Band Structure and Quantum Confined Stark Effect ... - ACS Publications

May 24, 2012 - Ultra-Broadband Optical Gain in III-Nitride Digital Alloys. Wei Sun , Chee-Keong Tan , Jonathan J. Wierer , Nelson Tansu. Scientific Re...
0 downloads 0 Views 572KB Size
Download PDF
Article pubs.acs.org/crystal

Band Structure and Quantum Confined Stark Effect in InN/GaN superlattices I. Gorczyca* and T. Suski Institute of High Pressures Physics, Polish Academy of Sciences, Warsaw, Poland

N. E. Christensen and A. Svane Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark ABSTRACT: InN/GaN superlattices offer an important way of band gap engineering in the blue-green range of the spectrum. This approach represents a more controlled method than the band gap tuning in quantum well systems by application of InGaN alloys. The electronic structures of short-period wurtzite InN/GaN(0001) superlattices are investigated, and the variation of the band gap with the thicknesses of the well and the barrier is discussed. Superlattices of the form mInN/nGaN with n ≥ m are simulated using band structure calculations in the Local Density Approximation with a semiempirical correction for the gap error. The calculated band gap shows a strong decrease with the thickness (m) of the InN well. In superlattices containing a single layer of InN (m = 1) the band gap increases weakly with the GaN barrier thickness n, reaching a saturation value around 2 eV. In superlattices with n = m and n > 5 the band gap closes and the systems become “metallic”. These effects are related to the existence of the built-in electric fields that strongly influence valence- and conduction-band profiles and thus determine effective band gap and emission energies of the superlattices. Varying the widths of the quantum wells and barriers one may tune band gaps over a wide spectral range, which provides flexibility in band gap engineering.

I. INTRODUCTION Optoelectronic nitride devices are based on nitride heterostructures: multi-quantum-well (MQW) or superlattice (SL) structures. The epitaxial growth of InN/GaN structures is difficult1−3 because of differences in lattice constants for the binary compounds, and research efforts have been focused on alloyed InGaN/GaN systems. With recent advances in growth technologies (fabrication of high quality nanoscale structures) the interest in InN/GaN short period SLs has increased. In MQW or SL structures the band gap is modified by the layer thicknesses rather than by composition as in the case of alloys. Changing the widths of the quantum wells and barriers, in case of SLs represented more adequately by the number of monolayers, one may tune the band gaps over a wide spectral range. Recent theoretical studies of nitride heterostructures have brought a deeper understanding of their properties.4−6 Wurtzite InN/GaN(0001) heterostructures possess large macroscopic polarization fields due to the spontaneous polarization along the [0001] polar axis and the piezoelectric effect caused by lattice mismatch between layers. The macroscopic polarization induces a built-in electric field. The ab initio studies of such SLs grown in the c-direction of the wurtzite structure supply information about these fields and the quantum confined stark effect (QCSE), which strongly influences the valence- and conduction-band profiles and thus determines the effective © 2012 American Chemical Society

band gap and emission energies of InN/GaN and AlN/GaN SLs.6,7 Due to the difficulty in obtaining InN/GaN SL systems with well-defined incorporated strain, measurements of the strength of the built-in electric fields are very scarce in these materials. Theoretical studies show a significant dependence on SL characteristics. The value predicted by Martin et al.8 via piezoelectric coefficients is 5.5 MV/cm for an InN(0001) film grown on a GaN substrate. Shieh et al.6 calculated values from 0.56 MV/cm to 9.25 MV/cm for the InN well of InN/GaN SLs, and from 1.08 MV/cm to 11.17 MV/cm for the GaN barrier of these SLs, depending on layer thicknesses and strain conditions. A second fundamental parameter characterizing the heterostructure is the valence band offset (VBO). Still there is no consensus on the VBO of wurtzite InN/GaN. A wide range of values have been reported so far. Experimental data range from 0.58 eV9 through 0.78 eV,10 0.8 ± 0.1 eV11 up to to 1.05 eV,8 while calculated values from 0.3 to 1.27 eV have been reported (see ref 12). The fabrication of very thin InN QWs in a GaN matrix with width from 1 to 5 monolayers of InN1−5,13,14 has stimulated the interest in InN/GaN SLs with thin layers of InN. Since the well Received: March 5, 2012 Revised: May 10, 2012 Published: May 24, 2012 3521

dx.doi.org/10.1021/cg300315r | Cryst. Growth Des. 2012, 12, 3521−3525

Crystal Growth & Design

Article

layer in these SLs is composed of a binary InN, problems encountered in the development of conventional InGaN/GaN QWs with high In contents, such as compositional fluctuation, can to a high extent be avoided. This motivated us to study the dependence of the band gap, Eg, on layer thicknesses in mInN/nGaN(0001) SLs with a few monolayers of InN (m = 1 to 6) and with the same or higher number of GaN monolayers, n ≥ m. We compare them with the band gaps of Inx Ga1−xN alloys. For the purpose of consistency, all the calculations are performed for fully relaxed structures mimicking a free-standing configuration.

II. CALCULATIONAL METHOD The electronic structures of the wurtzite nInN/mGaN(0001) SLs have been examined by self-consistent ab initio calculations. Approaches based on the Local Density Approximation (LDA) to density functional theory, with the Perdew−Zunger parametrization15 of the Ceperley−Alder exchange-correlation16 were used. The calculations were performed in two steps, applying two different computational schemes. In the first step the atomic coordinates were determined by minimization of the Hellmann−Feynmann forces. For this task we used pseudopotentials as implemented in the Vienna Ab-initio Simulation Package (VASP).17 A cutoff energy of 600 Ry for the plane wave basis set was sufficient to obtain converged results. We found that the resulting band gaps are very similar for cutoff energies of 400 and 600 Ry. The Ga-3d and In-4d semicore states are included in the band structure calculations. We impose “free-standing” conditions, assuming that in-plane lattice constants in the InN and GaN regions match completely. Then all the atomic positions and the in-plane lattice constant, a, as well as the c lattice parameter are free to vary until the system reaches a free-standing configuration. We have to note that this is not a “strain-free” configuration, because both InN and GaN regions are strained to match the equilibrium in-plane lattice constant. Subsequently, in the second step of calculations, the band structures were obtained by the linear-muffin-tin-orbital (LMTO) method in a full-potential (FP) version.18 The semicore cation d states of Ga(3d) and In(4d) were included as local orbitals.19 Further details of the LDA-LMTO calculations are given elsewhere.20,21 A semiempirical correction for the deficiency of LDA in predicting semiconductor gaps has been applied. The correction procedure (LDA+C) adds external potentials, which are sharply peaked at the nuclear sites and therefore produce an upshift in the s-like conduction-band-minimum state. The adjusting potentials are included in the full self-consistency cycles, and they affect total-energies only very little. (In fact they increase the predicted equilibrium volumes by typically 1−2%, that is, they reduce slightly the tendency to overbinding in the LDA). The parameters of the potentials are determined by fitting to experimental band parameters for binary compounds or to GW calculations.22 The parameters used in the sharply peaked potentials are specific for the atomic species and therefore transferable in the sense that they can be determined for the binary compounds and subsequently be applied to systems where the two compounds are combined, as in alloys, SLs, and heterojunctions. The parameters used in the present work were determined in previous works for GaN and InN by adjusting to experimental gap values, and subsequently applied to the InxGa1−xN and InxGa1−yAl1−x−yN alloy systems.23,24 They were kept unchanged while composition and volume were varied.23−27

Figure 1. Calculated band gaps, Eg, for mInN/nGaN SLs vs In cation fraction, x = m/(m + n), compared with calculations performed for InxGa1−xN alloys (asterisks and the dashed curve, “uniform In distribution”).24 Lines are spline fits to the calculated values. In the inset the gaps for 1InN/nGaN SLs are compared with the gaps obtained by the LDA without the band gap correcting potentials.

of the form InxGa1−xN, and therefore the Eg found here for the 1/1 SL lies on this dashed curve for x = 0.5. In the same series of calculations a further decrease of Eg for alloys with clustering of In atoms was demonstrated.23−26 The most pronounced effects of clustering were found, when every fourth cation hexagonal layer consisted entirely of In atoms.23−26 In this geometry the In0.25Ga 0.75N alloy is equivalent to the 1/3 SL. Therefore, in Figure 1 the band gap corresponding to this SL coincides with the band gap in the clustered In0.25 Ga 0.75N alloy (black dot in Figure 1). In the inset the band gaps for the sequence of 1/n SLs as calculated by the FP-LMTO method with LDA+C, are compared to the values obtained by VASP with pure LDA, in the first step of the calculations. The corrected gaps are larger than the LDA values by more than 1 eV, but the shift depends somewhat on x. Analyzing Figure 1, we observe that the SL band gap does not vary with the composition in a manner comparable to the case of alloys. In contrast, it depends on both well and barrier thickness, but differently: 1 In the case of 1/n SLs, the value of Eg increases slowly with increasing barrier thickness, converging to a value slightly above 2 eV. Further increase of the barrier width (above about 10 MLs) seems to have negligible effect on Eg. As we will show in the next section it is possible to determine a layer-resolved local band gap. In the case of the 1/n SLs the overall band gap corresponds to the local Eg of the InN ML. Contributions to the InN well wave functions coming from neighboring GaN layers cause a significant increase of the local gap from the value 0.65 eV (pertaining to bulk InN) to about 2.1 eV in the InN layer in these SLs. An increased influence of the GaN-like wave functions on the states related to the InN well can be seen up to m = 5. 2 In contrast, in SLs with more than one InN ML a decrease of the band gap with barrier thickness is found. For m = 2 the decrease is rather slow, but stronger for m = 3 and m = 4. 3 In particular, for the case of SLs with equal thickness of wells and barriers (m = n) band gaps decrease rapidly (starting from m = n = 2) with the increasing layer thickness and becomes zero for m = n > 5.

III. RESULTS AND DISCUSSION A. Band Gaps. The calculated band gaps of mInN/nGaN (denoted briefly as m/n) SLs for different numbers of well (m) and barrier (n) MLs, where n ≥ m, are presented on Figure 1. The results are compared with the previous calculations for InxGa1−xN alloys, where large bowing of the band gap was found.23−26 The dashed line in Figure 1 corresponds to what was called the “uniform cation distribution” in the case of alloys 3522

dx.doi.org/10.1021/cg300315r | Cryst. Growth Des. 2012, 12, 3521−3525

Crystal Growth & Design

Article

Figure 2. (a) Atomic positions in the 5InN/5GaN supercell. The numbers give the bond lengths in ångstrom. (b) Layer resolved density of states. The first two figures show the density of states for the layers at an A interface, last two similarly for the layers at either side of a B interface, while the middle figure shows the density of states for the layer in the middle of the InN well. The dashed line (E = 0) corresponds to the position of the lowest unoccupied state of the superlattice.

In conclusion, it emerges that the band gaps are more sensitive to the well thickness than to the barrier width and that the dependence on barrier thickness is different for SLs with different number of well MLs. To explain the behavior of Eg in m/n SLs, we analyzed the lattice relaxation, density of states and valence and conduction band profiles within each of the SLs. Below we discuss the results for the SL with the smallest nonzero band gap, that is, 5/5, aiming at explaining the SL “metallization” occurring in the 6/6 SL. B. Lattice Relaxation and Band Profiles. In Figure 2a, the atomic positions in the 5/5 SL are shown with two kinds of interfaces: type-A interface defined to be the plane (perpendicular to the [0001] direction), between the last nitrogen layer in the GaN barrier and the first In layer in the InN well, and a type-B interface defined to be the plane between the last N layer in the InN well and the adjacent first Ga layer in the GaN barrier. Comparing the bond lengths with the values for the bulk binaries (InN, 2.15 Å; GaN, 1.95 Å) it is seen that at the interface A bonds are shorter, and at the interface B longer than in the corresponding binaries. The layer resolved density of states functions are shown in Figure 2b for the 5/5 SL. Not all the layers are shown, only the layers on either side of the interface and the middle layer of the InN part of the supercell. At the interface A (layers GaN1 and InN1) the local conduction band minimum and valence band maximum are in the lowest position, whereas at the interface B (layers InN5 and GaN5) both band edges are in the highest positions. This situation is illustrated in Figure 3 where the corresponding local band edges are traced through the SL. Figure 3 shows a typical band edge profile of an SL with built-in electric field directed from the type B interface toward the type A interface in both well and barrier regions. The different characters of the A and B interfaces are clearly seen. Figure 3 demonstrates the strong effect of the built-in electric field. The values of the electric fields may be estimated by making a linear approximation to the edge profiles in the interior of the well and barrier layers. We find E = 7.7 MV/cm

Figure 3. Conduction and valence band profiles in the 5InN/5GaN superlattice. Two kinds of interface are indicated.

on the GaN side and E = 6.6 MV/cm on the InN side of the superlattice. The SL band gap is the difference between the overall conduction band minimum (which is found at the InN layer at interface B) and the overall valence band maximum (which is found at the InN layer at interface A). In this case the band gap (“indirect in real space”) is equal to Eg = 0.25 eV (see Figure 3). The VBO estimated from the differences in the valence band maxima in the central layers of InN and GaN in the 5/5 SL is around 0.5 eV. C. Determination of the Electric-Field and ValenceBand Offset. More precise values of the built-in electric fields and the VBO may be obtained using the N-1s core-levels as reference energies. These are the most localized s states and therefore the most sensitive to the gap adjusting potentials (sharply peaked at the nucleus21) in the LDA+C approach. Therefore we use the 1s levels without the adjusting potentials (computed in a separate calculation). In Figure 4 the core-level eigenvalues, E1s, within each layer of the 5/5 SL are plotted with respect to the distance along the [0001] direction (c-axis). The N-1s eigenvalues differ in successive MLs of the SL because of 3523

dx.doi.org/10.1021/cg300315r | Cryst. Growth Des. 2012, 12, 3521−3525

Crystal Growth & Design

Article

Figure 1 illustrates that for the InxGa1−xN alloys the band gap increases when the Ga fraction is increased. This trend is preserved in the 1/n SLs but reversed for SLs with m > 1. This is caused by internal electric field. Stronger than for 1/n SLs electric field for m/n SLs with m > 1 leads to smaller values of the band gap with increasing n just because the GaN barrier is becoming thicker: the slope of the band gap profile leads to a smaller band gap when the distance is longer. The VBOs were determined following the procedure described in ref 6 and the method used by Picozzi et al.28 The difference at the interface plane between the N 1s corestate energies in the InN and GaN regions is obtained by extrapolation, as illustrated in Figure 4. This determines the socalled “interface term”, which equals ΔEI = 0.46 eV in the case of the 5/5 SL. A smaller value is obtained for the 4/4 SL (ΔEI = 0.45 eV). To obtain the VBO we must add to the “interface term” the so-called “bulk term”, which denotes the difference between the N 1s binding energies with respect to the valence band maxima in bulk InN and GaN. For a given SL, the “bulk term” is calculated for the bulk material with the in-plane lattice constant a being that of the corresponding SL and with the c lattice constant allowed to relax. The calculated “bulk term” is approximately the same for the two SL structures considered, and given by ΔEB = 0.10 eV. Thus our calculated VBO for the 4InN/4GaN SL is ΔEVOB = ΔEI + ΔEB = 0.55 eV, and for the 5InN/5GaN SL, ΔEVOB = 0.56 eV. The obtained values of the VBOs are similar to those obtained by Moses et al.29 (0.62 eV) and by Shieh et al.6 (0.460 eV for the 4/4 SL and 0.468 eV for the 6/6 SL). Also, our VBO value for the 5/5 SL (0.56 eV) is in good agreement with the recent experimental values (0.58,9 0.78,10 and 0.8 ± 0.1 eV11).

Figure 4. N 1s energy in the 5InN/5GaN SL as a function of distance in the [0001] direction. For clarity, the GaN layer is shown twice.

the presence of the built-in electric fields. This results in a sawtooth like profile of E1s through the SL. The absolute value of the slope in the best straight-line fit in each region corresponds to the magnitude of the built-in electric field. We find E = 8.2 MV/cm on the GaN side and E = 7.0 MV/cm on the InN side of the superlattice, these values being slightly higher than those estimated from the band profiles (Figure 3). The electric fields estimated by the same method for other SLs depend strongly on the layer thicknesses. Examples of the extracted values of the electric field are given in Table 1. For the set of 1/n SLs the electric field decreases rapidly with increasing number of GaN MLs. The lowest value, obtained for 1/15, is about 0.6 MV/cm. For this set of SLs, the small values of the electric field do not influence the values of Eg significantly, especially for high values of n. On the other hand, very high values of the electric fields obtained for SLs with similar or equal number of well and barrier MLs, m−n, explain the strong decrease of the Eg with layers thickness. Due to the general trend that built-in electric fields decrease with increasing number of both well and barrier MLs, we expect in the 6/6 SL slightly lower values of the electric field than in the 5/5 SL, as is also confirmed by our rough estimate. Despite the weaker electric field, the overall band gap is closing in the 6/6 SL due to the thicker layers. Our estimated values of the electric field are higher than those obtained in the calculations of Shieh et al.6 (4.5 MV/cm (InN) and 5.8 MV/cm (GaN) for a 4/4 SL), but the general trends are similar. However, the conclusion that a decrease of the number of barrier layers barely changes the electric field in the barrier region6 may be valid only in the case of equal or similar numbers of well and barrier MLs. A decrease of the number (n) of GaN MLs in the 1/n and 2/n SLs strongly increases the electric field in the barrier region (from 0.6 MV/ cm for 1/15 SL to about 8 MV/cm for 2/4 SL). A similar dependence, but much weaker is found for the SLs with equal numbers of InN and GaN MLs (from 8.2 to 9.9 MV/cm in GaN and from 7 to 8.5 MV/cm in InN). In the asymmetric cases of 3/5 and 3/7 SLs extremely high electric fields are found in the InN side of the SL (11.4 and 13 MV/cm).

IV. CONCLUSIONS The present study shows that the band gaps in mInN/ nGaN(0001) SLs depend strongly on the numbers m and n of InN and GaN MLs. The strong built-in electric fields (i.e., QCSE) are responsible for the observed effect. The electric field strengths estimated from the N 1s core-states profiles depend on the layer thicknesses and relative widths of barriers and wells. For a 5/5 SL, they are about 7.0 MV/cm in the InN layer and about 8.2 MV/cm in the GaN layer. This leads to a very small gap value (Eg = 0.25 eV, “indirect” in real space) and to closing of the gap for the 6/6 SL. Thus, superlattices with n = m become metallic for n > 5, that is, electrons will readily tunnel across the SL layers. The same effect of metallization was found theoretically for mAlN/nGaN SLs by Cui et al.,30 where “metallization is predicted to occur at around n = m = 28”. This well width is much higher than for InN/GaN because of the large value of the AlN band gap. The internal electric field for the set of 1/n SLs decreases rapidly with increasing barrier thickness and becomes quite small (0.6 MV/cm) for the 1/15 SL, with a minimum influence on the overall band gap. The transition energies derived from photoluminescence experiments1,2,13 (∼3.25 eV for one InN ML embedded in

Table 1. Magnitude of the Electric Fields (in MV/cm) in Fully Relaxed mInN/nGaN(0001) SLs, as Determined from the N 1s Core State Energies (See the Discussion in the Text) m/n InN GaN

1/3

1/5

1/7

1/15

2/4

2/6

2/8

7.2

3.7

2.8

0.6

6.8

4.5

3.6

3524

3/3 8.5 9.9

3/5 11.4 7.6

3/7 13.0 5.9

4/4 7.3 8.6

5/5 7.0 8.2

dx.doi.org/10.1021/cg300315r | Cryst. Growth Des. 2012, 12, 3521−3525

Crystal Growth & Design

Article

(6) Shieh, C. C.; Cui, X. Y.; Delley, B.; Stampfl, C. J. Appl. Phys. 2011, 109, 083721. (7) Kamiya, K.; Ebihara, Y.; Shiraishi, K.; Kasu, M. Appl. Phys. Lett. 2011, 99, 151108. (8) Martin, G.; Botchkarev, A.; Rockett, A.; Morkoč, H. Appl. Phys. Lett. 1996, 68, 2541. (9) King, P. D. C.; Veal, T. D.; Kendrick, C. E.; Bailey, L. R.; Durbin, S. M.; McConville, C. F. Phys. Rev. B 2008, 78, 033308. (10) Wu, C.-L.; Lee, H.-M.; Kuo, C.-T.; Chen, C.-H.; Gwo, S. Appl. Phys. Lett. 2008, 92, 162106. (11) Kuo, C.-T.; Chang, K.-K.; Shiu, H.-W.; Liu, C.-R.; Chang, L.-Y.; Chen, C.-H.; Gwo, S. Appl. Phys. Lett. 2011, 99, 122101. (12) Wu, J. J. Appl. Phys. 2009, 106, 011101. (13) Yuki, A.; Watanabe, H.; Che, S.-B.; Ishitani, Y.; Yoshikawa, A. Phys. Status Solidi C 2009, 6, S417−S420. (14) Lin, W.; Li, S.; Kang, J. Appl. Phys. Lett. 2010, 96, 101115. (15) Perdew, J. P.; Zunger, A. Phys. Rev. B 1981, 23, 5048. (16) Ceperley, D. M.; Alder, B. J. Phys. Rev. Lett. 1980, 45, 566. (17) Kresse, G.; Furthmüller, J. Comput. Mater. Sci. 1996, 6, 15. (18) Methfessel, M. Phys. Rev. B 1988, 38, 1537. Methfessel, M.; Rodriguez, C .O.; Andersen, O. K. Phys. Rev. B 1989, 40, 2009. (19) Singh, D. Phys. Rev. B 1991, 43, 6388. (20) Christensen, N. E.; Gorczyca, I. Phys. Rev. B 1994, 50, 4397. (21) Christensen, N. E. Phys. Rev. B 1984, 30, 5753. (22) Svane, A.; Christensen, N. E.; Gorczyca, I.; van Schilfgaarde, M.; Chantis, A. N.; Kotani, T. Phys. Rev. B 2010, 82, 115102. (23) Gorczyca, I.; Łepkowski, S. P.; Suski, T.; Christensen, N. E.; Svane, A. Phys. Rev. B 2009, 80, 075202. (24) Gorczyca, I.; Suski, T.; Christensen, N. E.; Svane, A. Appl. Phys. Lett. 2011, 98, 241905. (25) Gorczyca, I.; Kaminska, A.; Staszczak, G.; Czernecki, R.; Łepkowski, S. P.; Suski, T.; Schenk, H. P. D.; Glauser, M.; Butté, R.; Carlin, J.-F.; Feltin, E; Grandjean, N.; Christensen, N. E.; Svane, A. Phys.Rev.B 2010, 81, 235206. (26) Gorczyca, I.; Suski, T.; Kaminska, A.; Staszczak, G.; Schenk, H. P. D.; Christensen, N. E.; Svane, A. Appl. Phys. Lett. 2010, 96, 101907. (27) The method has recently been used in several alloy calculations, see for example: Gorczyca, I; Suski, T; Christensen, N. E.; Svane, A. Phys. Rev. B 2011, 81, 153301. Carrier, P.; Wei, S. H. J. Appl. Phys. 2005, 97, 033707 used it to obtain the surprisingly small gap in InN by taking the adjusting potential parameters from GaN and InAs, a good demonstration of the transferability of the parameters.. (28) Picozzi, S.; Continenza, A.; Freeman, A. J. Phys. Rev. B 1997, 55, 13080. (29) Moses, P. G.; Miao, M.; Yan, Q.; Van de Walle, C. G. J. Chem. Phys. 2011, 134, 084703. (30) Cui, X. Y.; Carter, D. J.; Fuchs, M.; Delley, B.; Wei, S. H.; Freeman, A. J.; Stampfl, C. Phys. Rev. B 2010, 81, 155301.

thick (11 nm) GaN barriers) are significantly higher than the present calculated band gap energy (2.2 eV) for the 1/n SL in the limit of large n. There are several possible causes of this discrepancy: (1) The fabricated InN QW may contain some Ga atoms due to interdiffusion, in which case it does not correspond to the ideal system modeled in the present work. (2) The strain fields present in the experiments would be slightly different from those of the modeled free-standing SLs. (3) There may be free carriers in the samples (in InN usually a high concentration of electrons is present) that significantly screen the built-in electric field. (4) Excitonic effects have not been treated in the present theory. Optical transitions may be attributed to localized excitons in the barrier region, as it was suggested in ref 1 (excitons in GaN localized at InN well). It emerges that several important effects need to be examined before a detailed comparison between theory and experiments may be performed, and further investigations are required. The value of the VBO extracted for the 5/5 SL of ΔEVBO = 0.56 eV is close to other recent theoretical values and in very good agreement with the recently reported experimental value of 0.58 eV.9 Similar calculations of the electric-field and VBO were performed by Shieh et al.6 Comparing our method and that of ref 6, the latter uses the density-functional theory (DFT) bands derived in the LDA or generalized gradient approximation (GGA). This means that gaps are severely underestimated. We apply the DFT, but with the inclusion of gap-adjusting external potentials, which allows us to draw edge profiles for valence, as well as conduction bands enabling a more detailed comparison with experiments. Our main results concerning the dependence of the built-in electric fields and the band gaps on the SL geometry provide valuable insight to band gap engineering and consequently to processes of device optimization and design.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +48 604 196 155. Fax: +48 22 876 0314. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was partially supported by the European Union within European Regional Development Fund, through grant Innovative Economy (POIG.01.01.02-00-008/08) and the Polish Ministry of Science and Higher Education, grants NN202131339 and 2011/01/B/ST3/04353. Computational resources were partially funded by the Danish Centre for Scientific Computing (DCSC). NCW Hansen, Center for Scientific Computing in Aarhus (CSC-AA) is thanked for expert computing assistance.

■

REFERENCES

(1) Yoshikawa, A.; Che, S. B.; Yamaguchi, W.; Saito, H.; Wang, X. Q.; Ishitani, Y.; Hwang, E. S. Appl. Phys. Lett. 2007, 90, 073101. (2) Yoshikawa, A.; Che, S. B.; Hashimoto, N.; Saito, H.; Ishitani, Y.; Wang, X. Q. J. Vac. Sci. Technol. B 2008, 26, 1551. (3) Dimakis, E.; Nikiforov, A. Yu.; Thomidis, C.; Zhou, L.; Smith, D. J.; Abell, J.; Kao, C.−K.; Moustakas, T. D. Phys. Stat. Sol. (a) 2008, 205, 1070. (4) Taniyasu, Y.; Kasu, M. Appl. Phys. Lett. 2011, 99, 251112. (5) Cui, X. Y.; Delley, B.; Stampfl, C. J. Appl. Phys. 2010, 108, 103701. 3525

dx.doi.org/10.1021/cg300315r | Cryst. Growth Des. 2012, 12, 3521−3525