b) imaginary parts of <ε> as the best approximation of the bulk dielectric responses of InxAl1 xP alloys for x ¼ 0.000 (AlP), 0.186, 0.310, 0.475, 0.715, 0.831, and 1.000 (InP). Successive spectra are offset by 15. The spectra of InP and AlP are taken from Refs. [12] and [15], respectively.
dashed lines are the best fits to d2<ε1>/dE2 and d2<ε2>/dE2, respectively. For clarity, the number of points for d2<ε1>/dE2 is reduced appropriately and those for d2<ε2>/dE2 are not shown. As seen in Fig. 2(a) and (b), the E1, E00 , and E2 CP features are easily observed. However, in the second derivatives, additional CP structures are resolved, as indicated by arrows in Fig. 3. To confirm the Brillouin-zone origins of these CPs, we performed band-structure calculations using the LASTO method [10] within the local density approximation. Self-energy corrections are included by the empirical tight-binding formula,
lcv ðkÞ ¼ l0 þ l1
X i
Fig. 1. Imaginary part of <ε> of In0.475Al0.525P before (dotted line), during (dashed line), and after (solid line) chemical processing to remove overlayers, including nanoscopic roughness.
ð1Þ
where Ri neighbor
ð1Þ
ejk$Ri þ l2
ð2Þ
X
ð2Þ
ejk$Ri ;
(2)
i
and Ri run through the first and second nearestlattice sites, respectively. The three adjustable
T.J. Kim et al. / Current Applied Physics 14 (2014) 1273e1276
E2 E ' 0
E1+Δ1
A
E1
E 0'
E1+Δ1
E2
E2 E0'+Δ0'
5
E 2'
E (eV)
d2< >/dE2
E2'
B
500
x = 0.475 E1
InxAl1-xP
6
B E' 2 x = 0.186
E1
1275
4
E0'
A
E0' (0.205,0,0) E1+Δ1 E1
3
E1+Δ1
E0'
E2
E2' 2
x = 0.831
E0+Δ0 1
3
4
5
6
E0 0.0 0.2
0.4 0.6
0.8 1.0
In composition (x)
E (eV) Fig. 3. Open circles: numerically calculated second derivatives d2<ε1>/dE2. The solid and dashed lines are best fits to the results for d2<ε1>/dE2 and d2<ε2>/dE2 using Eq. (1). The CP energies so determined are marked by arrows.
Fig. 4. Energy band structure of InP calculated by the LASTO method. CPs giving rise to features seen in the optical spectra are indicated by arrows.
parameters l0, l1, and l2, are determined by requiring that lcv ðkÞ at the G-, L-, and X-points of the Brillouin zone agree with data. The parameters used in this calculation are 1.020, 0.973, and 0.892 for InP. The calculated band structure of InP and identifications of CPs
Fig. 5. Symbols: dependences of the CP energies on x, as determined from the <ε> data as described in the text. Lines: best fits of E(x) ¼ ax2 þ bx þ c to the results of the band-structure calculations.
including the E00 (0.205,0,0) saddle point are shown in Fig. 4. As a result, we could identify the five structures for x ¼ 0.475 and 0.831 as the E1, E1 þ D1, E00 , E2, and E20 CPs in Fig. 3. For the x ¼ 0.186 spectrum, seven features are identified from previously reported band structure calculations for AlP as the A, E1, E1 þ D1, E2, E00 , B, and E20 CPs [15]. The A and B structures seen in previous AlP work [15] are also clearly observed here for the x ¼ 0.186 spectrum. However, we were not able to determine their Brillouin-zone locations. Accordingly, we simply label these as A and B following the convention of Ref. [15]. All CPs are best represented by the excitonic lineshape (n ¼ 1), except for the E1 and E1 þ D1 CPs of In-rich InxAl1 xP (x ¼ 0.475, 0.715, 0.831, and 1.00), which are best represented by the two-dimensional lineshape (n ¼ 0) [16,17]. The CP energies and their uncertainties are listed in Table 1. Fig. 5 summarizes the experimental (dots) and theoretical (solid lines) results for the entire alloy series. The CP energies of AlP are from Ref. [15]. The A and B CPs are indicated by open triangles and squares, respectively. The energies of the E0 and E0 þ D0 CPs of InP are beyond our spectral range. Therefore, the energies of these CPs are taken from Ref. [17]. For device-design applications, the expected bowings (quadratic terms in the x dependence of the CP energies) were calculated by simulating In0.5Al0.5P in a special quasirandom-structures (SQS) calculation [18]. We used specifically the SQS-8 version. The atoms were allowed to relax to their equilibrium positions using the WIEN2k package [19]. The calculated
Table 1 Energies and uncertainties of the different CPs (eV). x
E1
0 0.186 0.310 0.475 0.715 0.831 1
4.57 4.26 3.92 3.65 3.44 3.28 3.13
± ± ± ± ± ± ±
0.002 0.02 0.02 0.02 0.01 0.004 0.001
E1 þ D1
E00
e 4.38 4.23 3.97 3.64 3.50 3.31
5.10 4.99 4.91 4.84 4.82 4.76 4.72
± ± ± ± ± ±
0.23 0.11 0.13 0.07 0.04 0.002
E20
E2 ± ± ± ± ± ± ±
0.02 0.04 0.02 0.01 0.02 0.01 0.001
4.82 4.84 4.83 4.95 4.93 4.89 4.97
± ± ± ± ± ± ±
0.005 0.01 0.01 0.02 0.04 0.03 0.003
5.90 5.61 5.54 5.63 5.52 5.47 5.57
± ± ± ± ± ± ±
0.03 0.05 0.05 0.04 0.35 0.06 0.03
A
B
4.11 ± 0.02 3.45 ± 0.04
5.35 ± 0.02 5.39 ± 0.08
1276
T.J. Kim et al. / Current Applied Physics 14 (2014) 1273e1276
Table 2 Values of the parameters a, b, and c obtained by fitting the compositional dependences of the CP energies calculated by the LASTO method to the quadratic E(x) ¼ ax2 þ bx þ c.
E0 E0 þ D0 E1 E1 þ D1 E00 E00 þ D00 E00 ð0:205; 0; 0Þ2p=a E2 E20
a
b
c
0.672 0.629 0.443 0.291 0.314 0.400 0.206 0.349 0.278
3.008 2.908 1.850 1.617 0.682 0.352 0.041 0.492 0.587
3.681 3.742 4.574 4.612 4.915 4.942 5.007 4.824 5.741
parameters a, b, and c of the quadratic expression E(x) ¼ ax2 þ bx þ c for the different CPs are given in Table 2. The results are shown as solid lines in Fig. 5. The agreement with the data is satisfactory, thereby supporting the validity of the theoretical calculations. However, the E00 CP energy of 4.71 eV for InP is between the lines of E00 and E00 (0.205,0,0). We estimate that the E00 CP structure of InP contains both CPs, and that they cannot be separately observed at room temperature [17]. 4. Conclusion We report pseudodielectric function spectra <ε> of InxAl1 xP alloys over the entire composition range 0 x 1, determined by SE from 1.5 to 6.0 eV. Real-time monitoring of the chemical removal of oxide and contamination overlayers and nanoscopic roughness ensures that these data are accurate representations of the actual bulk dielectric functions of these alloys. We determined CP energies of InxAl1 xP alloys from numerically calculated second energy derivatives of these data, and identified the origins of these features with electronic band-structure calculations using the LASTO method. These results will be useful in a number of contexts, including the design of optoelectronic devices based on InxAl1 xP for industrial purpose. A more detailed understanding of the band structure of these alloys has also been achieved. Acknowledgment This work was supported by a grant from Kyung Hee University in 2012 (KHU-20130161) and by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (2013016297). DEA was supported by the World Class University Program through the Korea Science and Education Foundation funded by the Ministry of Science, Engineering, and Technology under Grant R332008-0000-10118-0. JDS acknowledges support from the KIST
institutional program, partially by the Dream project, partially by Grant 2012K001280, and partially by the GRL Program through MEST. YCC was supported by grant NSC-101-2112-M-001-024-MY3 of the Republic of China. References [1] C.-T. Hung, S.-C. Huang, T.-C. Lu, Effect of passivation layers on characteristics of AlGaInP ridge waveguide laser diodes, Opt. Laser Technol. 59 (2014) 110e115. [2] H. Sugiyama, H. Yokoyama, H. Matsuzaki, T. Enoki, T. Kobayashi, Metal-organic vapor-phase epitaxy of pseudomorphic InAlP/InGaAs high electron mobility transistor wafers, Jpn. J. Appl. Phys. 44 (2005) 3798e3802. [3] Y. Gu, Y.G. Zhang, A.Z. Li, H. Li, Optical properties of gas source MBE grown AlInP on GaAs, Mater. Sci. Eng. B 139 (2007) 246e250. [4] G.O. Munns, W.L. Chen, M.E. Sherwin, G.I. Haddad, The growth of InA1P using trimethyl amine alane by chemical beam epitaxy, J. Cryst. Growth 127 (1993) 226e229. [5] H. Lee, M.V. Klein, D.E. Aspnes, C.P. Kuo, M. Peanasky, M.G. Craford, Optical study of (AlxGa1 x)0.5In0.5P/GaAs semiconductor alloys by spectroscopic ellipsometry, J. Appl. Phys. 73 (1993) 400e406. [6] S. Adachi, H. Kato, A. Moki, K. Ohtsuka, Refractive index of (AlxGa1 x)0.5In0.5P quaternary alloys, J. Appl. Phys. 75 (1994) 478e480. [7] H. Kato, S. Adachi, H. Nakanishi, K. Ohtsuka, Optical properties of (AlxGa1 x)0.5In0.5P quaternary alloys, Jpn. J. Appl. Phys. 33 (1994) 186e192. [8] M. Schubert, J.A. Woollam, G. Leibiger, B. Rheinl€ ander, I. Pietzonka, T. Saß, V. Gottschalch, Isotropic dielectric functions of highly disordered AlxGa1 xInP (0 x 1) lattice matched to GaAs, J. Appl. Phys. 86 (1999) 2025e2033. [9] T. Hofmann, V. Gottschalch, M. Schubert, Dielectric anisotropy and phonon modes of ordered indirect-gap Al0.52In0.48P studied by far-infrared ellipsometry, Appl. Phys. Lett. 91 (2007) 121908. [10] Y.-C. Chang, R.B. James, J.W. Davenport, Symmetrized-basis LASTO calculations of defects in CdTe and ZnTe, Phys. Rev. B 73 (2006) 035211. [11] D.E. Aspnes, A.A. Studna, High Precision Scanning ellipsometer, Appl. Opt. 14 (1975) 220e228. [12] D.E. Aspnes, A.A. Studna, Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV, Phys. Rev. B 27 (1983) 985e1009. [13] T.J. Kim, J.J. Yoon, S.Y. Hwang, D.E. Aspnes, Y.D. Kim, H.J. Kim, Y.C. Chang, J.D. Song, Interband transitions of InAsxSb1 x alloy films, Appl. Phys. Lett. 95 (2009) 111902. [14] Y.W. Jung, T.H. Ghong, Y.D. Kim, D.E. Aspnes, Effect of overlayers on criticalpoint parameters in the analysis of ellipsometric spectra, Appl. Phys. Lett. 91 (2007) 121903. [15] S.Y. Hwang, T.J. Kim, Y.W. Jung, N.S. Barange, H.G. Park, J.Y. Kim, Y.R. Kang, Y.D. Kim, S.H. Shin, J.D. Song, C.-T. Liang, Y.-C. Chang, Dielectric function and critical points of AlP determined by spectroscopic ellipsometry, J. Alloys Compd. 587 (2014) 361e364. [16] P. Lautenschlager, M. Garriga, M. Cardona, Temperature dependence of the interband critical-point parameters of InP, Phys. Rev. B 36 (1987) 4813e4820. [17] S.G. Choi, C.J. Palmstrøm, Y.D. Kim, D.E. Aspnes, H.J. Kim, Y.-C. Chang, Dielectric functions and electronic structure of InAsxP1 x films on InP, Appl. Phys. Lett. 91 (2007) 041917. [18] S.-H. Wei, L.G. Ferreira, J.E. Bernard, A. Zunger, Electronic properties of random alloys: special quasirandom structures, Phys. Rev. B 42 (1990) 9622e9649. [19] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, in: Karlheinz Schwarz (Ed.), WIEN2k, an Augmented Plane Wave þ Local Orbitals Program for Calculating Crystal Properties, Techn. Universitat Wien, Austria, 2001.
Contents lists available at ScienceDirect
Current Applied Physics journal homepage: www.elsevier.com/locate/cap
Dielectric functions and interband transitions of InxAl1
xP
alloys
T.J. Kim a, *, S.Y. Hwang a, J.S. Byun a, D.E. Aspnes b, E.H. Lee c, J.D. Song c, C.-T. Liang d, Y.-C. Chang d, H.G. Park a, J. Choi a, J.Y. Kim a, Y.R. Kang a, J.C. Park a, Y.D. Kim a, * a
Nano-Optical Property Laboratory and Department of Physics, Kyung Hee University, Seoul 130-701, Republic of Korea Department of Physics, North Carolina State University, Raleigh, NC 27695-8202, United States c Center for Opto-Electronic Convergence Systems, Korea Institute of Science and Technology, Seoul 136-791, Republic of Korea d Research Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan, ROC b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 April 2014 Received in revised form 23 June 2014 Accepted 29 June 2014 Available online 8 July 2014
We report pseudodielectric functions <ε> from 1.5 to 6.0 eV of InxAl1 xP ternary alloy films. Data were obtained by spectroscopic ellipsometry on 1.2 mm thick films grown on (001) GaAs substrates by molecular beam epitaxy. Artifacts were minimized by real-time assessment of overlayer removal, leading to accurate representations of the bulk dielectric responses of these materials. Critical-point (CP) energies were obtained from numerically calculated second energy derivatives, and their Brillouin-zone origins identified by band-structure calculations using the linear augmented Slater-type orbital method. © 2014 Elsevier B.V. All rights reserved.
Keywords: Ellipsometry InAlP Dielectric function Critical point
1. Introduction Phosphorus-based wide-bandgap III-V semiconductors are increasingly important for many applications such as highelectron-mobility transistors and optoelectronic devices [1e3]. Among III-V compounds, the InxAl1xP alloys are interesting because they exhibit the smallest refractive index and largest band gap of any arsenic- or phosphorus-based alloy that is lattice matched to GaAs [4]. As a result, InxAl1xP is a useful material for high-performance laser diodes and light-emitting diodes in the visible spectral range [1]. To properly design and understand devices, the optical properties of InxAl1 xP are needed. Several studies of these properties, determined by spectroscopic ellipsometry (SE), have been reported [5e9]. However, data obtained to date have In compositions limited to x ~ 0.5. Here, we report pseudodielectric function <ε> ¼ <ε1> þ i<ε2> spectra from 1.5 to 6.0 eV of InxAl1 xP alloys over the entire composition range 0 x 1. Measurements were done with SE, with overlayer removal assessed in real time to ensure that the <ε> data are as close as possible to the intrinsic dielectric responses of bulk material. We determined critical point (CP)
* Corresponding authors. E-mail addresses: [email protected] (T.J. Kim), [email protected] (Y.D. Kim). http://dx.doi.org/10.1016/j.cap.2014.06.026 1567-1739/© 2014 Elsevier B.V. All rights reserved.
energies from numerically calculated second derivatives, and identified the origin of these features from electronic bandstructure calculations done using the linear augmented Slatertype orbital (LASTO) method [10]. 2. Experimental InxAl1 xP ternary alloy films for x ¼ 0.186, 0.310, 0.475, 0.715, and 0.831 were grown on (001) GaAs substrates by molecular beam epitaxy. The layers are about 1.2 mm thick, well beyond their critical thicknesses, so the associated optical properties can be regarded as characteristic of strain-relaxed materials. The layers exhibited streaky reflection-high-energy-electron-diffraction (RHEED) patterns during growth, indicating that the growth was laminar. We used high-resolution X-ray diffraction to determine In compositions. <ε> spectra were obtained from 1.5 to 6.0 eV using a spectroscopic ellipsometer that has been described previously [11], now converted to rotating-compensator operation. The samples were at room temperature, and the angle of incidence was 67.08 . To obtain the best possible approximation to intrinsic dielectric responses, overlayer removal was assessed in real time while the vertically mounted samples were treated with various etchants. The samples were maintained in high-purity N2 during processing and measurement to prevent reoxidation and the accumulation of physisorbed overlayers.
1274
T.J. Kim et al. / Current Applied Physics 14 (2014) 1273e1276
3. Results and discussion
120
. d2 ε du2 ¼ nðn 1ÞAeif ðu E þ iGÞn2 ; ¼ Aeif ðu E þ iGÞ2 ;
n ¼ 0;
ns0;
(1)
100
E0'
E0
AlP
80
< 1>
x = 0.186
60
x = 0.310 x = 0.475
40 E0
x = 0.715
E1
20
x = 0.831
E0' 0
InP
E2 2
3
4
5
6
E (eV) 120
E1
(b)
E0' AlP
100
x = 0.186
E0
80
< 2>
Fig. 1 shows <ε> data obtained during the removal of overlayers by chemical processing, using In0.475Al0.525P as an example. We first applied methanol to remove physisorbed contamination. The highest value of <ε2> at the E2 peak indicates the closest approximation to an abrupt interface, that is, the elimination of oxide and contamination overlayers and nanoscopic roughness [12]. We achieved this with a treatment of Br in methanol followed by a NH4OH rinse [13]. Any residual overlayers, which for example could be nonstoichiometric residual oxides or surface roughness, have a negligible effect on the values of CP energies determined in the region of the E1 peaks, although the situation for the E2 region may be different [14]. While we cannot ensure that oxides and roughness have been eliminated completely, the fact that the peak values of <ε2> reached hard limits unaffected by additional processing for the different alloys is a strong indication that the most abrupt interface for our procedures had been attained. Fig. 2(a) and (b) show the full set of <ε1> and <ε2> spectra, with each spectrum offset successively by 15. The spectra for the binary end points InP and AlP were taken from Refs. [12] and [15], respectively. We can easily observe the redshift of the E1 and E00 CPs with increasing composition x. The oscillations below the E0 feature are interference effects involving light back-reflected at the substrateefilm interface as a result of the films being transparent below their fundamental absorption edges. To determine the CP energies of the CP features, we calculated second-energy-derivatives d2<ε>/dE2 of these spectra numerically, using appropriate smoothing [13,15] to reduce noise levels. We then fit the results to the standard analytic CP expression [16],
E1
(a)
x = 0.310 x = 0.475
60
x = 0.715
where E is the CP energy, G is its broadening parameter, A is its amplitude, and f is its phase angle. The exponent n is 1, 1/2, 0, and 1/2 for excitonic, one-, two-, and three-dimensional CPs, respectively. Results for several InxAl1 xP alloys are shown in Fig. 3. The open circles are values calculated from the <ε1> data, while the solid and
40
20
E0
x = 0.831
E0'
E1
E2 InP
0
2
3
4
5
6
E (eV) Fig. 2. (a) Real and (b) imaginary parts of <ε> as the best approximation of the bulk dielectric responses of InxAl1 xP alloys for x ¼ 0.000 (AlP), 0.186, 0.310, 0.475, 0.715, 0.831, and 1.000 (InP). Successive spectra are offset by 15. The spectra of InP and AlP are taken from Refs. [12] and [15], respectively.
dashed lines are the best fits to d2<ε1>/dE2 and d2<ε2>/dE2, respectively. For clarity, the number of points for d2<ε1>/dE2 is reduced appropriately and those for d2<ε2>/dE2 are not shown. As seen in Fig. 2(a) and (b), the E1, E00 , and E2 CP features are easily observed. However, in the second derivatives, additional CP structures are resolved, as indicated by arrows in Fig. 3. To confirm the Brillouin-zone origins of these CPs, we performed band-structure calculations using the LASTO method [10] within the local density approximation. Self-energy corrections are included by the empirical tight-binding formula,
lcv ðkÞ ¼ l0 þ l1
X i
Fig. 1. Imaginary part of <ε> of In0.475Al0.525P before (dotted line), during (dashed line), and after (solid line) chemical processing to remove overlayers, including nanoscopic roughness.
ð1Þ
where Ri neighbor
ð1Þ
ejk$Ri þ l2
ð2Þ
X
ð2Þ
ejk$Ri ;
(2)
i
and Ri run through the first and second nearestlattice sites, respectively. The three adjustable
T.J. Kim et al. / Current Applied Physics 14 (2014) 1273e1276
E2 E ' 0
E1+Δ1
A
E1
E 0'
E1+Δ1
E2
E2 E0'+Δ0'
5
E 2'
E (eV)
d2< >/dE2
E2'
B
500
x = 0.475 E1
InxAl1-xP
6
B E' 2 x = 0.186
E1
1275
4
E0'
A
E0' (0.205,0,0) E1+Δ1 E1
3
E1+Δ1
E0'
E2
E2' 2
x = 0.831
E0+Δ0 1
3
4
5
6
E0 0.0 0.2
0.4 0.6
0.8 1.0
In composition (x)
E (eV) Fig. 3. Open circles: numerically calculated second derivatives d2<ε1>/dE2. The solid and dashed lines are best fits to the results for d2<ε1>/dE2 and d2<ε2>/dE2 using Eq. (1). The CP energies so determined are marked by arrows.
Fig. 4. Energy band structure of InP calculated by the LASTO method. CPs giving rise to features seen in the optical spectra are indicated by arrows.
parameters l0, l1, and l2, are determined by requiring that lcv ðkÞ at the G-, L-, and X-points of the Brillouin zone agree with data. The parameters used in this calculation are 1.020, 0.973, and 0.892 for InP. The calculated band structure of InP and identifications of CPs
Fig. 5. Symbols: dependences of the CP energies on x, as determined from the <ε> data as described in the text. Lines: best fits of E(x) ¼ ax2 þ bx þ c to the results of the band-structure calculations.
including the E00 (0.205,0,0) saddle point are shown in Fig. 4. As a result, we could identify the five structures for x ¼ 0.475 and 0.831 as the E1, E1 þ D1, E00 , E2, and E20 CPs in Fig. 3. For the x ¼ 0.186 spectrum, seven features are identified from previously reported band structure calculations for AlP as the A, E1, E1 þ D1, E2, E00 , B, and E20 CPs [15]. The A and B structures seen in previous AlP work [15] are also clearly observed here for the x ¼ 0.186 spectrum. However, we were not able to determine their Brillouin-zone locations. Accordingly, we simply label these as A and B following the convention of Ref. [15]. All CPs are best represented by the excitonic lineshape (n ¼ 1), except for the E1 and E1 þ D1 CPs of In-rich InxAl1 xP (x ¼ 0.475, 0.715, 0.831, and 1.00), which are best represented by the two-dimensional lineshape (n ¼ 0) [16,17]. The CP energies and their uncertainties are listed in Table 1. Fig. 5 summarizes the experimental (dots) and theoretical (solid lines) results for the entire alloy series. The CP energies of AlP are from Ref. [15]. The A and B CPs are indicated by open triangles and squares, respectively. The energies of the E0 and E0 þ D0 CPs of InP are beyond our spectral range. Therefore, the energies of these CPs are taken from Ref. [17]. For device-design applications, the expected bowings (quadratic terms in the x dependence of the CP energies) were calculated by simulating In0.5Al0.5P in a special quasirandom-structures (SQS) calculation [18]. We used specifically the SQS-8 version. The atoms were allowed to relax to their equilibrium positions using the WIEN2k package [19]. The calculated
Table 1 Energies and uncertainties of the different CPs (eV). x
E1
0 0.186 0.310 0.475 0.715 0.831 1
4.57 4.26 3.92 3.65 3.44 3.28 3.13
± ± ± ± ± ± ±
0.002 0.02 0.02 0.02 0.01 0.004 0.001
E1 þ D1
E00
e 4.38 4.23 3.97 3.64 3.50 3.31
5.10 4.99 4.91 4.84 4.82 4.76 4.72
± ± ± ± ± ±
0.23 0.11 0.13 0.07 0.04 0.002
E20
E2 ± ± ± ± ± ± ±
0.02 0.04 0.02 0.01 0.02 0.01 0.001
4.82 4.84 4.83 4.95 4.93 4.89 4.97
± ± ± ± ± ± ±
0.005 0.01 0.01 0.02 0.04 0.03 0.003
5.90 5.61 5.54 5.63 5.52 5.47 5.57
± ± ± ± ± ± ±
0.03 0.05 0.05 0.04 0.35 0.06 0.03
A
B
4.11 ± 0.02 3.45 ± 0.04
5.35 ± 0.02 5.39 ± 0.08
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T.J. Kim et al. / Current Applied Physics 14 (2014) 1273e1276
Table 2 Values of the parameters a, b, and c obtained by fitting the compositional dependences of the CP energies calculated by the LASTO method to the quadratic E(x) ¼ ax2 þ bx þ c.
E0 E0 þ D0 E1 E1 þ D1 E00 E00 þ D00 E00 ð0:205; 0; 0Þ2p=a E2 E20
a
b
c
0.672 0.629 0.443 0.291 0.314 0.400 0.206 0.349 0.278
3.008 2.908 1.850 1.617 0.682 0.352 0.041 0.492 0.587
3.681 3.742 4.574 4.612 4.915 4.942 5.007 4.824 5.741
parameters a, b, and c of the quadratic expression E(x) ¼ ax2 þ bx þ c for the different CPs are given in Table 2. The results are shown as solid lines in Fig. 5. The agreement with the data is satisfactory, thereby supporting the validity of the theoretical calculations. However, the E00 CP energy of 4.71 eV for InP is between the lines of E00 and E00 (0.205,0,0). We estimate that the E00 CP structure of InP contains both CPs, and that they cannot be separately observed at room temperature [17]. 4. Conclusion We report pseudodielectric function spectra <ε> of InxAl1 xP alloys over the entire composition range 0 x 1, determined by SE from 1.5 to 6.0 eV. Real-time monitoring of the chemical removal of oxide and contamination overlayers and nanoscopic roughness ensures that these data are accurate representations of the actual bulk dielectric functions of these alloys. We determined CP energies of InxAl1 xP alloys from numerically calculated second energy derivatives of these data, and identified the origins of these features with electronic band-structure calculations using the LASTO method. These results will be useful in a number of contexts, including the design of optoelectronic devices based on InxAl1 xP for industrial purpose. A more detailed understanding of the band structure of these alloys has also been achieved. Acknowledgment This work was supported by a grant from Kyung Hee University in 2012 (KHU-20130161) and by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (2013016297). DEA was supported by the World Class University Program through the Korea Science and Education Foundation funded by the Ministry of Science, Engineering, and Technology under Grant R332008-0000-10118-0. JDS acknowledges support from the KIST
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