1. Introduction as to whether the calculations within LDA give correct effective masses is, therefore, still open.
Density-functional theory (DFT) is a powerful tool for There is even less explicit work on the correctness of the studying the electronic structure of solids. It is well momentum matrix elements (Ep) calculated within LDA, known that the band gаps of bulk solids calculated within which are often claimed to be accurate. However, the the theory are systematically underestimated compared validity of the statement has been less directly verified.
The standard argument is that the pseudo-wave-function, to experimentally determined ones  because of its in a pseudopotential approach, has a very high overlap failure in the description of the excited-state properties with the true wave function . The value of Ep for and the problem rests not just with the local-density GaAs calculated by Winkler , and Kageshima and approximation (LDA). Following k · p theory (see e. g.
Shiraishi  within LDA is about 1.7 times smaller Ref. ), however, one can say that if the fundamental than that determined experimentally [17,18]. Kageshima excitation gaps are incorrect, then band dispersions should and Shiraishi  concluded that the wave functions be too. This realization has been used by Cardona and calculated by the pseudopotential method lack a precise co-workers [3–6] in generating corrected band parameters description around the atomic core regions and momentum (Luttinger parameters, spin splittings) for a number of matrix elements cannot be directly estimated from these semiconductors. Nevertheless, there remains some dispute wave functions, because the functions are smoother regarding this issue . For instance, in 1992, Fiorentini and around the atomic cores, while the actual wave functions Baldereshi [8,9] using pseudopotential plane waves within oscillate greatly. To fix the error, a core-repair term was LDA found that conduction-band masses (mc) at the added. By including the correction, significant improvement point were very close to experimental values for GaAs, was indeed achieved for zinc-blende and wurtzite GaN.
AlAs and Ge. However, the values of mc found in Ref.  However, momentum matrix elements for poly-silane, significantly differ from those determined experimentally siloxene and GaAs calculated with the correction and for GaAs, GaSb, InP and InAs. Similarly, conduction-band without it differ from each other by only 5.43, 3.0 and 1.8%, effective masses calculated by Wang and Zunger  within respectively. Wang and Zunger , on the other hand, had LDA agree well with experimental data for Si, while for found a larger increase in the momentum matrix elements.
CdSe the masses were not very accurate and a semiTo complicate matters, Levine and Allan  found that, empirical modification led to somewhat better agreement.
even within the scissors approximation, the velocity operator Kane , on the other hand, found that it is not possible to gets renormalized.
get the correct band gap and the cyclotron masses in Si by a It should be noted that correctness of the calculated local static potential sush as the LDA. Fairly good agreement and experimentally determined values of Ep is also being with experiment was obtained by Wang and Klein  for debated. Efros and Rosen , for example, concluded conduction and valence band effective masses of GaP, GaAs, that Ep = 14.98 eV for bulk InP calculated by Fu, Wang ZnS and ZnSe using the linear combination of Gaussian and Zunger  using the direct diagonalization method is orbitals within LDA. Systematic study of effective masses smaller than the measured value 20.6 eV, which indicates for 32 semiconductors by Huang and Ching  using underestimation of the coupling between the conduction the semi-ab initio technic shows much better agreement.
and valence bands, and overestimation of the influence of Despite the importance of the effective masses, the question the remote bands. However, even the values of Ep extracted ¶ E-mail: email@example.com from experiments have an intrinsic scatter. For bulk InP, for 1 178 S.Zh. Karazhanov, L.C. Lew Yan Voon expample, the experimental values of Ep vary in the range group-II atoms have been generated by two ways: including from 16.6 (Ref. ) to 20.7 eV (Ref. ). So, the question the d-electrons, which are inside the valence shell (i) as to whether the momentum matrix elements calculated by into the valence complex, while keeping the semicore s LDA are correct is still open. and p states in the core, and (ii) into the core. For the It should also be noted that the above-mentioned latter case, we used Ecut = 30 Ry and nonlinear exchange problems are relatively less studied in AIIBVI correlation was included, which is known to give better semiconductors which have a cation d-band inside agreement with experimental data . For the former the main valence band playing a significant role in their case, convergent results were obtained for Ecut = 70 Ry.
electronic structures [24–29]. In AIIIBV semiconductors the The semicore d-electrons of AIIIBV compounds have been d-levels are below in energy to several eV than the lowest included into the core, because, as discussed in Section 1, sp valence band states . So the effect of the d-states on energy level of the electrons are much below the outermost electronic structure of AIIIBV compounds can be neglected. sp-levels  and the electrons are not expected to effect In the literature, the strong p-d coupling in the AIIBVI significantly on band structure of the compounds.
semiconductors has usually been taken into account by Band-structure calculations were performed using including the d-electrons (i) into the core, but including also the PEtot code developed by L.-W. Wang , which nonlinear core corrections for exchange and correlation, and uses the ab initio pseudopotential method within the (ii) into the valence complex [24–26], but keeping the s and LDA and neglects spin-orbit coupling. The Pulay–Kerker p semicore electrons in the core. Currently, description of scheme has been used for self-consistent potential the p-d coupling is still being improved. Despite numerous mixing. Also, g-space Kleinman–Bylander non-local studies, the question as to which of the approaches related pseudopotential implementations have been used with to the p-d coupling is correct and consistent with k · p mask function scheme, without the need for preprocessing band parameters is still open. of the pseudopotentials. The LDA exchange-correlation In this paper we provide the first systematic study contribution is accounted for by means of Perdew and of eigenvalues at, X and L, Kane momentum matrix Zunger’s parametrization  of the calculations by elements Ep and Ep (defined as 2P2/m0 in eV) Ceperley and Alder . The self-consistent solution of the corresponding to the fundamental direct p-s energy gap Eg one-electron Kohn–Sham equation has been performed by and the p-p gap Eg at point, conduction-band effective the planewave pseudopotential algorithm .
The potential for the unit cell considered was found masses m001, m011, m111, heavy-hole effective masses m001, c c c hh by performing self-consistent calculations using 10 special m011, m111, light-hole effective masses m001, m011, m111 and hh hh lh lh lh L L L k points in the Brillouin zone. Then, using the potential, Luttinger parameters 1, 2 and 3 and answer the question eigenenergies at the special k points were found by as to whether the band parameters calculated within the non-self-consistent calculations.
framework of LDA correct.
2.2. Band parameters 2. Computational details Carrier effective masses are defined as:
2.1. Local-density approximation 1 1 2E(k) = (1) Ab initio calculations were performed for AIIIBV (AlP, mc(k0) k2 k=kAlAs, AlSb, GaP, GaAs, GaP, InP, InAs, InSb) and some AIIBVI (ZnS, ZnSe, ZnTe, CdS, CdS, CdSe, CdTe) for a direction k about some point k0 in the Brillouin semiconductors of zinc-blende structure. We did not study zone. We studied effective masses along ,  and the AIIBVI compounds HgS, HgSe, and HgTe, because it is  directions in the vicinity of k0 (0, 0, 0) point. The well known that spin-orbit coupling (which is not accounted masses can, in principle, be calculated by the k · p theory for in this work) plays a significant role . The unit cell equation (see e. g. Ref. ). In this work, we calculate the considered consists of two atoms of group-III (II) at (0,0,0) band energies at a sequence of k points around and and an atom of group-V (VI) at (a/4, a/4, a/4), where a calculate mc from Eq. (1) directly.
is the lattice constant. We considered conduction-band effective masses m001, c Ab initio pseudopotentials have been generated using the m011, m111, heavy-hole effective masses m001, m011, c c hh hh method of Trouillier–Martins . We have considered s, p m111 and light-hole effective masses m001, m011, hh lh lh and d as valence states to build the pseudopotential for the m111. To calculate the effective masses, 11 k-points lh atoms of group-III and V using the p-potential as the local were used in the ranges from (2/a)/(0, 0, -1/20) potential, while the s and d are taken as the nonlocal parts. to (2/a)(0, 0, 1/20) for the direction , from We determined Ecut by requiring convergence of the total (2/a)(0, -1/20, -1/20) to (2/a)(0, 1/20, 1/20) for the energy Etot. For all the AIIIBV semiconductors considered, direction , and from (2/a)(-1/20, -1/20, -1/20) Ecut = 60 Ry was used. to (2/a)(1/20, 1/20, 1/20) for the direction .
To study the AIIBVI compounds, s, p and d states Momentum matrix elements have been determined at the were considered as valence states. Pseudopotentials for of point.
Физика и техника полупроводников, 2005, том 39, вып. Ab initio studies of band parameters of AIIIBV and AIIBVI zinc-blende semiconductors Table 1. Experimental (Ref. [37-40]) and LDA (a0, ad0) lattice constants () for the zinc-blende AIIIBV and AIIBVI semiconductors for d-electrons of group-II atoms included into the core (a0) and into the valence complex (ad0) AlP AlAs AlSb GaP GaAs GaSb InP InAs InSb a0 5.4131 5.6246 6.0788 5.2836 5.5073 5.9380 5.6591 5.8564 6.Experiment  5.4670 5.6600 6.1355 5.4512 5.6533 6.0959 5.8687 6.0583 6.ZnS ZnSe ZnTe CdS CdSe CdTe a0 4.8674 5.1706 5.6730 5.3038 5.5639 6.ad0 5.3404 5.6202 6.0072 5.8064 6.0572 6.Experiment  5.4102 5.6676 6.1037 5.8180 6.0520 6.Experiment  5.4110 5.6690 6.0890 5.8300 6.0840 6.Experiment  5.4100 5.6680 6.1000 5.8250 6.0520 6.Following Ref.  and using the calculated constant is in between the two experimental values. Since effective-mass values, Luttinger valence-band parameters for all other seimiconductors considered in this work, L L L 1, 2 and 3 have been calculated as: the calculated lattice constant is always smaller than the experimentally determined one, we used a = 6.084 A for 1 1 L CdSe from Ref.  for band-structure calculations using the 1 = +, (2) 2 m001 m001 experimental lattice constant. For other AIIBVI compounds, lh hh we used the experimental lattice constants given in Ref. .
1 1 L 2 = -, (3) 4 m001 mlh hh 3.2. Eigenvalues 1 1 1 L Self-consistent and non-consistent band-structure 3 = + -. (4) 4 m001 m001 mlh hh hh calculations have been performed using both calculated and experimentally determined lattice constants. Since Energies and momentum matrix elements are given in we neglected spin-orbit coupling, the valence band eV, and effective masses are given in unit of the electron maximum at point is triple degenerated for all rest mass m0 throughout the paper. All the calculations compounds considered. For AIIIBV and AIIBVI compounds, have been performed for two values of the lattice constants:
eigenvalues at, X and L are given in Table 2, (i) determined experimentally (to be called experimental and they are in general agreement with previous lattice constant hereafter), and (ii) determined by a search calculations [9,13,14,24–26,29,41–43]. Band gaps (Eg) of the total energy minimun (to be called theoretical lattice for AlP, AlAs, AlSb and GaP are indirect with valence constant hereafter).
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