Theoretical Studies of the Electronic Properties ... - Wiley Online Library

Electronic Structure of Ceramics --"-

Theoretical Studies of the Electronic Properties of Ceramic Materials W. V. Ching* Department of Physics, University of Missouri-Kansas City, Kansas City, Missouri 641 10 J. Am. Cerum. Soc.. 73 1111 3135-60 (1990)

The first-principles orthogonalized linear combination of atomic orbitals (OLCAO) method for electronic structure studies has been applied to a variety of complex inorganic crystals. The theory and the practice of the OLCAO method in the local density approximation are discussed in detail. Recent progress in the study of electronic and optical properties of a large list of ceramic systems are summarized. Eight selected topics on different ceramic crystals focusing on specific points of interest are presented as examples. The materials discussed are AIN, Cu20,p-Si3N4,Y203, LiB305,ferroelectric crystals, Fe-B compounds, and the YBa2Cu307superconductor. The results include the band structure, density of states, charge density distribution, spin density distribution, effective charges, total energy and totalenergy-derived results, optical absorption, positron annihilation spectra, and more. Extension of the band theoretical approach to the study of other areas of fundamental ceramic science is also discussed. [Key words: electronic structure, optical properties, atomic orbitals, calculations, local density approximation.]

~

R. H. French-contributing editor Manuscript No. 197624 Received April 23. 1990; approved August 28, 1990. Presented at the 92nd Annual Meeting oi the American Ceramic Society, Dallas, TX, April 23, 1990 (Electronic Structure of Cerarnlcs Symposium, Paper N O 9-SI-90). Supported by the Materials Science Division of the Office of Basic Energy Sciences of the U.S. Department of Energy under Grant No. DE-FG02-84ER-45170. 'Member, American Ceramic Society.

I . Introduction CERAMIC materials have numerous im-

portant applications in modern technology.' These applications can be roughly divided into overlapping categories according to different device functions such as electrical, optical, chemical, mechanical, thermal, magnetic, structural, nuclear, and even biological. Most ceramic materials are insulators, but semiconducting and metallic ceramic materials are also very common. The discovery of hightemperature superconductivityin ceramic oxides2.3 opened yet another very important area of application that has attracted thousands of researchers worldwide. Generally speaking, ceramic materials have much more complicated crystal structures than those of semiconductors, metals, or intermetallic compounds. Defects, impurities, and other forms of lattice imperfections are norms rather than exceptions, Furthermore, many of the applications are actually intrinsically dependent on the crystal imperfections themselves. As in any material, the behavior of electrons in the lattice and their interactions with the environment control the basic material properties. Thus, a fundamental understanding of ceramic materials at the atomic level is not only interesting in its own right, but is extremely important in improving materials performanceand in the search for or design of new materials for new applications. Although classical theories based on empirical or semiempirical means have provided valuable understanding about many properties of ceramic materials in the past, it is obvious that a quantum mechanical treatment is necessary for the study of the behavior of the electrons. The most fundamental property of a given crystal is its electronic structure, or the band structure. Although the band structures of most semiconductors and metals

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Wai-Yim Ching is a Curators' Professor of Physics at the University of MissouriKansas City (UMKC), Kansas City, Missouri. He received his Ph.D. in Physics from Louisiana State University in 1974 and his BSc. degree from University of Hong Kong in 1969. From 1974 to 1978, he was a research associate at the University of Wisconsin-Madison. Ching joined the Department of Physics of UMKC in 1978 as Assistant Professor and was promoted to Associate Professor in 1981 and Full Professor in 1984; he was named a Curators' Professor in 1988 and currently is the chairman of the Physics Department. Ching received the N. T. Veatch Award for Distinguished Research at UMKC in 1986 and was twice the recipient of the University of Kansas City Trustee Faculty Fellowship (1984 and 1990). A member of American Ceramic Society, American Physical Society, American Vacuum Society, and Materials Research Society, Ching has authored or coauthored more than 150 scientific papers. His research interest is in theoretical condensed matter physics, especially on the electronic, magnetic, optical, and structural properties of crystalline and noncrystalline solids. In recent years, his research has concentrated on the fundamental properties of ceramic materials including high-temperature superconductors.

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Journal of the American Ceramic Society-Ching have been extensively studied in the past and are well-known, the electronic structures of many important ceramic crystals have not been studied in detail, or at least not as rigorously as their counterparts in metals and semiconductors. For example, no information about the band structure of yttria ( Y 2 0 3 ) exists, not even at the most rudimentary level. The main reason for the paucity of rigorous quantum mechanical calculations of electronic structures in ceramics is the complexity of their crystal structures, which usually have many atoms in a unit cell and a low symmetry. The majority of the ceramic materials are oxides, carbides, or nitrides. In a planewave-based method, such as the pseudopotential method, the 2p states of 0 or N, which are not orthogonal to any core states, tend to result in a slow convergence of the basis expansion. This means that very large numbers of plane waves must be used, which further increases the computational demand. The low symmetry and the relatively open structure, coupled with a mixed type of bonding in most ceramic crystals also means that a muffin-tin type of approximation for the

Vol. 73, No. 11 crystal potential is less valid. Nevertheless, in the past few years significant progress has been made in the electronic structure studies of ceramic crystals based on the local density functional (LDF)theory,4-6 or simply the local density approximation (LDA).In the past two decades, the LDF theory has been extremely successful in explaining many physical properties of different condensed matter systems including bulk metals, semiconductors, and insulators as well as surfaces and interfaces. In some instances, highly accurate calculations can even predict properties before experimental verification. For example, the existence of several high-pressure phases of Si and their superconducting properties at very low temperature were predicted by the first-principles pseudopotential calculations,7.~ and were later verified by experiments.9 The rapid advance of computing technology has also facilitated largescale calculations on complex structures that were thought to be unrealistic not too long ago. This is best exemplified by the fact that soon after the structures of the newly discovered hightemperature superconductors were deter-

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Theoretical Studies of the Electronic Properties of Ceramic Materials

mined, their electronic structures were calculated immediately by different research groups using different state-of-the-artcomputational methods.~0-~4 In this paper, we will discuss the results of the electronic structures of different classes of inorganic crystals calculated by the first-principles orthogonalized linear combinations of atomic orbitals (OLCAO) method. For the reasons to be presented later, the OLCAO method has been particularly effective in giving highly accurate self-consistent band structures of complex ceramic compounds. The background information on the theory and practice of such a first-principles calculation of properties is outlined in five separate panels. The panels describe the LDF theory (I), the OLCAO method (II), physical quantities from band results (Ill), optical properties (IV), and total energy calculation o/). For the more introductory explanation of the

band theory, we refer to an excellent feature article by R. H. French.15 In the Section 11, we will review recent theoretical results on the electronic structures of ceramic compounds obtained by the OLCAO method. Because of the large number of crystals that have been studied, only a few selected examples focusing on a specific property or points of interest can be presented. Section Ill summarizes the strengths of the OLCAO method as applied to ceramics and emphasizes some existing theoretical difficulties that need to be addressed. In addition, future directions of research in ceramic materials using the band theoretical approach and the OLCAO method are discussed.

II. Review of Some Recent OLCAO Results In this section, we will review the results

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on the electronic structures of ceramic compounds obtained by using the OLCAO method in the past few years as well as the work currently in progress at the University of Missouri-KansasCity. In reviewing these results, we will loosely divide the materials into five groups which are listed in Table I: simple ceramics, complex ceramics, nonlinear optical materials, ferroelectric crystals and magnetic compounds, and high-temperature superconductors and related crystals. The definitions for each group are overlapping and very subjective. For example, many ferroelectric crystals are also nonlinear optical materials and the superconducting oxides are all complex ceramics. To facilitate discussion and tracing of publications, each material is also listed under four different headings: published or in press, research completed and manuscript in preparation (or unpublished results), research in progress, and targeted for study. Citation refer-

ences are given to all published or inpress entries. For a crystal with different phases, a small letter in parentheses follows to indicate the particular phase studied (c, t, m, h, and o represent cubic, tetragonal, monoclinic, hexagonal, and orthorhombic phases, respectively). If a system is listed under two different headings, it means that either the system has been recalculated for more accurate results or for additional information or a preliminary result has been published and a more detailed paper is being prepared. Most of the studies include the optical properties.Spin-polarizedcalculation has been conducted only with Fe-B compounds,87 but spin-polarizedcalculations have been successfully conducted on SczCu04by another group using a similar method.48 All the results described here are nonrelativistic. It is quite impossibleto give even a very brief account of each of the materials listed in Table I. Instead, we will select one

Table I. Electronic Structures of Some Ceramic Crystals Studied by the OLCAO Method Published or in press

Research completed

Research in progress

Research contemplated

Simple ceramics

Wurtzite: AIN (Ref. 47) Others: CuzO (Ref. 87), CuO (Ref. 88)

GaN, SIC, ZnO, ZnS, Be0 MgO, CuCI, 8203, L120,BN(c), BN(h)

a-Si02 (Ref. 93), other SiOz (Ref. 93) Stishovite (Ref. 93), a-Si3N4 (Ref. 92), p-Si3N4 (Refs. 93 and 94) Y203(c) (Ref. 97) Zr02(c,f,m) (Ref. 82)

a-SiO2, other SiOz Stishovite, /.-Si3N4 a-Al203, MgA1204 voz(m), VOe(t), vo, v 2 0 5

LiB305 (Ref. 101)

KTiOP04 (KTP)

CaFz, CaO TiOn

CUO AIP04 AlON

Complex ceramics

812, B5o Si2N20,a-Si3N4 ZrOz (c,t,m), other V oxides

Nonlinear optical materials

CSB~O~ KTiOAsO,, KH2P04

Ferroelectric or magnetic crystals

SrTi03(t) BaTi03(t) KNb03(t)

SrTiOz(c) (Ref. 105) BaTi03(c) (Ref. 105) KNb03(c) (Ref. 105) FeB, FezB, Fe3B (Ref. 87) High-temperature superconductors and related crystals

YBa2Cu307(123) (Refs. 12 and 13)

YBazCU306, La2cUo4

Substitution of 0 by F (Ref. 109) Substitution of Cu(1) by Zn (Ref. 110) Substitution of Cu(2) by Zn (Ref. 110) Substitution of Cu(1) by Ga (Ref. 110) Substitution of Cu(2) by Ga (Ref. 110) Substitution of Y by Zn (Ref. 110) Bi2Sr2Cu06(Ref. 50)

Y BaCu05

Bi~Sr2CaCu~O8 (Refs. 50 and 51) TIzBazCU06 (Refs. 50 and 52) Tl2Ba2CaCu208(Refs. 50 and 52) T1ZBazCazCu3010 (Refs. 50 and 52)

YBa2Cu408, Sr2Cu02CJ2 SrV02 Sr2V04

La2Ni 04

YBazcU306.75 YBazCU306.50 YBazCU306.z5

V substitutied in Bi-2212 TIBa2Cu05 TIBazCaCu2O7 TIBazCa2Cu3O9 TIBa2Ca3Cu401

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or two representative samples from each category to illustrate a particular point or points of interest either with regard to the physical quantities that can be obtained or in comparison with experiments. For more detailed discussions the published literature may be consulted. (1) Total Energy Calculation in AIN The application of the OLCAO method to the total energy calculation so far has been quite limited. In the case of Si, very accurate results have been obtained for 10 different phases,80 as described in Panel V. For ceramics, the calculation has been limited to a few cases with simple crystal structure, such as AIN with the wurtzite structure.47 Figure 2 shows the calculated total energy versus volume for AIN from which we obtained the equilibium lattice constant equal to 0.9994 of aexp,a bulk modulus 1.025 of ,E,l and its pressure derivatives in very good agreement with experiments. The smoothness of the data points on the EAV) curve indicates the stability of the calculation. Also calculated is the zone center optical phonon frequency and its pressure dependence.47This is done by calculating the total energy of the crystal as a function of the relative displacement amplitude u of the Al and N ions (with the center of mass position fixed) along the c direction. The ET(u) curve shown in Fig. 3 is fitted to a polynomial of order 4 and the vibrational frequency is extracted from the harmonic term. The process is repeated for different volumes (pressures) of the crystal to give the pressure dependence of this particular phonon frequency. We have obtained the A1 transverse optical phonon frequency of 19.60 THz as compared with the measured value of 19.77 THz. The calculated and measured pressure derivative for this phonon are 4.55 and 4.88 cm-lIGPa, respectively. A total energy calculation also has been attempted for the three different phases of Zr02-the monoclinic, tetragonal, and defect-stabilizedcubic phases-which are structurally much more complicated. This work is still in progress and will be reported elsewhere.@

(2) Optical Properties of Cu20 The band structure of C u p 0 calculated by the OLCAO method88 is shown in Fig. 4. The semiconductor band is characterized by a direct gap at r of 0.78 eV. This is only 35% of the often-quoted experimental value of about 2.2 eV.89 One would like to use this as an example of the failure of LDA in obtaining the correct band gap. However, a closer analysis of this problem shows that it is not that simple. Cu20 is the paradigm case of exciton formation in crystals and an enormous amount of literature exists on this subject.90 The experimental band gap is obtained from the fitting of the excitonic hydrogenic series to the exciton-

Fig. 1. Total energy versus volume for 10 different phases of Si: diamond, hexagonal (hex), body-centered cubic with 8 atoms per cell (bbc a), p-tin (b-tin), simple cubic (sc), simple hexagonal (sh), hexagonal close packed (hcp), body-centered cubic (bcc), and face-centered cubic (fcc). Data for St-12 (a hypothetical tetragonal phase with 12 atoms per cell) is not shown. See Ref. 80 for details.

Fig. 2. Total energy versus crystal volume in AIN. V, is the equilibrium volume of the cell.

Fig. 3. Total energy versus AI-N distances of separation (in units of the lattice constant c) in AIN for optical phonon calculation.


3 142

8

4

v

A P a,

0

!=

w

-4

-8

I

r

I

I

X

M

I

r

I

I

R

X

Wave vector Fig. 4. Band structure of CupO along high-symmetry lines of the BZ. Energy for the top of the valence band is set at zero. tnset shows the BZ of a simple cubic cell.

Vol. 73, No. 11 ic binding energy, and it is assumed that the band gap should be close to the major (n = 1) excitonic peak.89 lt is also taken for granted that the formation of the exciton probably does not significantly affect the absorption edge. To understand the band gap problem in CupO more clearly, the interband optical conductivity was calculated.The calculated real and imaginary parts of the dielectric function determined by the K-K conversion of u1 are shown in Fig. 5. Several interesting points merit comment. First, although the calculated intrinsic gap is only 0.78 eV, the optical absorption at r is symmetry forbidden, and, therefore, the absorption intensity becomes appreciable only for photon energies above 3.2 eV. At that energy, transitionsfrom the top of the valence band (VB) to the second conduction band (CB) become possible. This fact is not likely to change even if we go beyond the present LDA or improve the accuracy of the band calculation. Therefore, if we accept the current experimentalvalue of 2.2 eV as the intrinsic band gap for Cu20, then the optical absorption threshold is likely to be larger than 4.5 eV. All the absorption structures in the ~ ~ ( curve in Fig. 5 have to be shifted about 1.5 eV toward the higher photon energy, thus makina the absordion edae at about 4.7 eV. Thk is incompatible i i t h optical

0 )

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absorption data in the ultraviolet (UV) and vacuum ultraviolet (VUV) regions.91Second, the calculated static dielectric con, the effect of stant ~ ~ ( or0 E) ~ ~(without phonons) is very low, about 3.8. This will affect the exciton binding energy and the spectral analysis since the experimentally used value for E~ in CupO is about 7.8. To reconcile the above differences with the existing interpretation of the experiment, we conjecture that the exciton in Cu20 may be formed between a hole at the top of the VB and an electron at the minimum of the second CB, instead of the lowest CB. By using the estimated CB and VB effective masses to obtain the excitonic effective mass, we find that the analysis of the excitonic series is still con-

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sistent with the experimental data.88 Thus, we believe that, in the case of CunO, the band calculation gives the intrinsic gap, whereas the often-quoted experimental band gap is the optical gap to the second CB related to the absorption threshold. These two gaps are clearly not the same. Our reinterpretation of the excitonic spectra in CupO is presently only a conjecture and more theoretical work and experimentation in Cu20 are needed to resolve this puzzle.

(3) Electronic Structure and sonding in /3-Si3N4 The band structures of /3-Si3N4were first studied by the OLCAO method in 1981.92 Similar calculations also have

Fig. 5. Real (solid line) and imaginary (dashed line) parts of the dielectric function in Cu20.

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Journal of the American Ceramic Society-Ching been conducted for all polymorphic forms of Si02.93However, these calculation were not self-consistent. The selfconsistent OLCAO band was obtained in 1987 with some preliminary results on total energy calculations81 and the VB charge distribution.94 This band for /j-Si3N4is shown in Fig. 6. An indirect band gap of about 5 eV was obtained. The bottom of the CB is at r, but the top of the VB is at a point along r A (0.4 r A from r). The electronic charge distribution for P-Si3N4is shown in Fig. 7. It shows that the distribution is not spherical around either the Si or the N atom. There are significant amounts of charge residing on the open interstitial regions of the crystal. The bonding in Si3N4can probably be described as partially ionic and partially covalent.

Vol. 73, No. 11 Figure 8 shows the interband optical conductivity of p-Si3N4for photon energy up to 17 eV. We found no optical data on single-crystal /j-Si3N4, but data for amorphous Si3N4exist95 and are plotted in Fig. 8 for comparison. Since the shortrange order in amorphous and crystalline Si3N4are expected to be similar, their optical spectra are probably also very similar except for the fact that prominent structures may be washed out in the amorphous case. There is excellent agreement between theory and experiment in Fig. 8 as far as the general shape and the intensity profile of the absorption are concerned. Of special interest is the fact that the calculated and measured absorption thresholds are almost equal, about 6.0 eV. The calculated intrinsic band gap is only 5.0 eV. This is partly due

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to the fact that the optical calculation does not include phonon-assisted transitions and only direct transitions with the same k vectors are considered. However, the direct gap at r is about 5.3 eV. Thls is another example, as In Cu20, that the experimentally determined optical gap may differ considerably from the calculated intrinsic gap. A preliminary calculation of the total energy of p-Si3N4was presented earlier.8’ Satisfactory values of the equilibrium volume and bulk modulus were obtained despite the complexity of the crystal structure. Our calculated structural properties for p-Si3N4are in very good agreement with a very recent first-principles pseudopotential calculation.96We are currently attempting to improve the accuracy for the total energy of P-Si3N4and attempting a total energy calculation for the aSi3N4phase.

close to the fully ionic picture of YZ+O$-, Actual integration of the charge according to the distribution shown in Fig. 11 gives effective charges of 0.80 and 7.38 electrons for Y and 0, respectively, with 1.70 electrons per primitive cell (0.7%) in the interstitial region. If we divide the interstitial charge equally between Y and 0 atoms, the bonding in Y is likely to be described by the formula Y:.16+OA.44In Fig. 13, the calculated optical conductivity a and the JDOS are compared. Figure 13 shows that there are considerable differences between these two spectra, both in the peak positions and in their relative intensities. These differences indicate the importance of including matrix element effects explicitly in the optical calculation. The calcluated optical spectrum is in reasonable agreement with measure-

20

(4) Electronic Structure and Bonding

in Y203 Y203 is a useful ceramic oxide with an increasing number of applications. Very little is known about its electronic structure and bonding. We have recently applied the first-principles OLCAO method to study the band structure and the optical properties of single-crystal Y203.97 The calculated band and total DOS are shown in Figs. 9 and 10, respectively.Because the Y2O3 crystal contains 16 molecules in the cubic cell, the band structure is very complicated. The VB consists mostly of 0 2p orbitals, is about 3.8 eV wide, and has a total CB width of about 5.8 eV. The CB which results exclusively from the Y 4d orbitals is divided into two pieces separated by a gap of about 0.3 eV. This separation of the CB is related to the orbital symmetry of the d states and the crystal field in Y203. We find that most of the lower CB is from the Y 4dx2- and 4d3z2-p orbitals. the Y 4dxy,4&, and 4dzxstates are distributed rather evenly between the two pieces. The most interesting part of the electronic structure of Y2O3 is the rather peculiar type of bonding as illustrated by the charge density map shown in Fig. 11. Well-defined spherical regions of charge distribution for both Y and 0 with ionic radii of approximately 0.88 and 1.27 A (1 A = 0.1 nm), respectively, are discernible. (The ionic radii are determined by visual inspection of the contour map with the constraint that the total integrated charge in the spherical approximation be conserved.) However, within the Y sphere of charge, there is a ringsofzero charge distribution at about 0.3 A from the Y nucleus. This is more clearly shown in the linear plot of charge density along the Y - 0 bond shown in Fig. 12. This ring of zero charge in the Y atom is due to the coincidence of the nodes in the atomic Y 4d and 5s wave functions. If 0.3 A is used as the ionic radius of Y, then Y has very little charge remaining, and Y 2 0 3 is

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15

10

5

2 v

x o PJ a, c

Lu

-5

-10

-15

-20

+ r

I

I

I I

I I

I I

K

H

A

r

I

I

M

I

I

L

A

Wave vector Fig. 6. Band structure of /3-Si3N4along high-symmetry lines of the BZ. Energy for the top of the valence band IS set at zero. Inset shows the BZ for a hexagonal cell.


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Vol. 73, No. 11 ments on single crystals98 in terms of the overall shape and structure.The interpretation of the observed sharp peak at 6.0 eV as an excitonic peak is supported by our calculation because there is no such peak in ~ ( c o )On . the other hand, the calculated q(o)near the absorption edge is in closer agreement with the measurement on sputteredfilms,99 showing a twostep absorption increase near the edge. We refer the readers to Ref. 95 for more detaiIs.

Fig. 7. Valence charge density distribution in /?-Si3N4Contour lines are in units of electrons per the cube of atomic units Large circular contours enclose the N atoms and the smaller circles enclose the Si atoms

10

8 h

r

b 6 LD

0

2 0

I 0

2

4

6

8

10

12

14

16

Energy (eV) Fig. 8. Optical conductivity of fi-Si~N4.Data points are from Ref, 95 for amorphous Si3N4.

(5) Band Structure and Dielectric Function of LiB305 LiB30sis an inorganic crystal with excellent nonlinear optical properties.100 The discovery of the crystal was claimed to be guided by theoretical considerations.100 The crystal is fairly complex with four formula units per orthorhombic cell. The interesting part of the structure is that there are two types of B atoms, one threefold bonded and the other tetrahedrally bonded to the 0 atoms. The structurecan be more or less viewed as a B-0 network with Li atoms loosely situated at the open interstices of the network. We have used the OLCAO method to calculate the band structure and the DOS of this crystal,lOl which are shown in Figs. 14 and 15, respectively. A direct band gap of 7.8 eV at r is obtained, which is only slightly smaller than the reported experimental value.loOBoth the band structure and the DOS show many small mini-gapsamong the highly localized bands. This is typical of the electronic structure of a molecular crystal. The B-O3 or B-O4 units in LiB305 are rather localized units within the crystal. An effective charge calculation using simple Mulliken analysis shows a charge transfer from the B to the 0 atoms.101The average effective charge on the B, 0,and Li atoms are 2.65, 6.22, and 0.94 electrons, respectively. The fact that the Li atom has retained about 94% of its valence charge is consistent with the localized network-typeband structure discussed earlier. The Li atoms are basically not much affected, since they are located at the open interstitial areas. However, these effective charges were calculated using Mulliken analysis, which is not very accurate for ionic systems. A more accurate direct-space integration scheme as in Y203 may result in a more ionic picture for Li. The complex dielectric function of LiB30shas also been obtained from the optical calculation outlined in Panel IV. The 4 w ) curve up to 20 eV is shown in Fig. 16. The major structures in this curve compare well with a recent preliminary optical measurement using VUV techniques with a laser plasma light source.1Oz In Fig. 17, we show the three Cartesian components of the real part of ~ ~ ( in the 0 ) ) UV region obtained from the E ~ ( w of Fig. 16 by the K-K transformation. In this energy region, effects of lattice vibrations can be totally ignored; therefore,

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~?(w)=n;(w), where q indicates a cartesian component. The square of the measured refractive index data for q =x,y,z are shown in Fig. 17 for comparison. Note that the small anisotropy in the measured refractiveindex100 is faithfully reproduced by the calculation, as is the general frequency dependence of the data. The theoretical data are slightly above the measured data. This is accounted for by a slight deviation in the calculated band gap from the measured one. On the whole, the agreement between the measured and calculated optical properties is quite impressive. A similar calculation on another important nonlinear crystal KTiOP04 (KTP) has also been completed.103 The crystal structure of KTP is even more complicated. The orthorhombic cell contains a total of 68 atoms. Our calculation shows that the anisotropy in the refractive index for KTP is smaller than that for LiB305.

6

4

2 h

2 v

>r C

W

(6) Ferroelectric Crystals Ferroelectric crystals constitute a unique class of ceramic materials with many important applications in modern technology.104 Ferroelectric crystals display intrinsic electric polarization whose direction may be reversed by an external electric field. The simplest ferroelectric has an AB03 cubic perovskite structure which can be easily visualized as having the B atom at the center of the cube, the A atom at the corner of the cube, and the 0 atoms at the midpoints of the edges of the cube. Transformations of the cubic phase to the tetragonal, rhombohedral, and orthorhombic phases that take place at lower temperatures are very common in most ferroelectric crystals.104

-4

-2

0

2

3 147

0

-2

-4

I

H

I

N

I

I

I

r

P

N

Wave vector Fig. 9. Band structure of Y203 along high-symmetry lines of the BZ. Energy for the top of the valence band is set at zero. Inset shows the BZ of a body-centered cubic cell.

4

6

8

Energy (eV) Fig. 10. Total DOS of Y p 0 3 ,

Fig. 11. Contour map of valence charge distribution in Y203 on a plane containing the Y - 0 bond Contour lines are from a minimum of 0 01 to a maximum of 0 25 (in intervals of 0 005) in units of electrons per the cube of atomic units Two large circles enclose 0 atoms with a Y atom in between 0

2

5

8

11

14

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Journal of the American Ceramic Society-Ching We have recently used the OLCAO method to study the band structures and optical properties of SrTi03, BaTi03, and KNb03 in the cubic perovskite structure.105 BaTi03 and KNb03are ferroelectric and SrTi03 has also been suspected to be ferroelectric.1" The calculated band structure and the DOS for SrTi03 are shown in Figs. 18 and 19. Note that all ferroelectric crystals in the AB03 structure have very similar band structures characterized by a direct band gap at r and a split CB. The splitting of the CB into upper and lower parts is indicative of a strong crystal field which splits the d-orbital states of the A atom into r25and r12 states. The VB is mostly from the 0 2p with slight mixing of B and A orbitals. The highly ionic nature of the bonding in SrTi03 is illustrated in the charge density contour map shown in Fig. 19 for the SrTi03 crystal in the [I ,1,O]plane. Note that the central Sr atom lost almost all of its valence charge while the Ti ion retained some of its charge. A certain degree of covalent binding between Ti and 0 atoms can be argued. The calculated E ~ ( wcurve ) for SrTiO3 (Ref. 105) is shown in Fig. 20 with the optical data of Cardona.106 The agreement between the calculated and the measured spectra is very satisfactory both in the peak positionsand their relative intensities. This good agreement in optical spectra is indicativeof the accuracy of the calculated band structure.

(7) Spin-Polarized Calculation for the Fe-B Compounds The electronic structures of ferromagnetic FeB, Fe2B,and Fe3Bcrystals were recently studied by the spin-polarized OLCAO calculation.85 We present some limited results to illustrate the capability of the OLCAO method in studying the

Vol. 73, No. 1 1 magnetic properties of ceramic materials. In the spin-polarizedcalculation,the spinup and spin-down potentialsare coupled through the spin-dependentXC potential. The spin-up states correspond to the case in which the spins are aligned to the internal molecular field (assumed to be in the z direction). The spin-down states have spins pointed in the opposite direction. To start the calculation, the initial spin-up and spin-down potentials are given an artificial exchange splitting and the Schrodinger equation is solved for each spin case. The Fermi level is determined by summing both the spin-up and spindown bands to give the total number of electrons. The total charge density p(r) is the sum of the spin-up and spin-down charge densities and their difference gives the spin density ps(r). Spinpolarized band structure calculationscan be easily incorporated within the OLCAO method.48 The crystal potential is constructed according to the local spin density formalism. There are several popular spin-dependent XC potential forms. The most commonly used form is credited to von Barth-Hedin20 as modified by MoruzZI et al.707 A spin-polarizedcalculation is also necessary for the inclusion of relativistic spin-orbit interactions.49 Figure 21 shows the spin-projected DOS (positive values for the spin-up band and negative values for the spin-down band) for FeB, Fe2B, and Fe3B compounds. Also included is the result of bcc Fe. Spin magnetic moments at each site are obtained by resolving the DOS into the PDOS of the atoms and taking the difference between the spin-up and the spindown components. Our calculation shows the average moments per Fe site to be 2.15, 1.97, 1.95, and 1.26 & for Fe, Fe3B,Fe2B,and FeB crystals, respectively. The moments on the B atoms are

h

0.0 0.4 0.8 1.2

1.6

2.0 2.4

r(Y-0) (A) Fig. 12. Distribution of valence electron charge along a Y - 0 bond in Y$&. Y atom is at 0.0 A and the 0 atom is at 2.25 A.

0

2

4

6

8

10

Energy (eV) Fig. 13. Comparison of the calculated optical conductivity u (lower curve) and the JDOS (upper curve) in Y 2 0 3 For comparison wlth experimental data, see Ref 97

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opposite to those of Fe with values - 0.29, - 0.23, and - 0.10 p~ for Fe3B, Fe2B, and FeB, respectively. Thus, the average Fe moments in the Fe-B compounds do not necessarily scale with B concentration. Rather, it depends on the local atomic arrangements in each crystal.87 Figure 22 shows the contour maps of the spin density ps(r)=pt(r)-p&(r) of the three crystals on certain crystallographic planes which contain both Fe and B atoms. The negative contours shown by dotted lines around the B atoms are not spherical, whereas the positive contours of the Fe spin density are spherical. For antiferromagnetic crystals, the spin moments on the nearest-neighbor magnetic atoms are antiparallel. Spinpolarized calculations with antiferromagnetic crystals are usually more difficult than with ferromagnetic crystals because the magnetic unit cell may be 2 or even 4 times larger than t h e crystal unit cell. It is of interest to study the optical properties of magnetic crystals based on spin-polarizedband structures. By including the spin orbit interaction, one would be able to explore the magnetooptical properties of many interesting ceramic materials. Such a calculation requires highly accurate wave functions and the XC potential and has not been attempted with the OLCAO method. The spin wave stiffness constant in ferromagnetic Ni has been calculated using spinpolarized linear combinations of GTO wave functions including spin-orbit interaction.108 The result agrees very well with the experiment. (8) Electronic Structure of the YBa,Cu,O, Superconductor The OLCAO method was one of the first to give an accurate band structure of YBa2Cu307 (1 23) superconductor shortly after its discovery.12,13 Since then the method has been applied extensively to the study of various chemical substitutions in 123 (Refs. 109 and

I

r

I

r

S

X

z

Wave vector Fig. 14. Calculated band structure of LiB305along high-symmetry lines of the BZ. Energy for the top of the valence band is set at zero. Inset shows the BZ for an orthorhombic cell

4

3 h

-32 W"

1

0

-20

-15

-10

-5

0

Energy (eV) Fig. 15. Calculated DOS for LiB305.

5

10

10

0

5

10

15

20

Energy (eV) Fig. 16. Calculated imaginary dielectric function ~

~ ( for0LiB305. )

3150

Journal of the American Ceramic Society-Ching 110) and other high-T, ceramic superconductors5@52and related compounds. The materials that have been studied by the OLCAO method are listed in Table I. In this section, we restrict our discussion to the 123 compound exclusively. To facilitate discussion, the orthorhombic crystal structure of the 123 compound with 13 atoms in the unit cell is sketched in Fig. 23. It shows that the Cu(1) and 0(1) form Cu-0 chains and that Cu(2), 0 ( 2 ) , and O(3) form puckered Cu-0 planes. O(4) is in the apical position between the

Vol. 73, No. 1 1 Cu-0 chains and the Cu-0 planes. The Ba and Y atoms are located in interstices between the Cu-0 planes. There is now sufficient experimental evidence to show that superconductivity in the 123 compound is associated with the Cu(2)-0(3) plane. The band structure of 123 as calculated by the OLCAO method is shown in Fig. 24. Figure 25 shows the same band structure near the Fermi level along different symmetry lines of BZ shown in the inset. The details of these bands have been discussed widely in the published

10

8

6 4 I

I

I

I

I

1

2

3

4

5

$2

Energy (eV)

v

Fig. 17. Comparison between calculated d ( (solid line), &(a)(short dashed line), and &(W) (long dashed line) with the experimental data of n;(o), n; (0), and n$ (0) from Ref 100 for LiB305 crystal in the 1 to 5 eV range

~)

-2 -4

-6

r

r

M

X

X

R

Wave vector Fig. 18. Band structure of SrTi03 in the cubic perovskite structure along high-symmetry lines of the BZ. Energy for the top of the valence band IS set at zero Inset shows the BZ of a simple cubic cell

7 38

10

5 53

w”5

8

3 69

0 1 84

0

5

10

15

20

Energy (eV) Fig. 20. Calculated imaginary part of the dielectric function (solid line) Q(O) for SrTi03 Experimental data from Ref 106 (dashed line)

0 00 0 00

2 61

5 22

7 83

1044

Fig. 19. Charge density distribution on (1 101 plane of SrTi03. Sr ion is at the center and Ti ions are at the corners. 0 atoms are on the vertical lines between Ti atoms. Contour lines are from a minimum of 0.02 to a maximum of 0.25 in intervals of 0.05 electronkell.

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literature and thus will not be repeated,12.13 but we wish to emphasize three points: (1) Although a Fermi surface is present, there exist intrinsic hole states (4 holesicell) above the Fermi level with a significant gap separating a semiconductor-like VB and the CB. (2) The band structure is highly anisotropic and, therefore, two-dimensional in character. The bands along the k, direction are very flat, thus having large effective masses. There are four bands crossing the Fermi surface. (3) The overall occupied band width is 7.6 eV, and the unoccupied portion of the top of the VB is about 1.8 eV. These and other aspects of the band structure have been frequently used to argue for or against certain models of the mechanism for superconductivity, but no consensus has been reached thus far. For example, one of the suggested models is the excitonic enhancement model (EEM).111 In this model, all highT, oxides have either a semiconductorlike or a semimetal-like band structure with intrinsic holes at the top of the VB, such as those shown in Figs. 24 and 25. The EEM argues that these holes are missing electrons with a negative mass. By inverting the band, the system can be viewed as having p-type fermions filled from the top of the VB up to the Fermi level (with a "band" width of 1.8 ev), in anal-

ogy to the metallic superconductor in which the fermions are electrons. High T, is achieved by the simultaneous superconductivity and excitonic condensation of charge nonneutral excitons to form a collective long-range ordering. The charged excitons which are formed between the intrinsic holes and the electrons in the CB are due to strong Coulomb attrraction and are highly two-dimensional in character because of the crystalline anisotropy. Although this model has met with much skepticism, recent theoretical work on the exact solution of the twodimensional H atom problem112,113 seems to indicate that the conditions set by the EEM mechanism are not in violation of any fundamental physical principles. In any case, the band structure results are consistent with many of the measured normal-state properties, and they often provide much-needed insight concerning the superconducting mechanism in high-T, oxides. Figure 26 shows the total DOS, which is in reasonable agreement with recent photoemission experiments.114 In Figs. 27 to 30, the PDOS of Y, Ba, Cu(l), Cu(2), O(l), 0 ( 2 ) , 0 ( 3 ) ,and O(4) atoms in the cell and their orbital partial components are shown. This is probably the most detailed analysis of the DOS of the 123 compound.115 These PDOS diagrams are extremely useful in interpreting various experimental data involving electron-

4 2 0

-2

-4 25 h

f a

0

0

. L

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-25

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10

*

5

v)

8

0

-5 -10

10

5 0

-5 -1 0 -15

-10

-5

0

5

Energy (eV) Fig. 21. Spin-projected partial DOS in (a) Fe, (b) Fe3B,(c) Fe2B, and (d) FeB Upper panel for the spin-up band and the lower panel for the spin-down band

A

B

c

Fig. 22. Spin density distribution &(r) on various crystalline planes foi' (a) FeB, (b) FenB, and (c) Fe3B.Contours are from 0.002 to 0.04 in units of 0 001 (at units)-3 Broken lines indicate negative values Positive contours surround Fe atoms, whereas negative contours surround the B atoms


3152

a Fig. 23. Crystal structure of YBanCu307

ic states near the Ferrni level. As shown at the Fermi level, Cu(2) contributes more than Cu(l), and O(4) has the biggest contribution among the four types of 0. This leads us to believe that the Cu(2)-0(4) bond probably plays a rather significant role in properties related to the Fermi level.ll5 Based on the previously calculated band structure,13 the optical properties of 123 have also been studied. The direction-resolved optical conductivity curves are presented in Fig. 31, Of particular significance is the high anisotropy of the optical conductivity in the lowenergy region (

0.5

-5

h

cn .

2 0.c a

c (II c

I

c

1.c

0.0 1.0

(d

> 0.5 a

-2.0.:

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g

2 1.0 (d

cn. 0.0

cn

-

0s

c

cn

1 .c

0.5

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0.:

1

O.(

0-5

1 .(

0 .c 1 .c

0 .t

0.6

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0.5 0 .c

0.1 -8

-4

0

4

Energy (eV)

8

-8

-4

0

4

8

Energy (eV)

Fig. 27. Site-decomposed PDOS in YBa2Cu307and their orbital components. Left panel, Y, and right panel, Ba.

Vol. 73, No. 11 tion by extracting important physical parametersfrom the first-principlescalculations; other physical observables such as angle-resolved photoemission, X-ray absorption, positron annihilation, etc; realistic calculation of crystal fields;133 new structures and new materials predicted by total energy and force calculations;134 and problems associated with surfaces, interfaces, thin films, and artificial structures. Ceramic materials are diversified and complex. They can be categorized in different ways according to either structures, properties, or composition. Phase diagrams have played a central role in the processing and characterization of ceramic materials.Often, the end members of a phase diagram consist of simpler and better-knownceramic crystals, and the interior of the diagram contains the more complicated phases by mixing the appropriate amounts of the basic ones. One such phase diagram is shown in Fig. 34 as an illustration.135 The four end members in the Si-Y-0-N system are SiOn, Y2O3, YN, and Si3N4.Complex phases such as Y2Si207and Y4Si207N2reside on the edge and in the interior of the phase diagram. It is important to make systematic studies of the basic ceramic crystals which form the backbones of the more complex phases. Some of these ceramic phases are already on the list in Table I; therefore, the study of their electronic structures are either completed or in progress. Although the understanding of basic ceramic structures provides insight into more complex phases, there is no guarantee that a full account of the electronic structures of the complex phases can be obtained from the end members alone. For example, our recent study138 of the electronic structures and optical properties of MgO, a-Aln03,and MgAI2O4 using the OLCAO method shows that the density of states (DOS) of the spinel phase MgAI2O3cannot be approximated as a simple superposition of those of MgO and a-Aln03.To fully understandthe properties and the energetics of each phase and their relationship to the structures and compositions, each crystal must be studied individually by means of first-principlescalculations.Given the fact that considerable progress has already been made and that advances in the computing technology continue to appear, it is not unrealisticto launch an ambitious project of a full-scale investigation of the electronic structures and related properties of all technologically important ceramic materials. This will generate an important knowledge base in the ceramic sciences which has been overlooked in the past.

November 1990


Acknowledgments: I thank my colleagues, Drs. Y.-N. Xu, G.-L. Zhao, Xue-Fu Zhong, Z. Q. Gu, F. Zandiehnadem, B. N. Harmon, D. E. Ellis, K. W. Wong, and Chun C. Lin with whom I have collaborated over the years on different aspects of the theoretical calculation. I also acknowledge many helpful discussions and fruitful collaboration with many experimental colleagues, especially Drs. R. H. French, Y. C. Jean, J. Parker, D. J. Lam, and C. R . Aita. Special acknowledgment to Dr. YongNian Xu, who has been involved in most of the recent calculations. I also thank Professor D. L. Huber for a critical reading of this manuscript.

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3155

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82

g

3

%

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.o

.o

.o

o

a


3156

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6

6 Total O(4)

Total 0(1)

E

h

4

4

2

2

0 0.5

0 0.5

0.0 1.5

h h

E

0

0

c

m

1.0

2 7

0.5

c

m

v)

a,

al

m +

2 (I)

0

n

1.0

>, 7 0.5

v)

c

0.0 1.5

-I-

m

0.0 1.5

t .

2

cn

0

1.0

n

0.0 1.5 1.0

0.5

0.5

0.0 1.5

0.c 1.5

1.o

1 .c

0.5

0.5

0.0

0.C -8

-4

0

Energy (eV)

4

8

-8

-4

0

4

8

Energy (eV)

Fig. 29. Site-decomposed PDOS in YBa2Cu307and their orbital components. Left panel, 0(1), and right panel, O(4).

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Theoretical Studies of the Etectronic Properties of Ceramic Materials

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6

6 Total O(3)

Total O(2)

E

h

4

4

2

2

0 0.5

0.5

0

0.0

E

1.5

h

0

c

a

2 y

-

cn a,

c

m 2 v)

0

n

0.0 1.5

0

c

m

1.0

1.0

2 0.5 == cn

0.5

a,

c

0.0 1.5

c

0.0

2

1.5

1.0

c0n

1.0

0

"

0.5

0.5

0.0 1.5

0.c

1 .a

1 .c

0.5

0.5

0.c

0.c

1.5

-8

-4

0

Energy (eV)

4

8

-8

-4

0

4

8

Energy (eV)

Fig. 30. Site-decomposed PDOS in YBa2Cu30, and their orbital components. Left panel, 0(2), and right panel, O(3).


3158

74K. H. Weyrich, "Full-Potential Linear Muffin-Tin Orbital Method," Phys. Rev 8, Condens. Matter, 37, 10269-82 (1988). '5M.-H. Tsai, J D. Dow. and R. V. Kasowski, "Local-Density-Pseudofunction Theory of Bulk Si," Phys. Rev. 8, Condens Matter, 38,2176-78 (1988) 76H. J. F. Jansen, M Weinert, E. Wimmer. and A. J. Freeman, "Total Energy Full-PotentialLinearized Augmented-PlaneWaveMethod for Bulk Solid: Electronic Structure Properties of Tungsten," Phys Rev 5, Condens. Matter. 30, 561-69 (1984) 77B.Delley, A. J. Freeman, and D. E. Ellis, "MetalMetal Bonding in Cr-Cr and Mo-Mo Dimers: Another Success of Local Spin-Density Theory," Phys. Rev. Lett, 50, 488-91 (1983). 78J Hafner and W. Weber, "Total Energy Calculation for Intermetallic Compounds with a FirstPrinciples Linear Combination of Atomic Orbitals

5 h 7

in z0

4

v -c

=-

c

> .c

3

0

$

C

2

0 0

.0 c

1

0" O

I

I

0

1

I 2

I

I I -

3

4

5

6

7

8

Energy (eV) Fig. 31. The x , y.and z components of the calculated oDtical conductivity in YBazCu307

Vol. 73, No. 11 Method." Phys Rev. 8, Condens. Matter, 33, 747-54 (1986). 7% L. Zhao. T. C. Leung, B. N Harmon, M. Leil, M. Mullner, and W. Weber, "Electronic Origin of the Intermediate Phase of NiTi," Phys. Rev. B, Condens. Matter, 40, 7999-8001 (1990) 8oF. Zandiehnadem and W. Y. Ching, "Total Energy, Lattice Dynamics, and Structural Phase Transition of Si by the Orthogonalized Linear Combination of Atomic Orbitals Method." Phys. Rev. 6, Condens. Matter, 41. 12162-79 (1990). 8 l Y -N. Xu and W. Y. Ching, "Total EnergyCalculation in u-Si02and /3-Si3N4,"Mater. Res. Soc. Syrnp. Proc., 105, 181-86 (1988). 82(a) F. Zandiehnadem. R. A. Murray, and W. Y. Ching, "Electronic Structures of Three Phases of Zirconium Oxide." Physica 8 (Amsterdam), 150, 19-24 (1988).(b) F. Zandiehnadem and W. Y. Ching. "Total Energy Calculation on the Three Phases of ZrO2"; unpublished work. 83(a) B. I. Dunlap, J. W. D. Connolly, and J. R Sabin, "On Some Applications of Xu Theory." J Chem Phys., 71,3396-402 (1979) (b) B. I. Dunlap, J. W. D. Connolly, and J. R. Sabin. "On First-Row Diatomic Molecules and Local Density Models." J. Chem. Phys., 71, 4993-99 (1979). 84K Jackson and M. R. Pederson, "Accurate Forces in a Local Orbital Approach to the Local Density Approximation," Phys Rev. 5, Condens. Matfer, 42, 3276-81 (1990). 8 5 0 C. Allan and M. P Teter. "Nonlocal Pseudopotentials in Molecular-Dynamic-DensityFunctional Theory: Application to Si02," Phys. Rev. Lett.. 59, 1136-39 (1987). 86F. D. Murnaghan, "The Compressibility of Media under Extreme Pressures." Proc. Nati. Acad. Sci., U.S.A., 30, 244-47 (1944). 87W. Y. Ching, Y -N. Xu. B. N. Harmon, J Ye, and T. C. Leung. "Spin-Polarized Band Structures of FeB. Fe2B, and Fe3B Compounds Using First-Principles Spin-Polarized Calculations," Phys. Rev. 8, C o n dens Mafter, 42, 4460-70 (1990) 88W. Y. Ching, Y. N Xu, and K. W. Wong, "Ground-State and Optical Properties of CuzO and

0 0

Fig. 32. Calculated positron density distribution in YBa$u3O, by the OLCAO method (from Ref. 119): (a) [loo] plane, (b) [110] plane, and (c) [Ol 01 plane. Minimum contour line is 0.0001 e+/(at.units)3 at an increment of 0.005 e+/(at.units)3 with the maximum contour near the Cu(1) atoms.


November 1990

CuO Crystals." Phys. Rev 5. Condens. Matter, 40, 7684-95 (1989). 59C. Kittel. Introduction to Solid-state Physics; p. 341 Wiley, New York, 3972. gosee. for example, "EXCITONS," Modern Problems in Condensed Matter Sciences. Edited by E. I Rashba and M. D Sturge. North-Holland. New York, 1982 91M. Balkanski, Y Petroff. and D. Trivich, "Optical Propertiesof Cuprous Oxide in the Ultra-Violet," Sokd State Commun., 5,85-88 (1967). 9%. Y. Ren and W Y Ching. "Electronic Structures ot (I-and 8-Silicon Nitride," Phys. Rev. B,Condens. Mafter, 23, 5454-63 (1981). P. Li and W. Y. Ching. "Band Structures of All Poiycrystalline Forms of Silicon Dioxide," Phys. Rev. 3, Condens. Matter, 31, 2172-79 (1985). 94Y:N Xu and W. Y. Ching, "Comparison of Charge Density Distribution and Electronic Bonding in o-Si02and pSi3N4," Physica 5 (Amsterdam), 150, 32-36 (1988) 9% H. Phillips. "Optical Properties of Silicon Nitride," J. Necfrochem. Soc , 120 [2] 295-300 (1973). 9% Liu and M. L. Cohen. "Structural Properties and Electronic Structure of Low-Compressibility Materials: P-Si3N4and Hypothetical 8-C3N4," Phys. Rev 5, Condens. Matter, 41, 10727-34 (1990). 97W. Y. Ching and Y.-N. Xu, "Electronic and Optical Properties of Yttria," Phys. Rev. Lett., 64, 895-98 (1 990). 95(a) V. N. Abramov and A. I. Kuznetsov, "Fundamental Absorption of Y2O3 and YA103." Sov. Phys. Solid State (Engl. Trans/.),20 [3] 399-402 (1978). (b) T. Tomiki. J. Tamashiro, Y Tanahara. A. Yamada, H. Fukutani, T. Miyahara, H. Kato, S. Shin, and M Ishigame, "Optical Spectra of Y 2 0 3 Single Crystals in VUV." J. Phys. Soc Jpn., 55, 4543-49 (1986). SgC K Kwok. C R. Aita, and E. Kolawa, "Process Parameter-Growth Environment-Film Property Reltationships for Sputter-Deposited Yttrium-Oxygen System," J Vac. Sci. Techno/, A, 8,1300-34 (1990). 1oOC Chen, Y. Wu, A. Jiang. 8. Wu, G. You, R. Li. and S Lin, "New Nonlinear Optical Crystal: Li8305," J. Opt. SOC. Am. 6, 6 [4] 616-21 (1989). TOlY -N Xu and W Y. Ching, "Electronic Structure and Optical Properties of LiB305Crystal," Phys. Rev 5, Condens. Matter, 41, 5471 -74 (1990).

3159

H. French; private communication 303Y.-N.Xu and W. Y. Ching, "Theoretical Calculation of Linear Optical Properties of KTP Crystal"; unpublished work. 1MM. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials. Oxford University Press, Oxford, U.K., 1977. 105W. Y. Ching, Y.-N. Xu, and R. H. French, "SelfConsistent Band Structures and Optical Calculations in Cubic Ferroelectric Perovskites," ferroelectncs. 114, 000-00 (1990). 1°6M. Cardona, "Optical Properties and Band Structure of SrTi03 and BaTi03," Phys. Rev., 140. A651-A655 (1965). 107V. L. Moruni. J. F. Janak, and A R Williams, Calculated Electronic Properties of Metals, p. 15 Pergamon, New York, 1978 l08(a) J. Callaway and C S Wang, "Transverse Magnetic Susceptibility in the Local Exchange Approximation," J. Phys. f: Met. Phys., 5, 2119-24 (1975). (b) C. S Wang and J. Callaway, "Spin-Wave Stiffness Coefficient for Ni," Sohd State Commun , 20, 255-56 (1976). lW(a) Y.-N. Xu. W Y. Ching, and K. W. Wong, "ExploratoryStudy of Electronic Structure of the FluorineSubstituted YBazCu307," Phys. Rev. 5, Condens. Matter. 37,9773-76 (1988). (b) W. Y. Ching. Y.-N. Xu, and K W Wong, "Electronic Structure of Fluorine-SubstitutedYBa2Cu307Superconductor," Mafer. Res. SOC.Symp. Proc. 99, 403-406 (1988). 110Y -N. Xu, W. Y. Ching, and K. W Wong, "Electronic Structures of Ga- and Zn-Substituted YBa2Cu307Superconductors", pp. 41-46 in Materials Research Society Symposium M, HighTemperature Superconductors: Fundamental Properties and Novel Materials Processing. Edited by J. Narayan, C. W. Chu, and L. F. Schneemeyer. Materials Research Society, Pittsburgh, PA, 1990. 111(a) K. W. Wong and W. Y. Ching, "Theory of Simultaneous Excitonic and Superconductivity Condensation," Physica C (Amsterdam), 158, 1-14 (1989) (b) K. W. Wong and W. Y. Ching, "Theory of Simultaneous Excitonic and Superconductivity Condensation II. Experimental Evidences and Stoichiometric Interpretation," Physica C (Amsterdam), 158, 15-31 (1989). 1 1 % L. Yang. S.-H. Guo. F. T. Chan, K. W Wong. '02R.

~

Y,Si,O,

SiO,

Y,SiO,

0.4

0

90

Experimental OLCAO

--*-Atomic

0.3 ... c

80

u)

3

70

u 0

:0 2

60

._ -

E!

ON cl, 50

8

s

Z

0.1

40

30 0.0

20 0

5

10

15

Angle ( m a d )

Fig. 33. Comparison of calculated and measured momentum density in YBa2Cu307by the angular correlation of positron annihilation radiation (Ref. 125). OLCAO calculation is in much better agreement with experiment than the simpler atomic calculation.

10

Si,N4 10

20

30

40

50

60

70

80

90

%Y N Fig. 34. Phase diagram of the Si-Y-0-N system from Ref. 134.

"'

3160

Journal of the American Ceramic Society-Ching and W. Y. Ching, "Analytic Solution of TwoDimensional Hydrogen Atom. I Nonrelativistic Solution'' to be published in Phys. Rev. A, Gen. Phys 113F T Chan. S.-H.Guo, X L. Yang, K. W Wong, and W. Y. Ching, "Analytical Solution of TwoDimensional Hydrogen Atom 11. Relativistic Solution with Chern-Simon Action"; to be published in Phys. Rev. A, Gen. Phys. l14A.J. Ark0 et a/., "Large, Dispersive Photoelectron Fermi Edge and the Electronic Structure of YBa2C~305.9 Single Crystal Measured at 20 K," Phys. Rev. B, Condens Matter, 40,2268-77 (1990). 115W. Y. Ching. G. L. Zhao, Y.-N. Xu, and K. W. Wong. "Orbital-Resolved Pzrtial Density of States in YBa2Cu307",submitted to Phys. Rev. B, Condens. Matter 116K. Kamaras, C. D. Porter, M. G. Doss, S. L. Herr, D. B. Tanner, D. A. Bonn, J. E Greedan. A H. O'Reilly, C V. Stager, and T. Timusk, "Excitonic Absorption and Superconductivity in YBa2Cu307.,," Phys. Rev. Lett., 59, 919-22 (1987). 117J. Orenstien, G. A. Thomas, D H. Rapkine, C. G. Bethea. B F Levine, R.J. Cava, E. A. Rietman, and D. W. Johnson, Jr., "Normal-State Gap Transition in Cu-0 Superconductors," Phys. Rev. 6, Condens Matter. 36, 729-32 (1987). l16C S. Sundar, A. Bharathi. W Y Ching, Y C. Jean, P. H. Hor. R. L. Meng. Z. J. Huang. and C. W. Chu, "Positron Annihilation Studies on the TI-BaCa-Cu-0 Superconductors," Phys. Rev. 6, Condens. Matter. 42, 2193-99 (1990). 119A Bharathi. C. S Sundar, W. Y. Ching, Y. C. Jean, P H. Hor, Y. Y Xue. and C. W. Chu. "Positron Distribution in Understanding Annihilation Characteristics Across T, in High-Temperature Superconductors", to be published in Phys. Rev B, Condens. Matter, 42 (1990). 12OY. C. Jean, S. J. Wang. H. Nakanishi. W. N. Hardy, M. E Hayden, R. F Klefl. R. L. Meng. H. P. Hor, J. Z. Huang, and C. W. Chu, "Positron Annihilation in High-Temperature Superconductor YBa2cU306+d,"Phys. Rev 6, Condens Matter, 36, 3994-96 (1987) 121s.G Usmar, P Sferlazzo, K. G. Lynn, and A. R Moodenbaugh, "Temperature Dependence of Positron Annihilation Parameters in YBazCu307.,," Phys Rev B, Condens. Matter, 36,8854--57 (1987). '22Y. C. Jean, J. Kyle, H. Nakanishi, P E. A. Turchi. R. H. Howell, A L Wachs. M. J. Fluss, R. L. Meng. H P. Hor, J. Z. Huang. and C. W. Chu. "Evidence for a Common High-Temperature Superconducting Mechanism in La, ssSro.isCu04 and Y B ~ z C U ~ OPhys ~ . " Rev. Lett., 60, 1069-72 (1988). l23Y. C. Jean, C S. Sundar. A. Bharathi. J. Kyle, H. Nakanishi, P K Tseng. P. H. Hor. R. L. Meng. Z J Huang, C W. Chu. Z. Z. Wang, P. E A. Turchi. R.H Howell, A. L. Wachs, and M. J Fluss, "Local Charge Density Change and Superconductivity: A Positron Study," Phys Rev. Lett.. 64, 1593-96 (1990). '24J. Arponen and E Palanne, "Electron Liquid in

Vol. 73, No. 11 Collective Description3, Positron Annihilation," Ann. PhyS. (N.Y.), 121, 343-89 (1979). 125A Bharathi and 6. Chakraborty, "A Study of Positron Distributionand Annihilation Characteristics in YBazCu307.,," J. Phys. F Met. Phys.. 10, 363-75 (1988). 126A. Bharathi, L. Y. Hao. C. S. Sundar, W. Y. Ching, Y. C. Jean, P H Hor, Y Y. Xue, R. L.Meng, Z. J. Huang. and C W. Chu, "Angular Correlation Studies on YBa2CuJOr Superconductor"; presented at the Third Positron and Positronium International Conference, Milwaukee, WI, July 16-19, 1990. 327(a) R. A. Heaton, J. G Harrison, and C. C Lin, "Density Functional Theory with Self-Interaction Correction of the Electronic Energy Structure of Impurity Atoms in Insulator Crystals," Phys. Rev. B, Condens. Matter, 31,1077-89 (1985). (b) R Heaton and C C. Lin. "Self-Interaction-Correction Theory for Density Functional Calculation of Electronic Energy Bands for Lithium Chloride Crystal," J. Phys C: Solid State Phys , 17, 1853-66 (1984). 128(a) M. S. Hybertsen and S. G. Louie, "FirstPrinciples Theory of Quasiparticles: Calculation of Band Gaps in Semiconductors and Insulators," Phys. Rev Lett.. 55, 1418-21 (1985). (b) M S Hybertsen and S. G. Louie, "Electron Correlation in Semiconductors and Insulators: Band Gaps and Quasiparticle Energies," Phys. Rev 6, Condens. Matter, 34, 5390-413 (1986). 729N. Hamada, M. Hwang, and A. J. Freeman, "Self-Energy Correction for the Energy Bands of Si by the Full-PotentialLAPW Method," Phys. Rev. B, Condens Matter, 41. 3620-26 (1990). 130M. R Pederson and B. M. Klein, "Improved Theoretical Methods for Studies of Defects in Insulators: Application to the F-Center in LiF," Phys. Rev. B, Condens. Matter. 37,10319-28 (1988). 131R. C. Chaney and C. C. Lin, "Electronic Structure of the F-Center in a Lithium Fluoride Crystal by the Method of Linear Combination of Atomic Orbitals,'' Phys. Rev. B, Condens. Matter, 13, 843-51 (1976). 132J. G. Harrison, C C. Lin, and W Y Ching, "Electronic Structure of an Associated ImpurityVacancy Defect in Ionic Crystals, V2' in LiF." Phys Rev B, Condens. Matter, 24, 6060-73 (1981). 13% F. Zhong and W Y. Ching, "First-Principles Calculation of Crystal Field Parameters in Nd2Fe14B," Phys. Rev B. Condens Matter, 39, 12018-26 (1989) '%A. Y Lui and M. L Cohen. "Prediction of New Low-CompressibilitySolids," Science (Washington, DC), 245, 841-42 (1989) '35J. Weiss. "Silicon Nitride Ceramics: Composition, Fabrication Parameters, and Properties," Ann. Rev. Mater. SCI.. 11. 381-99 (1981). 136Y. N Xu and W Y. Ching, "Self-Consistent Band Structures, Charge Density Distributions,and Optical Absorptions in MgO, (I-AI~O~, and MgAI2O4"; submitted to Phys. Rev. B, Condens. Matter. 0