Journal of the Korean Physical Society, Vol. 65, No. 4, August 2014, pp. 457∼461
Effects of Crystal Orientation on the Optical Gain Characteristics of Blue AlInGaN/InGaN Quantum-well Structures Seoung-Hwan Park∗ Department of Electronics Engineering, Catholic University of Daegu, Hayang 712-702, Korea
Yong-Hee Cho, Mun-Bo Shim and Sungjin Kim CAE Group, Platform Technology Lab, SAIT, Samsung Electronics, Suwon 443-803, Korea
Chang Young Park and Y. Eugene Pak Advanced Institutes of Convergence Technology, Seoul National University, Suwon 443-270, Korea
Dhaneshwar Mishra and Seung-Hyun Yoo Department of Mechancial Engineering, Ajou University, Suwon 443-749, Korea
Keonwook Kang Department of Mechanical Engineering, Yonsei University, Seoul 120-749, Korea (Received 15 May 2014, in final form 28 May 2014) The effects of crystal orientation on the optical gain characteristics of blue AlInGaN/InGaN quantum-well (QW) structures with a reduced internal field were investigated by using the nonMarkovian model with many-body effects. The AlInGaN/InGaN system has a larger matrix element than the conventional InGaN/GaN system because the former has a smaller internal field than the latter for relatively small crystal angles. As a result, for QW structures with a relatively small crystal angle (θ = 30◦ ), the AlInGaN/InGaN system is shown to have a much larger optical gain than the conventional InGaN/GaN system. On the other hand, in the case of QW structures with a large crystal angle (θ = 90◦ ), the AlInGaN/InGaN system is shown to have a smaller optical gain than the conventional InGaN/GaN system. Hence, we expect that we can obtain high-efficiency optoelectronic devices by using quaternary AlInGaN QW for relatively small crystal angles. PACS numbers: 42.55.Px, 42.60.-v, 71.22.+i, 72.80.Ey Keywords: InAlGaN, InGaN, GaN, Quantum well, Laser, Optical gain DOI: 10.3938/jkps.65.457
I. INTRODUCTION
By now, several methods have been proposed in an effort to reduce the effect of the internal field due to polarizations [4–6]. Among them, the AlInGaN quaternary alloy has emerged as an interesting material for device applications [7–9]. The internal field in the well can be reduced by using an AlInGaN layer because the band gap, lattice constant, and spontaneous polarization of the alloy can be adjusted simultaneously [10–12]. Recent theoretical studies showed that the spontaneous emission coefficient of AlInGaN/InGaN QW structures with a small Al composition in the well was increased by 30% compared to that of the conventional InGaN/GaN system [13,14]. Also, the condition to give a zero internal field in the well is only possible when a quaternary AlInGaN layer is used as a well, not a barrier [13]. Thus, systematic studies of AlInGaN-based QW structures will be very important in order to obtain guidelines for device
The wide band-gap wurtzite (WZ) semiconductors have attracted much attention due to their potential applications for optoelectronic devices in the blue and the ultraviolet (UV) regions [1]. The (0001)-oriented WZ GaN-based quantum wells (QWs) are found to have large internal fields due to the strain-induced piezoelectric (PZ) and spontaneous (SP) polarizations [2,3]. As a result, the radiative recombination rate and the optical gain of these QW structures are reduced significantly due to the large spatial separation between the electron and the hole wavefunctions. Thus, reducing the internal field is essential if the internal efficiency in WZ GaN-based QWs is to be increased. ∗ E-mail:
[email protected]
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Journal of the Korean Physical Society, Vol. 65, No. 4, August 2014
design for providing improved quantum efficiency. However, despite its importance, many fundamental properties of the AlInGaN-based QW structure are not yet well understood because studies on these structures are in an early stage of development. In this research, we investigate the effects of crystal orientation on the optical gain of blue AlInGaN/InGaN QW structures. The optical gain spectra are calculated by using the non-Markovian model with many-body effects. The band structures and the wave functions are obtained by solving the Schr¨ odinger equation for electrons and the 6 × 6 Hamiltonian for holes [4,15,16]. We assume that AlInGaN/InGaN QW structures are grown on thick GaN substrates. These results are compared with those of conventional InGaN/GaN QW structures.
The Hamiltonian for an arbitrary crystal orientation can be obtained by using a rotation matrix: ⎞ ⎛ cos θ cos φ cos θ sin φ − sin θ cos φ 0 ⎠. U = ⎝ − sin φ (1) sin θ cos φ sin θ sin φ cos θ Rotations of the Euler angles θ and φ transform the physical quantities from (x, y, z) coordinates to (x , y , z ) co-
μo 2
e2 m2o ω
0
2π
dφ0
ki = Uiα kα , ij = Uiα Ujβ αβ , Cijkl = Uiα Ujβ Uiγ Ujδ Cαβγδ ,
(2)
where summation over repeated indices is indicated. The electric fields in the well and the barrier due to the SP and the PZ polarizations along the growth direction can be estimated from the periodic boundary condition for a superlattice structure as follows [17,18]: b w + PPb Z − PSP − PPwZ ) (PSP , w + b Lw /Lb Lw w = − F , Lb z
Fzw =
II. THEORY
g(ω) =
ordinates. The z-axis corresponds to the c-axis [0001], and the growth axis (defined as the z -axis) is normal to the QW plane (hkil). The relation between the coordinate systems for vectors and tensors is expressed as
0
∞
dk||
Fzb
(3)
where superscripts w and b denote the well and the barrier, respectively, and L and are the layer thickness and the static dielectric constant, respectively. The optical gain g(ω) with many-body effects, including the effects of anisotropy on the valence-band dispersion, is given by [19,20]
2k|| (2π)2 Lw
v · |Mlm (k|| , φ0 )|2 [flc (k|| , φ0 ) − fm (k|| , φ0 )]
·
(1 − ReQ(k , ω))ReL(Elm (k , ω)) − ImQ(k , ω)ImL(Elm (k , ω)) , (1 − ReQ(k , ω))2 + (ImQ(k , ω))2
where ω is the angular frequency, μo is the vacuum permeability, is the dielectric constant, e is the charge of an electron, mo is the free electron’s mass, k|| is the magnitude of the in-plane wave vector in the QW plane, Lw is the well width, and |Mlm |2 is the momentum matrix v element in the strained QW. flc and fm represent the Fermi functions for the conduction-band states and the valence-band states. The indices l and m denote the electron states in the conduction band and the heavy-hole (light-hole) subband states in the valence band, respectively. The many-body effects include plasma screening, band gap renormalization, and excitonic or Coulomb enhancement of the interband transition probability. The band gap renormalization is given by the screened ex-
(4)
change (SX) self-energy and the Coulomb-hole (CH) contributions. Coulomb interactions are calculated under the Hatree-Fock limit [21]. Also, Elm (k , ω) = Elc (k ) v (k ) + Eg + ΔESX + ΔECH − ω is the renor− Em malized transition energy between electrons and holes, where Eg is the band gap of the material, and ΔESX and ΔECH are the screened exchange and Coulomb-hole contributions to the band-gap renormalization, respectively. The factor Q(k , ω) accounts for the excitonic or the Coulomb enhancement of the interband transition probability [21,22]. The line-shape function is Gaussian for the simplest non-Markovian quantum kinetics and is given by [19,20]
Effects of Crystal Orientation on the Optical Gain· · · – Seoung-Hwan Park et al.
Re[L(Elm (k , ω))]
=
Im[L(Elm (k , ω))]
τc =
and
τin (k , ω)τc 2 πτin (k , ω)τc exp − E (k , ω) lm 22 22 0
∞
τc Elm (k , ω) τc 2 t sin t dt. exp − 2τin (k , ω)
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(5)
Fig. 1. (Color online) (a) Internal field as a function of the crystal angle θ between the growth direction and the c-axis and (b) the peak wavelengths of the 2.5-nm Al0.08 In0.25 Ga0.68 N/In0.13 Ga0.87 N QW structures. The peak wavelengths are obtained at a sheet carrier density of N2D = 10 × 1012 cm−2 . For comparison, results (dashed line) for the In0.23 Ga0.77 N/GaN QW structure are also plotted.
The intraband relaxation time τin and the correlation time τc are assumed to be constant. The τin and the τc used in the calculation are 25 fs and 10 fs, respectively. The material parameters for GaN, InN, and AlN used in the calculation were taken from Refs. [18] and [23] and references in there, except for the band gap. The band-gap expression for the AlInGaN materials obtained through a comparison with experimental results was used in the calculation [13]. Also, the spontaneous polarization constant for AlN was taken from Ref. [3].
III. RESULT AND DISCUSSIONS Figure 1 shows (a) the internal field as a function of the crystal angle θ between the growth direction and the c-axis and (b) the peak wavelengths of 2.5-nm Al0.08 In0.25 Ga0.68 N/In0.13 Ga0.87 N QW structures as a function of the crystal orientation. The peak wavelengths are obtained at a sheet carrier density of N2D = 10 × 1012 cm−2 . For comparison, results (dashed line) for the In0.23 Ga0.77 N/GaN QW structure are also plotted. For the quaternary
(6)
system, we consider the pseudo-polarization-matched AlInGaN/In0.13 Ga0.87 N QW structure with Al and In compositions of 0.08 and 0.25, which shows larger emission peak than the AlInGaN/InGaN system with zero internal field, although the internal field is not zero [13]. Both QW structures show that the internal field gradually decreases with increasing crystal angle and becomes zero at a given crystal angle. For example, the InGaN/GaN QW structure shows that the internal field becomes zero near a crystal angle of 55◦ . This is because the sum of the SP and the PZ polarizations in the barrier is equal to that in the well. On the other hand, for the AlInGaN/InGaN system, the internal field becomes zero near a crystal angle of 48◦ . The internal field change its sign when the crystal angle increases further. The transition wavelength of the AlInGaN/InGaN system increases slightly with increasing crystal angle. On the other hand, in the case of the conventional InGaN/GaN system, the variation in the transition wavelength is observed to be smaller than it is for the AlInGaN/InGaN system. Figure 2 shows the valence-band structures of 2.5-nm Al0.08 In0.25 Ga0.68 N/In0.13 Ga0.87 N QW structures with crystal angles of θ = (a) 0, (b) 30, and (c) 90◦ . The valence-band structures are obtained at a sheet carrier density of N2D = 10 × 1012 cm−2 . For comparison, results (dashed line) for the In0.23 Ga0.77 N/GaN QW structure are also plotted. The naming of the subbands for the QW structure follows the dominant composition of the wave function at the Γ point in terms of the |X , |Y , and |Z bases. The components i(=X ,Y ,Z ) X of each wave function are given by Pm Pm (3) (3) (6) (6) (1) (2) (1) Y = g m |g m + g m |g m , Pm = (g m + g m |g m (2) (4) (5) (4) (5) Z + g m + g m + g m |g m + g m )/2, and Pm = (2) (1) (2) (1) (4) (5) (4) (5) (g m − g m |g m − g m + g m − g m |g m − g m )/2. The valence-band structure of the QW structures with a general crystal orientation shows anisotropy in the QW plane, unlike the (0001)-oriented structure. In the case of θ = 0◦ , the subband energies of the conventional InGaN/GaN QW structure are shown to be redshifted because of the internal field being larger than that of the AlInGaN/InGaN QW structure. However, the redshift of the subbands is greatly reduced due to a reduction in the internal field for the QW structure with a large crystal angle. Also, for both QW structures, the effective masses of the first subbands are similar to each other.
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Journal of the Korean Physical Society, Vol. 65, No. 4, August 2014
Fig. 2. (Color online) Valence-band structures of 2.5-nm Al0.08 In0.25 Ga0.68 N/In0.13 Ga0.87 N QW structures with crystal angles of θ = (a) 0, (b) 30, and (c) 90◦ . The valence-band structures are obtained at a sheet carrier density of N2D = 10 × 1012 cm−2 .
Fig. 3. (Color online) y -polarized optical matrix elements of 2.5-nm Al0.08 In0.25 Ga0.68 N/In0.13 Ga0.87 N QW structures with crystal angles of θ = (a) 0, (b) 30, and (c) 90◦ .
Figure 3 shows the y -polarized optical matrix elements of the 2.5-nm Al0.08 In0.25 Ga0.68 N/In0.13 Ga0.87 N QW structures with crystal angles of θ = (a) 0, (b) 30, and (c) 90◦ . The optical matrix elements are calculated at a sheet carrier density of N2D = 10 × 1012 cm−2 and are averaged in the k x -k y plane because they show anisotropy in the QW plane for a general crystal orientation. For comparison, results (dashed line) for the In0.23 Ga0.77 N/GaN QW structure are also plotted. The y -polarized optical matrix elements increase greatly with increasing crystal angle. This is because the spatial separation between the electron and hole wave functions decreases with increasing the crystal orientation due to the reduced SP and PZ polarizations. For the QW structure with a crystal orientation of θ = 30◦ , the AlInGaN/InGaN system is shown to have a larger matrix element than the conventional InGaN/GaN system because the former has a smaller internal field than the latter. On the other hand, in the case of QW structures with a large crystal angle, the AlInGaN/InGaN system is shown to have smaller matrix element than the con-
Fig. 4. (Color online) y -polarized optical gain of 2.5-nm Al0.08 In0.25 Ga0.68 N/In0.13 Ga0.87 N QW structures with crystal angles of θ = (a) 0, (b) 30, and (c) 90◦ .
ventional InGaN/GaN system. Figure 4 shows the y -polarized optical gain of the 2.5nm Al0.08 In0.25 Ga0.68 N/In0.13 Ga0.87 N QW structures with crystal angles of θ = (a) 0, (b) 30, and (c) 90◦ . The optical gain spectra are calculated at a sheet carrier density of N2D = 10 × 1012 cm−2 and are averaged in the k x -k y plane because they show anisotropy in the QW plane for general crystal orientation. For comparison, results (dashed line) for the In0.23 Ga0.77 N/GaN QW structure are also plotted. The optical gain peak increases with increasing crystal angle because the y polarized optical matrix element increases with increasing crystal angle. For a crystal orientation of θ = 30◦ , the AlInGaN/InGaN system is shown to have a much larger optical gain than the conventional InGaN/GaN system. On the other hand, in the case of QW structures with a large crystal angle of θ = 90 ◦ , the AlInGaN/InGaN system is shown to have a smaller optical gain than the conventional InGaN/GaN system. Hence, we expect that we can obtain high-efficiency optoelectronic devices by using a quaternary AlInGaN layer for a relatively small
Effects of Crystal Orientation on the Optical Gain· · · – Seoung-Hwan Park et al.
crystal angles. In summary, the effects of crystal orientation on the optical gain characteristics of AlInGaN/InGaN QW structures with a reduced internal field were investigated by using the non-Markovian model with manybody effects. The conventional InGaN/GaN QW structure shows that the internal field becomes zero near a crystal angle of 55◦ . On the other hand, the AlInGaN/InGaN system shows that the internal field becomes zero near a crystal angle of 48◦ . For a crystal orientation of θ = 30◦ , the AlInGaN/InGaN system is shown to have a much larger optical gain than the conventional InGaN/GaN system. On the other hand, in the case of QW structures with a large crystal angle of θ = 90◦ , the AlInGaN/InGaN system is shown to have a smaller optical gain than the conventional InGaN/GaN system.
ACKNOWLEDGMENTS This work is supported by the Samsung Advanced Institute of Technology in Samsung Electronics Co., Ltd., under the Technology Collaboration program.
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