ENERGY BAND STRUCTURE OF ALUMINIUM BY THE AUGMENTED PLANE WAVE METHOD

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ENERGY BAND STRUCTURE OF ALUMINIUM BY THE AUGMENTED PLANE WAVE METHOD L. SMR6KA Institute of Solid State Physics, Czeehosl. Acad. Sci., Prague*) The band structure of metallic aluminium has been calculated by the augmented plane wave method. The energy values and wave functions were computed for the equivalent of 2048 points in the Brillouin zone and for energies ranging from the bottom of the conduction band (3s, 3p) to approximately 0'4 Ry above the Fermi energy. The density of states and the Fermi energy were determined using a variant of an accurate method developed by Gilat and Raubenhei mer (Phys. Rev. 144 (1966), 390). The results are discussed and compared with earlier results. 1. INTRODUCTION The band structure, Fermi surface and density of states of aluminium have been studied by several authors both theoretically and experimentally. A comprehensive review and discussion of previous results was given by Slater [2], Harrison [3], and recently by Snow [4]. Here, we must point out that with the exception of [4] all previous papers reporting on ab initio band structure calculations [5, 6, 7, 8] give only very few examples of the energy eigenvalues and eigenfunctions and therefore can hardly serve as a starting point for calculations of the physical properties of aluminium. In his paper [4], S n o w describes a very extensive selfconsistent calculation of energy bands for aluminium. Unfortunately, G r ei s e n [9] has found a systematic numerical error in this paper and Snow's results are therefore of uncertain value. In the present paper, we shall report on an augmented plane wave (APW) determination of electron energy values E(k) and eigenfunctions ~k in the 89 points, lying in the irreducible 1/48 part of the Brillouin zone (2048 points in the whole zone). In the APW calculation we assume that the ls, 2s and 2p electrons have the same wave functions both in the crystal as in a free atom (rigid-core approximation). The difference between these energies is determined by a simple tight binding method. The Hartree-Fock-Slater approximation is used to construct the crystal potential and in APW calculations. No attempt is made to treat the problem self-consistently (see section 3). 2. THE CRYSTAL POTENTIAL In APW calculations the crystal potential is usuauy approximated by the "muffintin" potential. This potential is spherically symmetrical inside the APW spheres surrounding each atom and is assumed to be constant between these spheres. In the *) Cukrovarnick6 10, Praha 6, Czechoslovakia. Czech. J. Phys. B 20 (1970) 291

L. Smr~ka Table 1 Crystal potential. r -- V(r) r -- V(r) r -- V(r) 0'005 0'010 0"015 0.020 0.025 0.030 0.035 0.040 0.045 0'050 0.055 0.060 0.065 0.070 0-075 0.080 0'085 0"090 0'095 0'100 0'110 0.120 0'130 0.140 0.150 0'160 0'170 0.180 0.190 0"200 0-210 0.220 0-230 0.240 0-250 0.260 0-270 0"280 0.290 0.300 0.320 0.340 0.360 0.380 5141'673 2540.684 1673"304 1239.472 979"162 805.668 681-818 589.026 516"854 459.390 412'388 373.310 340.325 3t2.129 287-764 266.506 247.805 231.240 216'464 203'209 180'440 161"598 145"776 132'323 120-763 110.738 101"980 94'272 87'452 81-381 75"952 71.072 66'667 62-675 59'040 55'724 52"685 49.890 47.313 44'932 40.678 36.993 33-780 30"961 0-400 0.420 0.440 0"460 0.480 0.500 0'520 0"540 0'560 0-580 0.600 0.620 0.640 0.660 0"680 0.700 0-720 0.740 0.760 0.780 0.800 0'820 0.840 0-860 0.880 0.900 0"920 0.940 0-960 0.980 1-000 1.020 1'040 1'060 1"080 1.100 1.120 1"140 1-160 1.180 1'200 1.220 1.240 1-260 28-473 26.268 24"304 22"549 20.975 19-558 18"277 17.120 16"068 15-109 14.234 13.433 12"698 12.022 11'399 10.822 10.289 9.793 9.333 8'904 8.504 8.130 7"779 7.451 7-143 6.853 6.580 6.323 6.081 5'852 5.636 5-432 5-239 5.056 4.883 4.720 4.565 4.419 4.280 4-149 4.024 3-906 3'794 3.688 1-280 1-300 1"320 1-340 1"360 1-380 1.400 1-420 1"440 1"460 1-480 1-500 1.540 1-580 1-620 1.660 1.700 1"740 1"780 1.820 1"860 1'900 1.940 1'980 2.020 2.060 2-100 2-140 2-180 2.220 2.260 2-300 2.340 2.380 2.420 2.460 2.500 2.540 2.580 2.620 2.660 2.700*) 2.740 2.780 3'588 3.492 3"402 3-316 3"234 3'156 3.082 3.011 2-944 2.880 2.819 2.760 2.650 2.550 2"457 2"371 2.291 2.217 2.148 2'083 2.023 1'966 1"913 1'864 1'817 1.774 1.733 1"695 1"660 1.626 1.596 1.567 1.540 1.516 1.493 1.473 1'454 1-437 1"421 1"408 1.396 1-385 1"376 1-368 Note: r is in Bohrs, V(r) is in Rydbergs. V e = --1'36244 Ry. *) Radius of APW sphere. For the lattice constant a = 7.6529 Bohrs the proper radius equals 2.7057 Bohrs.

Energy band structure of aluminium by the augmented plane wave method present calculation the radius of the APW sphere is chosen equal to half the distance to the nearest neighbouring atom. The constant potential Vc is approximated by an average value of the potential in the region between radii of APW and Wigner-Seitz spheres. 0 C3 CU O Ck -10-15 - 20 Fig. 1. Crystal potential 0 1 2 3 RADIUS [BOHRS], atomic potential... RS is radius of APW sphere. The "muffin-tin" potential was constructed by the method proposed by Mattheiss [10]. In this method the exchange and Coulombic contributions to the "muffin-tin" potential are treated separately. The total Coulomb potential is represented by the Coulomb potential of the neutral atom plus spherically symmetrical contributions from Coulomb potentials of neighbouring atoms. An analogous method is used to calculate the total charge density Q(r). The exchange potential is then represented by Slater's free-electron approximation In this paper thirteen shells of neighbouring atoms were used when constructing the crystal potential.the atomic Coulomb potential andatomiccharge density were obtained by means of self-consistent atomic calculations described by Herman and Skillman [11]. The resulting crystal potential is tabulated in Table 1 and shown in Fig. 1. Czech. J. Phys. B 20 (1970) 293

L. Smrdka 3. APW CALCULATION The APW method, originally proposed in 1937 by Slater [12], has been described in most detail in the recent book by Loucks [13]. In the APW method the wave function for an electronic state k is assumed ~o be a linear combination of the APW function. An augmented plane wave consists of a plane wave in the region of the constant potential and a sum of atomic orbitals inside the APW spheres. The coefficients in the atomic-orbital expansion are chosen in such a way that the discontinuity in the value of APW at the spheres' boundaries is eliminated. The remaining discontinuity in the slope is taken into account by including appropriate surface integrals in the variational expression for the energy. The resulting secular determinant is then used for the determination of E(k) eigenvalues and the corresponding eigenfunctions. Here, the linear combination of 32 APW functions is used and the expansion in spherical harmonics was taken up to l = 12. The computer programs used for the calculation were written by the author for MINSK 22 and IBM 7040 computers1). 4. THE DENSITY OF STATES The density of states is defined by the expression o f ds (1) = IvkEl' where fj is the volume of the unit cell and the integral is taken over a surface of constant energy, The present method of calculation the n(e) curve resembles that described by Gilat and Raubenheimer [1]. In our method the irreducible 1/48 part of the Brillouin zone is divided into a large number of small cubes in which the energy is supposed to be a linear function of wave vector k. Under this assumption expression (1) can be easily evaluated in each cube separately and the total density of states is then obtained by the summation over all cubes. In our case the cubes with the side length 1/72 of FX distance are used near the zone boundaries and with side length 1/48 of FX distance in the remaining part of the Brillouin zone. It is necessary to know E(k) and VkE in the centre of each cube and therefore we must interpolate between APW eigenvalues. This was done by fitting the pseudopotential to our data. Small differences between the APW and pseudopotential eigenvalues are removed by multiplying the pseudopotential spectrum by a appropriate constant in the neighbourhood of each APW eigenvalue. 1) We tested our computer programs by recalculating the energy levels with the help of incorrect Snow's potential. The results agree very well with those obtained by Greisen with the same potential. 294 Czech. J. Phys. B 20 (1970)

Energy band structure of aluminium by the augmented plane wave method Table 2 Energy bands in aluminium. 4(a/n) k Band 1 Band 2 Band 3 Band 4 r (ooo) A (OLO) A (020),~ (o30) A (040).4 (050).4 (060) ~J (070) x (o8o) 2? (1 I0) (120) (130) (140) (15o) (16o) (170) z (18o) 27 (220) (230). (240) (250) (260) (270) z (280) 27 (330) (340) (350) (360) (370) z (380) 27 (440) (450) (460) (470) W (480) z (550) (560) (570) K (660) A (111) (121) (131) I --0'19027 1 --0.18003 1 --0-14934 1 --0.09834 1 --0.02728 1 0'06344 1 0"17329 1 0"30087 4' 0"41714 1 --0"16979 + --0'13912.4- --0-08816 -]- --0'01716 4-0"07349.4-0"18325.4-0'31071 3 0"42671 1 --0.10852.4- --0.05768.4-0'01315.4-0.10358.4-0"21307 4-0.34016 3 0.45543 1 --0.00704 4-0.06350.4-0.15358.4-0.26262.4-0.38908 3 0'50333 1 0.13365.4-0.22323.4-0.33161.4-0.45704 3 0.57064 1 0.31214.4-0'41944 -- 0'51291 3 0-49374 1 --0'15956 + --0"12891.4- --0"07799 1 0-80540 1 0.62367 1 0"48931 -- 0"97258 q- 0-81456.4-0-63325 1 0.49923 - - 1.00094 -- 0.91525 - - 0-84842 - - 0.80063.4-0"66163 1 0-52873 3 0.89607 - - 0"81001 -- 0.74300 - - 0'69511 -- 0"66637 I 0'57651 3 0"72378 - - 0.65669 -- 0'60877 - - 0.58002 3 0'57064 3 0'58957 - - 0'54165.4-0.54072 1 0'52218 -- 0.92505 - - 0-89649 4 0-88696 + 0-98793 + 0.92208 + 0.83801 -- 0.77193 4 0.76236 1 0-97193 -4-0.88850 4-0.82207 4-0.77280 4-0.70318 4 0.65679 1 0.80405 + 0'73694 + 0.68818 7-0.65730 2' 0.62359 1 0"66939 + 0.62076.4-0.59424 1 0'57571 4-0-98609 + 0'95838 1 0"94914 -+- 0.88286 4-0'85016 1 0.84037 4-0-89416 4-0"75538 1 0"74036 + 0'95795 4-0.78241 1 0'67544 4-0"86530 Czech. J. Phys. B 20 (1970) 295

L. Smrdka Table 2 (Continued) 4(a/n) k Band 1 Band 2 Band 3 Band 4 (141) (151) (161) (171) S (181) (221) (231) (241) (251) (261) (271) (281) (331) (341) (351) (361) (371) (381) (441) (451) (461) Q (471) (551) (561) A (222) (232) (242) (252) (262) (272) U (282) (332) (342) (352) (362) (372) (442) (452) Q (462) (552) A (333) (343) (353) + --0-00705 + 0-08353 + 0.19320 + 0.32053 3 0.43628 + --O.09834 --0-04754 0.02323 0.11359 0.22299 0.34994 - - 0.46501 + 0.00306 0'07354 0'16355 O.27248 0.39869 - - 0.51291 + 0.14364 0.23313 0.34128 - - 0.45823 + 0.32183 0.41974 1 --0.06783 + --0.01715 + 0.05344 + 0.14359 + 0.25270 + 0.37913 3 0.49374 + 0.03332 0.10363 0.19341 0-30192 0-41973 + 0-17355 0.26269 - - 0.36184 + 0.34247 1 0.08359 + 0-15362 + 0.24291 + 0'92708 + 0'86042 + 0"80775 + 0'64258 1 O.509O4 + O.99471 0.89117 0.80550 0-73836 0.68993 0.65400 + 0-53773 + 0.78678 0-70050 0.63298 0.58453 0.55519 + O.54O72 + 0.61379 0.54600 0.49758 + 0.47688 + 0.47817 0.43905 I 0.87378 + 0.76925 + 0.68284 + 0.61519 q- 0.56663 -+- 0.53719 1 0.52217 + 0.66394 0.57693 0.50887 0.46026 0.43905 + 0.48948 0.42125 + 0.38119 + O.36152 1 0-55785 + 0.47025 + 0-40189 + 0.83022 + 0'78544 1 0'77565 0"85379 0"68039 + 0'65403 0'94843 0"88988 0-72118 + 0-59424 + 0'93094 0-86491 0-81677 -- 0-77438 -1-0-79857 0'75059 + 0'88249 + 0'70291 1 0"57571 0.92929 0.75058 - - 0'98537 + 0'97643 1"00060 0"97160 0'96327 0.91372 0.87508 + 0"86531 0.96751 q- 0.80410 296 Czech. J. Phys. B 20 (1970)

Energy band structure of aluminium by the augmented plane wave method Table 2 (Continued) 4(a/n) k Band I Band 2 Band 3 Band 4 (363) (443) Q (453) L (444) + 0-34247 + 0.22314 -- 0.30370 2" 0-28430 + 0.36152 + 0.38230 + 0-32186 1 0-30187 + 0-97640 I Note: Energies are given in Rydbergs and are referred to zero constant potential between spheres. The eigenvalues are identified using the notation of Bouckaert, Smoluchowski, and Wigner [16]. 5. RESULTS AND DISCUSSION The energy eigenvalues resulting from our APW calculation are given in Table 2 with respect to V~ = 0. The value of the constant potential between spheres was found to be -1.36244 Ry (see section 2.) and must be added to the tabulated values to (000) (080) (#80) (444) (000) (660 (0801 lol... II'''ll ' ' ' II ' ' ' It... ] 2 i 0.2-2 0 - - 0,2 1 1 F X W L r ~KtU X Fig. 2. Energy bands in directions of high symmetry. obtain correct energies. The E(k) curves in the directions of high symmetry are shown in Fig. 2. The comparison with Heine's [7] and Segall's [8] eigenvalues for the points of high symmetry is given in Table 3. Besides this, the results of calculations by Grei sen [9] based on Snow's [4] self-consistent potential are included. It was found that our bands are in general narrower than the bands reported by previous authors, Czech. J. Phys. B 20 (1970) 297

L. Smr~ka but have a similar shape. The very small difference between our bands and those obtained by Grei sen shows that the error in Snow's paper does not influence very much the shape of his potential, and that the selfconsistency is not very important in the band structure calculation of aluminium. Table 3 Comparison of energy eigenvalues for points of high symmetry. i Heine a) Segall b) Present Greisen e) rl x~ xl wl w3 z~ L1 K3 K1 K1 0.000 0.592 0"717 0.949 0.826 0.774 0' 699 0.742 1"075 0'000 0"622 0'698 0"923 0"819 0-776 0'483 0-512 0"699 0"723 0"802 0"000 0"607 0"680 0-866 0"814 0"761 0'475 0'492 0"684 0"712 0"766 0'000 0-607 0'678 0'856 0'818 0"760 0'477 0'488 0"684 0"714 0-762 Note: a) as listed by Segall [8], b) reference [8], c) reference [9]. The pseudopotential interpolation scheme usually used for aluminium bands [14] gives in our case the following values for adjustable parameters The core-states energy levels Vlll= = 0-949031 0.0129625 Ry, V200 = 0.0394375 Ry. Ej~ = - 113.767 Ry --- (- 1547.80 ev), E2~ = - 8.834 Ry = (- 120.19eV), Ezv= - 6.066Ry =(- 82.53eV) were determined by the tight binding method. The unsmoothed plot of the density of states is drawn in Fig. 3. Because of our narrower bands the present n(e) curve is slightly higher than the free electron parabola, but still in good agreement with it in the low energy region. In the reg!on of higher energies, peaks and cusps due to critical points occur and disturb the parabolic char- 298 Czech. J. Phys. B20 (1970)

Energy hand structure of aluminium by the augmented plane wave method acter of n(e). Most of them correspond to the energies in the high symmetry points. This is not so for the peak at 0.545 Ry. The crossing of bands is probably the reason for this. The small wriggles on the graph are due to computational scatter and the finite size of cubes.? tad c~ 6 5.,< qe 4 (/} tu r u... (3 3 2 7 [ /J ~/ I..J/~ 1 S" ', j" ' i I 0 f i i i I! *l Ih I t IJ,]J I J -0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ENERGY frydbergs2 Fig. 3. The density of states, present... free electron approximation. The Fermi energy E/was found to lie at 0"61912 Ry (E s - E(F1) = 0.80939 Ry = = ii.0117 ev), which is about 0.003 Ry below the W~ eigenvalue. For this reason, the model of the Fermi surface proposed by Ashcroft [15] rather than the Harrison's original model [3] is supported by our results. 6. CONCLUSIONS The present APW energy band calculations for aluminium based on a more exact crystal potential gives slightly narrower bands than those reported previopsly. The density of states curve is determined accurately enough to show quantitatively the effect of various critical points. Received 2. 6. I969. References [1] Gilat G., Raubenheimer L. J.: Phys. Rev. 144 (1966), 390. [2] Slater J. C.: Quantum Theory of Molecules and Solids, Volume 2. McGraw-Hill Book Company, New York 1965. [3] Harrison W. A.: Pseudopotentials in the Theory of Metals. W. A. Benjamin, Inc., New York 1966. Czech. J. Phys. B 20 (1970) 299

L.Smrdka: Ener#y band structure of aluminium by the augmented plane wave method [4] Snow E. C.: Phys. Rev. 158 (1967), 683. [5] Maty~i~ Z.: Phil. Mag. 30 (1948), 429. [6] Anton~ik E.: Czech. J. Phys. 2 (1953), 18. [7] Heine V.: Proc. Roy. Soc. A240 (1957), 340, 354, 361. [8] Segall B.: Phys. Rev. 124 (1961), 1797. [9] Greisen F. C.: Phys. stat. sol. 25 (1968), 753. [10] Mattheiss L. F.: Phys. Rev. 133 (1964), A1399. [11] Herman F., Skillman S.: Atomic Structure Calculations. Prentice-Hall Inc., Englewood Cliffs, New Jersey 1963. [12] Slater J. C.: Phys. Rev. 51 (1937), 151. [13] Loucks T. L.: Augmented Plane Wave Method. W. A. Benjamin, Inc., New York 1967. [14] Phillips J. C.: Phys. Rev. 112 (1958), 685. [15] Ashcroft N. W.: Phil. Mag. 8 (1963), 2055. [16] Bouckaert L. P., Smoluchowski R., Wigner E.: Phys. Rev. 50 (1936), 58. 300 Czech. J. Phys. B 20 (1970)

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