Self-consistent Band Structures, Charge Distributions, And Optical-Absorption Spectra in MgO,...

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PHYSICAL REVIEW B VOLUME 43, NUMBER 5 15 FEBRUARY 1991-I Self-consistent band structures, charge distributions, and optical-absorption spectra in Mgo, a-Alamo„and MgAlz04 Yong-Nian Xu and W. Y. Ching Department of Physics, University of Missouri Ka— nsas City, Kansas City, Missouri 64110 (Received 13 September 1990) The electronic structures, the charge-density distributions, and the optical-absorption spectra in MgO, a-A1, 03, and MgA1204 crystals are studied by means of the first-principles self-consistent or- thogonalized linear combination of atomic orbitals method. The calculated band structures of MgO and n-Al&03 are compared with experimental data and previous calculations. It is concluded that the electronic structure and optical properties of the spinel MgA1204 cannot be taken as a simple average of that of MgO and a-A120, . By direct space integration of the calculated charge, it is shown that the ionic nature of these crystals should be best described by the formulas Mg" +0", A12' +03' ', and Mg' +Alq '+04" . The charge-density maps also reveal that there are open channels of very low electron density in MgA1204, which may be important in ionic conductivity. The calculated optical dielectric functions for the three crystals are compared with recent vacuum ultraviolet data for an energy range up to 40 eV and show good overall agree- ment. The existence of an excitonic peak near the absorption edge tends to sharpen the edge, there- by increasing the gap as compared with the one-electron local-density calculations. For absorption spectra beyond the excitonic peak, most of the experimentally resolved structures are reproduced by the calculations without the need of an adjustment to widen the gap. It appears that the high conduction-band states in these three crystals, calculated by using the local-density approximation, are reasonably accurate. I. INTRODUCTION MgO and a-A1203 are important ceramic oxides that have been subjected to numerous experimental and theoretical studies. ' They have high melting points, low coe%cients of expansion, and small thermal conductivi- ties. There is also an increasing trend in using MgO and n-A1203 as substrate materials for thin-film growth and as optical materials. MgO is highly ionic and has a sim- ple NaC1-type structure. Its band structure has been studied by various methods ' and is reasonably well es- tablished. Still, there are difterences with regards to the size of the band gap, valence-band (VB) width, and values of effective charges on each ion. ' o.-Alz03, on the other hand, has a much more complicated corundum structure. Several calculations were performed in the past on its band structure. ' ' Most of these calculations were non-self-consistent or semiempirical in nature. n-A12O3 has also been studied in the form of small clusters. ' A recent study on the band structure of u-A1203 used a self-consistent pseudofunction method. ' That study con- centrated on the interpretation of experimental reAectivity data in terms of critical point analysis and does not provide detailed analysis of the electronic- structure and charge-density distribution. A closely re- lated crystal to MgO and A1203 is the spinel MgAlz04 with a complex cubic structure. Spinel oxides form a large class of inorganic solids with similar structures. We are not aware of any electronic-structure study on the spinel phase. In this paper we present the results of calculations on the electronic structures of MgO, n-Alz03, and MgA1204 using the first-principles orthogonalized linear combina- tion of atomic orbitals method (OLCAO) in the local- density approximation (LDA) within the density- functional formalism. The band structures, total and par- tial density of states (DOS), charge-density distributions, and effective charges on each ion are studied in detail. We have calculated the interband optical conductivities, or equivalently the linear dielectric functions of the three crystals up to a high photon energy of 40 eV. The results are compared with recent experimental data from vacu- um ultraviolet (VUV) measurements, ' using a laser plas- ma light source. Such comparisons provide a direct as- sessment on the accuracy of the band structures especial- ly for the higher conduction bands (CB). We also show that the electronic and the optical properties of the spinel phase MgA1204 cannot be taken as a simple weighted su- perposition of that of the MgO and e-A1203 crystals. This paper is organized as follows: In Sec. II, the structures of the three crystals are briefly discussed focus- ing on the relative short-range orders and interatomic distances. This is followed by a brief outline of the method of calculation in Sec. III. Results on band struc- tures and DOS are presented in Sec. IV. Detailed analysis on the charge-density distributions and optical absorptions are presented and discussed in Secs. V and VI, respectively. Some discussion and concluding re- marks are given in Sec. VII. II. CRYSTAL STRUCTURES The structures of the three crystals used in the present calculation and their interionic distances are summarized 43 1991 The American Physical Society

description

Estruturas de bandas da safira e outros óxidos.

Transcript of Self-consistent Band Structures, Charge Distributions, And Optical-Absorption Spectra in MgO,...

  • PHYSICAL REVIEW B VOLUME 43, NUMBER 5 15 FEBRUARY 1991-I

    Self-consistent band structures, charge distributions, and optical-absorption spectrain Mgo, a-Alamoand MgAlz04

    Yong-Nian Xu and W. Y. ChingDepartment ofPhysics, University ofMissouri Kansas City, Kansas City, Missouri 64110

    (Received 13 September 1990)The electronic structures, the charge-density distributions, and the optical-absorption spectra in

    MgO, a-A1,03, and MgA1204 crystals are studied by means of the first-principles self-consistent or-thogonalized linear combination of atomic orbitals method. The calculated band structures of MgOand n-Al&03 are compared with experimental data and previous calculations. It is concluded thatthe electronic structure and optical properties of the spinel MgA1204 cannot be taken as a simpleaverage of that of MgO and a-A120, . By direct space integration of the calculated charge, it isshown that the ionic nature of these crystals should be best described by the formulasMg" +0",A12' +03' ', and Mg' +Alq '+04" . The charge-density maps also revealthat there are open channels of very low electron density in MgA1204, which may be important inionic conductivity. The calculated optical dielectric functions for the three crystals are comparedwith recent vacuum ultraviolet data for an energy range up to 40 eV and show good overall agree-ment. The existence of an excitonic peak near the absorption edge tends to sharpen the edge, there-by increasing the gap as compared with the one-electron local-density calculations. For absorptionspectra beyond the excitonic peak, most of the experimentally resolved structures are reproduced bythe calculations without the need of an adjustment to widen the gap. It appears that the highconduction-band states in these three crystals, calculated by using the local-density approximation,are reasonably accurate.

    I. INTRODUCTION

    MgO and a-A1203 are important ceramic oxides thathave been subjected to numerous experimental andtheoretical studies. ' They have high melting points, lowcoe%cients of expansion, and small thermal conductivi-ties. There is also an increasing trend in using MgO andn-A1203 as substrate materials for thin-film growth andas optical materials. MgO is highly ionic and has a sim-ple NaC1-type structure. Its band structure has beenstudied by various methods ' and is reasonably well es-tablished. Still, there are difterences with regards to thesize of the band gap, valence-band (VB) width, and valuesof effective charges on each ion. ' o.-Alz03, on the otherhand, has a much more complicated corundum structure.Several calculations were performed in the past on itsband structure. ' ' Most of these calculations werenon-self-consistent or semiempirical in nature. n-A12O3has also been studied in the form of small clusters. 'A recent study on the band structure of u-A1203 used aself-consistent pseudofunction method. ' That study con-centrated on the interpretation of experimentalreAectivity data in terms of critical point analysis anddoes not provide detailed analysis of the electronic-structure and charge-density distribution. A closely re-lated crystal to MgO and A1203 is the spinel MgAlz04with a complex cubic structure. Spinel oxides form alarge class of inorganic solids with similar structures.We are not aware of any electronic-structure study on thespinel phase.

    In this paper we present the results of calculations on

    the electronic structures of MgO, n-Alz03, and MgA1204using the first-principles orthogonalized linear combina-tion of atomic orbitals method (OLCAO) in the local-density approximation (LDA) within the density-functional formalism. The band structures, total and par-tial density of states (DOS), charge-density distributions,and effective charges on each ion are studied in detail.We have calculated the interband optical conductivities,or equivalently the linear dielectric functions of the threecrystals up to a high photon energy of 40 eV. The resultsare compared with recent experimental data from vacu-um ultraviolet (VUV) measurements, ' using a laser plas-ma light source. Such comparisons provide a direct as-sessment on the accuracy of the band structures especial-ly for the higher conduction bands (CB). We also showthat the electronic and the optical properties of the spinelphase MgA1204 cannot be taken as a simple weighted su-perposition of that of the MgO and e-A1203 crystals.

    This paper is organized as follows: In Sec. II, thestructures of the three crystals are briefly discussed focus-ing on the relative short-range orders and interatomicdistances. This is followed by a brief outline of themethod of calculation in Sec. III. Results on band struc-tures and DOS are presented in Sec. IV. Detailedanalysis on the charge-density distributions and opticalabsorptions are presented and discussed in Secs. V andVI, respectively. Some discussion and concluding re-marks are given in Sec. VII.

    II. CRYSTAL STRUCTURESThe structures of the three crystals used in the present

    calculation and their interionic distances are summarized

    43 1991 The American Physical Society

  • 434462 YONG-NIAN XU AND W. Y. CHING

    0 has a very simple rocksalt struc-gconsists of an interpenetrating cc

    d f r any discussion. Fore

    ' ' ' f the crystal structurethere is no nee or an

    detailed description o t e cu-A}203, a e-Al 0 has a corundum

    i a h bohedral) unit cell con-iven by Batra. 0;- 2 3

    structure wi aith a trigonal (or rhom o e1 . The coordinates of thetaining two 2 3o Al 0 molecules. e co

    rrounded by six Othat an Al atoms is surrounatoms are such thai hbor (NN) distancesf two different nearest-neig or

    s four NN Al atoms The corundumstructure can als o be visualized as a exag

    f close-packed 0 atoms wit e22h d 11 coordinated holes.

    h h 11 11d inside the octa e ra y c

    ~ ~

    re the unit cell for t e exagpbe three times a ges as lar e as the r om o e

    h structure of a largeinel hase represents t e s ruc" It h s a face-centered-cubic

    ace roup 0 wit eight formula units per cu-f ceramic oxides. It as a ac

    the cell, each Mg ion isour 0 atoms while eac

    six octahedrally distri uteis best described as a cu e con

    yp I and type II as s own iontain Mg ions in tetra e ratype-I octants co

    n in Fi . 1(b); and the type- oc anh i Fi . 1().hedral coordination as s owAl ions in octahed

    0 the divalent ions inel such as MgAlz 4, eIn a normal spind the trivalent iont e-I octant an e

    e 0 the type-I octants contain a oas NiF z 4,tnvalent ions while the type-II octan s co

    h lf f the trivalent ions. Thed the other ha ocations can actua y e indistribution of cation

    1 and the inverse spine s o gi1 to ive a ran-s inel characterized by a parameterdom spine

    '1 bl trivalent-ion sites in type-f the 16 avai a e r'd b divalent ions. ranII octants occupie by

    inel. In many1 s inel to 0.5 for the inverse spine .i"s d trivalent ion to formids can be treated as a rivasystems, voids d r

    alled a defect inverse spine such h ",.f"--.-Al 0 . The nc varie

    diff' 1s inels gives rise to very idff 1tronic an md magnetic properties among i e

    22compounds.

    III. METHOD OF CALCULATION'

    tent OLCAO method for the electronic-The self-consistentf solids has een escri estructure calculation o

    r)

    'Q~ +

    (c)g~s OAt C)0

    Al 0 . (a) Cubic cell showingFIG. 1. Crystal structure of MgA z 4.b) T e-I octant with a Mg ion

    ' ht onstants of type I and II. ( yp-c) t e-II octant witin tetrahedral coordination with 0; (c yp-

    ions in octahedral coordination with 0.

    ub ' We will only outline theseveral recent pubublications.The basis func-'n the resent calculation.major steps in the p

    sion of the crysta oc1 Bl h functions aref t bt 1 f O

    ( ls2s2p, p, , 3opriate for each crys-3dy, ~3dz~~ 2 zi 3 2

    as the linear com-orbitals are expresse as etal. The atomic or i(GTO). This basis set- ypn-t e orbitals

    1 referred to as a full basis because i c-sh 1 f th atoms in the crystal

    1 Th 1-shell orbitals o t e a

    plus another s e p yf em t orbita s. e cnstructed according to t e usua

    h }1 Wformalism wit t ea for correlation correction. e

    and charge densi 'nsities are eac expressei h the coefficients ofaussian functions wit e1'fi'hermined by linear y ing

    a. The uality of t e ing 'y . qensure an accurate represen a 'monitored to en

    '

    1 b the fitting functions.charge density and pd the otentia y ethe matrix ele-To reduce t e size of the secular equation, e

    formation of MgO, a-A1,03, and MgA1204.TABLE I. Crystal-structure information o g, a- 2 3,

    Crystal structureLattice constant

    Space groupNearest-neighbor

    0distance (A)Mg-0Al-0

    Mg0NaC1

    a =4.213 A

    2.1015 (6)

    O, -A1203

    Corunduma =5.128 AP=55.333'

    6

    1.9692 (3)1.8570 (3)

    MgA1204

    Spinela =8.063 A

    1.9187 (4)1.9293 (6)

  • 43 SELF-CONSISTENT BAND STRUCTURES, CHARGE. . . 4463

    ments of the Hamiltonian and the overlap involving coreorbitals are eliminated by the orthogonalization pro-cedure. Self-consistency is generally achieved in less than20 iterations with the eigenvalues stabilized to less than0.0001 eV. After the self-consistency in the potential hasbeen obtained, the eigenvalues and the eigenvectors of theband secular equations are solved at 89, 308, and 89 kpoints in the irreducible portion of the Brillouin zone(BZ) for MgO, o.-AIz03, and MgA120~, respectively. Theenergies and wave functions at these k points are used toevaluate the DOS and the optical conductivities using thelinear analytic tetrahedron method. The calculation of31optical conductivity includes the efFects of the k-dependent moment matrix elements. The method for theoptical conductivity calculation in oxides has been de-scribed before. 27, 28, 32

    IV. BAND STRUCTURE AND DENSITY OF STATES

    A. MgO

    4-

    3

    2Q

    QN 0 i

    2Q

    M

    V3O

    2-

    Total

    r $ I ~ I ~ I ~ ~ ~ ) ~ ~ ~

    0

    The calculated band structure and the DOS for MgOare shown in Figs. 2 and 3, respectively. Also shown arethe projected partial DOS (PDOS) for the 0 and Mgatoms. The calculated band gap of 4.19 eV is direct at I,smaller than the reported experimental gap of 7.5 eV, andclose to the other recent calculations (see Table II). Theupper VB has a two-peak structure, a sharp one at 4. 5eV and a broader one centered at 1.2 eV. The totalwidth of the upper VB is about 5.4 eV. This is in goodagreement with experiment. ' Other recent calcula-tions gave a narrower bandwidth than our result. Thelower 0 2s band is 1.9 eV wide and peaks at 15.4 eV atthe leading edge of the band. The CB also has a two-peak structure, one at 8.9 eV and a sharper one at 12.3eV. There is a substantial degree of hybridization be-tween Mg and 0 states. The two CB structures are the

    0 ~ g g-20 15 10

    ~ i ~ $ ~ ~ ~ r / ~ ~ ~ ~ / ~ ~ ~ ~

    -5 0 5 10 15ENERGY (eV)

    FIG 3 DOS and PDOS of MgO

    B. a-A1203

    antibonding counterparts of the two bonding structuresin the VB. The effective masses of the CB and the VB arealso estimated from the band curvatures. The isotropicelectron effective mass m * at I is 0.35m, . The top of theVB is triply degenerate, so the hole effective masses arenot isotropic. Their components are listed in Table IIwith an average value of 2. 60m, for the heavy hole and0.33m, for the light hole.

    The calculated band structure for o.-Alz03 in the trigo-nal lattice is shown in Fig. 4. The corresponding DOS

    12-

    10-

    B-(0

    4-2-

    0

    4-

    r X KWAVE VECTOR

    FIG. 2. Band structures of MgO.

    l.

    Z A I't)II('AVE VECTOR

    FIG. 4. Band structure of O.-A1,03.

    A

  • 4464 YONG-NIAN XU AND W. Y. CHING 43

    and PDOS are shown in Fig. 5. Unlike MgO, the calcu-lated band gap for n-A120~ is indirect and equal to 6.29eV. This value is almost identical to the 6.3 eV obtainedby the pseudofunction calculation. ' The bottom of theCB is at I and the top of the VB is at a point of 0.20 ofI X from I . However, the energy at I is only 0.06 eVlower than the top of the VB. So for any practical pur-poses, we may regard o.-AlzO& as a direct gap insulator.The reported experimental gap value for o.'-AlzO& is 8.7eV. ' There are many more structures in the DOS curverejecting an increased complexity of the crystal struc-ture. The width of the upper VB is 7.3 eV which issignificantly wider than that of Mg0. The lower 0 2sbandwidth is as wide as 3.3 eV which peaks at the leadingedge of the band at 16 eV. Also, shown in Fig. 4 asdotted lines, is the VB x-ray photoemission spectroscopy(XPS) data of Ref. 35. The agreement is quite good ex-cept that there was a shoulderlike structure at about 10eV in the data that is not present in the calculated DOS.This structure is likely to be due to surface contaminationin the sample. More recent data by polarized x-ray-emission spectroscopy shows no such structure.

    The effective-mass components of the CB alongdifferent directions of BZ edges are listed in Table II withan average value of about 0.35m, . The hole effectivemasses are huge because of the very Oat top of the VB.At the I point, the top of the VB consists of hybridizedorbitals of 0 2p and Al 3p. The lowest CB is nondegen-erate with mixed Al 3s, 0 2s, and 0 3s characters. We donot find any significant participation of the Al 3d orbitalsfor the CB states with energy less than 11 eV, i.e., about 5eV from the CB minimum.

    C. MgA12O4

    The band structure and the DOS for the spinelMgAlz04 are displayed in Figs. 6 and 7, respectively.Like o.

    -A120&, the calculated band gap is indirect andequal to 6.51 eV. The quoted experimental gap is 7.8eV. The bottom of the CB is at I but the top of the VBis at a point 0.20 along I -K. The energy difference be-tween this point and I is even smaller, less than 0.004 eV.

    . So the top of the VB is again very Aat. The CB minimumat I is nondegenerate and composed of the s orbitals of

    a-A120&

    TABLE II. Calculated band-structure information on Mg0, a-A120&, and MgA1, 0~.MgO MgA1204

    Band gap (eV)Calc. {present)

    (others)Experiment

    direct (I )4.19

    4.65 '4. 50 4. 87'7 58 d7 8f

    indirect6.29

    6.3, , 8.0'8.7 9 4

    indirect6.51

    Upper VB width (eV)Calc. (present)

    (others)Experiment

    5.505.1,'4.48',4.80,4.75'

    6.5', 5 6

    7.398.1, ,6.0'

    5.32

    0 2s bandwidth (eV)Calc. (present)

    1.961.96

    3.263.26

    2.552.55

    Gap between upper VBand 0 2s bands (eV) 9.81 8.53 9.52

    Effective mass (m, )Electron

    Heavy hole

    Light hole

    "'Reference 11.Reference 9.

    'Reference 8.Reference 1.

    'Reference 16.Reference 40.

    0.35 (I ~L)0.35 (I ~X)0.35 (I ~K)2.77 (I ~L)1.60 (I ~X)

    4. 10, 1.80 (I ~K)0.31 (I ~L)0.35 ( I ~X)0.32 (I ~K)

    4.28

    ~Reference 2."Reference 43.'Reference 13.'Reference 34."Reference 33.

    o16 (r x)o45 (r z)o.4o (r w)0.38 ( 1"~D)

    large

    0.44 (l ~L)0.44 (r-x)o.44 (r-K)

    large

    6.07

  • 43 SELF-CONSISTENT BAND STRUCTURES, CHARGE. . .

    PO

    Total

    15

    10

    5-Q

    Q 0~NQ

    ~

    ~

    ~ ~ ~ ~

    ~

    ~ ~ ~ s~

    ~ s ~ ~

    0

    V3C3

    10

    pj(J~ ~ ~ ~ J ~ I l 1 $ I l ~ ~

    -ZO 15 10 -5 0ENERGY (eV}

    ~ ~ ~ ( ~ ~ ~ ~

    5 10

    FIG. 5. DOS and PDOS of o.'-Alz03.

    all the atoms in the cell. The electron effective mass is0.44mslightly larger than that in MgO and the holeeffective mass is large similar to o.-A1203. The upper VBhas a total width of about 5.3 eV which is smaller thanthat of MgO and much smaller than u-Alz03. The DOSfor this band has many structures due to the complexityof the crystal with major peaks at 0.5, 1.2, 2.4,

    3.7 and 4. 5 eV. The major feature of the CB DOS isa peak at 13.5 eV and a very deep minimum at 12.3 eV.The general shape of the DOS appears to be intermediatebetween MgO and o.'-Alz03. The lower O 2s band hastwo pieces. The upper one is very sharp and peaks at15.7 eV, resembling that in MgO; the lower piece isabout 1.5 eV wide and has two peaks at 17.5 and16.9 eV. Analysis of the wave function at I indicatesthat the lowest band involves hybridization of O 2s andAl 3s orbitals; the next lowest band involves the 0 2s andMg 3s interaction; while the much more localized 0 2sbands constitute the sharp peak in the DOS at 15.7 eV.

    The calculated information on the band structures forthe three crystals is listed in Table II.

    It is tempting to see if the electronic structure of acomplex oxide such as the spinel MgA1&04 can be mim-icked by that of MgO and u-Alz03. To this end, we plotthe broadened DOS of MgA1204 and compare it with thesum of the DOS of MgO and a-A1203 in Fig. 8. Al-though the general shapes of the two curves are similar,there is a lack of close agreement. MgAlz04 has a muchnarrower VB width. This is most likely due to thedifference in the NN distances. The Mg0 bond length(1.919 A) in MgAlzO& is shorter than that in MgO (2.102A), implying a stronger bond thus a narrower band. cz-A1203 has two Al0 lengths of 1.969 and 1.857 A versusonly one type (1.929 A) in MgA1204, which accounts fora broader VB width in e-A1203. The structures in the CBare also different. Most prominently, the deep minimum

    30,Total

    p, p-

    15-

    10

    10-

    0-

    Q) p ~ ~ ~

    Q 1P-05

    0'

    0

    ~ I ~ ~ ~ ~ ~ ~ ~ ( r ~ ~ I $ ~ ~ ~ ~

    Al

    -10- 5.

    5-

    r X KWAVE VECTOR

    0- ~/hi I I0

    ENERGY (eV)10

    FIG. 6. Band structure of MgA12O4. FIG. 7. DOS and PDOS of MgA1204.

  • 4466 YONG-NIAN XU AND W. Y. CHING 43

    25Q)

    +~ 20-

    ~~ 15-

    ~ 10-CGP3CO

    Total

    y Ig I

    i~

    g

    y II

    I 'II ~rj

    II

    I0 r20

    ~ I I I r I I ~ ~ I ~ ~ ~

    15 10 -5ENERGY (eV)

    5 10

    FIG. 8. Comparison of broadened total DOS: solid line, ascalculated for MgA1, 04,' dashed line, superposition of MgO andu-A1~0&.

    at 12.3 eV in MgA1204 is totally absent in the combinedDQS of Mg0 and O.-A1203. Even for the much more lo-calized 0 2s band, MgA1204 has a sharper leading edgeand a narrower bandwidth than the superimposed curve.We therefore conclude that for a precise determination ofthe electronic structure of a complex oxide, it is necessaryto perform a first-principles calculation for the phase con-sidered, instead of deducing it from the electronic struc-tures of the other simpler structures.

    V. CHARGE DISTRIBUTION

    The charge-density distribution in oxide compounds isan important subject because it relates directly to theeffective ionic charges on the ions which play a majorrole in ionic conductivity. The oxidation state of the ionsin a compound is also related to the effective ioniccharge. However, the concept of effective ionic charge isnot a precisely defined one because of the difficulty inpartitioning the charge distribution in real space of thecrystalline unit cell. In an empirical type of study and inpoint-defect models, it is generally taken for granted thatthe ions take full integral values of charge such as Mg +,Al +, 0, etc. , which are probably more appropriate forthe description of oxidation states. There has been muchdispute on the effective charges on Mg and 0 in MgOfrom various calculations ranging from the pure ionicpicture of Mg +0 (Refs. 5 and 6) to a more covalentpicture of Mg'+ 0'

    In the LCAQ type of calculation, it has been very pop-ular to use the Mulliken population-analysis scheme toestimate the effective charges on each atom. However,the Mulliken scheme is only approximate since it assumesan equal sharing of the overlap integrals between twodifferent atoms. The scheme works quite well for systemsin which the atomic wave functions used in the expansionare not too extended such as in minimal-basis-type calcu-lations. However, in-a more accurate calculation, a fullbasis and beyond is more likely to be used. In this case,the Mulliken analysis can lead to erroneous results be-cause its deficiency becomes more acute due to the ex-tended nature of the basis functions. The effective Mul-liken charges calculated for MgO, a-A1203, and MgAlz04

    TABLE III. Charge-density distribution in MgO, a-A1203,and MgA1~04.

    Mulliken charge (elec. )Mg

    Al0

    Integrated charge (elec. )Mg

    Al0

    Interstitial charge (elec. )Total charge in cell (elec.)

    O

    Ionic radius (A)MgAl0

    MgO

    2.64

    5.36

    0.06

    7.710.238

    0.358

    1.243

    Q, -A1,0,

    2.706.20

    0.197.571.82

    48

    0.6781.170

    MgA1204

    1.963.016.01

    0.090.257.641.70

    64

    0.3450.6951.220

    in the present case using a full basis set are listed in TableIII. These numbers show very little charge transfer fromthe cations to anions and are obviously not correct inrejecting the ionic nature of these crystals. In MgO, itwrongly predicts a slight charge transfer from the 0 tothe Mg ion. Batra's calculation' on n-A12Q3 using theMulliken scheme also showed very small charge transferfrom Al to 0. Since the projection of PDOS into partialcomponents uses the Mulliken scheme, the PDOSpresented in Sec. VII are qualitative at most.

    In this paper, we show that a much more reliable wayto obtain the effective charges on each ion is to perform areal-space integration on the charge densities obtainedfrom the self-consistent band calculations. In theQLCAO method, the potential function has no artificialspherical boundary which has been the source of arbitrar-iness in determining the effective charges in some othermethods. Also, the final self-consistent charge density isin a convenient form of a sum of atom-centered Gaussianfunctions. Still, this process is also approximate becausethere is no precise way of partitioning the charge in thereal space. We assume an average spherical distributionof charge for each ion and integrate the p(r) up to aneffective ionic radius determined by inspecting the two-dimensional contour charge plots. We impose the con-straint that the integrated charges in spheres plus a smallamount of interstitial charge must add up to the totalcharge in the unit cell. The charge density for MgOalong a Mg0 bond and on the (001) plane are shown inFigs. 9(a) and 9(b). We determine the eft'ective ionic ra-dius for the Mg and 0 ions to be 0.358 and 1.243 A, re-spectively, which give the integrated effective charges onMg and 0 ions to be 0.06 and 7.71 electrons, respectively.This indicates that MgO is highly ionic with almost all ofthe valence electrons on Mg transferred to Q. This as-signment leaves 0.23 electrons per unit cell in the intersti-tial region of the crystal. Dividing this interstitial chargeequally among Mg and 0 ions, we may write the ionicdescription for MgO as Mg' +0'

    For e-A1203 in the rhombohedral lattice, there is nosimple crystalline plane that contains both the Al and O

  • AND STRUCTU R.ES CHAR. GE-SELF-CQNSISTEN

    the formula oatQms ln' iso jllustra

    the cell lea os the lm-63+ I .75

    This calculat ot to determine theQf having the

    '

    1 r situation haspQf tan e

    h pns A slmh accuf a

    fad jj for tnd jn

    effectivY O3 compou

    u to bentered in the 2 3

    h ionic fpencoun

    jves t. ~ jcture

    h r e integrationa full ionized p1.44

    1-ather23+ 2 39

    ated crysta stfuc-O3

    re comp icaMgA1204bution. Althougd charge distr '

    aller than that

    jlar to those1 to the + y p]ane a

    quite Slmllane parallel o

    . d shows thatoes not cont 'nnel along

    agponal of the P ad O resembles t aa,nbonding betwe

    43

    hprt AlOit alpng the o1 A

    h charge density 0 and one Aions W p lane which con;e charge of Fig

    e 1pt t e ctains twobond and on p

    ' lp- From the'

    at p.4 Aown 'n 'g'

    '

    m in p(1Qn as show

    the minimum~

    lue as the~

    a ears ttake thjs v

    IO(a) ') ance we tna&

    the contourfrom the A

    Qf Al By in P'

    radius of

    1 n1 as ecting t ejonic radius o

    effectjve ionic raeffectl e

    see that the eh discernible

    . &O(b) e s'

    canse of t ePlot of F g

    r than 0.4 A becd the Al ions

    to be largerar e aroun

    Al hasibution of c a g

    diagram we~erlcal dlstrl

    innear-sp, .er

    .

    d s Accor lngd 0 jpns in o'-

    .4-A ra lu '" Al and

    eyond the' ionic rndll fPr

    The in-te the effectiv o

    A respectivelyestimate

    678 and 1. '1 trons for Al

    to be o.h nP&9e

    A12O3harges are t e

    t ccounts fortegrated eff

    or O. Ts assig'

    cell, leaving

    'ective ci nmen a57 electfon& fpr '

    s ln the unit ca,nd

    lence electronsf the cell. This

    962%%uo. h jnterstli

    of the 48 va1 re ion

    t 1 ionized an -interstitial ch glar equal partitioning

    IO ()1.0-M

    " 0.6-Q

    0.4-

    2

    1.0-

    & 0.8-Q& 0.6-V

    0.4-

    0.0 I0.0 0.7

    r(Mg-0) (A)

    0.4i

    1.6 2.0

    it ' ' ' n in MgO: (a) along aCharge-density gFIG. 9.102 A; (b) in the

    h' 'n""'1 fMg

    001 to 0 25 in t e i1' es are fromin)3electron/(a. u.

    t ' ' -Al 0; (a) alongin o;-A2~ g-e yo dof1.9

    . The contour in3val of 0.005 in units o e ec

  • 4468 YONG-NIAN XU AND W. Y. CHING 43

    FIG. 11. Charge-density distribution in MgA1204 in a planeparallel to the x-y plane with z = containing only Al and 0atoms. The contour lines are from 0.01 to 0.25 in the interval of0.005 in units of electron/(a. u. ) .

    The next plane with z =0 contains only Mg ions at thecenter and the corners of the plane. The charge densityon this plane is shown in Fig. 12. The more or less sym-metric pattern of the charge distribution away from theMg ions comes from the Al and 0 atoms on the adjacentplanes. In Fig. 13 we plot the charge contour on the

    FIG. 13. Charge-density distribution in MgA120~ in the [110]plane. The dark circles at the lower part of the plane near theopen channel represent the Mg ions. A layer of Al ions is be-tween two layers of 0 ions represented by large circles. Thecontour lines are from 0.01 to 0.25 in the interval of 0.005 inunits of electron/(a. u. )'.

    (110) plane which contains all three types of ions. In thelower half of this plane, there is a huge empty channelwith almost no or little charge. We expect the small Mgion to be easily diffused along such a channel. No similarempty channels exist in MgO or u-A1203 crystals. FromFigs. 11 and 12, we estimate the effective ionic radii forMg, Al, and 0 ions in MgA1204 to be 0.345, 0.695, and1.220 A, respectively. The corresponding integratedcharges for each ion are 0.09, 0.25, and 7.64 electrons, re-spectively. This leaves 1.70 electrons out of a total of 64electrons in the interstitial region. Again, by equal parti-tioning of the interstitial charges among the 14 atoms inthe primitive cell, we write the ionic formula for thespinel to be Mg' +Al +0 '

    VI. OPTICAL ABSORPTION

    FIG. 12. Charge-density distribution in MgA1204 in a planeparallel to the x-y plane with z =

    ~

    containing only the Mgatoms which are at the center and the corners of the plane. Thecontour lines are from 0.01 to 0.25 in the interval of 0.005 inunits of electron/(a. u. ) .

    The optical-absorption measurements on MgO and u-Alz03 have been carried out by several groups in thepast. " Only very recently, using a laser plasma lightsource and the VUV technique, the reAectivity data inwide-gap insulators can be measured up to 40 eV andhigher, ' which ensures a maximum accuracy in convert-ing optical data using the Kramers-Kronig relation.Furthermore, this new experimental setup provides amuch higher energy resolution such that smaller struc-tures in the spectrum can be resolved. The availability ofhigh-quality optical data on MgO, o.-A1203, andMgA12O4 is part of the motivation for the presenttheoretical calculation. With experimental data mea-sured up to a high-photon frequency, the interband tran-sitions from very low-lying 0 2s bands to the very highCB states can be studied. Previous experimental mea-surements usually limited the photon energy of less than10 eV (Ref. 43) because the windowed VUV sources stopat 11 eV.

    The first-principles optical calculation within theOLCAO method has been described in considerable de-

  • 43 SELF-CONSISTENT BAND STRUCTURES, CHARGE. . . 4469

    tail in our published work on other materials.Generally, the momentum matrix elements between eachpair of the occupied VB and the empty CB are calculatedat each k point in the irreducible portion of the BZ usingthe Bloch functions from the band calculation. The in-terband optical conductivity in the dipole approximationis then evaluated by considering all possible pairs of tran-sitions. The summation over the BZ is facilitated by us-ing the linear analytic tetrahedron method. ' The inter-band optical conductivity is related to the imaginary partof the dielectric function e2(co) through ez(co)=(4'/co)o. &(co). The real part, e&(co), can be obtainednumerically by Kramers-Kronig transformation. Thezero-frequency limit of e, (co) gives the electronic part ofthe static dielectric constant eo, a parameter of funda-mental importance in many aspects of materials proper-ties.

    The calculated imaginary part of the dielectric func-tions e2( co ) for the three crystals MgO, Alzo3, andMgA12O4 are shown in Figs. 14, 15, and 16, respectively.Also shown are the experimental VUV data from Ref. 2.

    It must be pointed out that in comparing the experi-mental and the theoretical curves, no shift in the energyscale has been applied. It has been long recognized thatthe LDA scheme underestimates the band gap in semi-conductors and insulators, sometimes up to 50%. Acommon practice has been to apply an operator to widenthe gap so as to match the experimentally reported bandgap and shift all the CB states accordingly. This ap-proach has some drawbacks. First, it assumes that themeasured gap, usually from some kind of optical experi-ments, is to be identified with the calculated intrinsic gap.In many instances, this is not the case either because thethreshold absorption is symmetry forbidden or is of sucha low intensity that it cannot be detected experimental-ly. Second, it also assumes that all the CB states will beunderestimated in energy relative to the VB top by thesame amount. The best way to check the validity of thisassumption is to compare the calculated and the mea-sured optical transition spectrum with the experimental

    5-

    2-

    05

    I

    10I

    15I

    20I

    25 30 35 40ENERGY (eV)

    FIG. 15. Imaginary part of the dielectric function ez(co) forcx-A1203. (a) experimental data from Ref. 2; (b) present calcula-tion.

    data up to a very-high-photon energy. Third, in manywide-gap insulators, existence of excitons near the ab-sorption edge greatly complicates the absorption spec-trum itself. Since the formation of an exciton is part ofthe many-body interaction in crystals which is not prop-erly accounted for by the one-electron calculation, thediscrepancy in the gap value cannot be totally attributedto the shortcomings of LDA. In the present study, weare surprised to find that for the three crystals we stud-ied, there is no need to apply such a shift to align the ma-jor structures beyond the threshold in the absorptioncurves. Apparently, the formation of an exciton has

    I

    10I

    15I

    20I

    25I

    30I

    35 40I

    10I

    15I

    20I

    25I

    30I

    35 40ENERGY (eV)

    ENERGY (eV)

    FIG. 14. Imaginary part of the dielectric function e~(co) forMgO: (a) experimental data from Ref. 2; (b) present calculation.

    FIG. 16. Imaginary part of the dielectric function e2(co) forMgA12O&. (a) experimental data from Ref. 2; (b) present calcu-lation.

  • 4470 YONG-NIAN XU AND W. Y. CHING 43

    somehow modified the absorption-edge profile, making itmuch sharper than predicated by the one-electron calcu-lation. However, the major absorption features beyondthe excitonic peak are quite well accounted for by theone-electron LDA calculation.

    For MgO, the experimental e2(co) curve in Fig. 14(a)shows an excitonic peak at 7.7 eV with a very sharp ab-sorption edge. The main spectrum shows three majorpeaks at 10.6, 13.3, and 17.1 eV with smaller structures at14.9, 19.2, 20.8, and 27.0 eV. These structures are gen-erally well reproduced by the calculation except relativeintensities of the structures are somewhat off. There areno other prominent absorption structures beyond 25 eV.The overall magnitude of the e~(co) curve is also in goodagreement with experiment. The new data of Ref. 2 onMgO are actually quite similar to the much earlier mea-surement of Roessler and Walker. The main differenceis that the leading peak at 10.6 eV is now higher than thesecond peak near 13 eV. We also note that our assign-ment of the peak structures is different from that of Fong,Saslow, and Cohen, who used the empirical pseudopo-tential band structures to calculate the ez(co) curve.

    For o;-A1203, the excitonic peak is at 9.2 eV and the ex-perimental curve shows a single major peak at 11.8 eV.Some weak shoulder structures at 12.6, 14.8, and 16.9 eVare also discernible. Beyond 20 eV, there is very little ab-sorption. The calculated curve reproduces these featuresvery well except for the fact that the calculated profileappears to be a bit too broad. But again, the overall mag-nitude is in very good agreement with the experimentalcurve. The experimental optical gap for o.'-Alz03 wasquoted as 8.8 eV as determined from the cutoff of thetransmission data. ' However, the absorption coefficientcurve actually shows the threshold to be at 6.4 eV, veryclose to the calculated gap. French has attributed this asarising from the charge-transfer transitions from Cr im-purities. ' Arakawa and Williams had also measured theoptical properties of single-crystal n-Alz03 in thevacuum-ultraviolet region. Although the general featuresof the spectrum are quite similar, their data are much lessresolved than that of Ref. 2.

    For MgA1204, a sharp excitonic peak exists at 8.2 eV.Unlike MgO, there appears to be a small absorptionstructure at 6.4 eV below the excitonic peak. It is notclear whether this is due to the interband absorption orsome other mechanisms such as phonon-assisted transi-tions to impurity or defects states in the gap. The experi-mental ez(co) curve for MgA1204 shows three very broadpeaks at 11.1, 14.2, and 19.2 eV. Beyond 25, eV, the ab-sorption is negligible. These features are again wellreproduced by the calculation except the relative intensi-ties of the peaks are in lesser agreement. It is also clearthat the ez(co) curve for MgA120has no resemblance tothe average e2(co) curves of MgO and a-A1~03.

    Apart from the complications due to the existence ofthe excitonic peaks near the absorption edges, the overallagreement between theory and experiment for the opticalabsorptions up to 40 eV for all three crystals is quitegood. The calculation does show more structures in thespectra. However, it should be remembered that the cal-culation is essentially for a crystal at zero temperature,

    30

    20 -.15-10 -.

    15 -.20

    25.20. (b)15.10

    1040.30 -.

    20 -.

    10 .20 -.30 -.40

    0I

    10ENERGY (eV)

    1

    15 20

    FICz. 17. Real part of the dielectric function el(~) for (a)MgO, (b) a-A1203, and (c) MgA1~04.

    while the real measurement may be affected by the pres-ence of phonons at finite temperature such that some ofthe finer structures may not be well resolved.

    In Fig. 17 we display the real part of the dielectricfunction for the same three crystals as obtained frome2(co) by the Kramers-Kronig transformation. All threecurves have very similar features. The most importantinformation is the value of e, (co) in the limit of zero fre-quency. This corresponds to the electronic part of thestatic dielectric constant of the material. Our calculatedvalues of E'p for MgO, a-A1203, and MgA1204 are 4.28,3.86, and 6.07, respectively, which are also listed in TableII. Our calculation shows that a-A1203 has the lowestdielectric constant and MgA1204 the highest among thethree crystals. Thus ep for MgA1204 does not interpolatebetween the corresponding values from MgO and a-Alz03. However, the calculated value of ep will also besensitive to the gap value. If the band gap is increased, Epwill decrease and vice versa. In a capacitance type ofmeasurement to determine the dielectric constant of amaterial, ep values will be significantly affected byelectron-phonon interactions, and the measured ep values

  • 43 SELF-CONSISTENT BAND STRUCTURES, CHARGE. . . 4471

    will be larger than the calculated ones which accountsonly for the electronic part alone.

    VII. DISCUSSION

    We have presented the results of first-principles studiesof the electronic and optical properties of three importantoxide crystals, namely, Mg0, cx-A1203, and MgAlz04.This is the first time the electronic structure of the spinelhas been studied, together with MgO and n-Alz03 usingthe same computational method. A number of importantelectronic parameters such as the bandwidth, band gaps,effective masses, etc. , using the LDA are obtained. Forall three crystals, the electron effective mass is of the or-der of less than 0.5mcomparable to most semiconduc-tors and is much smaller than the hole effective mass.This indicates that electronic conduction in these crystalsis possible, if somehow the electrons can be promoted tothe CB by overcoming the large band gap.

    We have also made a detailed analysis of the chargedistributions in the three crystals. It is shown that theMulliken analysis is not adequate for ionic crystals and adirect-space integration scheme gives much moreaccurate results. MgO is found to be almost fullyionized; a-Alz03 has a slight covalent character andMgA1204 is somewhat intermediate. Ionicformulas of Mg' +O' Al 0 ' andMg' A12 +04' are deduced. Detailed contourmaps on MgA1204 reveal some open channels of very lowelectron density in the cubic cell of the spinel structureand it is argued that such open channels may be very im-portant in ionic transport in MgA1204.

    The interband optical calculation produced results insurprisingly good agreement with recent data. The pres-ence of excitonic levels in all three crystals has greatlymodified the absorption profile near the threshold. Thiscould partly be due to transitions from the VB to the res-onant states of the exciton in the CB continuum. Theexperimental data in the threshold region are furthercomplicated by the possible transitions to the charge-transfer bands due to the trace of impurities which al-ways exist in these oxides. However, for absorption inthe energy range beyond the excitonic peak, the agree-ments between theory and experiments are very goodwithout the need of applying a finite shift in the energyscale. This seems to suggest that the high CB states fromthe OLCAO-LDA calculation are reasonably accurate.It is a generally accepted notion that the LDA approxi-mation is incapable of giving accurate CB states becauseit is strictly valid only for the ground state. And thishas been the main reason why the band gaps for semicon-ductors and insulators are always underestimated by the

    LDA calculation. Another equally valid argument is thatthe underestimation of the band gap in wide-gap insula-tors is due to the incomplete cancellation in the self-energy terms in the Coulomb and exchange potentials inthe LDA. ' Application of the self-interaction correc-tion to the band-structure calculation of LiF increases thegap value considerably. Our surprisingly good resultson MgO, a-A1203, and MgA104 may lead to further in-vestigation as to whether this prevailed notion is univer-sally correct, or there are exceptions to this such that theone-electron LDA calculation of the CB in some systemsmay actually turn out to be more valid than generally be-lieved. It can also mean that the band-gap estimation insome ionic compounds in which the excitons are foundnear the band edge may be fundamentally different fromthat in the covalently bonded semiconductors.

    We have completed a similar first-principles study ofthe electronic and optical properties of another very im-portant and related oxide system, silicon dioxide. Theband structures of all polymorphic forms of Si02 werestudied before by Li and Ching ' using the OLCAOmethod. That calculation, however, was not self-consistent. The self-consistent OLCAO band structurefor o. quartz was presented some time ago, sho~ing muchimprovement. The self-consistent band structures andoptical absorptions in all polymorphic forms of Si02 havebeen calculated. The correlation of electronic structureand optical properties to the different short-range ordersin various polymorphic Si02 turns out to be highly in-teresting and that work will be published separately.

    A final comment we wish to make is that ceramic ox-ides are usually contaminated with defects and impuritiesand the properties measured are most likely at room tem-perature or even higher. Our calculation, on the otherhand, is for a perfect crystal at zero temperature.Temperature-dependent optical measurements show thestructures in e2(co) become more broadened and shift inenergy as temperature is increased. Our results showthat a state-of-the-art quantum-mechanical calculationon a highly idealized situation can already give results ingood agreement with measurements on real samples.Further progress is therefore possible by extending thepresent work to more realistic systems which containcrystal imperfections as well as the effects of finite tem-peratures. Such studies will greatly aid the fundamentalunderstanding of ceramic materials and enlarge theirscope of applications.

    ACKNOWLEDGMENTS

    This work is supported by the U. S. Department of En-ergy under Cxrant No. DE-FG02-84ER45170.

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