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Density Functional Theory (DFT) Simulations and Polarization Analysis of the Electric Field in InN/GaN Multiple Quantum Wells (MQWs) Zbigniew Romanowski,† Pawel Kempisty,† Konrad Sakowski,† Pawel Stra¸k,† and Stanislaw Krukowski*,†,‡ Institute of High Pressure Physics, Polish Academy of Sciences, Sokołowska 29/37, 01-142 Warsaw, Poland, and Interdisciplinary Centre for Materials Modeling, Warsaw UniVersity, Pawin´skiego 5a, 02-106 Warsaw, Poland ReceiVed: May 15, 2010; ReVised Manuscript ReceiVed: July 21, 2010
The ab initio DFT simulations of InN/GaN multiquantum wells (MQW) were used to obtain electric potential profile in the system that, after appropriate averaging procedure, reveal electric field in the wells and barriers and electric potential jumps at the interfaces. The field changes and the potential jumps were used to obtain the density of the polarization charges and the dipole layer at InN/GaN interfaces, respectively. It was shown that polarization dipoles are confined within the one double atomic layers, proving that they have different nature from the dipole layers emerging at the semiconductor surfaces or within p-n junctions. In parallel, a new formulation of polarization analysis, based on the energy minimum principle, was used to determine the electric field in polar InN/GaN multiquantum wells, purely within electrostatic framework, without resorting to experimental data. The obtained fields depend on both the well and the barrier thicknesses. DFT data are in good agreement with the continuum polarization analysis results that were obtained accounting the DFT determined potential jumps and using the standard polarization parameters. 1. Introduction Gallium, aluminum, and indium nitrides are presently considered principal materials for constructing multiple quantum wells (MQWs) based short wavelength optoelectronic devicess laser diodes (LDs) and light-emitting diodes (LEDs)sand accordingly, they are the focal point of present day semiconductor research.1 In spite of considerable progress in understanding of physical properties and in technological implementation of these materials, some fundamental problems remain to be solved to open routes for further progress. Since the overwhelming majority of the optoelectronic devices are constructed on the polar GaN(0001) surface, a natural consequence of such construction is the presence of built-in and strain-induced electric fields.2,3 The electric field changes energy levels of both type of carriers, electrons and holes, the phenomenon known as quantum confined Stark effect (QCSE).5,6 Yet far more sinister consequences are related to the fact that the electrons and holes are shuffled to the opposite ends of the quantum well.7-9 This effect significantly reduces overlap of their wave functions and lowers radiative recombination rates and, accordingly, the efficiency of optoelectronic devices, both LDs and LEDs.10-13 The negative influence of QCSE effect may be enhanced by Auger recombination or carrier leakage for high carrier density at high injection currents.14-17 That could lead to decrease of the device efficiency for higher injection currents, the phenomenon nicknamed as the “efficiency droop”.18 Even in typical blue LD and LEDs, based on 10 at. % indium QWs, the QCSE is supposed to cause a considerable separation of the carriers and dramatic reduction of the efficiency of the * Corresponding author: phone 48 22 888 02 44; fax 48 22 632 42 18; e-mail
[email protected]. † Polish Academy of Sciences. ‡ Warsaw University.
devices. An even more drastic influence is expected to exist in more In-rich green LEDs and LDs which are currently being developed. It is understandable that the QCSE was studied intensively using both theoretical and experimental methods. The experimental investigations were directed mostly toward size dependence of the emission from MQW structures.6,8 It is commonly assumed that a fingerprint of the field separation of the carriers is a red shift of the emission line and dramatic increase of the carrier lifetime for wider wells.9,10 A second possible indication of the QCSE is the screening effect, which can be controlled by the doping of the barriers of the QW system.19 Typically, n-type doping by Si donors is used for that purpose. A different route to alleviate QCSE would be to design polarization-matched GaInN/AlGaN MQW in order to reduce carrier leakage and to remove the efficiency droop altogether.20 It is known that these recipes are based on simplified treatment of the problem and identification of the main trends in the system. Since the electro-optical emitting MQW system in both LEDs and LDs is located in the vicinity of the p-n junction, the additional field can, in principle, affect the properties of the system and change observed trends in the optically emitting system. Therefore, an interesting solution is to use optical pumping of the wells, using limited or high optical power. This could help to identify the presence of screening effects in the observed spectra. Among other methods that could be additionally used to investigate the piezoelectric effects, the pressure investigations play an important role.21 Yet another solution is to grow MQW on semipolar planes which could potentially remove the piezoelectric contributions completely.22 In summary, the progress in understanding of these systems is slow because the researchers need to cope with additional, potentially strong effects which could affect the results, such as the presence of the extended defects within the well system or the segregation of indium. The latter could, in principle,
10.1021/jp104438y 2010 American Chemical Society Published on Web 08/09/2010
Electric Field in InN/GaN Multiple Quantum Wells transform 2D translationally invariant QW system into a collection of quantum dots, having drastically different symmetry and consequently the electrical and optical properties. Therefore, interpretation of the results of the MQWs measurements is a fairly nontrivial task, and additional insight from the theoretical analysis could be very helpful. Several theoretical approaches were used to shed some light onto various aspects of the problem. Initially, the detailed discussion of various heterostructures with the identification of monopole and dipole electric charges was given by Franciosi and Van de Walle.23 Since the QW system was originally considered difficult to tackle by ab initio methods due to its shear size, the approximate step by step methods were developed. A most systematic approach along this line was to obtain the spontaneous polarization and piezoelectric tensor of bulk material, i.e., GaN or InN which, via the deformation tensor, can be used to determine the magnitude of polarization in the strained thin layer. Such an approach is typically used in estimation of the magnitude of the effect in mixed GaInN/GaN QWs. An extension of this approach was to use self-consistent tight-binding calculations of various nitride MQWs.24 The first attempts to use DFT calculations to nonpolar GaAs/AlAs heterostructures resulted in limited success.25-29 This was related to small size of simulated systems and to relative small charge effects in nonpolar semiconductors. Afterward, first direct DFT studies of polar nitride In0.2Ga0.8N MQWs were undertaken by Buongiorno Nardelli et al.30 They found that the strain-related polarization charge at GaN/GaInN heterointerfaces contributed significantly to the valence and conduction band offsets.30 They have also proved that no interface related states exist in the bandgap.30 Much later, similar efforts was undertaken by Junquera et al. which also brought limited success due to difficulties in extracting physically relevant quantities from strongly oscillation charge distributions.31 In parallel, DFT calculations were used to polar AlN/GaN heterostructures.32,33 These investigations determined band offsets, polarization, and formation enthalpies of polar and nonpolar AlN/GaN heterostructures. In the present paper we extend this approach, applying DFT ab initio methods to simulations of the electric properties of the GaN/InN QW system. The InN/GaN MQWs are more difficult technologically than standard GaN/GaInN MQW design. Nevertheless, considerable technological advances were made which make possible investigations of the InN/GaN MQW systems by both electrical and optical measurements.34,35 The InN/GaN are still of relatively poor quality, but their construction opens the route to potential investigation of these systems. As InN/GaN MQWs are relatively easy to be tackled by ab initio simulations, they offer a possibility of comparison of experimental data and DFT simulations. 2. Calculation Method A freely accessible DFT SIESTA code, used in the reported calculations, combines norm conserving pseudopotentials with the local basis functions.36-38 The pseudopotentials for Ga, In, and N atoms were generated, using ATOM program for allelectron calculations.39-41 Gallium and indium 3d electrons were explicitly included in the valence electron set. For Ga, In, and N atoms, the double ζ local basis set was used with polarization. Local density approximation (LDA) Ceperley Alder (CA) formulation with two orbital basis set were used in the tests: Sankey confinement potential and soft confinement potential, with the following values for the lattice constants of bulk crystals: GaN, a ) 3.181 Å and c ) 5.146 Å; InN, a ) 3.527 Å and c ) 5.713 Å. These values are in good agreement with
J. Phys. Chem. C, Vol. 114, No. 34, 2010 14411 the experimental data for GaN, a ) 3.189 Å and c ) 5.185 Å, and for InN, a ) 3.52 Å and c ) 5.72 Å. The results reported below were obtained using LDA CA approximation and orbitals basis generated with soft confinement potential. All additional data on the simulation procedure may be found in ref 42. 3. Piezoelectric Analysis In typical blue or green LED or LDs, the MQW system could be located in the p-n junction electric field region or positioned outside this area.18 In the first case, the electric conditions change, depending on the external electric field applied to the device. In the second, the electric potential distribution is universal, reflecting the fact that the Fermi energy is controlled by dominant point defects in the bulk of semiconductor; i.e., it is located at the same energy level at both sides of MQW system. The latter case is considered below. Accordingly, the electric potential change across the entire MQW system, located far from the p-n junction, is zero; i.e., contributions of polarization and screening charges cancel each other. Therefore, standard periodic boundary conditions (PBC) at the edges of the simulated area can be adopted for electric potential. This is particularly fortunate because PBC is currently used in the overwhelming majority of DFT simulations. The PBC were also used in the DFT investigations of AlN/GaN superlattices fairly recently.44 In principle, the nonzero electric field can be also investigated within DFT framework. Recently, an efficient Laplace correction scheme was presented which allows us to alleviate PBC condition, preserving efficiency of the calculations and stability of self-consistent-field (SCF) iterative solution scheme.43 This allows to set up a preserved electric potential difference still using highly efficient fast Fourier transform (FFT) to solve the Poisson equation. The zero field assumption is in agreement with Landau and Lifshitz identification of minimal energy state for insulating solids.45 They argued that, in large scale, zero electric field corresponds to global energy minimum not only for metals and semiconductors but also for insulators. The zero field condition, in the case of insulators, could take even several hours to attain.45,46 It is worth to stress out that such a state is attained only for the distances much larger than screening length. The zero field assumption is the foundation of the Berry phase approach used in determination of polarization of insulating solids.46-48 The simulated (1 × 1) supercell consists of InN well and the GaN barrier of the thickness zInN and zGaN, respectively, being a single block of the GaN/InN MQWs system. Assuming that the wells are located in the region outside the p-n junction, the zero potential difference condition across the entire MQW system is fulfilled, as the Fermi level is identically fixed with respect to the band energies at both sides. Neglecting small effects at the edges of MQWs system, the PBC for potential can be applied to individual block which imposes following bound for z-components of the electric fields in InN and GaN layers:
E3,GaNzGaN + E3,InNzInN + ∆V1 + ∆V2 ) 0
(1)
where, as it will be shown below, ∆V1 and ∆V2 are the potential jumps due to existence of dipole layers in the GaN-InN and InN-GaN polar heterostructures. These dipole layers have not been employed in the description of nitride devices.49-51 As argued by Landau and Lifshitz, for large size systems, the
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screening field in the insulator (semiconductor) interior compensates the field, created by spontaneous polarization. This is not fulfilled in the case of MQWs system where the layers are much thinner than the screening length, and therefore the field is different: the screening long distance field is absent, and the minimal energy state should be attained for the uniaxial electric field, created by spontaneous polarization and by piezoelectric effects. In the case of MQWs system, the spontaneous piezoelectric fields in both wells and barriers are incompatible with PBC condition for potential, expressed by eq 1. In order to comply with eq 1, the system has to create the electric fields in the well and the barrier having the opposing sign. This leads to emergence of polarization charges that were determined in DFT calculations by Buongiorno Nardelli et al.30 The field modification costs additional electric energy, and the system attains the state which minimizes that contribution. In principle, the energetically stable electric state of the system could be calculated either counting the Coulomb energy of polarization charges or by integration of the electric field density, both approaches giving the same result.45 For the uniaxial fields the latter approach is much simpler, giving the condition that the fields in the well E3,InN and the barrier E3,GaN are such that minimize the excess electric field energy functional, ∆f:
in GaN barrier: E3,GaN )
(0) (0) (εGaNE3,GaN - εInNE3,InN )zInN - εInN(∆V1 + ∆V2) εInNzGaN + εGaNzInN
(4b) The potential jumps at interfaces ∆V1 and ∆V2 cannot be derived from the polarization approach alone. There is no theoretical tool to obtain these quantities within such approach; therefore, they have to be obtained from DFT simulations. It has to be noted that the above-derived expressions for the electric field in the wells and in the barriers are different from those derived by Pabla et al. for InGaAs/GaAs MQWs structures.52-54 Simplified to the presently assumed conditions, i.e., under no bias and outside the junction field, their expressions are
for the well: E3,GaN )
zInNE(0) 3 zInN + zGaN
(5a)
zInNE(0) 3 zInN + zGaN
(5b)
for the barrier: E3,InN ) -
∆f )
ε0 2
∫ d2r[ ∫0z
InN
∞ (0) dz [ε33,InN (E3,InN - E3,InN )]2 +
∫0z
GaN
∞ (0) dz [ε33,GaN (E3,GaN - E3,GaN )]2
]
(2)
where E3(0) is the piezoelectric field, defined as difference of the field in the well and in the barrier, i.e.
E(0) 3 ) E3,InN - E3,GaN (0) where E(0) 3,InN and E3,GaN are the fields arising from the spontaneous and strain-induced polarization in strained bulk InN and GaN,2,4 given by
E(0) 3 )
D(0) 3 ε0ε∞33
)
P3 ε0ε∞33
)
Peq + e333 + 2e311 ε0ε∞33
(3)
In eq 3, the index denoting material (GaN or InN) was omitted for simplicity (notation of Bernardini et al. was used3). The InN and GaN spontaneous polarization and piezoelectric constants were obtained from ref 3. These values are as follows: for GaN, eq ) -0.029 C/m2, e33,GaN ) 0.73 C/m2, and e31,GaN ) -0.49 PGaN eq ) -0.032 C/m2, e33,InN ) 0.97 C/m2, and C/m2; for InN, PInN e31,InN ) -0.57 C/m2. The strain values 3 and 1 (Voigt notation is used, i.e., index “1” denotes “11”, etc.) were derived from the average positions of atoms in the supercell after relaxation. Equation 1 was used to remove the electric field in the barrier, i.e., E3,GaN from the energy minimization procedure. Employing minimum condition to the excess energy functional in eq 3, subject to constraint given by eq 1, the following electric fields were obtained:
in InN well: E3,InN )
(0) (0) (εInNE3,InN - εGaNE3,GaN )zGaN - εGaN(∆V1 + ∆V2) εInNzGaN + εGaNzInN
(4a)
(5c)
Without additional input these expressions could not provide the magnitude of the fields. Therefore, the authors supplemented them by the field value obtained from elastic and piezoelectric properties of GaAs and InAs. For the calculation purposes they considered 15% In well, obtaining the polarization field E3(0) ) 215 kV/cm.50 In their formulation dipole related potential jumps ∆V1 and ∆V2 were not used. Accordingly, their formalism correctly recovered the condition in eq 1, while the field magnitude was supplemented using other arguments, which poses some doubt on the validity of the procedure. Later this formulation was modified and applied to InGaN/GaN quantum wells by Takeuchi et al.10 They replaced the polarization field definition, given by eq 5c, by the value obtained from fitting to photoluminescence (PL) data. From analysis of the PL of In0.13Ga0.87N/GaN structure they obtained the polarization field 1.1 MV/cm. A similar procedure was used by Jho et al. to slightly different In0.15Ga0.85N/GaN structure, from which the authors obtained polarization field equal to 2.1 ( 0.2 MV/cm.54,55 These authors added the contribution of the finite screening length in determination of the polarization field. Nevertheless, so large variations of the field cannot be explained by different In content only. Identical formulation was recently applied to other InGaN/ GaN MQWs systems without significant modifications.57,58 In contrast to the above approach, the formulation presented in eqs 1-4, based on the minimum-energy principle, allows us to obtain the magnitude of the field from the electrostatic arguments without resorting to adjustment to experimentally determined PL data. The new contribution which was added was the dipole layer potential jumps ∆V1 and ∆V2, which was not accounted for in the Pabla et al. model. This new contribution cannot be obtained for piezoelectric analysis; it has to be derived from DFT calculations. It has to be noted also that these
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Figure 1. Average potential profiles for the wells of 8 (a), 6 (b) and 4 (c) In-N ALs and for the barrier of 16 Ga-N ALs, obtained from DFT SIESTA simulations: dashed black line, horizontally averaged potential; thin solid red line, potential adjacent averaged over 2.5 Å in z direction; thick solid blue line, linear approximation to averaged profile.
Figure 2. Average potential profiles for the wells of six In-N ALs and the barriers of 6 (a), 10 (b), and 16 (c) Ga-N ALs, obtained from DFT SIESTA simulations: dashed black line, horizontally averaged potential; thin solid red line, potential adjacent averaged over 2.5 Å in z direction; thick solid blue line, linear approximation to averaged profile.
potential jumps were not used in refs 54-58; therefore, the determination of the polarization fields may have been seriously affected by the dipole contribution. The magnitude of these contributions is determined in the next section.
of ∼2.5 nm. The InN quantum wells consisting of 4, 6, and 8 In-N ALs were simulated. The diagrams show the potential profiles within periodic the InN-GaN system, obtained by horizontal averaging of electric potential derived from solution of Poisson equation in SCF loop, then the adjacent averaged potential over 2.5 Å in the vertical direction, and finally the linear fit to such adjacent averaged potential profile, allowing
4. DFT Results The DFT results, presented in Figure 1, were obtained for constant barrier width of 16 Ga-N atomic layers (ALs), i.e.,
TABLE 1: Elastic Deformations, Electric Potential Jumps (in V), and Electric Fields (in MV/cm) Obtained from Polarization Analysis and DFT Simulations for 16 AL GaN Barrier and Different Thickness of InN QWs ALs GaN-16
1(InN)
3(InN)
1(GaN)
3(GaN)
V1
V2
E3,InN(pol)
E3,InN(DFT)
E3,GaN(pol)
E3,GaN(DFT)
InN-8 InN-6 InN-4
-0.073 -0.058 -0.082
0.033 0.036 0.050
0.028 0.045 0.019
-0.018 -0.036 -0.056
-1.64 -1.49 -1.88
1.33 1.23 1.28
-7.04 -8.20 -7.31
-10.17 -11.17 -17.31
4.69 4.04 3.43
6.43 5.28 6.18
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TABLE 2: Elastic Deformations, Electric Potential Jumps (in V), and Electric Fields (in MV/cm) Obtained from Polarization Analysis and DFT Simulations for Six AL InN QWs and Different Thickness of GaN Barriers ALs InN-6
1(InN)
3(InN)
1(GaN)
3(GaN)
V1
V2
E3,InN(pol)
E3,InN(DFT)
E3,GaN(pol)
E3,GaN(DFT)
GaN-6 GaN-10 GaN-16
-0.051 -0.058 -0.058
0.018 0.028 0.036
0.053 0.045 0.045
-0.059 -0.029 -0.036
-2.01 -1.21 -1.49
0.88 0.64 1.23
-6.38 -6.60 -8.20
-2.24 -11.11 -11.17
13.28 5.49 4.02
9.15 7.85 5.27
us to obtain macroscopic values of electric field and potential jumps. As it is shown here, the procedure of adjacent averaging leads to physically coherent picture of the field within the well and the barrier with reasonably good approximation to macroscopic potential within the structure. It is worth noting that in spite of graphic similarity the diagram presented in Figure 1 shows the electric field, and not the band profiles, which cannot be derived from DFT calculations directly. From these diagrams it follows that the electric field in the well is different for various well widths. Note that the field difference is not related to the screening, which is absent in our model; it stems from the PBC conditions, as expressed in eq 1. The numerical data containing comparison of the fields obtained from polarization analysis and DFT simulations are presented in Table 1. It has to be stressed that, in addition to the fields in the barriers and in the wells, considerable potential jumps are observed at interfaces. Generally these potential jumps are huge, attaining more than 1.5 V in single QW. As the fields, the potential differences at both edges of the well are not identical. This is a new effect which stems from the dipole layers on both GaN-InN and InN-GaN interfaces (heterojunctions). As it follows from the above data, the potential jumps are considerable, exceeding 1 V in all cases, making InN well considerably shallower. The localization of the carriers in QW is therefore less efficient and that allows for their easier escape from the wells. Another interesting aspect is the field dependence on the barrier thickness. The DFT data, obtained for six AL thick InN well and for different thickness of GaN barrier, presented in Figure 2, show large variation of the field in the wells. Thus, the fields in the well depend on the barrier thickness (see Table 2). It is also evident that for the lowest thickness of the barrier the field in the well is not uniform. This may be related to the additional charge located at the interfaces. For larger barrier thickness, the field is virtually constant (the potential profile is linear). The magnitude of the field is large summing up to the difference of potential of about 1.4 V, giving the largest value for the thickest barrier, which confirms analysis given above. The detailed comparison of the polarization and DFT results, presented in Table 2, indicates a reasonable agreement of these two approaches. As it is shown below, the significant difference is obtained for the thin layers and barriers only where the continuum approximation, being basis of polarization analysis, is burdened by largest errors. Another interesting feature is related to direct verification of the presence of the dipole layer. It is worth noting that, according to eq 4, in the absence of the interface dipole layer contribution (i.e., ∆V1 ) ∆V2 ) 0), the field in the wells and in the barriers depends on the ratio of the thicknesses InN of GaN layers only. Similarly, in the formulation of Pabla et al., the fields depend on the wall to barrier thickness ratio.50-54 Thus, a straightforward verification of these two approaches can be made for the identical thickness of both layers, i.e., for zInN/zGaN ) 1, changing its value only. The results are shown in Figure 3. From the results shown in Figure 3, it follows that these fields are evidently different, confirming our analysis, being in
Figure 3. Average potential profiles for the identical number of ALs in both InN wells and GaN barriers: 4 (a), 6 (b), and 8 (c) obtained from DFT SIESTA simulations: dashed black line, horizontally averaged potential; thin solid red line, potential adjacent averaged over 2.5 Å in z direction; thick solid blue line, linear approximation to averaged profile.
contradiction with the prediction of Pabla et al.52 It has to be added that the 4 AL case does not recover the linear distribution of the potential, which is most likely due to the overlap of the charges and dipoles from both interfaces for the thin case. These simulation results and polarization analysis are summarized in Table 3. The data in the tables show that the DFT and polarization analysis results are in reasonable agreement. The only exception
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TABLE 3: Elastic Deformations, Electric Potential Jumps (in V), and Electric Fields (in MV/cm) Obtained from Polarization Analysis and DFT Simulations for Identical Number of Atomic Layers (ALs) in InN QW and GaN Barriers ALs
1(InN)
3(InN)
1(GaN)
3(GaN)
V1
V2
E3,InN(pol)
E3,InN(DFT)
E3,GaN(pol)
E3,GaN(DFT)
GaN-4 and InN-4 GaN-6 and InN-6 GaN-8 and InN-8
-0.051 -0.051 -0.050
0.019 0.019 0.031
0.053 0.053 0.054
-0.044 -0.055 -0.038
-1.79 -2.01 -1.25
0.77 0.88 1.34
-5.41 -6.38 -6.81
4.70 -2.24 -7.86
14.58 13.29 6.61
4.60 9.15 7.70
is the 4 AL case when the continuous approximation is probably not applicable. Altogether, the above results confirm also the validity of the energy minimization approach, in conjunction with the polarization results of Bernardini et al. and also prove applicability of DFT direct simulations to quantum well systems.3 It has to be added that the above results neglect the role of screening by mobile charge and charged point defects; i.e., it corresponds undoped MQW system. The assumption is verified by direct plot of the band structure of GaN-8 and InN-8 system, shown in Figure 4. As it follows from this diagram, the system has no mobile charge, i.e., no electrons or holes in InN and GaN layers, confirming that the above results corresponds to pure polarization charge field in MQWs.
reduced permitting to adjust linear profiles and consequently the fields in the layers and potential jumps at the interfaces. This led to determination of the polarization charge and to identification of dipole layer at the GaN-InN interfaces. The effect considerably changes the depth of InN quantum wells as the potential jumps exceed 1 V which detrimentally affects localization of the carriers in InN QWs. This effect may seriously hamper carrier localization in InN/GaN and, accordingly, in similar InGaN/GaN QWs. Both approaches were mutually compared which resulted in coherent picture of the fields in the layers, creating new possibilities in the modeling of semiconductor two-dimensional quantum structures, such as quantum wells by atomic level ab initio simulations, based on application of DFT formalism.
5. Summary
Acknowledgment. The research was supported by the European Union within European Regional Development Fund through grant Innovative Economy (POIG.01.01.02-00-008/08). The calculations were made using computing facilities of Interdisciplinary Centre for Mathematical and Computational Modelling of Warsaw University (ICM UW).
The newly formulated, minimum excess energy approach opens the route to calculation of the electric fields in twodimensional quantum well structures exclusively within the electrostatic theory, i.e., without resorting to PL data, used in earlier reports. The polarization analysis indicates that the fields in the well depend on the thickness of both barriers and wells. The obtained fields are very high leading to the energy difference above 1.5 eV in six ALs thick InN quantum well, influencing carrier distribution within the wells and negatively affecting localization of carriers in InN/GaN MQS. It is expected that such effect should be also observed in standard InGaN/GaN MQWs, i.e., in optically active part of blue and green LDs and LEDs. Ab initio DFT simulations were used to obtain the electric potential distribution in the MQWs system. The electric potential profile in the direction perpendicular to the structure was obtained by averaging in the plane parallel to the structure. By an appropriate averaging procedure, these variations were
Figure 4. Band diagram (left) and density of states (DOS) obtained for GaN-8 and InN-8 system. The total DOS is denoted by black dashed line; DOS associated with Ga, In, and N atoms are denoted by green shade, red, and black dotted line, respectively.
References and Notes (1) Nakamura, S.; Fasol, G.; Pe Arton, S. J. The Blue Laser Diodes; Springer: New York, 2000. (2) Bernardini, F.; Fiorentini, V.; Vanderbilt, D. Phys. ReV. Lett. 1997, 79, 3958. (3) Bernardini, F.; Fiorentini, V.; Vanderbilt, D. Phys. ReV. B 1997, 56, R10024. (4) Bernardini, F.; Fiorentini, V. Phys. Status Solidi B 1999, 216, 391. (5) Miller, D. A. B.; Chemla, D. S.; Damen, T. C.; Gossard, A. C.; Wiegmann, W.; Wood, T. H.; Burrus, C. A. Phys. ReV. Lett. 1984, 53, 2173. (6) Halsall, M. P.; Nicholls, J. E.; Davies, J. J.; Cockayne, B.; Wright, P. J. J. Appl. Phys. 1992, 71, 907. (7) Buongiorno Nardelli, M.; Rapcewicz, K.; Bernholc, J. Appl. Phys. Lett. 1997, 71, 3135. (8) Vaschenko, G.; Patel, D.; Menoni, C. S.; Keller, S.; Mishra, U. K.; Denbaars, S. P. Appl. Phys. Lett. 2001, 78, 640. (9) Łepkowski, S. P.; Teissyere, H.; Suski, T.; Perlin, P.; Grandjean, N.; Massies, J. Appl. Phys. Lett. 2001, 79, 1483. (10) Takeuchi, T.; Wetzel, C.; Yamaguchi, S.; Sakai, H.; Amano, H.; Akasaki, I.; Kaneko, Y.; Nakagawa, S.; Yamaoka, Y.; Yamada, N. Appl. Phys. Lett. 1998, 73, 1691. (11) Langer, R.; Simon, J.; Ortiz, V.; Pelekanos, N. T.; Barski, A; Ander, R.; Godlewski, M. Appl. Phys. Lett. 1999, 74, 3827. (12) Seo, I. J.; Kollmer, H.; Off, J.; Sohmer, A.; Scholz, F.; Hangleiter, A. Phys. ReV. B 1998, 57, R9435. (13) Lefebvre, P.; Allegre, J.; Gil, B.; Mathieu, H.; Grandjean, N.; Leroux, M.; Massies, J.; Bigenwald, P. Phys. ReV. B 1999, 59, 15363. (14) Shen, Y C.; Muller, G O.; Watanabe, S.; Gardner, N F.; Munkholm, A.; Kramers, M R. Appl. Phys. Lett. 2007, 91, 141101. (15) Rozhansky, I. V.; Zakeheim, D. A. Phys. Status Solidi C 2006, 3, 2160. (16) Schubert, M. F.; Chajed, S.; Kim, J. K.; Schubert, E. F.; Koleske, D. D.; Crawford, M. F.; Lee, S. R.; Fisher, A. J.; Thaler, G.; Banas, M. A. Appl. Phys. Lett. 2007, 91, 231114. (17) Xie, J.; Ni, X.; Fan, Q.; Schimada, R.; Ozgor, U.; Morkoc, H. Appl. Phys. Lett. 2008, 93, 121107. (18) Morkoc, H. Handbook of Nitride Semiconductors and DeVices; Wiley-VCH: Berlin, 2008; Vol. 3. (19) Ambacher, O.; Smart, J.; Shealy, J. R.; Weimann, N. G.; Chu, K.; Murphy, M.; Schaff, W. J.; Eastman, L. F.; Dimitrov, R.; Wittmer, L.; Stutzmann, M.; Rieger, W.; Hilsenbeck, J. J. Appl. Phys. 1999, 85, 3222.
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