AD-AO 180 CALIFORNIA UNIV DAVIS DEPT OF APPLIED … · 2021. 7. 16. · ad-ao" 180 california univ davis dept of applied science f/s 20/12 exnpprimental and th4eoretical study of

AD-AO" 180 CALIFORNIA UNIV DAVIS DEPT OF APPLIED SCIENCE F/S 20/12EXNPPRIMENTAL AND TH4EORETICAL STUDY OF THE FEASIBILITY OF THE GU--ETCfU)

MAR I1 C FONG, FWOOTEN AFOSR-77-3390

UNCLASSIFIED AFOSRTR-81-0452 ML

I

-,ECURITY CLASSIFICATION OF THIS PAGE (Whe n d

REPORT [(W UMENTATIt Pr WL IBEFORE COMPLETING FORVO

11 REPORT NUMBER -C3. RECIPIENT'S CATALOU NUMBLR

XFOSR-TR. )45!~ 1-EXPERIMIENTAL AN' THEORETICAL STUDY 5. TYPE OF REPORT A PERIOD CnV

OF THE FEASIBILITY OF THE CUIhN EFFZCT IN FinalBiSC1, BiSBr, BiSI, BiSeI, BiSeBr, and 6 EFRIGOG EOTNMEBiSeCi .PROMN lG EOTNME

7. AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(s)

C. Y. Fong and F. Wooten AFOSR-77-3390

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10 PROGRAM ELEMENT, PROJECT. TASK

AREA & WORK UNIT NUMBERS

SDepartment of Applied Science 2 3I6/B2''University of California 61 102FSDavis, CA 95616

'-"it, CON OVLLING OFFICE NAME A ND ADDRESS 12. RFPnPT

DATF1

Air 'Orce O ffice of Scientific Research March 1981Boiling AFB, Bldg. 410 13. NUMBER OF PAGES

jh Washington, D.C. 20332 351.MONITORING AGENCY NAME I ADORESS(:ildilff... Iron, Conutrolli Ofice) 15. SECURITY CLASS. (of this r port)

Unclassified0 I-.-ECLASSIFICATION DOWNGRADING

mt 16. DISTRIBUTION STATEMENT (of this Report) SHDL

Approved f or pubico release;distribution unlimited.

IS. SUPPLEMENTARY NOTES

19.KE WODS(Cotiue n eveseaid i nce~say ed denif byblck umer

20. ABSTWRCT (Continue onl reverse aide it necessary and identify by block number)

-The op~tical reflectance of BiSI has been measured with photon energy, from .2~e. to 3.6 eV. The band structures of BiSBr, BiSCI, BiSebr, BiSeCl and BiSel have0.... been calculated by using the self-consistent pseudopotential mcthod. These re-

sults.show that: (i) they are indirect gap semiconductors; (ii) the top ofthe valence band is located at some k-point along rZ (the z-direction) withfinite curvature (small effective mass); (iii) the highest valence band at r(the center of the Brillouin zone) is only a few tenths of an eV lower than

__ the eSVergy of the top of the valence band and is flat_(large effective mass)..

DDFOR 47 EDITION OF I NOV 615 IS OBSOLETE Ucasfe

SECURITY CLASSIFICATION OF THIS PAGE (*?.oen Data Entfered)

S(dURITY CLASSIFICATION OF THIS PAGE(4?un Data Entered)

1 especially in the-directions perpendicular to z.. (iv) in addition to the above1listed features, the minimrum of the conduction band fo&'BiSCI and BiSe~l ilocated at some k-point along rX, the x direction and is a few tenths of an eVlower than the Lowest conduction band at r.

These results suggest that: All the six compounds studied in this project willexhibit the Curio effect if they are properly doped. They are potential candi'dates for microwave oscillators.

SECuRITY CLASSIFICATIOW OF I'11 PAGE(When Val. Fnr...d)

jgAF *s-" 8 1 - 0 4 5 2

/ Tg

" PERIMENTAL ANDTIEORETICAL_,TUDY OF THE IASIBILITYOF TII-U EFFECT IN BI tlBiSBr, BISI, BiSel, WTSeBr ANWD BiSeCI*

q ri1na Scientific Repot7- /t ,

.by

J. . .........

Ancison ror

IVC. Y. Fong zftF. Wooten rT2C3

Department of Applied Science -_University of California I

Davis, CA 95616 I .

*Supported by US Air Force Office of Scientific Research under Grant No.

,,'il ter 1 ,38 1 ')7 * ,, ,, ,..0 ,)7t tkt4

ik

-1-

ABSTRACT

The optical reflectance of BiSI has been measured with photon energy, lio,from 2.2 to 3.6 eV. The band structures of BiSBr, BiSCI, BiSI, BiSeBr,BiSeCI and BiSel have been calculated by using the self-consistent pseud:-potential method. These results show that: (i) they are indirect gapsemiconductors; (ii) the top of the valence band is located at some k-pointalong rz (the 2-direction) with finite curvature (small effective mass);(iii) the highest valence band at ' (the center of the Brillouin zone) isonly a few tenths of an eV lower than the evergy of the top of the valenceband and is flat (large effective mass), especially in the directionsperpendicular to z. (iv) in addition to the above listed features, theminimum of the conduction band for BiSCl and BiSeCl is located at somek-point along rX, the 2-direction and is a few tenths of an eV lower thanthe lowest conduction band at r.

These results suggest that: All the six compounds studied in this projectwill exhibit the Gunn effect if they areproerly doped. They are potentialcandTdats for i mcrowave oScTlTators.

The dopings in BiSBr, BiSeBr, BiSI and BiSel should be restricted to p-type.Effective doping to exhibit the Gunn effect will be to substitute either thegroup IV elements for Bi or the group V elements for Se. For BiSCI and BiSeCl,both h and p-type dopings are possible. The p-type doping should be restrictedto the same elements as suggested for the other four crystals. The bestresults for n-type doping should be obtained by substituting the Bi by thegroup VI elements.

The electric field used for the Gunn effect should be applied along the z-direction for^the p-type dopings. It is necessary to apply the electricfield in the x-direction in order to exhibit the Gunn effect in n-typecrystals.

Comparisons of the other available experimental gap energies with the theoreti-cal results are made. Discrepancy between the measured and calculated trendof gap values in I- and Br- compounds is discussed and a possible explanationis provided.

The bonding properties of these crystals are presented.

AIR FORCE OFFICE OF SCIENTIFIC RESEARCH (AFSCjXOTICE OF TRA su1TTAL TO DDThis technical report haJ uvtn roviowed an4 isapproved for public roieaio lAW AYR 190-12 (7b).

Distribution is uwlimtied.A. D. BLOSETeoulocal I forut oa Offloor

-2-

I. INTRODUCTION

The objective of this grant (No. 77-3390) is to show whether BiSBr,BiSCI, BiSI, BiSeBr, BiSeCl and BiSel are candidates for microwaveoscillators.

The criterion for a crystal to be used as a microwave oscillator isthat it should exhibit the Gunn effect. Consider a degenerate n-dopedsemiconductor, then there are electrons in the conduction band evenat T = O°K. The current density under the influence of an externallyapplied dc field is given by

a = (1)

where F is the electric field, and a is the conductivity. Let p and Tbe the density and the life time of the electrons, then

pe2Tm* (2)

where m* is the effective mass of the electrons. From the analogy of

the free electron case 1 ,the value of m* is inversely

proportional to the curvature of the energy with respect to the crystalmomentum of an electron.

As the electric field is increased, not only does it accelerate theelectrons but also it causes scattering of the electrons. The scatteredelectrons can be in states such that their effective masses increase.Consequently a decreases and so does the current. The typical J vs Ecurve is shown in Fig. 1. The region A indicates that current decreasesas the strength of the applied field increases. The sample exhibits anegative conductivity. This is the Gunn effect.

If one recognizes that positive a represents power consumption, it isclear that negative a means that the sample supplies power, and can beused as an oscillator.

From the above discussions, it is apparent that we need to find the bandstructure, E(K), for the electrons in a semiconductor, then decidewhether it can be used as an oscillator. We believe that this is thefirst study of this kind for the Bi-compounds.

-3-

In Section II, we discuss the experimental set up for the reflectancemeasurements. The method used for calculating the band structure willbe described in Section II1. The results and the discussions aboutthe basic properties and applications will be presented in Section IV.Finally, in Section V concluding remarks will be given.

II. EXPERIMENTAL ASPECT

Reflectance data were obtained for samples of the crystals BiSI.Originally, the crystal samples were in a matted form with smallquantities of reactant materials filling the interstitial space betweenthe actual crystals of the material of interest. The samples werewashed with a 56% solution of HI, then methanol, then acetone severaltimes each. The cleaned crystal samples were then mounted on analuminum sample holder and placed in the reflectance chamber.

The range of the scans was from 3350 A to 5400 A. The light sourceused was a tungsten filament lamp. The light detector was an uncoatedphotomultiplier tube (number 9843B from EMI GENCOM) with an S-11response, operated between 800 and 1000 volts. The tube was electro-statically and magnetically shielded and all the electronics shareda common ground to reduce noise as much as possible. A signal tonoise ratio of 25 was typical. Typical dark count was 0.2 microamps.A more complete description of the reflectance chamber and the opticalsystem has been presented elsewhere.1 A schematic diagram is shownin Fig. 2.

The measurements were performed at atmospheric pressure since absorption-in the visible range is not a problem, and no filter was used for thefinal runs as there was no discernible difference either way.

The reflectance data are the relative reflectance of the samples. Theellipsoidal mirror assembly was adjusted to occult the reference beamin such a way as to obtain optimal reflectances. If the full intensityof the reference beam were used, the reflected signal would be relativelytoo weak, produciog a small and noisy relative reflectance curve. Themirror position was adjusted for each sample to maximize the relativereflectance and to minimize noise.

0

To perform a data scan, the monochrinator was set at 3350 A, the incre-ment (gA) and the number of data po'ints (410) were entered into thereflectance program, which carried out the scan automaticdlly. The datawere then plotted as a graph of relative reflectance versus energy orwavelength.

I1. METHOD OF CALCULATIONS

The self-consistent pseudopotential method is used to calculate the bandstructures uf the 6 Bi-compounds. The procedures involved in the methodare summarized schematically in the block diagram (Fig. 3).

-4-

The ionic pseudopotentials of Bi, Se, I, Br were de ermined by thelocal scheme proposed by Starkloff and Joannopoulos. The essentialfeature of the scheme is to multiply the self-consistent atomicpotential3 by a function which has the following form

l - X

1 + e 'X(r-rc) (3)

This function weakens the potential for r<rc and approach 1 for r>rc.The A and rc are determined by fitting the valence state energies tothe atomic calculations. 3 Thus, the resulting ionic pseudopotential(effective potential) gives the same atomic energies as in the fullself-consistent calculations.

For S and Cl, the extrema of the outermost s and p states are veryclose together (%O.l A) as given in Ref. 3. It is difficult to usethe f-function, Eq. (3), to fit simultaneously the s and p statesenergies. By making use of the non-uniqueness of the pseudopotentialin the core region, we added a repulsive s-potential of Gaussian formin the core region. The Gaussian potential has the form:

V (r) = Ar2 e-ctr (4)

The. charge densities used for the initial construction of the electron-electron interaction -- Coulomb and exchange potentials -- were obtained.from earlier empirical pseudopotential calculations of the Sb-compounds.4

We used the so called Xct - exchange of SlaterS, and set x= 1.

The self-consistent cycles were carried out at one special k point (0.25a;0.25b, O.117c) where b is the longest and c is the shortest latticeconstant. The charge densities associated with this V point do not over-lap with the ones of 4 nearest unit cells 6.

To assure the convergence of the energy eigenvalue with respect to thenumber of plane waves, we used about 380 planes waves as basis functions.In addition, there are another 380 plane waves included via the Lbwdinperturbation7 .

It took about 8 cycles in each crystal for the potential to attain self-consistency. The resulting electron-electron interaction and the ionicpseudopotentials are used to calculate the band structure, E vs k.

-5-

IV. RESULTS AND DISCUSSION

The measured reflectance of BiSI for photon energy, 41w, between 2.2to 3.5 eV is shown in Fig. 4. The structures at 2.3 and 2.6 eV inthe present results correspond to the double structure of 2.2 and2.4 eV in SbSI. Tentative identification of the origins of the twostructures will be presented later.

Let's turn our attention to the results of the self-consistantcalculations. It is necessary to show how good the ionic pseudo-potentials are. We choose I as an example to show the comparisonof the ionic pseudopotential and the full self-consistent (coreincluded) ionic potential. 3 The two potentials are compared inFig. 5a. For r > 3.a.u., perfect agreement of the two potentialsis obtained. The pseudopotential in the region r<3, is considerablyweaker than Vionic. The pseudo wavefunctions of the s and the pstates are compared with the full self-consistent results in Fig. 5b.Outside rc, the pseudo wavefunctions agree reasonably with those ofthe full self-consistent calculations. In the core region, thepseudo wavefunctions are smoother, i.e. no wiggling, than the exactwavefunctions. The ionic pseudopotentials of other elements showsimilar features. So, we simply plotted the pseudopotentials of Bi,S, Se, Br and Cl, in Fig. 6a-6e. The values of rc and X for Bi, Se,Br, I are listed in Table I along with the comparisons of the fittedvalence states energies and those given by full self-consistent cal-culations using the Herman-Skillman programs. Parameters characteriz-ing potentials for S and Cl are given in Table I. These potentialswill also be useful for future calculations of semiconducting solidswith these elements as constituents. The Se results have been checkedwith'the ones determined by Joannopoulos. 8

The band structures of the six compounds are plotted along the crystalaxes in Fig. 7a-7f. Since our present interests are those bands nearthe fundamental gap, only 6 bands below and 4 bands above the gap aregiven.

There are several common features in these calculations. They are:

(i) All of these crystals are indirect gap semiconductors.

(ii) The top of the valence band is located at some V-point, o, alongFZ (z-direction) and is a few tenths of an eV above the highestvalence band energy at r (R=o).

(iii The valence band along the a (x) and b (y) directions have lessdispersion (curvature) than the ones along the c (z) axis.

-6-

Feature (i) agrees qualitatively with our earlier empirical calcula-tions of Sb-compounds.4 Most of the Sb-compounds, however, have thetop of the valence bands at 1'. The joint features of (i) and (ii)indicate that the curvature of the top of the valence band is greaterthan that at F. Consequently, the effective mass of the hole atr is larger than that at the valence band edge. Coupled with thesmall energy difference at "ko and F, these crystals should exhibitthe Gunn effect if they are p-dopped (e.g. S or Se to be replaced byP, or As). Therefore, we find that these six crystals are potentialcandidates for microwave oscillators. Finally, the third feature,the difference in dispersion along the different axes, is a mani-festation of the anisotropy in the crystal structure of these compounds.

The characteristics of the conduction bands show more variation forthe 6 crystals than those exhibited in the valence bands. The Br andI-compounds have the minimum of the conduction bands at P, while thetwo Cl-semiconductors exhibit the minimum at some k-point along FX(x-direction). Therefore, the Br and I crystals have the smallestdirect gap at F and the gap for both BiSCI and BiSeCl appears along rX.The values of the lattice constants, the fundamental gaps and thesmallest direct gaps for these Bi-compounds are summarized in TableIII accompanied by a few measured gap values.

The difference in the conduction bands of the Cl-crystals from theones of the other compounds makes the possibility of exhibiting theGunn effect in n-type BiSCI and BiSeCl. In these cases, the electricfield should be applied along the x-direction.

From Table III, one notices that BiSCl and BiSeCl have the same latticeconstants. This deserves some explanation. We are unable to find thelattice constants of BiSeCl in the literature. As a first trial, thelattice constants of BiSel were used. The self-consistent calculationsshow that BiSeCl is a semimetal. Next, we tried the lattice constantsof BiSeBr. A similar result was obtained. This suggests BiSeCl may bea semimetal. However, we reject this possibility based on the calcu-lated total valence bandwidth. Both results show that the width isabout 24 eV, which is roughly a factor of 2 wider than the other fivecompounds. There does not exist any plausible physical mechanism tocause the large width in the BiSeCl except that the lattice constantsused for the calculations are too large. Finally, we used the latticeconstants of BiSCI. Results seem in line with the others. However,as seen in Table Ill, it has still the smallest fundamental gap. Onewould expect that the lattice constants of BiSeCl should be larger thanthe ones of BiSCI based simply on the sizes of Se and S. The difficultyin obtaining a reasonable band structure of BiSeCl suggests that thecrystal may not be stable under normal conditions (atmospheric pressure).Our calculations correlate the absence of crystallographic data for thisparticular crystal.

-7-

Another point given in Table III is worth explanation. We see that thecalculated band gaps of BiSel and BiSeBr are comparable. Because thesame experimental group has reported two different gap values of BiSel at

different times, it is difficult to argue that either of the experi-mental values are more accurate than the theoretical prediction. We,then, turn our attention to examine the charge density of these twocrystals. Intuitively, one would say that since Er is more electro-negative than 1, BiSeBr should have a larger gap than BiSel. However,we found that this is too simple a picture. In Fig. 8a-Sb we comparethe total valence charge densities of these two crystals in a sectionconsisting of Bi, Se, and a halide. The atoms are labeled in thefigures. Comparing the charge densities around the halides, we seethat the Br has more charge than the 1. This result is consistentwith the intuitive feeling. Now, let us look at the charge densitybetween the Bi-Se bonds. The I-crystal has more bond charge than theother. We can conclude that BiSeBr is more ionic between the Bi-Brbond but less covalent in the Bi-Se bond, whereas SiSeI is just thereverse. How does this bonding property influence the value of theenergy gap? We examine the charge densities from the four highestvalence bands. In order to show more clearly the features, we choosea section (11 in Fig. 9) consisting of the sare Bi and halides but theSe forming the horizontal bond with the Bi. In Fig. lOa and lob, thevalence charge densities of BiSeBr and BiSel in Section II are plotted.Both BiSeBr and BiSel exhibit more charge around the Bi and Se thanaround the halides. So, the ionic character of the I and Br shouldbe manifested by the charges associated with lower energy valencestates. Because of the stronger ionic nature of the Br, the bondbetween theBiand the Se is less covalent. As a result, the BiSeBrhas a smaller gap than BiSel. Furthermore, from the charge densities shownin Fig. 10, we suggest that the p-doping of these crystals will notbe effective for exhibiting the Gunn effect, if the halides aresubstituted by group VI elements. It is necessary to substituteeither the Bi by group IV elements or the chalcogenides by group Velements. For n-doping, only substitution of Bi by group VI elementswill be effective. The charge of the lowest conduction band should beassociated with the Bi-atom, because as shown in Fi-s. lla, b the corre-sponding charge distributions are around the metal ion.

Using the band structure of BiSI, and the earlier interpretation of theSb-compounds, the structures at 2.4 and 2.6 eV in the reflectance areinferred to be mainly caused by the transitions from states near '8 tostates near r6 along rZ.

IV. CONCLUSION

We have measured the reflectance of BiSI and calculated the band structureof tiie six Bi-compounds. The self-consistent pseuopotential method isused for the calculations. The results show:

A. All the six Bi-compounds are potential candidates for microwaveoscillators, if (a) all of them are p-dopped; (b) the BiSClend the BiSeCI are n-dopped; (c) BiSeCl cannot be grown underatmospheric pressure.

B. Smaller lattice constants are necessary for the semiconductingbehavior of BiSeCl. Apparently, there does not exist anycrystallographic data of BiSeCl. This is consistent with thedifficulty in getting reasonable band structure.

C. The doping should be specific in these crystals. For p-doping,substituting either the group IV elements for Bi or the group Velements for the chalcogenides is the effective way. The Gunneffect is not expected to be observed if one substitutes forthe halides. For n-type doping, the only substitution to exhibitthe Gunn effect is to replace the Bi by the group VI elements.

D. for p-type doping, the d.c. field is to be applied along thez-direction, i.e. the direction of the needle. For n-type doping,the d.c. field should be applied along the x-direction.

In addition to the above findings for the microwave oscillators, we

also cite the following basic properties:

E. These six compounds are indirect semiconductors.

F. The bands along the z-directionare in general more dispersivethan those along the x and the y-directions.

G.- There are two different types of bonding in these crystals. Ionicbonding appears between Bi and the halides. These ionic statesform the lower energy valence bands. The bonding between the Biand the chalcogenides is substantially of covalent character. Thestates associated with this bonding contribute to the bands nearthe fundamental gap.

H. The value of the fundamental gap is determined by the interplay ofthe ionic and the covalent bondings. More ionic bonding meansless charge is available to form the covalent bonding. Consequently,the value of the gap is smaller.

-9-

References

1. T. Huen, G. B. Irani and F. Wooten, Applied Optics 10, 552 (1971).

2. Th. Starkloff and J. D. Joannopoulos, Phys. Rev. B 19, 1077 (1979).

3. F. Herman and S. Skillman, Atomic Structure Calculations, PrenticeHall, New Jersey, 1963.

4. J. Alward, C. Y. Fong, 1M. El-Batanouny and F. Wooten, Solid StateComm. 25, 307 (1978).

5. J. C. Slater, Quantum Theory of frolecules and Solids, Vol. 4, p. 35,McGraw-Hill, New York, 1974.

6. J. D. Joannopoulos and i. L. Cohen, J. Phys. C: Solid St. Phys. 6, 1572 (1973).

7. For examples, D. Brust, Phys. Rev. 134A, 1337 (1964).

8. J. D. Joannopoulos, private communication.

9. D. V. Chepur, D. M1. Bercha, I. D. Turyaritsa and V. YU Slivka, Phys.Status Solids, 30, 461 (1968).

10. 0. V. Luksha, A. P. Zhdankin, L. M. Suslikov, N. I. Davogsher andV. Yu Slivka, Soviet Phys. J. If, 724 (1975).

Table Captions

Table I. The values of k and rc for Bi, Se, Br, I and the comparisonof the fitted and the reference atomic energies.

Table II. The parameters to characterize the ionic pseudopotentials of Cland S.

Table III. The lattice constants, the values of the fundamental gaps, thesmallest direct gaps and the available experimental o:p values.

-10-

Figure Captions

Fig. 1. Typical J vs. c curve for the Gunn effect.

Fig. 2. Schematic diagram of system for reflectance measurements.

Fig. 3. The block diagram of the self-consistent pseudopotential method.

Fig. 4. Reflectance of BiSI.

Fig. 5a. Comparison of the ionic pseudopotential and full self-consistentpotential of I.

5b. Comparison of the pseudo wavefunctions and the full self-consistent (exact) wavefi.nctions of the s and p states of I.

Fig. 6a. Ionic pseudopotentials of Bi.

6b. Ionic pseudopotential of S.

6c. Ionic pseudopotential of Se.

6d. Ionic pseudopotential of Br.

6e. Ionic pseudopotential of Cl.

Fig. 7a. The band structure of BiSBr

7b. The band structure of BiSeBr

7c. The band structure of BiSC

7d. The band structure of BiSeCl

7e. The band structure of BiSI

7f. The band structure of BiSeI

Fig. 8a. The total valence charge densities of BiSeBr.

8b. The total valence charge densities of BiSel.

Fig. 9. Atomic configuration in a unit cell.

Fig. lOa. The charge densities of 4 highest valence bands of BiSeBr.

lob. The charge densities of 4 highest valence bands of BiSel.

Fig. fla. The charge densities of the lowest conduction band of BiSeBr.

11b. The charge densities of the lowest conduction band of BiSel.

LO

m ~ C) C'.J

4.J cu CZ C) C3 C)

(U

4-- 0

LO C' r- r-

cn C') C; "

S- C0 a) C) Cr) k) co

-4- 4--)i

C A

0) ~t (\ ) n(>I-l c. C) co

0o ) C - z-S.- C r -

c 0)

0)to C) 0

0 Ch0n(fl, LO) t

l< - .- -o coS

'U 0) 10 C)CC

co V ) C)

-S-2

-W r .4- LO

to l W A C ))0. 0.0:

(A co LAcu

cr

-P 4-)tu C 4-.$

4-3 a O-~n ~ co LA

9.-'0 Co C

W UCoj Coj

0 0

LO

oo 'Len fLA

co 0C~j -

C) C)

00 O n LO m. O O.LU

-o

S-.

r' - co egv jCiY- 0 0) LO m~ LO

4(0LA LA LA r- LO

124-) to- -

- S.-

-Do

0L~ < - .

I- ~-*0ci

oc

f- co co co OD 0

0 Q

tn V) s.D -) Cs2 co o N- -ca -

Figure I

-Jr

C)

CD

-- 0

C> ii

da-C-CC W-

U- tx c

-16-

Calculate the wave functions Obtain the ionic potentials,of the valence bands at a the energies of the valence s,special k-point p states for the constitutents

of the crystal from self-consistent atomic calculations

Calculate the initial Hartree Pseudize the ionic potentialsand exchange potential by multiplying each potential

-Ara function f l-e

1+e-X(r-rc)where A, rc are determined byfitting the energies of thevalence states

Using the pseudized ionicpotentials and the initialHartree and exchange potentials,calculate new Hartree and ex-change potentials until self-consistency is achieved

Figure 3

-17--

ij

zLUJ

I-C)CMj

tO 0 LO00

0 0 0

Lnn

4w4

- 0i

40 ('J 0 to I

4.P4

C3 I, L

(pti W

I

-*1 9-

r(a~) H

0 .1 2 30 K

I.Bi

~0

-5

* -

-20-

-

C~I)

r') LE

0~ Se

-10L

-22-

c)

C\j

-23-

V(ryd)

CD CD

rn co (10 (0 rn oIOc

rO CO(0 (A@) OC-

_v3

U-)

ao ao '. ro '.D Coc

Lu

C\J CIA-

(A)

C-)J

NO(D00ro 00(4)(

<Jo

LUJ

oq CJ r %,%

C11 -3 C*.j

(A@ I

-9z-

- A

r~~C% (ric (0 r)D

C-,J

(

C~C

LLJJ

III

~(A@ I

co co (D (.r4-)o (.00 4-

<_ C5

OV Co C'J

-6,

-30-

C)00

C)

ri

C.D

Li U)(D Ln W

Ul =) CC) 43C) C)

0000

ODC)

CL)Oo CD

4- coCo

CDC3

C) W

"Ln

v

N)

CD

1~

kc D >1t7

1

I

/ t.

-

/p

P q~s -~ * - * - ~33

ll'k/

II

.1

.t7

all:5cc

rn

..........Ln

(D LI)-4 rn Lf) co

CD Ln

M Ln('00

rnco

CDo

Ln0

~ZIf)

C-C,

1-4-

C;r

- t14

7CCD

L0r'

.7)

AC

I