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PHYSICAL REVIEW B VOLUME 43, NUMBER 5 15 FEBRUARY 1991I
Selfconsistent band structures, charge distributions, and opticalabsorption spectrain Mgo, aAlamoand MgAlz04
YongNian Xu and W. Y. ChingDepartment ofPhysics, University ofMissouri Kansas City, Kansas City, Missouri 64110
(Received 13 September 1990)The electronic structures, the chargedensity distributions, and the opticalabsorption spectra in
MgO, aA1,03, and MgA1204 crystals are studied by means of the firstprinciples selfconsistent orthogonalized linear combination of atomic orbitals method. The calculated band structures of MgOand nAl&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 aA120, . 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 chargedensity 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 agreement. The existence of an excitonic peak near the absorption edge tends to sharpen the edge, thereby increasing the gap as compared with the oneelectron localdensity 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 highconductionband states in these three crystals, calculated by using the localdensity approximation,are reasonably accurate.
I. INTRODUCTION
MgO and aA1203 are important ceramic oxides thathave been subjected to numerous experimental andtheoretical studies. ' They have high melting points, lowcoe%cients of expansion, and small thermal conductivities. There is also an increasing trend in using MgO andnA1203 as substrate materials for thinfilm growth andas optical materials. MgO is highly ionic and has a simple NaC1type structure. Its band structure has beenstudied by various methods ' and is reasonably well established. Still, there are difterences with regards to thesize of the band gap, valenceband (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 werenonselfconsistent or semiempirical in nature. nA12O3has also been studied in the form of small clusters. 'A recent study on the band structure of uA1203 used aselfconsistent pseudofunction method. ' That study concentrated on the interpretation of experimentalreAectivity data in terms of critical point analysis anddoes not provide detailed analysis of the electronicstructure and chargedensity distribution. A closely related 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 electronicstructure study on thespinel phase.
In this paper we present the results of calculations on
the electronic structures of MgO, nAlz03, and MgA1204using the firstprinciples orthogonalized linear combination of atomic orbitals method (OLCAO) in the localdensity approximation (LDA) within the densityfunctional formalism. The band structures, total and partial density of states (DOS), chargedensity 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 vacuum ultraviolet (VUV) measurements, ' using a laser plasma light source. Such comparisons provide a direct assessment on the accuracy of the band structures especially 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 superposition of that of the MgO and eA1203 crystals.
This paper is organized as follows: In Sec. II, thestructures of the three crystals are briefly discussed focusing on the relative shortrange orders and interatomicdistances. This is followed by a brief outline of themethod of calculation in Sec. III. Results on band structures and DOS are presented in Sec. IV. Detailedanalysis on the chargedensity distributions and opticalabsorptions are presented and discussed in Secs. V andVI, respectively. Some discussion and concluding remarks 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 YONGNIAN XU AND W. Y. CHING
0 has a very simple rocksalt strucgconsists of an interpenetrating cc
d f r any discussion. Fore
' ' ' f the crystal structurethere is no nee or an
detailed description o t e cuA}203, a eAl 0 has a corundum
i a h bohedral) unit cell coniven 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 nearestneig or
s four NN Al atoms The corundumstructure can als o be visualized as a exag
f closepacked 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 facecenteredcubic
ace roup 0 wit eight formula units per cuf 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 ratypeI 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 eI octant an e
e 0 the typeI octants contain a oas NiF z 4,tnvalent ions while the typeII 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 rans inel characterized by a parameterdom spine
'1 bl trivalention sites in typef 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 electronicThe selfconsistentf 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 eI octant with a Mg ion
' ht onstants of type I and II. ( ypc) t eII 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 crys3dy, ~3dz~~ 2 zi 3 2
as the linear comorbitals are expresse as etal. The atomic or i(GTO). This basis set ypnt e orbitals
1 referred to as a full basis because i csh 1 f th atoms in the crystal
1 Th 1shell 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 eleTo reduce t e size of the secular equation, e
formation of MgO, aA1,03, and MgA1204.TABLE I. Crystalstructure information o g, a 2 3,
Crystal structureLattice constant
Space groupNearestneighbor
0distance (A)Mg0Al0
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 SELFCONSISTENT BAND STRUCTURES, CHARGE. . . 4463
ments of the Hamiltonian and the overlap involving coreorbitals are eliminated by the orthogonalization procedure. Selfconsistency is generally achieved in less than20 iterations with the eigenvalues stabilized to less than0.0001 eV. After the selfconsistency 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 kdependent moment matrix elements. The method for theoptical conductivity calculation in oxides has been described 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 twopeak 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 calculations 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 twopeak structure, one at 8.9 eV and a sharper one at 12.3eV. There is a substantial degree of hybridization between Mg and 0 states. The two CB structures are the
0 ~ g g20 15 10
~ i ~ $ ~ ~ ~ r / ~ ~ ~ ~ / ~ ~ ~ ~
5 0 5 10 15ENERGY (eV)
FIG 3 DOS and PDOS of MgO
B. aA1203
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 trigonal lattice is shown in Fig. 4. The corresponding DOS
12
10
B(0
42
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 YONGNIAN XU AND W. Y. CHING 43
and PDOS are shown in Fig. 5. Unlike MgO, the calculated band gap for nA120~ 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 purposes, 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 structure. 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 xray photoemission spectroscopy(XPS) data of Ref. 35. The agreement is quite good except 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 xrayemission spectroscopy shows no such structure.
The effectivemass 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 nondegenerate 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 between 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
aA120&
TABLE II. Calculated bandstructure information on Mg0, aA120&, 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 (rx)o.44 (rK)
large
6.07

43 SELFCONSISTENT BAND STRUCTURES, CHARGE. . .
PO
Total
15
10
5Q
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 uAlz03. 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 mimicked by that of MgO and uAlz03. To this end, we plotthe broadened DOS of MgA1204 and compare it with thesum of the DOS of MgO and aA1203 in Fig. 8. Although 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. czA1203 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 eA1203. The structures in the CBare also different. Most prominently, the deep minimum
30,Total
p, p
15
10
10
0
Q) p ~ ~ ~
Q 1P05
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 YONGNIAN XU AND W. Y. CHING 43
25Q)
+~ 20
~~ 15
~ 10CGP3CO
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 anduA1~0&.
at 12.3 eV in MgA1204 is totally absent in the combinedDQS of Mg0 and O.A1203. Even for the much more localized 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 firstprinciples calculation for the phase considered, instead of deducing it from the electronic structures of the other simpler structures.
V. CHARGE DISTRIBUTION
The chargedensity 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 inpointdefect 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 popular to use the Mulliken populationanalysis 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 minimalbasistype calculations. However, ina more accurate calculation, a fullbasis and beyond is more likely to be used. In this case,the Mulliken analysis can lead to erroneous results because its deficiency becomes more acute due to the extended nature of the basis functions. The effective Mulliken charges calculated for MgO, aA1203, and MgAlz04
TABLE III. Chargedensity distribution in MgO, aA1203,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 nA12Q3 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 arealspace integration on the charge densities obtainedfrom the selfconsistent band calculations. In theQLCAO method, the potential function has no artificialspherical boundary which has been the source of arbitrariness in determining the effective charges in some othermethods. Also, the final selfconsistent charge density isin a convenient form of a sum of atomcentered 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 twodimensional contour charge plots. We impose the constraint 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 radius for the Mg and 0 ions to be 0.358 and 1.243 A, respectively, 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 assignment leaves 0.23 electrons per unit cell in the interstitial region of the crystal. Dividing this interstitial chargeequally among Mg and 0 ions, we may write the ionicdescription for MgO as Mg' +0'
For eA1203 in the rhombohedral lattice, there is nosimple crystalline plane that contains both the Al and O

AND STRUCTU R.ES CHAR. GESELFCQNSISTEN
the formula oatQms ln' iso jllustra
the cell lea os the lm63+ 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
1ather23+ 2 39
ated crysta stfucO3
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
innearsp, .er
.
d s Accor lngd 0 jpns in o'
.4A ra lu '" Al and
eyond the' ionic rndll fPr
The inte 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 ()1.0M
" 0.6Q
0.4
2
1.0
& 0.8Q& 0.6V
0.4
0.0 I0.0 0.7
r(Mg0) (A)
0.4i
1.6 2.0
it ' ' ' n in MgO: (a) along aChargedensity 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~ ge yo dof1.9
. The contour in3val of 0.005 in units o e ec

4468 YONGNIAN XU AND W. Y. CHING 43
FIG. 11. Chargedensity distribution in MgA1204 in a planeparallel to the xy 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 symmetric 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. Chargedensity 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 between 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 uA1203 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, respectively. This leaves 1.70 electrons out of a total of 64electrons in the interstitial region. Again, by equal partitioning 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. Chargedensity distribution in MgA1204 in a planeparallel to the xy 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 opticalabsorption measurements on MgO and uAlz03 have been carried out by several groups in thepast. " Only very recently, using a laser plasma lightsource and the VUV technique, the reAectivity data inwidegap insulators can be measured up to 40 eV andhigher, ' which ensures a maximum accuracy in converting optical data using the KramersKronig relation.Furthermore, this new experimental setup provides amuch higher energy resolution such that smaller structures in the spectrum can be resolved. The availability ofhighquality optical data on MgO, o.A1203, andMgA12O4 is part of the motivation for the presenttheoretical calculation. With experimental data measured up to a highphoton frequency, the interband transitions from very lowlying 0 2s bands to the very highCB states can be studied. Previous experimental measurements usually limited the photon energy of less than10 eV (Ref. 43) because the windowed VUV sources stopat 11 eV.
The firstprinciples optical calculation within theOLCAO method has been described in considerable de

43 SELFCONSISTENT 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 interband optical conductivity in the dipole approximationis then evaluated by considering all possible pairs of transitions. The summation over the BZ is facilitated by using the linear analytic tetrahedron method. ' The interband 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 KramersKronig transformation. Thezerofrequency limit of e, (co) gives the electronic part ofthe static dielectric constant eo, a parameter of fundamental importance in many aspects of materials properties.
The calculated imaginary part of the dielectric functions 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 experimental 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 semiconductors 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 approach has some drawbacks. First, it assumes that themeasured gap, usually from some kind of optical experiments, 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 experimentally. 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 measured 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) forcxA1203. (a) experimental data from Ref. 2; (b) present calculation.
data up to a veryhighphoton energy. Third, in manywidegap insulators, existence of excitons near the absorption edge greatly complicates the absorption spectrum itself. Since the formation of an exciton is part ofthe manybody interaction in crystals which is not properly accounted for by the oneelectron 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 studied, there is no need to apply such a shift to align the major 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 calculation.

4470 YONGNIAN XU AND W. Y. CHING 43
somehow modified the absorptionedge profile, making itmuch sharper than predicated by the oneelectron calculation. However, the major absorption features beyondthe excitonic peak are quite well accounted for by theoneelectron 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 absorption 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 generally 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 measurement 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 assignment of the peak structures is different from that of Fong,Saslow, and Cohen, who used the empirical pseudopotential band structures to calculate the ez(co) curve.
For o;A1203, the excitonic peak is at 9.2 eV and the experimental 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 absorption. 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 magnitude 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 chargetransfer transitions from Cr impurities. ' Arakawa and Williams had also measured theoptical properties of singlecrystal nAlz03 in thevacuumultraviolet 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 phononassisted transitions to impurity or defects states in the gap. The experimental ez(co) curve for MgA1204 shows three very broadpeaks at 11.1, 14.2, and 19.2 eV. Beyond 25, eV, the absorption is negligible. These features are again wellreproduced by the calculation except the relative intensities 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 aA1~03.
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 calculation is essentially for a crystal at zero temperature,
30
20 .1510 .
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) aA1203, and (c) MgA1~04.
while the real measurement may be affected by the presence 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 KramersKronig transformation. All threecurves have very similar features. The most importantinformation is the value of e, (co) in the limit of zero frequency. This corresponds to the electronic part of thestatic dielectric constant of the material. Our calculatedvalues of E'p for MgO, aA1203, and MgA1204 are 4.28,3.86, and 6.07, respectively, which are also listed in TableII. Our calculation shows that aA1203 has the lowestdielectric constant and MgA1204 the highest among thethree crystals. Thus ep for MgA1204 does not interpolatebetween the corresponding values from MgO and aAlz03. 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 byelectronphonon interactions, and the measured ep values

43 SELFCONSISTENT 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 firstprinciples studiesof the electronic and optical properties of three importantoxide crystals, namely, Mg0, cxA1203, and MgAlz04.This is the first time the electronic structure of the spinelhas been studied, together with MgO and nAlz03 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 order of less than 0.5mcomparable to most semiconductors 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 adirectspace integration scheme gives much moreaccurate results. MgO is found to be almost fullyionized; aAlz03 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 important in ionic transport in MgA1204.
The interband optical calculation produced results insurprisingly good agreement with recent data. The presence 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 resonant states of the exciton in the CB continuum. Theexperimental data in the threshold region are furthercomplicated by the possible transitions to the chargetransfer bands due to the trace of impurities which always exist in these oxides. However, for absorption inthe energy range beyond the excitonic peak, the agreements 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 OLCAOLDA calculation are reasonably accurate.It is a generally accepted notion that the LDA approximation 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 semiconductors and insulators are always underestimated by the
LDA calculation. Another equally valid argument is thatthe underestimation of the band gap in widegap insulators is due to the incomplete cancellation in the selfenergy terms in the Coulomb and exchange potentials inthe LDA. ' Application of the selfinteraction correction to the bandstructure calculation of LiF increases thegap value considerably. Our surprisingly good resultson MgO, aA1203, and MgA104 may lead to further investigation as to whether this prevailed notion is universally correct, or there are exceptions to this such that theoneelectron LDA calculation of the CB in some systemsmay actually turn out to be more valid than generally believed. It can also mean that the bandgap 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 firstprinciples study ofthe electronic and optical properties of another very important 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 selfconsistent. The selfconsistent OLCAO band structurefor o. quartz was presented some time ago, sho~ing muchimprovement. The selfconsistent band structures andoptical absorptions in all polymorphic forms of Si02 havebeen calculated. The correlation of electronic structureand optical properties to the different shortrange ordersin various polymorphic Si02 turns out to be highly interesting and that work will be published separately.
A final comment we wish to make is that ceramic oxides are usually contaminated with defects and impuritiesand the properties measured are most likely at room temperature or even higher. Our calculation, on the otherhand, is for a perfect crystal at zero temperature.Temperaturedependent optical measurements show thestructures in e2(co) become more broadened and shift inenergy as temperature is increased. Our results showthat a stateoftheart quantummechanical 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 temperatures. 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 Energy under Cxrant No. DEFG0284ER45170.
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