NASA-CR-196109

FINAL TECHNICAL REPORT

f o r

Cooperative Agreement NCC 3-55

w i th

j NASA - Lewis Research Center

Theoretical Studies of the Transport Propertiesin Compound Semiconductors

Principal Investigator: Benjamin Segall

Department of PhysicsCase Western Reserve University

i (NASA-CR-196109) THEORETICAL N94-37501j STUDIES OF THE TRANSPORT PROPERTIES

IN COMPOUND SEMICONDUCTORS Finalj Technical Report (Case Western Unclas

Reserve Univ.) 60 p

I G3/33 0014047

https://ntrs.nasa.gov/search.jsp?R=19940032990 2020-03-24T03:21:10+00:00Z

INTRODUCTION

This final report is an overview of the work done on Cooperative Agreement

NCC 3-55 with the Solid State Technology Branch of the NASA-Lewis Research Center

(LeRC). Over the period of time that the agreement was in effect, the principal

investigator - and in the last three year - the co-principal investigator (Walter^Rr

Lambneeht) worked on a significant number of projects and interacted with members

of the Solid State Technology (SST) branch in a number of different ways. For the

purpose of this report, these efforts will be divided into five categories: 1) work

directly with experimental electrical transport studies conducted by members of the

SST branch; 2) theoretical work on electrical transport in compound semiconductors;

3) electronic structure calculations which are relevant to the electrical transport in

polytypes of SiC and SiC-AIN alloys; 4) the electronic structure calculations of polar

interfaces; and 5) consultative and supportive activities related to experiments and

other studies carried out by SST branch members. Work in these categories will be

briefly discussed fefeTow. Much of the past work was discussed in more detail in the

various proposals for this grant.

Copies of references [4], [5], [6] and [7] are attached to this report. Copies

of references [1], [2] and [3], which also acknowledge support by this grant, are not

included as the monitor of this grant is a co-author of those publications and thus has

copies of them.

WORK CARRIED OUT UNDER THE AGREEMENT

(1). Work Directly Connected with Experimental Electrical

Transport Studies Conducted by Members of the SST branch:

Because of SiC potentialities in a wide range of applications, in particular,r

semiconductor applications at high temperature, high fields and high frequencies, this

material has attracted considerable interest over the last decade. In the early 1980s,

researchers at LeRC were the first to grow epitaxially large area crystal films. This

was done by the chemical vapor deposition (CVD) method. Thus, there was

considerable interest at LeRC to determine the electrical transport properties of

those films. We note here that while SiC can form in a large number of

crystallographic structures (poiytypes), those films had the 3C (the cubic) structure.

The electrical properties are obtained by so-called Hall measurements. A series of

such measurements on several films produced at LeRC was initiated by Drs. E. J.

Haugland and S. A. Alterovitz of the SST branch. The PI worked with these

researchers to interpret the data in terms of the band structure of 3C SiC. The

results of these investigations appeared in Applied Physics Letters [1].

About the time of this work, two other groups of researchers published the

results of measurements on similar SiC films. Their analysis lead to different

interpretations than ours. Our analysis of their results,which was also published [2],

backed our interpretation, which is now the widely accepted interpretation.

Another project in this category was also related to experimental work by Drs.

Haugland and Alterovitz along with that of a visiting researcher, Dr. P. P. Szydlik. This

involved the study of the properties of the two-dimensional electron gas in a ln0.iGa0.9

As/GaAs heterostructure. In this structure the electrons reside in the InGaAs alloy,

which has a smaller band gap than the GaAs. The experiments consisted of Hall and

Shubnikov de Haas (SdH) measurements. The latter involves oscillations in the

magnetoresistance. From analyses of the SdH oscillations as a function of

temperature and as a function of magnetic field, it is possible to determine an

effective mass m* and an electron scattering time T SdH- From the Hall mobility -

determined in the Hall measurements - another electron scattering TH could be

determined. It was found that the ratio TH /TsdH was about 5 to 6. The fact that the

ratio was considerably larger than one was explained by the fact that while small

angle scattering does not contribute appreciably to TH, it does contribute to TSCJH-

Relevant scattering mechanisms which could contribute to the difference were

identified and the value of m* was discussed in terms of known values for the pure

compounds. A paper on this work was published [3].

(2). Theoretical Work on Transport in Compound Semiconductors:

The aim of this work was to set up a reliable means for calculating the

transport properties, in particular, the Hall mobility and the so-called Hall coefficient

rH(T) (which is given by RH = rH/en where RH is the Hall constant and n(T) is electron

density). Both quantities, especially H.H, depend on the various scattering mechanisms

that are operative and on the nature of the electronic energies in the vicinity of the

relevant band extremum, e.g., conduction band minimum (or minima). For SiC, and

other semiconductors of current interest, both the mechanisms and the details of the

band edges are not known with accuracy and certainty. The goal is to compare the

calculated ^IH(T) with suitable experimental results over a reasonable range of T and

thereby pin down the relevant scattering mechanisms and certain aspects of the band

edges.

The PI, with the help of two CWRU undergraduate physics students, has set up

programs to numerically solve the Boltzmann transport equations. All the principal

scattering mechanisms relevant for compound semiconductors are included. The

programs have been checked extensively. Of most immediate interest to us is the

high temperature mobility of 3C SiC. Subsequent to that will be the study of the

mobilities in the other polytypes; that will probably be a somewhat more involved

undertaking.

(3). Electronic Structure Calculations Relevant to Electronic

Transport in Polytypes of SiC and Related Materials:

A knowledge of the band structure is important for the transport because

transport is effected by the nature of the band edge: whether it is degenerate or

nondegenerate, whether it is at the center of the Brilloun Zone (BZ) and hence is a

single extremum, or is at some other k value in the BZ and hence is a "multi-vallied"

extremum. In addition to having a large density of states, a multi-vallied band edge

introduces the possibility of "intervalley scattering". Also important is the particular

location of the extremum and the effective mass of the band edge.

The study of the band structure(s) of SiC is a significant project because, as

noted earlier, SiC forms in numerous different structures, or polytypes. The different

structures can be described by different orderings of the close-packed layers of

atoms, these layers being the (111) plane of the cubic or the (0001) planes of the

various hexagonal or rhombohedral structures. For our present purposes the 3C, 2H,

4H and 6H polytypes are the most important. The latter two are the most common

and easily grown forms of SiC while the 2H (wurtzite) form is difficult to grow. The

various structures differ in the frequency and ordering of the "cubic" [ jf ) and

"hexagonal"/ jp J stackings of the successive layers. With 3C having all the cubic and

2H all the hexagonal stacking, the sequence of the four polytypes in terms of

increasing "hexagonality" is 3C, 6H, 4H and 2H.

Since the position of the valence band maximum of all the binary compound

-*semiconductors, with the exception of AIM, are at k=0, it is the conduction band

minima which are of most concern. (All the values of the parameters describing the

degenerate valence band maxima are unknown and thus, are of interest.) We first

studied the 3C and 2H, the simplest forms of SiC. In all of our self-consistent

calculations, we obtain the "energetic" properties (lattice constants, bulk moduli, and

cohesive energies) in addition to the band structure. It was found that the band gaps

were indirect in both structures with the minimum for 3C being at the point X (of the

fee BZ) and that for 2H being at point K of the hexagonal BZ with a state at M being

close in energy. We note that the points K and M are not the hexagonal equivalents of

the point X when 2 BZs are appropriately superimposed. In addition to the "shifting"

of the position of the conduction band edge, the band gap for 2H was found to be

almost 1 eV larger than for 3C. To obtain a qualitative understanding of the above

results, calculations for the closely related compound semiconductors AIM and GaN

were carried out. An understanding was achieved and was described in a publication

[4] acknowledging this grant.

More recent work included calculations of the 4H and 6H polytypes. With the

results of these studies, we were able to determine the trends in the various band

gaps including the minimum gaps of the aforementioned four polytypes as a function

of hexagonality. This among other things was addressed in a publication [5]

acknowledging the grant.

Further work studying the details of the conduction band edges is in progress.

We note that recent transport studies conducted at the CREE Corporation have

shown that the mobility of 4H SiC is higher and much more isotropic than that for the

6H form. These results can have significant implications for device applications, and

represent one "hottest" unsettled issue in the SiC community. Tentative results from

our calculations may have some bearing on this question and will be pursued

vigorously in the near future.

Finally, we note the work on alloys of SiC and the closely related compound AIM

which is closely latticed matched to SiC. These alloys are of interest because AIN has

a significantly larger band gap than SiC (for wurtzite AIN, Eg = 6.3 eV) so that the

alloys will allow us to "tailor" the band gap over a large range. Moreover, for the

equilibrium (wurtzite) form, AIN is direct. Thus, there is a crossover from an indirect

to a direct band gap. A direct gap is advantageous for optical emission. Work on

these alloys was briefly discussed in a report to the 1994 International Conference on

SiC [6]. A more complete report will be written shortly.

(4). Electronic Structure Calculations of Polar Interfaces

Since semiconductor heterojunctions are now playing an important role in

electronic technology and in basic physics, the study of the electronic structure at

interfaces between semiconductors hardly needs justification. Such studies are

generally done using the supercell approach, i.e., by studying a periodic superlattice of

the two constituent semiconductors with a suitable (generally large) unit cell.

However, when the interfaces are charged (the condition which characterizes "polar"

interfaces), the calculated interfacial energy is the sum of the energies of the two

oppositely charged interfaces. Since two interfaces are inequivalent, it is not possible

to obtain the energies associated with the individual interfaces. Similarly, because of

8

the existence of an electrical field associated with the interfacial charges, it is also not

possible to separate out the band offsets at the two interfaces. A scheme for

circumventing this problem was worked out and published [7]. The approach was

applied to polar inversion domain boundaries in GaAs and NaCI.

(5). Consultative and Supportive Activities

The PI has consulted with several members of the SST branch in connection

with some of their projects. We briefly discusses a number of interactions.

The most frequent interactions were those with Dr. E. J. Haugland. The PI had

numerous discussions with him about anticipated transport studies particularly those

for SiC at relatively high temperatures. There were also numerous discussions with

Dr. Haugland about his recent Hall data on 4H and 6H films.

The PI has, over the past several years, sought to obtain samples of 3C SiC of

reasonably good quality for Dr. Haugland to study in the high temperature Hall

apparatus that he has set up. Recently, we have finally succeeded in obtaining such

samples from a research group in the Naval Research Laboratory. Hopefully, this will

allow a definitive determination of the high temperature mobility of the cubic form.

This will be compared with our theoretical calculations.

There were also discussions with Dr. Haugland about the interpretation of his

electrical transport measurements of a number of GaAs crystals grown in space (on

the STS-40 flight, June 1994). The properties of these samples differed from those of

earth-grown samples, which were otherwise as similar as possible to their space-

grown counterparts. Dr. Haugland has reported on the data and on the (tentative)

interpretation of them in internal NASA meetings

The PI also consulted with Rafael Mena on the changes in the electrical

transport properties of the two-dimensional electron gas at the interface of

AIGaAs/lnGaAs heterojunctions produced by illumination. Among other issues, the

effect of electron occupancy of the second and possibly higher subbands was

discussed.

Finally, there were several discussions with Dr. David Rosenfeld on some

transport problems in SiGe alloys in certain transistors.

10

REFERENCES

Articles Published Under the Grant

1. B. Segall, S. A. Alterovitz, E. J. Haugland and L G. Matus, Appl. Phys. Lett., 49,

584, (1986).

2. B. Segall, S. A. Alterovitz, E. J. Haugland and L G. Matus, Appl. Phys. Lett., 50,

1533, (1987).

3. P. P. Szydlik, S. A. Alterovitz, E. J. Haugland, B. Segall, T. S. Henderson, J. Klem

and H. Morkoc, Superlattices and Microstructures, 4, 619, (1988).

4. W. R. Lambrecht and B. Segall, Mat. Res. Soc. Symp. Proc. Vol. 242, p367;

1992 Materials Research Society.

5. W. R. Lambrecht, C. H. Lee, K. Klein, A. E. Petukhov, E. A. Albanesi and B. Segall,

Proc. NATO Advanced Workshop on Wide Band Gap Electronic Materials, Mensk,

Belarus, May 4-6, 1994. To be published.

6. W. R. Lambrecht, E. A. Albanesi and B. Segall, Proc. International Conference on

Silicon Carbide and Related Materials, Washington, DC, November 1993, Inst. of

Phys. Publ., Ltd., Bristol, UK. Accepted for publication.

7. W. R. Lambrecht, C. Amador and B. Segall, Mat. Res. Soc. Symp. Proc. Vol. 253,

p381; 1992 Materials Research Society.

11

PAGE BLANK NOT FH.HWD

. or o> o>3

«.. nz a5E.a«Sa gsio E5>— *»4> **7JaUiOa ^

itIf,14

rt n fi

ftrt f «— ' r« <*•rte«

»

ft <e> toZ^ -T «"•

•7 "~ ""*V

20.2

(20.

7*)

199(

207*

)11

.3

ag =1M > N

- Z

£ -e

o> f — ti-j -r — >

~- *»i-i =? -i

— -« — !IN r<|*• "i

_ J

ti t»|— 0) «|"™ -i ~"••• ^ \— sat2 "1

|1

£ £o oo n2.?i ^_ ^ ,-

s sN - N533

Z.19

; t le

ole

latt

^ « C _ , 2 ~ - o a ! l * ; - 2 a^5 = 5 5 > - " » « ' 0 - « / ^.s41 = i^si^-tiS S r - s S t - f - S T N f i - ac l S j s s S i ^ f e w s e S Ss f s . - s f j j S - g ^ j j :iia-H-yi^ij2 .y s « s a » . ! - s - -! - s 2 : 5 2 ? ^ S . ; - £ § =

S »

^ ? i " ^ ~ i - r

~ •=i .1 _= -2 = £ t * ,».;~ ~ . s ? ^ = £ ~

<2 = ? ;£J "» -s .gJ> .

"5i1 If == ££ "5 £''3£ Sill * l-rl-S

s S •S P

- jl^ss-a- l«^-2 -5 r^ = S § S f r ; = ^ g < S 5 5 «« o o 2 » _ f " » > - ;»>.T

» « ST i *=i

i illl s***s*fii*HiI i f iS-Jf- i f^ iJ-s-**"*• if ^s ™ •• — " *• *^ «• ™ ^* **™ 9 £

1 *si* Jii^lfiiU!

OWQ1NAL PAGE 18OF POOR QUALITY

QZ«<03aQ

2?£0IiN <Q Z

N (B

S22«& S£3— Ui

HUb. 3O asz H

o"1

to50.Soo<

AB

ST

HA

CT

f.

^

"!c

is

i

*3H*

7

x

Ic

£Zz(C

5 4-

- 5= *•

M ~—

^ <c" *

.= i5 1**tl ~

I tZ; 5.? i

. * 'j

U3 ^

J^1^3- s* .'§~ ti "f« «•y;

-a £S "^*5 4,

I5:i tl•" I

M s«r a.i-s! i? s

•coC

j

V

•?If

r

INT

RO

DU

CT

ION

* ~4

i *

ll

il** §

l-a-- s= o" 5.- -

- «

Is& «<I"- A

£ 71i .1(i ^

* 725 '/>

« i

«" s

•f '§I i

! 1i -s§ i- ±

<•;

i

<c

«r

*•=

jlj

12i*'

1

4>

•j>

.£

5

s"•s=•

£=w=«

al

I lic

re hat

mil

vet

been

a

sy

ste

ma

tic

elm

ly of

(lie

Ii.

iinl-M

ni. n

n

nin

U ilir

<a

ine

ap

pro

ach

. S

urli

a sliu

ly.

uliicli ha*

llie

a<

liAuui;«

-

>iii|

>ai

i.sou

lie

lupei)

Ibe

va

1101

13 m

aie

rials

and

Elr

urt

iire

s m

u

lie n

iai

availa

lile.

it

ap

pe

ar* I

Ilii*

rla

it o

f m

ate

iial.s

a unifo

rmly

arr

ura

lr i

pip

Avnlrtl b

i>re

.

?i-2 -^

K t,

f. =i *12 *s* ™

Z —1H =

o ^? "«~' 3

— <*S4, S

-•" w— rc: o

•— <-•i« 3

•= 5r !?| -3

1 *£ z

— ««

" -r*. *vw "^i 'S</: -"(C S

— ft>(C •

|ls jl-

.*: *

Ala

liiliiei)

in

H

ie z

iticU

eiiilp

|Z

D)

Mriirln

re l

iy <

-|>

ila\h

i| (t

imvili

. \\

dy b

een il.-

niu

nsuaie

il (o

r liaN |l|. il

is M

ill a

ilia

llon

K,-

f, M

A

IM. T

in-

lo A

IN p

rrcip

ilale

t (o

inie

il iiiM

ile a

n f

rr A

l m

atr

ix l

>y N

-im

i in

i|i|a

iihililiiniiii siiu

aiio

ii.

UIIK

. llie

<|iit>

Mio

ii ii'liH

lier

llie

iM

iiil

ap

is ilir

erl

or

iiiilin>rl

i\

vor

v in

* F r~ ^ ti

t!iii!>. ii NIf:i _j §_

rs— "75

• I. «~'-53 g-

1 Jr*

tb3

£

.2

O

5

JT

^s

*c

i•e*

«a

^

"5.€>

rs*•;

. M

ET

HO

D

•w

CO

MP

UT

AT

ION

A

— 3— .s

~K »

r s.s _- O

*~ CS>. — i

S 7•S51 S*" S* *S

Ii•5 Z

1JU-tt «^

s^it O» _)

A a•= .2"3 •

'— *5* cz =.=• s-of a

|l

Z -f^ 3

4

a

15:c(•

i

Iij='5<01-e

«E

i£n

"i

h IIM

-I| In llie

alu

iiiir

«

* *£

^ >.% ^

7 T~ ?<• e2^N K

•: =•o =5^5 &ta <e* 5

Is*c •>

II

il=iL 5o §•

S XC £

S~i

Inonlp

r In

apply

ill

IIIIIM

lirin

lim

liirnl

m i

I f

"* "~

*f J

"•e <L

T *— o.a. —

^. —

"11 £•I 2Iis" i« .

It•S £JT-S>. 3*

="«r *S e

n" i* 4M .-0. —

— R— W

— i. zs —

If

^ 55 !T

Tc ,£ E.1 '5

i

~

?

*•

ID

-*5«*VsO

Of

if

Su3

M

l.

-=w

i

v>C

«<

"Cs

*fflvtvJ

«

r <^

r £

^ *'- T0. M

? i_C :=* wf« •"•- (V

0 ^^? •£ S~ or i

!i— «

ll

1 •-

x ^o> —-S —7" "™.s '5 t• •X 9,

t '?

!]•»• &•: *

•»• •="*t *r

-•»•- s

> i

fl

ine\

a

rniia

ry i

n ilir t

ota

l energ

y re

sults

.uere

ronve

rKe<l

lo <

1

RE

SU

LT

S

Euu&

tiouo o

f sta

le

renpiii

our

rpxull*

for

Ibe e

iierg

y a

s a

functio

n o

f volu

inp.

i.o.

HIP

piju

a

if th

e r

«lc

ula

le<

l re

tulla

lo (lie

llo

w-b

niit

b-r

'rrr

aiile

et|»aiiu

n o

f »l

ai<»

l. S

oc.

Sym

p. P

roc.

Vol

. 24

2. '

1992

Mal«

rl«li

fUuarc

h S

ocie

ty

III lliln

M>rll

on.

UP

of kla

lc.

UV

IIM

H!

a lit

Mil.

R

: ,• y" X

\( ("f? /j \XJN»

> 0

-1- It

=U. Jt

\ IL .5

.DA DA xpc .Q a x

•1

J3

= «

ii« -i

POORPAGE 'is

a

^*2"

^L

in"o

;^ -•- $-i S, a u 1.'- .= Z-v> S

I 5,1 f| £ •-3GO0) ^^

o ^

4l f =

; o 'j »

3 *

-B -S5a- « ^

< - »

. - : • * _ : = a : . = = : £ , c= = ^ - n o _ . • — s . s

= .

c - -; a ^ i" "5 '"" • o" -J;

- .*• afi

.s

I -= 4 2 'a es~ 2

inoo

8<atn

[ 4 . . . . . . . . ^ . . . . . .

OWGINAL PAGE *Of POOR QUALITY

£ ^ ^» C

ELECTRONIC STRUCTURE AND RELATED PROPERTIES

OF TETRAHEDRALLY BONDED WIDE-BAND-GAP MATE-

RIALS CONTAINING EARLY ELEMENTS OF THE PERIO-

DIC TABLE

\V. R. L. LAMBRECHT. C. H. LEE. K. KIM. A. G. PETUKHOV.E. A. ALBANESI AND B. SEGALLDepartment of PhysicsCase Western Reserve UniversityCleveland. OH-44W6-7079

1. Introduction

The tetrahedrally bonded materials involving early elements of the periodictable, such as Be. B. C, and N have rather extreme properties comparedto the conventional tetrahedrally bonded semiconductors. These uniqueproperties make them interesting for a variety of applications, includingtheir use as hard coatings for mechanical tools, metal/ceramic compos-ites, heat-sinks, and electronics. The latter are the most demanding type ofapplications because they require semiconductor grade purity single crys-talline material in order to realize the superior performance these materialspromise to have. But their extreme properties also makes them difficultmaterials to grow in the form of pure single crystals. Their properties arethus not yet very well known. In order to make a rational choice of materialfor specific applications, it is necessary to understand the interrelationshipsbetween the intrinsic materials properties and the trends in these propertieswith atomic number, crystal structure, and so on. Virtually all materialsproperties can in ultimate instance be related to the underlying electronicstructure. The purpose of this paper is to describe the basic trends in elec-tronic structure of these materials and some of their related properties.The results presented here are based on a systematic study of the elec-tronic structure of these materials carried out over a number of years inour research group. Although many of the results discussed here were pre-

2 W. R. L. Lambreciit et ai.

seated elsewhere they were not included in an overall description of thetrends as presented here. Some new results are included as well. Extensivereferences are provided as a guide to our previous work not covered here indetail.

We first describe some of the relationships between intrinsic propertiesand their importance for various types of applications. Next, we briefly de-scribe our computational approach. Then, we present calculated results forthe groundstate properties such as the equilibrium lattice constants, bondstrength, elastic constants, as well as the underlying electronic structure —in particular the band gaps — for a number of materials systems.

We classify the materials as follows:

— materials containing only elements from the second row of the periodictable: diamond. BN. BeO

— related Be compounds: BeCN-? and Be2C- poly types of SiC:- B-compounds: BN. BP. BAs- the group Ill-Nitrides: BN. A1N. GaN. InN:- related 1I-IV-N2 materials:- alloys among III-N's and heterovalent alloys.

2. Relations among intrinsic properties

Wide-band-gap semiconductors have a number of advantages for electronicapplications over conventional semiconductors. Among these are largerbreakdown fields, resistance to harsh environments, ability to operate athigh-temperatures, and high-thermal conductivity. These various proper-ties are important for a variety of applications where conventional semi-conductors fail. For example, in conventional semiconductors, intrinsic car-rier concentration at a few 100°C wipe out the effects of doping and makesemiconductor devices inoperable. Obviously, since intrinsic carrier gener-ation varies exponentially with the gap, wide band-gap materials have animportant advantage in this respect. Intrinsic carrier generation also playsan important role in the time-decay of space charges induced by opticalexcitation of e.g. an MIS or Schottky diode. This may enable new kinds ofsemipermanent memory and UV detetectors.

At high temperatures, the formation of various types of defects alsostart to degrade the material. For example, dislocation movement proceedsby kink formation and migration. Vacancy and interstitial production alsohave a thermal activation energy. All these processes require bond breakingand thus the stronger the covalent bond the more resistant the materialsare to high-temperature degradation of the crystalline quality. The wide-band-gap materials we will consider here derive their wide gaps from strong

TETRAHEDRALLY BONDED WIDE-BAND-GAP MATERIALS 3

covaieut bonding and not exclusively from high ionicity. They are thusalso strongly bonded. This is also what gives these materials resistance toradiation damage.

The above effects are not only important tor high-temperature operationbut also for high-power and high-frequency applications. In fact, both otthese lead to generation of heat. The thermal conductivity is thus anotherimportant performance parameter for a semiconductor. The wide-band-gap tetrahedrally bonded materials have a high phonon-mediated thermalconductivity because of their high sound velocities, which in turn are closelyrelated to the stiffness of the bonds. The latter can best be described interms of bond strengths, elastic constants and the force constants involvedin certain phonon modes.

Another important parameter for high-speed electronics is the staticdielectric constant. This is because in a very general sense, electronic devicesare based on the transfer of charges from one spatial area to another andthus involve capacitance. One clearly wants as low a dielectric constant aspossible. Also good insulators with low dielectric constant are required forisolating various active parts of a device from each other. Thus, as one startsto use wider gap semiconductors, one also needs even wider gap insulatorsfor packaging applications. The electronic dielectric constant at frequenciesabove vibrational frequencies €00 is roughly inversally proportional to theband gap. In partially ionic materials, the static dielectric constant involvesalso a contribution from the ionic motions. Thus ionicity is an importantparameter. Ionicity can be denned in terms of the ratio of the atomic energylevel difference to the effective gap between bonding and antibonding states.Phillips' ionicity /, [1] is related to Harrison's [2] polarity ap by /, = apwith

aP = \Ee - E a / \ / ^ \ E c - £a|2 + 40*. (1)

Here Ec and En are cation and anion atomic energy levels and i3 is thecovalent interaction between sp3 orbitals pointing towards each other. Thequantity EG = \/\Ec - £a|'

2 + 4d2 can be interpreted as an average effectivegap between cation and auioii levels. Effective values for these levels canbe extracted from our first- principles calculations [3] and define an ionicityscale. The ionicities of some of the materials discussed here are given inTable 1. •

The underlying reason for the unique properties of the materials involv-ing early elements of the periodic table can be understood on the basisof the simple molecular bond picture embodied in Eq.(l). Because of theabsence of lower energy p-like orbitals in the core, the 2p levels are anoma-lously deep compared to 3p levels. Also. 2s levels feel the nucleus morestrongly than 3s electrons because the nucleus is less effectively screenedby the small core consisting only of Is electrons. This relatively compact

W. R. L. Lambrecht et al.

TABLE !. Polarities calculated fromLMTO-ASA calculations for equal cation and an-ion sphere sizes

SiC U.473

BeO 0.818

BNBPBAsBeC'

0.475U.018U.0190.588

A1NGaNInNBeN'

0.8070.7710.7920.701

Individual bond polarities in BeCN?.

nature of the 2s and 2p orbitals is responsible for the small lattice constantand strong covaleiit bonding. The strong covalent interaction J leads to alarge effective gap Erj.

When atoms of the second row are combined with atoms of the thirdrow. there is also an anomalous difference in atomic energy levels. Thisleads to a strong ionic component to the bonding for SiC (even though thelatter is a IV-IV compound) and A1N and a surprisingly low ionicity forBP. The latter occurs because the 2p level of B is sufficiently deep thatits position approaches that of the P 3p level. By Eq.(l) that implies lowionicity. In fact, taking into account that the effective charge on the V-eleinent is given by q = 1 - 4ap, one finds that it-actually has positivecharge when ap < 1/4. This is the case for BP and BAs which as a resulthave inverted ion character: P or As behave as cation while B behaves asanion despite the fact that normally the group-Ill element behaves as cationand the group-V element as aiiion. On the other hand, the enhancementof ionicity due to the deep 2s and 2p levels for SiC and III-N compoundscontributes significantly to their large gaps.

The relative extent of 2p versus 2s orbitals compared to those of 3sand 3p has another important effect. It implies that there is less advan-tage in promoting a 2s electron to a 2p orbital to form directed bonds. Infact, strong bonds can also be realized by means of IT bonding between 2p-orbitals directly. This means that for elements from the second row of theperiodic table there tends to be a competition between tetrahedral bondingand planar sp2 plus TT bonding. Tims. C and BN have competing layeredstructures: graphite and h-BN. BeO does not exhibit such structures [4]which is related to the higher ionicity of this II-VI compound. Indeed ion-icity tends to favor higher coordination.

Ionicity also plays a role in the relative stability of cubic versus hexag-onal stacking of tetrahedrally bonded layers [5]. Thus, the relatively lowionicity compounds BP. BAs and BN and (zero ionicity ) elemental C- fa-

TETRAHEDRALLV BONDED WIDE-BAND-GAP MATERIALS 5

vor the cubic structure while the Ill-nitrides have wurtzit ic structures andSiC exhibits polytypism. lonicity plays also an important roie in electronictransport. The polar mode scattering increases with increasing effectiveciiarges.

Wide band gaps have another important application in optoelectronicswhere they extend the range of light emission frequencies into the blue andUV. In this context, it is important whether the gap is direct or indirect.Other important parameters for electronic applications are effective masses,deformation potentials, and other band structure details. Some exampleswill be discussed below.

Finally, an important aspect of potential semiconductor materials is thepossibility of doping the material n and p-type. Diamond has proved to bequite difficult to dope n-type. GaN was difficult to dope />type althougheffective p-type doping with Mg was recently achieved using post-dopingannealing and Low energy electron beam irradiation I LEEBI) treatments[6]. A1N has sofar resisted effective doping of either type. BN has beenreported to be both n and ;>type dopable ["]. Although this is an importantissue, we will not discuss it further because we have not done studies of thisaspect in our group.

Ill summary, we have here given some examples of how the electronic,structural and bonding properties are interrelated and how they impactdevice applications. In the remainder of the paper we focus on the electronicstructure.

3. Computational Method

The theoretical framework of our first-principles calculations is densityfunctional theory [8]. This approach basically reduces the many-body prob-lem of interacting electrons to that of independent electrons in an effectivepotential with the same ground state electron density as the real system.The total energy of the system is expressed in terms of the Coulombic en-ergy of the electron charge density and the positively charged nuclei, thekinetic energy of the independent model electron system and a remainderwhich is called the exhauge- cor relation energy. The method is in principlerestricted to the study of groundstate properties. The local density approxi-mation (LDA) allows one to incorporate effects of exchange and correlationin a mean-field approximation. It assumes the same functional form for theexchange-correlation energy as in the homogeneous electron gas but ap-plies it locally to each volume element of the real inhomogeneous system:EIC = / erc[ra(r)]ri(r)d3r. We use the parameterization of the exchange andcorrelation functional by Hedin and Luudqvuist [9].

An important point to realize about this theory is that the eigenvalues

'J VV. R. L. Lambreciu et ai.

f , ik of the effective one-eiectrou Schrodiueer equation

/I2

[- — ^ + ! ' f /( r) + 'v-trjuv-ir) = ( n k V n k ( r ) . ( '*)

where r// is the Hartree potential combining the electron and nuclear elec-trostatic potential, and vrc is the exchange-correlation potential, do nothave the meaning of single particle excitations. For a periodic solid the€nk constitute the band structure. In other words the band structures wecalculate in this theory form a useful construct to understand total energyproperties and bonding but do not represent the energies required to ex-tract electrons from the system as measured for example by photoemissionor as involved in optical excitations (excluding excitonic effects). The truequasiparticle excitations are given be the related equation:

with I.IC(r,r'.Enk) the non-local and energy dependent exchange-correlation self-energy operator. In practice, however, the wavefunctionsof the two equations are quite close to each other and the effects of replac-ing the iv<: by Src can be treated by perturbation theory. The self-energyoperator can be approximated by Hedin's GW approximation [10]. Car-rying out this calculation in practice is rather cumbersome and has beendone for only a few systems [11, 12. 13, 14]. The main rinding, however, isthat the spectrum of Enk and enjt mainly differ by a gap correction whichis roughly k-independent and energy independent. The latter can actuallybe estimated easily by means of an approximate treatment of the abovetheory proposed by Bechstedt and Del Sole [15].

The fact that LDA band structures underestimate the gap is well knownas the "gap problem". It is actually not a failure of LDA but a questionof appropriately interpreting the theory. The GW approximation predictsthe correction to be only weakly dependent on crystal structure. Thus LDAband structures fnk can be used to investigate changes in band structurewith crystal structure. The GW calculations actually also predict a smallrigid shift of the bands with respect to the average electrostatic potential.This is irrelevant for most purposes, because one is usually interested onlyin iuterband differences, but it does have an effect on the band-alignment ata heterojuiictiou. The magnitude of this shift appears to depend somewhaton the actual choice of parametrization of the LDA exchange correlationpotential and to that extent is not yet well established theoretically [12].While for wide bauds the only important correction appears to be the abovedescribed gap correction, narrow bands such as core and semi-core levels

TETRAHEDRALLY BONDED WIDE-BAND-GAP MATERIALS 7

are subject to additional self-energy corrections. This is e.g. important lorGa 3d states in GaN [16].

The method we use to calculate the energy bands, i.e. to solve Eq. (2) andobtain the self-consistent potential, is the linear muffin-tin orbital method[17]. Most of the results reported here were obtained in the so-called atomicsphere approximation (ASA) in which Wiguer-Seitz spheres replace the ac-tual Wigner Seitz cells and a spherical approximation is made to the chargedensity and potential inside each sphere. This method is in very good agree-ment with the so-called full-potential calculations which do not make thisapproximation except for quantities which involve small energy differencesor in cases where significant distortions of the local spherical symmetryoccur. For example, cohesive energies, lattice constants, bulk moduli andband structures are usually given quite accurately by the ASA. Surfaceenergies and workfunctions. however, are not because of the strong non-spherical dipole potential near a surface. In fact, the main problem wi ththe ASA is in the spherical approximation to the charge density rather thanto the potential. The errors induced by using slightly overlapping sphericalcharge densities can to a good extent be corrected by means of the so-calledEwald correction [17] in which one corrects for the overlap by redistribut-ing the overlap charge into a homogeneous background charge. We haveused this approach successfully ill our structural energy minimizations forthe chalcopyrite II-IV-N2 compounds [18]. For the calculation of opticalplionon frequencies and elastic constants, which require the calculation ofvery small energy differences in order to stay within the linear responseregime with respect to the strain or phonon distortion perturbation, weneed full-potential calculations. The latter were done using the approach ofMethfessel [19].

Our approach for dealing with disorder in alloys and other specifics willbe described along with the results in the appropriate sections.

4. Results

4.1. DIAMOND, BN, AND BEO.

In Table 2, we present calculated lattice constants, bulk moduli, bondstrengths (defined as 1/4 of the cohesive energy per zincblende unit cell withrespect to free (spin-polarized) atoms), and band gaps for diamond. BN.and BeO. We include calculations both for the cubic and hexagonal crys-tal structures: i.e. cubic diamond and lonsdaleite diamond and zincblendeand wurtzite for the other binary compounds. Our results for hexagonaldiamond and BN are are in good agreement with those of Salehpour andSatapathy [21] and Gorczyca and Christenseu [22] respectively, but aremore complete. (Not all quantities given here are provided in the above ref-

# W. R. L. Lambreciu et al.

erences.) We uoce that the groundstate equilibrium properties are similar inthe hexagonal and cubic forms. The present ASA calculations, however, arenot sufficiently accurate to determine the reiaiive stability of the wurtziteand zincblende structures, in addition, special care has to be taken in us-ing equivalent k-point sets and in relaxing the internal degree of freedomto establish this small energy difference. That would require full-potentialcalculations. The structure determination was not the primary purpose ofthe present study. We also note that typical errors for bulk moduli are 10%. The ~10 % difference in bulk modulus between cubic and hexagonal dia-mond and c-BN is probably overestimated in part because we did not relaxthe structural degrees of freedom, i.e. the relative position of the two sub-lattices and the c/a ratio in hexagonal structure, which, are not determinedby symmetry. The overestimate of the bulk modulus is consistent with theunderestimate of the lattice constant and overestimate of the bond energy.The band gaps of the hexagonal crystals, however are expected to be quiteaccurate.

TABLE 2. Properties of diamond. BN. BeO and BeCNj: lattice constants (tic for cubic, anand c/a for hexagonal) (A), bulk modulus B (GPa), bond energy Ei, (eV), and minimum bandgap Eg (eV).

Diamond

BN

BeO

a e ( A )

B (GPa)Eb (eV)Eg (eV)

ac (A)

B (GPa)Efc (eV)Eg (eV)

oc (A)

fl (GPa)£6 (eV)E, (eV)

cubic

theory3.58

4344.24.4

3.62

3813.84.5

3.80

2243.46.9

expt.3.567

4423.75.5

3.616

290-4653.36.4

3.79

3.0(9.3)6

hexagonal

theoryan (A)c/aB (GPa)Eb (eV)

T-A Eg

an (A]c/aB (GPa)Et, (eV)

T-.V £9 (eV)

an (A)c/a5 (GPa)Eb (eV)

T-.V £, (eV)

expt2.46

1.633°5064.73.6

2.51

4254.25.7

2.641.633°

2483.78.3

2.522°

(4.716

2.561.633"

(7.6)"

2.6641.623

224-2493.010.7

r- A-

r- A'

r - r0 assuming ideal c/a = >/8/3

6 estimate assuming gap correction to be independent of structure

TETRAHEDRALLY BONDED WIDE-BAND-GAP MATERIALS 9

The baud gaps depend significantly on the structure. Even the k-pointlocation of the conduction band minimum changes. In Fig. i we show howsome of the conduction-band eigenvalues relative to the valence band max-imum change across the series. One may note that the s-iike F( states inboth cubic and hexagonal structures come down in the series while ther« state of hexagonal and F^ state of cubic move up. Mk and h\ statesof the hexagonal structure also move up toward BeO. One can relate thek-points of the cubic fee and hexagonal hep Brillouiii zones by assumingthe ( l l l j c direction to be equivalent to the [OOOll/i direction and the [110]c

direction to be equivalent to the [1120]/, direction [23. 24]. The cubic A"point corresponds to the point 2/3 along M-L/, and is different from A"/t.\Ve note that in all cases going from cubic to hexagonal the A conduc-tion band minimum moves up in energy and the A" minimum comes down.This is thus a structure related effect. In the case of BeO. however, thef minimum is lower in energy than Kt, and thus leads to a direct gapmaterial. We also note that the gap in hexagonal diamond is lower thanin cubic while the inverse is true for BN and BeO. The bond energy andbulk moduli decrease across the series diamond-BN-BeO while the latticeconstant increases. Both hexagonal diamond and wurtzitic BN appear tobe metastable phases. It is not clear whether there is a stability regionin the phase diagram or whether their occurence is due only to kinetics.The latter appears to be the case since hexagonal diamond has sofar onlybeen formed under shock compression conditions. BeO is only known in thewurtzitic phase. This is consistent with its high ionicity. The cubic phaseis significantly higher in energy (0.14 eV/atom according to Chang et al.[25]) and is unlikely to be formed.

Relation to graphitic structures \Ve note that diamond and c-BN havecompeting layered structures graphite and h-BN. which are actually theequilibrium phases at ambient pressures. As shown by Fahy et al. [28.29] theenergy differences between the layered forms and the tetrahedrally bondedforms are small but there is a large energy barrier which impedes the tran-sition from one phase to the other. In their models, the transition pathfrom rhombohedral graphite to diamond, or h-BN to wurtzite BN corre-sponds to a gradual decrease of the distance between the layers which thenstart to buckle. Layered phases of BeO have been shown to be unstable byWentzcovitch et al. [4].

Diamond tetrahedral bonding, however, can also be stabilized by sur-face hydrogen. Hydrogenation of aromatic molecules yields buckled tetra-hedrally bonded rings. Similarly, graphite sheets start to form buckled ringswhen hydrogeiiated from the edges inwards. This leads to a curling or bend-ing of the planes [30] and to the formation of a diamond-like structure on

PAQ€OF POOR

R. L. Lambreciu et ai.

20

3N BeO

15

•Si.

UQCLU•z.LU

10 -

5 j-i\i

3.5 3.6 3.7 3.8effective cubic lattice constant (A}

Figure I. Conduction band eigenvalues of diamond, BN and BeO versus effective cubiclattice constant. Full lines and symmetry notation on the left refer to the hexagonalstructure, dotted lines and symmetry notation on the right refer to the cubic structure.

the prism planes of graphite. This is suggested by the open circles in Fig. 2which shows the calculated structure at the prism plane interface betweengraphite and diamond. That relaxed structure was calculated [31] using thesemi-empirical Tersoff potential (321. It was conjectured by Lambrecht etal. [31] to form a critical step in the nucleation of diamond.

Interfaces We have also studied interfaces between some of these materi-als: diamoiid/c-BN [20]. diamond/BeO [33] and diainoud/Cu [34]. We foundthat the diamond/c-BN offset is of type II. i.e. the valence and conductionband of diamond are both above those of c-BN. This also has importancefor alloys, as will be discussed below. The bonding at these interfaces isquite strong because there are no dangling bonds and the valence bands ofthe materials overlap each other in a wide energy range. In contrast, thebonding at Cu/diamond interfaces is much weaker because the Cu3d-bandis narrow and essentially only the Cu4s bands form bonds with the diamonddangling bonds. An important quantity is the adhesion energy IT,,, which

TETRAHEDRALLY BONDED WIDE-BAND-GAP MATERIALS

[00011

C

B

B

A

Figure 2. Structure of the prism plane interface between graphite and diamond. Threelayers of diamond almost perfectly match two layers of graphite. Hydrogen atoms (smallcircles) satisfy the dangling bonds of the "non-bonded" diamond layers. The structureat the interface (open circles) indicates that the curling of buckled rings forms an initialdiamond nucleus.

is given by

= 7.4 T7B - (4)

where 7.4, 75 are the surface energies of the materials brought together atthe interface and 7, is the interface energy. For diamond/C'u. the adhesionis much weaker than for the interfaces of diamond with c-BN and BeO.mainly because the Cu surface energy is so much smaller than that ofthese tetrahedrally bonded materials. Surface energies basically scale withcohesive energies because they involve bond breaking. Table 3 gives thecalculated surface energies and adhesion energies. Details can be found in[20, 33. 34].

The suitability of c-BN and BeO as substrates for heteroepitaxial dia-mond growth was confirmed by several experimental studies [35. 36. 37).While fully developed epitaxial films could be grown 011 c-BN [36]. only ori-ented growth of particles of size up to about 10 /tin in diameter and withslight misorientations of a few degrees has been realized sofar on BeO. Thisis consistent with the better lattice match and adhesion energy to c-BN.

W. R. L. Lambreciu et al.

TABLE 3. Surface and adhesion energies i in J / m 2 1

diamond ideal (110) surface energy 6.5c-BN ideal 1110) surface energy 5.2zincblende BeO ideal (110) surface energy 4.8diamond/c-BN adhesion energy 5.4diamond/BeO adhesion energy 4.6diamond 2x1 (111) surface energy 5.5Cu (111) surface energy 2.0diamond/Cu adhesion energy 2.3

4.2. RELATED BE COMPOUNDS: BE2C, BE3N2 AND BECNj

Besides the elemental IV. III-V and II-VI materials of the second row of theperiodic table, three other compounds with tetrahedral or closely relatedbonding are of interest: BeCNj, Be7C. and BegN?. The main propertiesobtained in our calculations for the first two of these are given in Table 4.

TABLE 4. Properties of BeCN? and Be2C.

StructureLattice constantc/aw6

Bulk modulus BCohesive energyBand gap (eV)

a (A)

(GPa)(eV/atomi

BeCN2

chalcopyrite3.711.960.303337.7

4.2 (6.3)c

direct T - T

. Be,Cantifluorite

4.29 (4.342)"

2136.0

1.15 (2.15)c

indirect F - A"

0 expt. value in parenthesesb u is the internal displacement parameter of the chalcopyrite structure [18]

c With estimated gap correction based on interpolation for related materials.

Beryllium carbonitride BeCNz is todate a hypothetical material [26, 18],which is here assumed to have the chalcopyrite structure in analogy withother IMV-V2 compounds such as ZnGePz. One may think of it as beingderived from BN by replacing every other B (group-Ill) by a Be (group-II)or a C (group-IV) atom. This substitution maintains the local stoichiometrybecause each N atom is surrounded bv two Be and two C atoms. The

TETRAHEDRALLV BONDED WIDE-BAND-GAP MATERIALS 13

Figure J. Ckalcopyrite structure assumed for the hypothetical material BeCNj. Thestructure represented here is idealized. Bond lengths relax by a relative motion of the Nsublattice towards C and away from Be. If Be and C atoms are both replaced by B weobtain the zincblende stucture of c-BN.

structure is shown in Fig. 3. The bonding in this material is tetrahedral.There are several interesting things about this material.

— It has a lattice constant intermediate between those of c-BN and BeOand may thus form a good bufferlayer for diamond or c-BN growth.We note that it is more ionic than c-BN and may thus, as BeO. bemore likely to avoid layered sp- bonded structures.

— It has a pseudodirect band gap in the sense that the "directness" re-sults from the fact that the zincblende A" point is folded onto F inthe chalcopyrite structure. A strong mixing of F and A" states, how-ever, is expected to occur due to the significant difference between Beand C potentials and. in addition, due to the lattice distortions. Thisshould lead to a significant dipole matrix element and thus stronglight-emission.

- Related materials such as ZnGeP? are promising for non-linear opticsapplications [27] due to the occurence of IV-V bonds. If. as is expected,similar non-linear optical advantages of the C-N bonds occur here, theywould be of particular interest because the band gap is already in theUV.

H W. R. L. Lambreclu et ai.

/-?£•><!' A material with bonding cioseiy related to tetrahedrai bonding isDe->C. It has the cubic aiuifluorite structure. The latter consists of a feesublattice of carbon atoms with Be atoms occupying ail tetrahedrai inter-stitices. In the zincblende structure by contrast, only half of the tetrahedraiinterstices of the fee lattice are filled with a counter ion. We have calculatedthe ground state properties and baud structure of this material [33]. Themain results are given in Table 4. The material is an indirect semiconductorwith band gap and lattice constant close to those of 3C-SiC.

Formation energies Under atmospheric conditions Be2C is unstable to-wards formation of Be(OH)2 by reaction with water vapor, or BeO withOa. This indicates that Be2C is less stable than Be(OH)2 or BeO. Neverthe-less, a small amount of Be^C formation was detected in growth exprimentsof diamond on BeO [37]. This indicates that in spite of the much smallerformation energy of Be-^C than that of BeO. Be2C can form under the veryreducing conditions of diamond growth. This is significant for the prospectsof synthesising BeCN2- Our calculations predict a much higher energy offormation for this material, close to that of BeO. The energies of formation,defined with respect to the elements in their standard state and being pos-itive in the case of exothermic reactions, i.e. when the compound has lowerenergy than the sum of the energies of the elements, are given in Table 5.They were calculated by combining our LDA calculated cohesive energiesof the respective solids with LDA binding energies of the molecules 02 andN-2 given by Jones and Gunnarssou [39]. In this way, the systematic LDAoverestimate of bonding is cancelled out. The energy of carbon was takento be that of diamond instead of graphite, the latter being almost equal inenergy. Zero point motion corrections were not included in our cohesive en-ergies because for the present purpose, we only need a qualitative estimate.We note that a large part of the stabilization energy of BeCNs is due tothe effective relaxation energy of 1.2 eV/atom. The related material Bes^is included in this table. Its structure is an ordered vacancy structure ofthe antifluorite structure. One may see that per equivalent amount of Beatoms, i.e. 1/3 mole of Be3N2, its energy of 47 kcal is also much smallerthan that of BeCN2. This indicates that under reaction conditions whereeither Be2C plus N2, or Be3N2 plus graphite, or BeCN2 could form, thelast is most likely in a thermodynamic sense. It indicates that when Beis exposed to C and N simultaneously, BeCN2 may form exothermicaUy.This calculation, being strictly thermodynamic. of course, does not tell usanything about possible reaction barriers.

TETRAHEDRALLV BONDED WIDE-BAND-GAP MATERIALS I-5

TABLE 5. Exothermic ener-gies of formation liu kcal/moieiof some Be compounds.

BeO1/2 Be,C1/3 Be.?N?BeCN2

theory

15814.5

156

expt.

1451447

4.3. SILICON CARBIDE

Silicon carbide is unique among the wide-band-gap semiconductors in thesense that it exhibits poiytypism. Over a hundred different crystal struc-tures called polytypes have been discovered [40]. All are tetrahedrailybonded but differ in the stacking of what are the {111} planes in cubicSiC (called 3C-SiC or J-SiC) and the basal planes {0001} in hexagonal orrhombohedral SiC, sometimes collectively denoted a-SiC. The polytypescan most easily be described as a periodic structure of twin boundaries incubic SiC. When a twin boundary occurs every 3 (2) layers, we obtain (3)((2)) which is the Zhdanov notation [41] for 6H-S1C (4H-SiC). When theyoccur every layer we obtain the wurtzite structure which is purely hexag-onal and denoted 2H-SiC or (1). Rhomobohedral polytypes exhibit morecomplex patterns such as (32) or 15R.

The relative stability of the various polytypes has been studied exten-sively by Heine et al. [42]. Some of the main conclusions of their work isthat polytypes with bands of three and two layers between twin boundaries(e.g. 6H. 4H. 15R) are most stable, followed by 3C SiC, while least stableis 2H-S1C. On the basis of their total energy calculations, they proposed amodel in terms of interlayer interactions. The origin of the poiytypism inthis context is that the first nearest neighbor layer interactions prefer tostack cubically with respect to each other while the second nearest neigh-bor interaction favors hexagonal stacking and has almost exactly half themagnitude of the first nearest neighbor interaction. The third and higherneighbor layer interactions are found to be negligible. Their model explainsthe preference for polytypes with bands of two and three successive cubi-cally stacked layers. It also explains why in surface growth, if complete 3Dequilibrium is not reached, there is a tendency to form cubic stacking. Thisoccurs because for the surface layer there is only one next neighbor layerinteraction to be considered instead of two in the bulk. 'Finally.- small dif-ferences in the phonon contribution to the free energy were claimed to be

16 \V. R. L. Lambreciu et ai.

the origin oi'different temperature stability regions for 4H and GH. Experi-mentally, it is found that poiytype selection can be influenced by a numberof kinetic factors besides thermodynamic equilibrium. Growth on vicinalsurfaces tends to stabilize the same poiytype as the substrate on which it isgrown through a ledge growth mechanism, imperfections on large terracestend to promote independent nucleation and cubic growth. Lower temper-atures also tend to lead to cubic growth. All of this is consistent with thenotion that there must be sufficient mobility of species on the surface tomove to ledges to continue the substrate poiytype and that otherwise cubicnuclei tend to form. In case of cubic growth in the [111] orientation one is of-ten faced with the problem of so-called double position boundaries (DPS).The latter correspond to the two equivalent orientations of a cubic nucleuswith respect to the substrate. Stress can induce polytpic transformations[43]. Also, ion implantation can influence polytypism after recrystallizationof the damaged area [44]. Finally, doping and growth rates may have aninfluence on which poly types form during growth [44].

Our own work has focused on the relationship between polytypism andproperties. Integrated properties such as elastic constants [45] are onlyweakly dependent on poiytype while band structure depends dramaticallyon poiytype. This is because baud structure essentially corresponds to theformation of electronic standing waves and is thus very sensitive to thestructure [24]. The changes in band structure also lead to important changesin optical reflectivity [46].

Fig. 4 shows the dependence of the minimum gap and some specific indi-rect gaps on poiytype. The polytypes are here ordered in order of increasinghexagonality, i.e. the proportion of layers that are stacked hexagonally. Onemay see that changes in the position of the conduction band minimum oc-cur. While the low hexagonality polytypes have their minimum along thehexagonal M — L-axis. including the cubic A"-minimum. 2H-S1C has itsminimum at K similar to hexagonal diamond and BN. One may also no-tice that the specific gaps do not vary monotonously with hexagouality.Details on how the band structures are related to each other can be foundin [24, 23].

Fig. 5 shows the calculated UV-reflectivity spectra of some SiC poly-types compared to experimental data. We note that a single k-independentand poiytype independent gap correction of 1 eV as applied. The latternot only provides good agreement for the various peak positions in the re-flectivity spectra but also for the minimum gaps given above. This lendssupport to our statements in the section on computational method that thegap corrections beyond the LDA are relatively structure independent.

The changes in band structure between the polytypes leads to differenteffective masses, and hence are important for transport [24]. For example.

TETRAHEDRALLY B O N D E D \ V I D E - B A N D - ( i A P MATERIALS

5.0 r

3C 6H '5R -H 2H

v.T-Kgap

T-U gap

3C

2.020 40 60

HEXAGONALITY (%)80 100

Figure 4- Band gaps of SiC polytypes. Squares: experiment, solid circle: thoereticalminimum gap: diamonds, upward and downward triangles, specific k-point gaps as indi-cated: dashed line, nearly linear T - U gap along U = M - L line of hexagonal Brillouinzone.

it is found that (iH SiC has twelve ef[uivaieut minima along the M — L axesclose to M. which are elongated in the c-direction and with only small bar-rier between them. This means that at larger temperature the two minimadisplaced along the c-axis can effectively be thought of as one minimumwith a very large mass anisotropy. This is consistent with recent measure-ments of a large anisotropy in mobilities in 6H-SiC [47].

We have also studied several interfaces involving SiC. SiC/AIN andSiC/BP interfaces [48] will be briefly discussed in the section on alloys be-cause of the close relationship among these topics and the new results wepresent here on the corresponding alloys. We also studied SiC/TiC inter-faces which are of interest for metallization of SiC arid in connection withceramic reinforcement of metals [49]. Finally, we studied so-called inversiondomain boundaries in SiC [50]. These are planar defects that occur oftenduring growth on stepped Si {001}.

ORIGINAL PAGC f5OF POOR QUALITY

\V. R. L. LaniDreciit ei ai.

RE

FLE

CT

IVIT

Y

0.5 r 2H ° B

0.4 \-A ,

0.3 ;-

0 " t , 1 ,• * - 1 ' I ' - - -

0 . 5 r 4 H D a

0.4 r /v-x' — -.

0.3 1- ^X

0 "» i — T-7"

0.5 i- 15R

0.4 h Dx-X^"--^X

0 " I 'T; - i.1. j i i - - - i . . .

0-5 h 6H D. B

0.4 - D /D--O

0.3 - .S

O.t. \ ^ ^ i t

0.5 h 3CB ;

0.4 h X^xA / \

0.3 h /— ^

n 9 i . -r-rr

C .

C

>-*K

\ ~5

C- r^^-

C !

1

C ".

4 5 6 7 8 9PHOTON ENERGY (eV)

10

Figure 5. Reflectivity of SiC poiytypes. Solid lines experiment, dotted lines theory.

4.4. BORON COMPOUNDS: BN. BP AND BAS

Table 6 presents an overview of boron compounds c-BN. BP and BAs as

TETRAHEDRALLY BONDED WIDE-BAND-GAP MATERIALS

TABLE 6. Properties of tetrahedrallv bonded boron compounds: <ilattice constant. B bulk modulus. Eb bond energy, £, minimum bandgap, and conduction-band minimum location km ,n .

BN

BP

BAs

presentppk

Expt. a-b

presentpp6

Expt."'6

presentppc

Expt.a'c

a (A)

3.623.6063.6154.514.5584.5384.79

4.7774.777

D (GPa)

381367

'290-4(55172165

173-265130145-

Eh ( e V )

3.83.63.32.82.92.62.32.6-

E, ( e V )

4.54.2ti.41.351.22.41.061.2

2.1d

A' mm

.V

X

AA

AA

Experimental values from Landolt-Bornstem iables [53].6 Pseudopotentiai calculation by VVentzcovitch et al. [51]c Pseudopotentiai calculation by VVentzcovitch et al. [52]

Our estimate assuming correction same as for BP. Expt. values are uncertain.

calculated using the ASA-LMTO method. We have included data on BN,which given above to facilitate the comparison to BP and BAs. The reasonwhy BP in the table below has a slightly underestimated lattice constantwhile the others have sligthly overestimated values is that the calculationof [48] for BP did not include the so-called combined correction. For thepresent purposes this is not relevant. Our results are in good agreementwith the previous calculations by VVentzcovitch et al. [51, 52], although out-values for the gaps differ somewhat. We note the usual "softening" as we goto heavier elements. Due to the very low ionicity mentioned earlier, the BPand BAs compounds have a band structure somewhat similar to Si withthe minimum along the <A axis close to A". Experimental band gaps arenot well known. We estimate a gap correction of ~1 eV for both BP andBAs on the basis of their similarity to SiC and Bechstedt and del Sole 'smodel [15]. We emphasize that the low ionicity is expected to reduce polarscattering which may give these materials an advantage in mobilities.

4.5. GROUP-III NITRIDES

We have performed total energy and band structure calculations for the en-tire series of Group-Ill nitrides in both the zincblende and wurtzite struc-ture. The two forms are close in energy and can both be stabilized undersuitable epitaxial growth conditions. This has already been realized for GaNand InN but not for A1N. At high pressure the nitrides exhibit a phase tran-

PAGE BOF POOR QUAUW

•JO W. R. L. Lamoreciu et ai.

-"ition 10 the rocksait s t ructure. This is a well known phenomenon ciue to! lie increase in ionicity under compression and the tendency towards higher'.•uorciinatiou for ionic compounds. It has been studied theoretically by Gur-• zyca and Christensen i22! and by Muiioz and Kunc |54] and experimentallyl>y Perlin et al. [55]. Volstadt et ai. [56] and Ueno et ai. [57]. Our calculatedgroundstate properties can be found in 1231 while full details on the bandstructures are given in [58].

15

CCLUZLU

10 I-

s -

-5BN A!N GaN InN

Figure o. Trends in selected band states of III-nitiides with atomic number in zincblendeand wnrtzite structure.

Trends in the band structures. The trends of some eigenvalues with, cationin the series of nitrides are shown in Fig. 6. It includes the important eigen-values near the gap, i.e. the valence band maximum fj'5 for ZB or f \ and Fgfor wurtzite. the conduction band state FI and A"| for WZ and A'f for ZBwhich "compete" for the miiiumum. The conduction band T\ states in bothwurtzite and zincblende are mostly s-like (actually purely s-like in ZB) and.being antibonding states, have important catiou-s components. One maysee that these states show the largest variation with atomic number, and.in particular, are monotonically decreasing with increasing atomic number.

TETRAHEDRALLY BONDED WIDE-BAND-GAP MATERIALS Jl

Of course, the above statement requires that we compare the different baudstructures with respect to the same reference ievei. The natural referencelevel used here is the ASA zero of energy [59] which coresponds to the av-erage of the point charge electrostatic potential and is close to the averagepotential in the interstitial region. This is not sufficiently accurate to de-termine band-offsets because that requires consideration of charge transfer.But it is adequate for our present qualitative discussion. The reason forthe downward trend with atomic number is well known and also ocurs forexample in the series C-Si-Ge-Sn. It is simply due to the fact that s-orbitalshave a non-zero value at the nucleus and as such feel the —Z/ r potential.Thus, heavier elements have lower s-ievels with respect to correspondingp-levels. Since the FI state is s-like and the A" f-level in zincblende (or dia-mond) is a mixture of s and p-like components and the A"| state in wurtziteor lonsdaleite is purely p-like. this explains why diamond. SiC and BN areindirect gap materials in either zincblende or wurtzite structure while GaNand InN are direct in both. A1N turns out to be indirect in the zincblendestructure and direct in the wurtzite. Even in WZ. however, the indirectand direct gap in A1N are close to each other in energy. Fig. 6 shows thatt he variation of FI is linear from Al to In but has a different slope betweenB and Al. This emphasizes again the peculiar role of the second row ele-ments. In addition, the F^ state is lower in c-BN than the F£ state. Thisagain is a manifestation of the relative energies of p and s-levels and thusa result, once again'of the absence of a p-like core for 2p wave functions.We note that the levels shown are LDA levels which are underestimated forthe conduction band states as discussed above. Again, however, the LDA isadequate for the study of the structurally induced changes in band gap. Itis noteworthy that for zb-InN the LDA gap is actually negative while thereal gap is 1.9 eV. The gap in this material is thus almost entirely due tothe self-energy correction.

lonicity explains why the bandgap variation is so much stronger in theIII-N series than in the III-As or III-P series. The reason for this is that thehigh ionicity makes the cation component of the FI state relatively moreimportant. The lower ionicity of GaP compared to GaN. for example, isconsitent with GaP having an indirect gap. Indeed, the FI state havinga smaller Ga (heavy element) component is less sensitive to the nuclearpotential and lies relatively higher with respect to the Gap-P, like A'f state.By the time we get to GaAs. the anion is also a heavier element and thus italso tends to bring the F^ state down with respect to p-iike states. lonicityalso explains why SiC is indirect in both structures while A1N is only directin wurtzite. The reason is that SiC being less ionic has a relatively smallercomponent of the Si (i.e. "heavier" element) in the conduction band Ffstate. We note that even for Si. the p-iike F^ state lies below the s-like FI

22 W. R. L. Lambrecht et al.

state.

Elastic constants. \Ve have recently also performed full-potential calcu-lations of the zincblende Ill-nitrides including uniaxiai strain distortions.This allowed us to obtain the full set of cubic elastic constants. From thelatter, the hexagonal elastic constants can be obtained by a simple tensortransformation operation [CO] under the assumption that the bonding issimilar in both materials. The changes due to the twinning of the tetrahe-drons can be incorporated in this transformation [60].

TABLE 7. Elastic constants for Ill-nitrides in GPa. (expt.values in parentheses)

cubic

hexagonal

B

C1,,C,2

C,3

Css

A1N

2U2

504152199

398 (345)°140 (125)70 (120)

468 (395)96(118)

GaN

201

296154206

396 (296 )6

144 (130)64(158)

476 (267)91 (24)

InN

139

184116177

271 (190)"124(104)21 (121)

375 (182)46 (99)

0 Tsubouchi and Mikoshiba [61]k S avast en ko and Sheleg [62]

Table 7 gives our calculated elastic constants for cubic and hexagonalIll-nitrides [63]. The deviations from the experimental values are of theorder of 20 % for A1N — except for the C\s which is difficult to measure— and worse for the other two materials. Particularly disturbing is thedeviation by almost a factor 2 for the Cn and Cy± in GaN and InN. Theselarge elastic constants are closely related to the Youngs moduli in the c-plane and perpendicular to it and should be more easy to determine thanthe shear moduli. These discrepancies are much worse than typical for LDAresults and suggest problems with the experimental values. We note that forSi [19], Diamond [19], and SiC [45], the elastic constants obtained by thismethod are to within 10 % of the experimental values. Only for A1N are theexperimental values given here based on sound velocity measurements. Inthe other cases, they are based on analyses of X-ray linewdiths in powders,which is a rather indirect and probably inaccurate method for obtaining

TETRAHEDRALLY BONDED WIDE-BAND-GAP MATERIALS -3

elastic constants. The poor quality and small sizes of crystals available att he time of the measurements is probably responsible for these experimentalproblems.

GaJd and In4d semicore states A noteworthy aspect of the GaN and InNbandstructures is the effects of the Ga3d and In4d semicorelevels. Becausethe lattice constants of these materials are fairly small, the Ga or in atomsare close to each other and their semicore d states have significant inter-actions. Also, they are close in energy to the deep N2s states. Indirectinteractions through hybridization with N2s thus further broadens thesebauds. The LDA eigenvalues of Ga3d and In4d actually overlap with theN2s bands. This means that the Ga3d and In4d participate in the bonding.It is found necessary to include these d-states as valence bands in order toobtain accurate values for the ground state properties (16. 23. G4i . Oa theother hand, in X-ray photoelectron spectroscopy (XPS). the Ga 3d levelsare found to be situated well below the N2s. This is another manifestationof the difference between LDA eigenvalues and the true quasiparticle exci-tations of the system. These effects are more pronounced for narrow bandsand increase with increasing energy separation from the Fermi level. Wefound that the XPS spectrum of Ga3d can be well accounted for by meansof the so-called ASCF approach [16].

Interfaces We have calculated band-offsets among III-N semiconductors[65. 66]. The calculations were performed for the zincblende crystals and forthe {110} interface. Since the valence-band maxima are similar in nature inboth strucures, one may expect that valence band offsets would be similarfor the wurtzite. We can then obtain the conduction band offsets by addingthe experimental gaps.

TABLE 8. Valence-bandA£u = £f - £;* (eV) and conduc-tion-band AEC = E? — EC offsetswith A£* for zincblende andfor wurtzite.

A£« A£e'

AIN/GaN 0.85 0.81 1.T8GaN/InN 0.51 1.07 1.09AIN/InN 1.09 2.15 3.14

-~^ W. R. L. Lambreciit ec ai.

The vaience-band offset ot" AIN/GaN measured by XPS l O T j is U.8 ±0.3'.?V in good agreement with our results. Interlace orientation effects need tobe studied separately and are expected to be of the order of a few 0.1 eV.

10 15 20PHOTON ENERGY (eV)

25 30

Figure 7. UV-reftectivity of GaN. The bottom panel shows theory and experiment forwurtzite and E L c: the second panel shows the theoretically predicted polarizationdependence for wurtzite: dashed line E \\ c, full line £ J. c; the top panel shows thecalculated reflectivity for zincblende.

Optical properties We have calculated optical response functions for A1Nand GaN. Our results for GaN are shown in Fig. 7. As for SiC polytypes.we find that good agreement with all major peak positions in experimentalreflectivity can be achieved with a single constant gap correction or —1.0

PAQC fg

QUALITY

TETRAHEDRALLY BONDED WIDE-BAND-GAP MATERIALS -'5

"V. There is. however, a significant discrepancy in absoiuce intensi ty wuichincreases with increasing energy. A possible explanation for this is diffusescattering in the experiment. Our calculations predict significant differencesin spectra between zincblende and wur tz i te and significant anisotropy in thecase of wurtzite. These remain to be confirmed by experiment. For wur tz i teGaN. our calculated results and the experiments by Rife et al. [681. shownhere are in good agreement with those of Olson et al. [69]. A more detailedanalysis of these spectra will be given elsewhere.

0.5

0.4

> 0.3

OLUill 0.201cc

0.1

0.0

WZ-AIN

10 15PHOTON ENERGY (eV)

20 25

Figure 8. Calculated refletivity of wiirtzite A1N: full line. E L c. dashed line. £ || c.AuIDA gap correction of 1.72 eV was included.

Our calculated reflectivity for wurtzite A1N is shown in Fig. 3. Tkeresults are in good agreement with those of Loughin et al. [70] as far aspeak positions and general shape of the spectrum is concerned in the energyregion below ~ 10 eV. As for GaN. our calculations predict a significantlystronger reflectivity at high energies than observed in the experiment.

4.6. II-IV-N2 MATERIALS

An interesting class of wide-band-gap semiconductors, which has sofar re-ceived very little attention, is formed by the II-IV-V2 materials. These arederived from tile-corresponding III-V compound by substituting half of theIII elements by a group II and half by a group IV. This does not completelyspecify the crystal structure. We have performed studies of thes materials

OFPAQ€ RQUALITY

26 \V. R. L. Lambreclu et al.

assuming tiie chalcopyrite structure because the latter is a simple super-structure of zincbleiide and is the equilibrium form of II-IV-V2 materialsother than nitrides such as ZnGeP?, ZiiGeAs?, AlgSiP-j. hi this structure,each group-V atom is tetrahedrally coordinated with two group-II atomsand two group-IV atoms. We have studied the compounds with 11= Be orMg and IV= Si and C and also MgSiP-j [181. We have already briefly dis-cussed BeCN-2 along with diamond. c-BN and BeO. Among these nitrides,only the silico-nitrides with IV=Si have sofar been synthesized. They havea crystal structure that is derived from wurtzite in a manner similar to howchalcopyrite is derived from zincblende. We have sofar only studied the sim-pler but related chalcopyrite structure. Our main results are summarizedin Table 9.

TABLE 9. Properties of II-IV-Vo materials: lattice constant a. bond lengths </.bond energy Eh. bulk modulus B, band gap E,,.

BeCN?MgCN,BeSiN7

MgSiNj

MgSiP,

a (A)

!.714.114.10

4.44

5.64

div-v (A)

1.501.581.701.76*1.771.76*2.26

<// / -v ( A )

1.711.921.88

1.76*2.08

2.08*2.59

Eh (eV)

3.82.53.0

2.8

1.9

B (GPa)

333210240

195

75

E, ( eV l

4.62 (6.3iu

3.69 (5.014.47 (5.7)

4.03 (5.3)

1.77 (2.5)

0 Including estimated correction, see Petukhov et al. [18]b Experimental values, see Petukhov et al. [18] for original refs.

Generally speaking (except BeCNj), these materials have higher gapsthan the corresponding Ill-nitrides, which is probably related to theirhigher ionicity. The folding effects of the chalcopyrite lead to conduction-band minima at F, except for MgSiPj. Except for BeCNj, however, thevalence-band maxima are displaced from F. Although the gaps are thusindirect, except in BeCN-^, they are very close to the corresponding directgaps. In Fig. 9 we show the gaps versus lattice constant along with thoseof the related nitrides, diamond, and BeO. The figure also includes a calcu-lated gap for A1BN2 in the (idealized, i.e. unrelaxed) chalcopyrite structure.The lines are only guides for the eye, and should not be interpreted as gapsof alloys since the latter may exhibit non-linear behavior with lattice con-stant. One may appreciate from tlu's figure that these materials all havegaps similar to those of diamond and c-BN but extend over a lattice con-stant range where no binary tetrahedrally bonded compounds occur. Theythus fill the gap between the lattice constants of the tetrahedrally bonded

TETRAHEDRALLV BONDED \VIDE-BAND-GAP MATERIALS

12 r

10 I-

CL

O 8 !-

<CQ

6 I-

4 L-

BeO

z-BeO

w.

c-BN"

Od-C

BeCN. AIM

MgSiN2

AIBN

3.4 3.6 3.8 4.0 4.2a (Angstrom)

4.4

Figure H. Band gaps of II-IV-N? materials and related binary compounds versus effectivecubic lattice constant a (A). The squares indicate wurtzitic compounds, the circles cubicor closely related structures. Note that for BeO and A1N the equilibrium phase is wurtzitewhile the zincblende phases indicated by z- are hypothetical.

materials made from elements of the second row (B, C. N) only and thosemade from mixtures of the elements of the second and third row (SIC, BP.A1N). This makes them possibly useful as bufferlayers or substrates for thegrowth of c-BN and diamond. The figure suggests that even higher gapscan be expected for the corresponding wurtzitic compounds.

An interesting question is whether the carbonitrides which sofar havenot been synthesized can be stabilized. From the above cohesive energiesand our calculated cohesive energy of SiC and diamond it follows that thereactions

MgSiN2 + 2C - MgCNo + SiC.

BeSiN2 + 2C - BeCN2 + SiC.

= llSkcal/mole (5)

= -84kcal/mole (6)

are respectively endothermic and exotherimc. The relative instability ofMgCN2 with respect to MgSiN2 follows from its larger mismatch in atomic

'-'3 W. R. L. Lambreciu et ai.

radii. Similarly BeCN-2 has less strain to overcome than BeSiN-j. The syn-thesis of BeCNj, which is the most interesting of this class of materialsbecause its properties are closest related to those of diamond and becauseit is a direct gap material in contrat to its parent compound c-BN. wouldseem to be synthesizable by the substitution of Si by C in the existing com-pound BeSiN2. It remains an open question whether alternative /r-bondedcrystal structures are possible for these materials. Their higher ioiiicitywhich favors higher coordination, however, disfavors such structures. Thishas been argued earlier in this paper to be the reason why such structuresdo not appear for BeO. Since BeCN2 is intermediate in properties betweenc-BN and BeO. we are presently unable to predict whether it will behavelike BN and exhibit TT bonding, or whether it will behave like BeO and notshow TT- bonding.

1.7. ALLOYS

Several alloy systems among the wide-band-gap semiconductors are of in-terest. The Ill-nitrides are known to form continuous alloys. Of these. \vehave sofar only studied the GaxAl t_ rX system [71].

xN alloys The baud gaps were found to behave nearly linearlywith concentration and a cross-over takes place between direct and indirectgaps in the zincblende alloys at about 40 % GaN. The relatively low energyof formation ( A£/ « 0.02 eV/atom and hence the good miscibility we foundfor this alloy system is expected on the basis of the chemical similarity ofAl and Ga and the close lattice match.

.4/ rjBi_ r,V alloys Alloys with of Ill-nitrides with c-BN suffer from a muchlarger atomic size mismatch. We have not yet performed a systematic studyof them. Only the 50 % compound with the chalcopyrite structure wasinvestigated in connection with the related II-IV-N2 compounds. Our cal-culated cohesive energies for AlBNa, c-BN and A1N yield a formation en-ergy of A.E/ « 2 eV/atom, i.e. two orders of magnitude higher than forAlrGai_rN. The cohesive energy used here does not include relaxation ofthe bond lengths and is thus an overestimate. Nevertheless, it is clear thatthe Al rBi_ rN alloy system will have a much lower solubility. The bandgap we obtained for AlBNj is significantly less than the average of those ofz-AIN and c-BN as can be seen in Fig. 9. The baiidgap bowing is conven-tionally described by means of a bowing parameter b defined by

E,(x) = ~Eg + A£3(x - 1/2) - bx(l- x),

TETRAHEDRALLY BONDED WIDE-BAND-GAP MATERIALS -9

wi th I., = {£.,(0) + £",(1)1/2 the average gap and A£3 = £,( 1) - £0 ,(0)'he gap difference between the endpoints .4 and B. Using our LDA gapslor c-BN. chalcopyrite A1BN? and z-AIN of respectively 4.73. 3.71 and 3.44'A", we find b = 1.5 eV. Including the estimated gap corrections, we obtaintor these numbers respectively: fi.4. 5.0. 4.7. b = 2.2 eV. Bond length relax-ations are expected to influence this bowing strongly. The present numbersare thus only an indication that bowing is large in this system. A similarsituation is expected to hold for B rGai_.rN alloys.

[feterovaient alloys The heterovalent alloys among group IV compounds,or. elemental solids, and III-V compounds present an even more challengingproblem. We have carried out a study of diamond-c-BN alloys [72] and ofi.lie alloys of SIC with A1N and DP. All of these are reasonably latticematched alloys systems. \Ve have thus neglected bond-length relaxationssofar. Details of our calculations for diamond-c-BN were presented in [72].Our results on the SiC-AIN and SiC'-BP were reported at the MRS SpringMeeting 1993 but have not yet been published, except for a brief mentionof the SiC-AIN results in [73. 58).

First, we discuss the short-range order in these systems and our compu-tational approach. Because of the donor and acceptor character of IV- V andIV-III bonds respectively, there is a significant energetic advantage in com-pensating the two types of cation-anion bonds by placing them in close prox-imity and by incorporating them in equal amounts. -Charge transfer fromone to the other will satisfy each bond with two electrons and thus yield again in band-structure energy [74]. The charging of the bonds that accom-panies this, however, leads to a mutual attraction. We thus assume thatlow-energy configurations have nearest neighbor IV-V and IV-III bonds.For a similar reason, we exclude cation-cation and anion-anion "wrong"bonds completely as these nearest-neighbor interactions would lead to highenergies. We are then left with a simple pseudobinary alloy problem inwhich we consider only disorder on one of the sublattices and place theaccompanying anions so as to satisfy the above short-range order chargeneutrality or compensation rules. In the case of SiC/BP. it is necessary totake into account the fact that in BP. B behaves as anion and P as cation.Thus we assume Si-P and C-B bonds. The other structures were foiund tohave sufficiently high energies that they could be ruled out [48].

This does not completely specify the relevant local structures but guidesthe choice of physically plausible structures. We then choose the Li2 order-ing for 25 % or 75 % and Llo ordering for 50 % cation mixtures on the feesublattice of the zincblende structure. These form the basis of the so-calledConnolly-Williams method [75] in which one considers fee nearest neighbortetrahedrons of the type AnB4,n with A and B the two types of cations and

OftJGSNAL PAG€OF POOR

•50 W. R. L. Lambreciu et al.

a ranging from 0 to 4. Next, v.-e perform rlrst-pnnciples calculations oi therotal energies and band structures of these mixed ordered compounds. \\erhen assume that the properties of the disordered system can be obtainedas a suitable average over these "basis-states". In principle, one should ob-tain the distribution of clusters in the alloy at a given concentration andtemperature by the minimization of the free energy. The latter includes theenergy of formation, the configurational entropy, and eventually the vibra-tional free energy and elastic strain energy contributions. This approach isknown as the cluster variation method.

For the alloys considered. we find that the energy of formation is nearlya parabolic function of the concentration and is high. For 50 % SiC-AINit is about 0.25 eV/atom. This is high and implies that there is onlya. very limited thermodynamic miscibility- The near parabolkity impliesthat the Connolly- Williams cluster-expansion model can be mapped ontothe ferromagnetic Ising model [72 j. This allows as to make an immedi-ate and simple estimate of the miscibility temperature, which is given by0.816 x •2±Ef(l/2)/kB with kg the Boltzman constant and A£/(l /2) theenergy of formation of the 50 % compound. This has a value of ~oOOO Kin the present case. Lattice relaxation effects and inclusion of longe-rangecorrelations may easily reduce this number by a factor of two. The presentcalculation thus only gives a rough estimate. The experimental phase dia-gram [76] of the SiC-AIN alloy system indicates a miscibility temperatureof ~2000°C. Furthermore, it indicates that alloys preferentially form in thewurtzite structure except near the SiC rich end where other polytypes arestabilized. In any case, for the present purposes, it suffices to point out thatthese heterovalent alloys are likely to be metastable towards phase sepa-ration up to rather high temperatures. Nevertheless, once they have beenformed by non-equilibrium processes, e.g. from a melt or gas phase mix-ture. the phase separation process may be slow because it would requiremassive diffusion. The overall conclusion of this is that thermodynamicequilbirum considerations may be rather irrelevant to the actual structureof these alloys. Thus, instead of calculating the thermodynamic equilibriumdistributions of the various types of clusters, we can equally well assume arandom distribution.

Pn= i' '(l-x)4-n , (8)

of the 5 types of basic tetrahedron clusters. We use this procedure to obtaina first approximation to the band-gap behavior in the disordered alloys.Before presenting those results, we note that inclusion of configurations ofthe type we ruled out from the start would drive up the formation energyeven higher and reinforce our conclusions. Although such configurations

TETRAHEDRALLY BONDED WIDE-BAND-GAP MATERIALS 31

will probably appear in small amounts due to random errors durins growth,i hey can be considered as defects instead of the average structure. In somesense, they are similar to anitisite defects in a simple binary compound.Our calculations for the interfaces i'43l which considered "'.vrong bonds" ofthe type Si-Al. C-N. showed that these give rise to defect states in the gap.

Figs. 10 and 11 show our calculated band structures of the ordered com-pounds of SiC-AIN and SiC-BP respectively. The Brillouin zone notationfollows Bradley and Cracknel! [77]. Figs. 12 and 13 show the band gaps ofthe ordered compounds we used as "basis" for the cluster expansion as wellas the disordered alloy average obtained using a random probability distri-bution of the clusters. The bowing coefficients are included in the figure.We give both the LDA values and the corrected gaps for the ordered com-pounds. We assume here that the correction can be linearly interpolatedbetween the known corrections for the end compounds.

We note that the A""6 point where the conduction band-minimum ocursfor SiC and A1N is folded onto F in the ordered intermediate compounds.For the 50 % and 75 % A1N compounds, however, the valence-band max-imum does not ocur at F. This is related to the very flat valence-bandmaximum along the F - A" direction in the tetragonal Lla structure whichcorrepoiids to F - A/ in the simple cubic LI? structure. The valence bandmaximum in both cases occurs at R. Both the minimum gap and the small-est direct gap are indicated in Fig. 12. The smallest direct gap, in any caseis only pseudodirect because it corresponds to a transition form the highestvalence state at F to the folded X*b minimum. Since the perturbation ofthe parent band structures is strong in these materials, the matrix elementsfor these pseudodirect transitions may be appreciable. The curve for thedisordered alloy gap is based on the minimum gaps. In SiC-BP. the alloyswith 50 % or more BP have negative LDA gaps.

We note that in both cases a very large bowing is present. The band-gaps in SiC-BP are predicted to be smaller than the almost equal gaps atthe end points over almost the entire composition range. As in diamond-c-BN. [72] this is related to the type-II offset in this system. Indeed, theband-offset at heterojunctions among these materials is shown in Fig. 14and indicates that the gap of the overall SiC-BP heterojucntion system issmaller than that of the two pure compounds. This is thus proposed to bethe primary cause of the large bowing. SiC-AIN has a type-I offset, i.e. anoffset with the larger gap enclosing the smaller gap. Nevertheless, a strongbowing is also apparent here. In both cases the origin of this phenomenonmay be argued to be the large charge transfers taking place between theconstituent semiconductors. If one thinks of the alloy as a molecular levelmixture of regions of BP (or A1N) and SiC. it is clear that charge transferprocesses similar to those found at the heterojunction must occur. To what

OWGSNAl. PAGE fSOF POOR QUALITY

\V. R. L. Lambreciu et ai.

(SiC)(AL\h

- I 8 L

Figure 10. Band structures of ordered compounds of (SiC)3(AlN), (SiC)(AlN) and(SiC)(AlNh. The gap is indicated by shading.

(SiC)3(BP) (SiC)(BP) (SiC)(BP)3

Figure ll. Band structures of ordered compounds of (SiCb(BP). (SiC)(BP) and(SiO(BP)j.The gap is indicated by shading.

TETRAHEDRALLY BONDED WIDE-BAND-GAP MATERIALS 33

SiC-AiN

a. 3.5 h<'C - r

§ 15 t< -,

1.5

b=3.7eV ,--_/'

0.50.25 0.5 0.75

x

Figure /i2. Band gap versus concentration in zincbiende (SiC)i-riAINlj ailoys: tri-angles. LDA minimum gaps for ordered compounds: pluses. LDA pseudodireci gaps 01the ordered compounds: squares, corrected gaps for the ordered compounds: solid line,corrected gaps for the disordered alloys. 6 is the bowing parameter.

2.5 4SiC-BP

b=4.5 eV

OQ

0.5 h

0

-0.50.5x

0.75

Figure 13. Band gap versus concentration in zincbiende (SiC)i-x(BP) I alloys, triangles.LDA minimum gaps of the ordered compounds: squares, corrected gaps for the orderedcompounds: solid line, corrected gap for the disordered alloys. 6 is the bowing parameter.

extent one can transfer this idea to the homogeneous ailoys considered inour calculations is not clear. Qualitatively, however, it is clear that similarB-Si (N-Si) and P-C (C-A1) bonds occur and that the balance betweenthe charge transfers in these bonds determines the electrostatic dipole off-setting the potentials in BP (A1N) units from the ones in SiC units.

We note that for SiC-AIN allovs of wurtzitic structure, a large bowing

OSJQiNAL PA6C KOF POOR QUALITY

34 W. R. L. Lambrecht et al.

oDCLU

4 r

> 2 F

1 h

-2 -

BP

v SiC

AIM

Figure 14. Calculated band offsets at SiC/AIN and SiC/BP zincblende heterojunctions.The valence band maximum £„ of SiC is arbitrarily chosen as zero. If transitivity isassumed the figure also gives an estimate of the A1N/BP offset.

has indeed been found to exist [78] both for the direct T - T gap and theindirect T - A" gap. Although our present calculations are for zincblendederived alloys, corresponding charge transfer effects are expected for htewurtzite alloys. Thus the experimental results provide indirect confirmationof the validity of our results. Work on the wurtzite alloys is in progress.

5. Conclusions

lu this paper, we have reviewed the electronic structure and related proper-ties of a wide range of tetrahedrally bonded semiconductors whose commonfeature is that they include early elements of the periodic table, i.e. Be. B.C. N and 0. This survey includes diamond, SiC, the binary compoundsof formula III-N with the group-Ill element including B. Al, Ga. In: thecompounds of formula B-V with the group-V element being N, P, and As:and the IMV-N2 compounds with IV being Si or C and II being Be andMg. It also includes the II-VI compound of the second row: BeO. The Be2Cand BeaNj compounds, which have a bonding type that is closely relatedto tetrahedral bonding, were also included as results for them are relevant

TETRAHEDRALLY BONDED WIDE-BAND-GAP MATERIALS 35

for the discussion* of the synthesis of BeCN2- Bandgaps, lattice constants,bond energies or cohesive energies, bulk moduli and, in the case of the ni-trides, other elastic constants were presented. The strong covalent bondingand the deep N2p levels were shown to be the basic reason for the wideband gaps occuring in these materials. The interrelationship between vari-ous properties, and, in particular, between crystal structure and electronicstructure was discussed in detail. Special attention was given to SiC whichforms in a large number of poly types. The optical reflectivity of some of themost important polytypes of SiC and of GaN and A1N were discussed. Theyprovide an important experimental check of the calculated band structures.We have revealed the trends with atomic number and ionicity. Related is-sues such as the electronic structure of alloys and interfaces were discussed.Besides an extensive discussion of previously published material, the paperincludes new results on SiC-BP and SiC-AIN alloys, new results on hexag-onal modifications of diamond, BN and BeO, and on the related materialBe2C. We have analyzed the prospects for synthesis of a new matrial BeCN2on the basis of our calculated energies of formation.

Acknowledgements

We thank M. Alouani, M. van Schilfgaarde and M. Methfessel for pro-viding computer programs, J. C. Angus, A. Argoitia for collaborations ondiamond, W. J. Choyke and R. P. Devaty and coworkers for collaborationson SiC reflectivity, and J. Rife, W. R. Hunter and D. K. Wickenden for col-laborations on GaN reflectivity. The nitride work was supported by NSFDMR-92-22387, the diamond work by the NSF-MRG DMR-89-03527 andONR N-00014-89-J-1631 and the SiC work by Wright-Laboratories (Con-tract No. F33615-93-C-5347) and NASA Lewis Research Center (NCC 3-25). Part of the computations were done a the Ohio Supercomputer Center.

References

1. J. C. Phillips, Bonds and Bands in Semiconductors, (Academic, New York, 1973).2. W. A. Harrison, Electronic Structure and Properties of Solids, (Dover, New York

1989).3. W. R. L. Lambrecht and B. Segall, Phys. Rev. B 41, 2832 (1990).4. R. M. Wentzcovitch, A. Continenza, and A. J. Freeman, in Diamond, Silicon Car-

bide and Related Wide Band gap Semiconductors, ed. J. T. Glass, R. Messier, and N.Fujimori, Mater. Res. Soc. Symp. Proc. Vol. 162, (MRS, Pittsburgh 1990), p. 611.

5. C.-Y. Yeh, Z. W. Lu, S. Froyen and A. Zunger Phys. Rev. B 46, 10086 (1992).6. H. Amano, M. Kito, K. Hiramatsu, Jpn. J. Appl. Phvs. 28, L2112 (1989).7. R. H. Wentorf, Jr., J. Chem. Phys. 26, 956 (1957); 'ibid. 34, 809 (1961); ibid. 36,

1990 (1962).8. P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964); W. Kohn and L. J. Sham

ibid. 140, A1133 (1965).9. L. Hedin and B. I. Lundqvist, J. Phys. C 4, 2064 (1971).

36 W. R. L. Lambrecht et al.

10. L. Hedin, Phvs. Rev. 139. A796, (1965).11. M. S. Hybertsen and S. G. Louie, Phys. Rev. B 34, 5390 (1986).12. R. W. Godby, M. Schliiter, and L. J. Sham, Phys. Rev. B 37, 10159 (1988).13. M. Rohlfing, P. Kruger, and J. PoUmann, Phys. Rev. B 48, 17791 (1993).14. M. Palummo, L. Reining, R. W. Godby, and C. M. Bertoni in Proc. 21 st Int. Conf.

on the Phvsics of Semiconductors, held in Beijing, Aug. 1992, ed. Ping Jiang, andHou-Zhi Zheng, (World Scientific, Singapore 1993), p. 89.

15. F. Bechstedt and R. Del Sole, Phys. Rev. B 38, 7710 (1988).16. W. R. L. Lambrecht, S. Strite, G. Martin, A. Agarwal, H. Morkog, and A. Rockett,

Phys. Rev. B (1994), submitted.17. O. K. Andersen, Phys. Rev. B 123060 (1975); O. K. Andersen, O. Jepsen, and

M. Sob, in Electronic Band Structure and its Applications, edited by M. Yussouff,(Springer, Heidelberg, 1987).

18. A. G. Petukhov, W. R. L. Lambrecht and B. Segall, Phys. .Rev. B 49, 4549 (1993).19. M. Methfessel, Phys. Rev. B 38, 1537 (1988); M. Methfessel, C. O. Rodriguez, and

O. K. Andersen, Phys. Rev. B 40, 2009 (1989); H. M. Polatoglou and M. Methfessel,ibid. 41, 5898 (1990).

20. VV. R. L. Lambrecht and B. Segall, Phys. Rev. B 40, 9909 (1989); 41, 5409 (1990).21. M. R. Salehpour and S. Satpathy, Phys. Rev. B 41, 3048 (1990).22. I. Gorczyca and N. E. Christensen, Physica B 185, 410 (1993).23. W. R. L. Lambrecht and B. Segall, in Wide Band Gap Semiconductors, ed. T. D.

Moustakas, J. I. Pankove, and Y. Hamakawa, Mater. Res. Soc. Symp. Proc. Vol. 242(MRS, Pittsburg, 1992) p. 367.

24. W. R. L. Lambrecht in Diamond, Silicon Carbide and Nitride Wide-Bandgap Semi-conductors, ed. C. H. Carter, Jr., G. Gildenblat, S. Nakamura, and R. J. Nemanich,Mater. Res. Soc. Symp. Proc. Vol. 339 (1994), to be published.

25. K. J. Chang, S. Froyen, and M. L. Cohen, J. Phys. C 16, 3475 (1983); K. J. Changand M. L. Cohen, Solid State Commun. 50, 487 (1984).

26. W. R. L. Lambrecht and B. Segall, Phvs. Rev. B 45, 1485 (1992).27. B. F. Levine, Phys.Rev. B 7, 2600 (1973).28. S. Fahy, S. G. Louie, and M. L. Cohen, Phys. Rev. B 34, 1192 (1986); ibid. 35, 7623

(1987).29. R. Wentzcovitch, S. Fahy, M. L. Cohen, and S. G. Louie, Phys. Rev. B 38, 6191

(1988).30. S. P. Mehandru, A. B. Anderson, and J. C. Angus, J. Phys. Cheni. 96, 10978 (1992).31. W. R. L. Lambrecht, C. H. Lee, B. Segall, J. C. Angus, Z. Li, and M. Sunkara,

Nature 364, 607 (1993).32. J. Tersoff, Phys. Rev. Lett. 61, 2879 (1988).33. W. R. L. Lambrecht and B. Segall, J. Muter. Res. 7, 696 (1992).34. W. R. L. Lambrecht, Physica B 185, 512 (1993).35. S. Koizumi, T. Murakami, T. Inuzuka, and K. Suzuki, AppJ. Phys. Lett. 57, 563

(1990).36. A. Argoitia, J. C. Angus, J. S. Ma, L. Wang, P. Pirouz, and W. R. L. Lambrecht,

J. Mater. Res. (1994), in press; L. Wang, P. Pirouz, A. Argoitia, J. S. Ma, and J. C.Angus, Appl. Phys. Lett. 63 1336 (1993).

37. A. Argoitia, J. C. Angus, L. Wang, X. I. Ning, and P. Pirouz, J. AppJ. Phys., 73,4305 (1993).

38. C. H. Lee, W. R. L. Lambrecht, and B. Segall, unpublished.39. R. O. Jones and O. Gunnarsson, Rev. Mod. Phys. 61, 689 (1989).40. A. R. Verma and P. Krishna, Polymorphism and Polytypism in Crystals, (Wiley,

New York, 1966).41. G. S. Zhdanov, C. R. Acad. SCi. USSR, 48, 43 (1945).42. V. Heine, C. Cheng, G. E. Engel, and R. J. Needs, in Wide Band Gap Semicondcu-

tors, edited by T. D. Moustakas, J. I. Pankove, Y. Hamakawa, Mater. Res. Soc. Symp.

TETRAHEDRALLY BONDED WIDE-BAND-GAP MATERIALS 37

Proc. Vol. 242 (MRS, Pittsburgh, 1992) p. 507; C. Cheng, R. J. Needs, and V. Heine,J. Phys. C 21, 1049 (1988); J. J. A. Shaw and V. Heine, J. Phys.: Condens. Matter2 4351 (1990); C. Cheng, V. Heine and I. L. Jones, ibid., 5097; C. Cheng, V. Heineand R. J. Needs, ibid. 5115. N. Churcher, K. Kunc, and V. Heine, J. Phys. C 19, 4413(1986).

43. P. Pirouz, Scripts Met. 23, 401 (1989); P. Pirouz and J. W. Wang, Ultramicroscopy51, 189 (1993); P. Pirouz, Inst. of Physics Conf. Ser. No. 104, p. 49 (1989).

44: J. Pezoldt, A. A. Kalnin, D. R. Moskwina, and W. D. Savelyev, JVucJ. Instr. andMeth. in Physics Res., B80/81, 943 (1993). A. A. Kalnin, F. Neubert, J. Pezoldt,in Proc. 4th European Conference on Diamond. Diamond-like and Related Materials,AJbufeira, Portugal, Sepember 20-24, 1993, ed. P. K. Bachman, I. M. Buckley-Colder,J. T. Glass, M. Kamo, (Elsevier Sequoia, Lausanne 1994), p. 346; also published inDiamond and Related Materials Vol. 3, Nos.4-6.

45. W. R. L. Lambrecht, B. Segall, M. Methfessel, and M. van Schilfgaarde, Phys. Rev.B44, 3685 (1991).

46. W. R. L. Lambrecht, B. Segall, W. Suttrop, M. Yonagathan, R. P. Devaty, W. J.C'hoyke, J. A. Edmond, J. A. Powell, and M. Alouani, AppJ. Phys. Lett. 63, 2747(1993); and unpublished.

47. W. J. Schaffer, G. H. Negley, K. G. Irvine, and J. W. Palmour, in Diamond, Si/-icon Carbide and Nitride Wide-Bandgap Semiconductors, ed. C. H. Carter, Jr., G.Gildenblat, S. Nakamura, and R. J. Nemanich, Mater. Res. Soc. Symp. Proc. Vol. 339(1994), to be published.

48. VV. R. L. Lambrecht and B. Segall, Phys. Rev. B 43 7070 (1991).49. W. R.. L. Lambrecht. and B. Segall, Acta metal, mater. 40, S17 (1992).50. VV. R. L. Lambrecht and B. Segall, Phys. Rev. B 41, 8353 (1991); ibid. 42, 1462

(1990); W. R. L. Lambrecht, C. H. Lee and B. Segall, in Atomic Scale Structure ofInterfaces, edited by R. D. Bringans, R. M. Feenstra and J. M. Gibson, Mater. Res.Soc. Symp. Proc., Vol. 159 (MRS, Pittsburgh 1990), p. 377-82; VV. R. L. Lambrecht,C. H. Lee, M. Methfessel, M. van Schilfgaarde, C. Amador, B. Segall, in Defects inMaterials, Mater. Res. Soc. Symp. Proc.,Vol. 209, edited by P. D. Bristowe, J. E.Epperson, J. E. Griffith and Z. Liliental-Weber, (MRS, Pittsburgh 1991), p. 667.

51. R. M. Wentzcovitch, K. J. Chang, and M. L. Cohen, Phys. rev. B 34, 1071 (1986).52. R. M. Wentzcovitch, and M. L. Cohen, J. Phys. C 19, 6791 (1986).53. Landolt-Bornstein: Numerical Data and Functional Relationships in Science and

Technology, New Series, Group III, Vol. 17a, edited by O. Madelung, (Springer, Berlin1982).

54. A. Munoz and K. Kunc, Physica B 185, 422, (1993).55. P. Perlin, C. Jauberthie-Carillon, J. P. Ithie, A. San Miguel, I. Grzegory, and A.

Polian, Phys. Rev. B 45, 83 (1992).56. H. Volstadt, E. Ito, M. Akaishi, S. Akimoto, and O. Fukunara, Proc. Jpn. Acad. B

66 7 (1991).57. M. Ueno, M. Yoshida, A. Onodera, O. Shimomura, and K. Takemura, in Proc. 5th

Int. Conf. High pressure in Semiconductor Physics, Kyoto (1992), Jpn. J. Appl. Phys.32, Suppl. 32-1, 42 (1993).

58. W. R. L. Lambrecht and B. Segall, in Properties of the Group-Ill Nitrides, editedby J. H. Edgar, EMIS Data Review Series, (IEE, Stevenage Herts, UK 1994), in press.

59. W. R. L. Lambrecht, B. Segall, and O. K. Andersen, Phys. Rev. B 41, 2813 (1990).60. R. M. Martin, Phys. Rev. B 6, 4546 (1972).61. K. Tsubouchi and N. Mikoshiba, IEEE Trans. Sonics Ultrason. SU-32, 634 (1985).62. V. A. Savastenko and A. U. Sheleg, Phys. Stat. Sol. (a) 48 K135 (1978); A. U.

Sheleg and V. A. Savastenko, Inorg. Mater. 15 1257 (1979).63. K. Kim, W. R. L. Lambrecht, and B. Segall, Phys. Rev. B (1994), in press.64. V. Fiorentini, M. Methfessel, and M. Scheffler, Phys. Rev. B 47, 13353 (1993).65. E. A. Albanesi, W. R. L. Lambrecht, and B. Segall, J. Vac. Sci. Techno]. B (1994),

in press (PCSI-21 Proceedings).

38 W. R. L. Lambrecht et al.

66. E. A. Albanesi, W. R. L. Lambrecht, and B. Segail, in Diamond. Silicon Carbideand Nitride Wide-Bandgap Semiconductors, ed. C. H. Carter, Jr., G. Gildenblat, S.Nakamura, and R. J. Nemanich, Mater. Res. Soc. Symp. Proc. Vol. 339 (1994), to bepublished.

67. G. Martin, S. Strite, A. Botchkarev, A. S. Agarwal, A. Rockett, W. R. L. Lambrecht,B. Segail, and H. Morkoc, Appl. Phys. Lett.. (1994), to be published.

68. J. Rife, W. R. Hunter, W. R. L. Lambrecht, B. Segail, and D. K. Wickenden, Bull.Am. Phvs. Soc, 39, 211 (1994); and unpublished.

69. C. G. Olson, D. W. Lynch, and A. Zehe, Phys. Rev. B 24, 4629 (1981).70. S. Loughin, R. H. French, W. Y. Ching, Y. N. Xu, and G. A. Slack, Appl. Phys. Lett.

63, 1182 (1993); and S. Loughin, Ph. D. Thesis, Univ. of Pennsylvania, Philadelphia1992.

71. E. A. Albanesi, W. R. L. Lambrecht, B. Segail, Phys. Rev. B 48, 17841 (1993).72. W. R. L. Lambrecht and B. Segail, Phys. Rev. B 47 , 9289 (1993)73. VV. R. L. Lambrecht, E. A. Albanesi, and B. Segail, in Proc. International Conference

on Silicon Carbide and Related Materials, Washington DC, November 1993, in press.74. W. R. L. Lambrecht, C. Amador, and B. Segail, Phvs. Rev. Lett. 68, 1363 (1992).75. J. W. D. Connolly and A. R. Williams, Phys. .Rev. B 27, 5169 (1983).76. A. Zangvil and R. Run, J. Am. Ceram. Soc. 71, 884 (1988).77. W. C. Bradley and A. P. Cracknell, The Mathematical theory of Symmetry in Solids:

Representation Theory (or Point Groups and Space Groups (Clarendon Press, Oxford1972).

78. G. K. Safaraliev, Yu M. Tairov, and V. F. Tsvetkov, Sov. Phys. Semicond, 25, 865(1991).

Theoretical Study of the Electronic, Optical, Interface, and AlloyProperties of SiC and the Group-III-NitridesVV R L Lanihreclit. E A Albanesi. and B Segall

Department of Physics. Case Western Reserve University, Cleveland. 011-44106-7079

ABSTRACT: We present a brief overview of the electronic structure studies of SiCand the I l l -ni t r ides performed or in progress in our group. The trends in the

. ,.;: band structure across the series and the dependence on polytype are discussed.. a.s ;. well as the relation between baud-offsets at interfaces and baud-gap bowing-. • •'

1. INTRODUCTION '• • : > >• i ;- • f ' • -,• . ' ' '

SiC and the Group-Ill Nitrides form a closely related class of wide-band-gap semiconductors.Here \VP present a brief overview of their electronic structure and related properties. This paperis mainly meant as a guide to previous" and 'on-going work carried out in our group. A. fullaccount has been or will be presented elsewhere. Details of the computational approach can:be found in the references. Briefly, we use the linear muffin-tin orbital (LMTO) method (seeAndersen t.l til. 1087 for a recent description) combined with local density functional theory.Most <if our work used (l ie atomic-sphere-approximation (ASA). The full-potential version (FP-LMTO) was used in some cases as indicated.2. W'LK ELECTRONIC STRUCTURE ;~ ----- ' -------------- ...... ••- - — .-•- -.,.-'

..^AYe^ have calculated the electronic band structure for SiC'. BN. A1N. GaN. and InN in both(lie zinc-blende- (zli) atid wtirtzite (wz) , structures (Lambrecht and Segall 1992. 1993a)._ The. -min imum baiul gaps- at the experimental lattice constant are summarized in Table 1. .

'- " ' "'• ' ' ' ' "* '' ' ' ' • - ""*' :'.

- f- -Table I: Calculated minimum band gaps for wurtzite and zincblende SiC and group-III-uitridesw i t h experimental values in parentheses (in eV). The numbers in square brackets are obtained 'from n'-LMTO. others wi th in ASA. " . •- • - .

SiC'

BN-A1NGaNInN

r.r, - .V,

r.r, - .v,r.r, - AT-Ti5 - r,TIS - TI

1.4(2.417)° r«-.U,

4.:} (0.4)c

~ 3.4"2.0 (3.5)'

r-0.1

r« - A'-2- r, - r,

TI - rtr r,'1 6 — A 1

2.7 [2.0]2.8 [2.2] (3.30)fr

5.74.0 (0.28)''2.4(3.6)'0.3 (2.05)3

" H u m p h r e y s R G ."Bimbefg D . ancTC'hoyke W J 1981, Solid State Coiumun. 39 103'' I 'alr ick L. Hamilton D R. and Choyke W J 1966. FA vs. Rev. 143 526' Chreuko R M 1974. Solid State Coimuun. 14 511'' I'. B. Perry and R. F. Rutz 1978. Appl. Phys. Lett, 33: :J9" St.rite S. Riian J. Li Z. Salvador A. Chen H. Smith D J. Choyke W J. and MorkocJI

. 1990. .l'..Va.c. Sci. Teclnioi' D 9 ;92- ^ " ''; Dingle HTSeH D D. Stoko\v-skT S E". Dean P -Land Zetterstonu R"B 1971. Phys. Rev. B 3"

.,497 ,,'..„.._,>J Tyagai V A . Evstigneev A . M . Krasiko A N. Andreeva A F. and Malakhov V Ya 1977.

.S'ov. /'/M's. Semicond. 11 1257 '. . . -., .

The ASA-LMTO calculated equilibrium lattice volumes, cohesive energies, bulk moduliand (heir pressure derivatives for zb and wz SiC. A1N. GaN and zb-BN were reported by

ORIGINAL PAGE ISOF POOR QUALITY

i ; I IH| Segall ( 1092.1989) and agree well \v i t l i CJorczyra and C'hristensen (1903) and•i t her c a l c u l a t i o n s referenced there.

U'e I I O I P t h a t , the local dens i ty approximation ( L D A ) underestimates the gap by 1-2 eV ini ln-se sys tems, in agreement w i t h simple estimates of the self-energy correction introduced byl l cc l i s i cd t and Del Solo ( I 9 X X ) and applied to these materials by Lambrecht and Segall ( 1902).I i om our ca l cu l a t i ons ol op t ica l reflectance discussed below, the correction is found to be nearly.1 c o n s t a n t s h i f t of the conduct ion bands and to be independent of polytvpe. For 211-SiC. wem i t e i ! i ; i i the gaps at A and M are very close w i t h i n ASA while FP-LMTO correctly e ;ves thei i i i i i i i i i n i i i i > a p to be at A. The K-}C s t a t e must thus be par t icular ly sensitive to non-spherical( inn r i b n i j o i i s to the p o t e n t i a l . Similar FP corrections are t h u s expected for the n i t r i d e s but. donot change the conclusions about the location of the minimum bandgap.

Al though! the different polytypes have nearly the same total energies, they have quited i f f e r e n t band s t ructures , including different gaps and different locations of the conduction-band m i n i m a . The origin of these changes was explained by Lambrecht and Segall i 190*2) bv' • < > m p a r i n g t he band s t ruc tures of ZB and VVZ in the same Brillouin zone ( DZ). The conductionb a n d - m i n i m a at A",,.. ( .V-; ,) move up in the zincblende ( w u r t z i t e ) s t ructure . This is due to (.liel a c i t h a t A',,.. (A'-;,) has lower symmetry in the zincblende ( w u r t z i t e ) s t r u c t u r e . These effectschange the locat ion of the- conduction band min imum in BN. AIM. and Si(.' but not in ( i a N andI n N . because in the l a t t e r , the m i n i m u m always stays at f as a result of the deeper potentiallor ( a i i o i i - s ( i r b i t a l s for the higher Z-Hements.

2H-SIC 4H-SiC

Figure 1: Band s t ruc ture of 2H and 411 Sit'.

1 he baud-s t ruc ture of the intermediate 4H polytvpe can be understood in terms of folding-" f l f c t s of I | I P '2([ band s t r u c t u r e . The band structures of 211 (using FP) and 411 SK' (usingA S A ) arc shown in Fig. I. The bands on the surface of the 2II-BZ (L — A — II) are doubly( h - g i M i i - i a t i - in 211 because of the presence of a six-fold screw axis along c. These bauds are'•ssptitially folded onto the M - f - 7i" plane of the 4H BZ. the degeneracy being lifted in Thelower symmetry - I I I s t r u c t u r e . The bauds now become doubly degereuate on the 411 L — A —'H'plane corresponding to the six-fold screw axis with double pitch.

The op t i ca l response f u n c t i o n s (e^(u;) . f i ( - - ) . and the reflectance) have been calculated using

OFf§

QUAUW

i lie m u f f i n - t i n orbital basis set for 3C. 211 and 411 SiC. Tlie results for 3C and 411 SiC werepresented in .(Lambrecht rt nl. 1993b) and (Suttrop ft al. 1993). Satisfactory agreementbetween theory and experiment is obtained for the major peak positions of 3C and 411 up to 10<j\ inc luding a constant LDA correction. Preliminary results on the nitrides indicate a similar(•(inclusion.

:{. S T U A I N EFFECTS

A separate investigation using the FP-LMTO method was made of the strain effects on theb u i l d - s t r u c t u r e of 3C-SK' (Lambrecht et nl. 199ia). This study included a calculation of the fullsci of clast ic constants. The elastic constants tensor is only weakly dependent on the poly type.Hence. those for one poly type can be obtained (approximately) from those for another by anappropriate Coordinate transformation. We thus obtained good agreement with experimentali l a l a lor the elastic constants of hexagonal or «-SiC. no distinction being made here between( h e various -|iexag6tiarj)olyYypes. • • • • . . . . .

I'lio strain effects on the band-structure were reported in the form of deformation potentialsI Kambrecht 'r / nl. I99la). The important deformation potentials, in Pikus and Bir's notationami iii eV. for the 'band gap' edges a re ' • -. , .: .

T;y ,r-'' •• -fr>i , i h ,/iS \\.\ £ . . ; . , , . ; • . . , .1.3 -2.1 -<(.() -2.1.5 S.9

Id our knowledge the only deformation potential for which there are exper imental dataavailable- Is i he acoustic deformation potential E. for the conduction baud minimum at A". Itwas determined to be 22. eV by analyzing the:\vidth of cylcotron resonance lines as a func t ionnl temperature ami - 'assuming. a' r -x r~:J/'2 behavior (Kaplan ft nl. 1985). The range of thed a t a , however. is too narrow to determine this behavior unambiguously. An al ternat ive (andeven be t t e r ) f i t . to the data , shown in Fig. . 2.. is obtained if we assume t h a t in addition tolM'\*lff'i'°'n|i«»'-i6ri "p 'otenfial scattering, there is a significant impurity scattering, increasing withincreasing' temperature' and- approximately described by an inverse lifetime r~' = l.:j(T'"-2 t i ) I O ">~ . Al present, we do not have a well developed theory for the linear dependence ofi he i m p u r i t y scattering cont r ibut ion . However, an increasing r"1 with increasing tcmperai'tirewould be expected for scat ter ing from neutral impurity complexes, a situation which is expectedl < > r h ighly compensated mater ia l and likely to be the case in SiC samples available al t h a t time.\\e IK, !<• i hat Kaplan el al. ( 1993) reported a new value of 12.6 eV for the deformation potentiallor samples of higher q u a l i t y . __ ^ ''

120

<-£"- 100oHI

'"~ 80UJ-.

60

40'CO.--

5£ 20

" " ' 0

. phonon: X=22.phonon: X=9 —-

impurity •our fit

o

i. o-

40 50-. 60 ,70v 80 90. 100TEMPERATURE (K)

Figure 2: Fits of the inverse lifetime, see text.

PAGE ISOf >OOR

4. HETEROJ UNCTIONS AND ALLOYSThe electronic s t ruc ture of alloys was studied using the cluster-expansion approach as de-

scribed by Lambrecht and Segall (1993c) The valence-baud offsets have been determined forSiC/AIN and SiC/BP {110} heterojunetions (Lambrecli t and Segall 19911)) to be l.r, and 1.2c\' respectively. For SiC/BP I IIP offset is of typo II (staggered). In t h a t case. I ho effective gap<>f MIC system is smaller t han the smaller of the two. Consequently, a large bowing is expectedfor the alloy. Indeed, we find a bowing parameter b = 4.4 eV. with b defined by -

where E<; = [Er,-(0] 4- EV;( 1 )]/2 is the average gap and A£>; = E V v ( l ) - EVy(O) is the gapdilferoiice between the endpoints. Even for SiC/AIN. however, we find a large bowing of 3.1oV. For SiC/AIN. experimental data were reported for the 211-polytype (Safaraiiev ft <il. 1991).In q u a l i t a t i v e agreement w i t h our calculations for ziiicbleiule. a large bowing of 5.2 eV wasobtained for Mie indirect minimum bandgap F - K as well as for the directT - F band gap(h = I.I oY.) One may speculate t h a t the large bowing for'SiC/AIN and SiC/BP is related tothe hotorovalent nature of these alloys which leads to important charge re-arrangements andhence a large dipole at the heterojunetions and large deviations from a "virtual crystal" picturew i t h averaged cations and anions in the alloy.

Wo also studied the band-gap bowing in ziiicbleiule Al.,.(iai_.,.N alloys (Albanesi d at. 1993).In t h a t case, wo determined the cross-over from direct to indirect band gap to take pi are at.1 =; ().(i. The band-gap bowing coefficient in th i s system was found to be negligibly smalland s l i gh t ly upward (i.e. b < ()) if bond-length relaxations are included. For the direct gap.I ' i r . i . - I i,. the nearly l inear behavior is in q u a l i t a t i v e agreement w i t h the exper imenta l data for.MIC w n r t z i l e l'|,. — F|t. gap. which is expected to behave similarly. . • - , > « f^'... (

Acknowledgement: We acknowledge support from N'SF. ONR and NASA Lewis Research^Con I or du r ing various stages of t h i s work. E A Albanesi acknowledges support from CONICET.i;;( A r g e n t i n a ) . ' ; . . - . . - • • • . . •;,- • - . , , . . . . .

R E F E R E N C E S : " '

Alhanos i F. A . Lambrechl VV R L and Segall B 1993.' Phys. Rev. fl.'in pressAndersen () K. Jopsen (). and Sob M 19S7. Electronic Band Structure ami its A

<•<!. Yiissonlf M . (Springer. Heidelberg) p i • > » « ' - . * ' » •Bodisiodt F and Del Sole R I9SS. Phys. Rev. B 38 7710 >- u«: . .Corcxyca I and Chrislensen N E 1993. Physica B 185410K a p l a n R. Wagner R .1. Kim II -I . and Davis R J 19N">. Solid State Connunn. 55 InK a p l a n R. Moore VV .1. Freitas J A . Altaiskii Yu M. Zuev V" L. and Ivanova L M 1993. th i s Conf. Proc.Lambrecht. VV R L and Segall B 1989. Phys. Rev. B 40 9909Lambrecht VV R L. Segall B. Methfessel M. and van Schilfgaarde M 1991a. Phys. Rev. B 44-3685Lambrecht VV R L and Segall B 1991b. Phys. Rev. B 43 7070Lambrecht VV R L and Segall B 1992. Mater. Res. Soc. Proc. Vol. 242. p 367 :

Lambrechl VV R L and Segall B I993a. in Properties of the Croup-IH Nitrides, ed. Edgar J II.Data Review Series of the Electronic Materials Information Service (EMIS) of the British Institutionof Electrical Engineers ( I E E ) . (IEE. Stevenage Herts. England), to be published

Lambrecht VV R L . Segall B . Sut t rop VV . Yonagathan M. Devaty R P. Choyke VV .1. Ediuoiid .1 A.Powell A .1 . and .Alouani M 19931). Appl. Phys. Lett. 63 2747

Lambrecht VV R L and Segall B 1993c. Phys. Rev. B 47 9289Safaraiiev (\ K . Tairov Yii M. and Tsvetkov Y F 1991.. Sor. Phys. Semicond. 25 8(>.r>Suttrop VV. Yonagathaii M. Devaty R P. Choyke VV .]. Edmond J A. Powell A J. Lambrecht VV R L.

Segall B. and Alouani M 1993. this Conf. Proc.

m

POOR QUALITY

1LANK WOT

V *0 9 b _J &| £ 1 5 | £"5 Z * . * o•5 i j :§ -a 5-S- i -1"- ^ S '5 3 «

• "3 3 c a S 3

S 2 Ss=

i~ 00 »* S u

Ck

„• --s z — -<--o 2 « « «.1 3 T « 2 =" i «. a. «

E u c CL *.= .2 ° 5 * .S

•• — w - -~ •» ~5- 5. 5 s y =5-fi.j ~ .=. C

1-= W:E 5 —IK

-. s KIs W-IA n"SI .SH > » > « J•5 <i• £

1*J g

14o •:

TJ * sU <-S

PAQ€ ISOF POOR QUALITY

b.oenZ

U

0uH<.JSu.u

«<UKU-l

B^ay52siSru =z"»uuW

U.{SuHZ

es<uOM

Z

u

"*"SZV

.

,- 5• _*«

&J

"~'*"w.

. Ij .?-< £

C3 5

S|..a «^ -" in i

QJ sO-ca o< -5^-^•J.s

" r" "".

Ita fa —*™ ^«< -_; s

-» 1as «

SS<I S.£Ilillis ^ «~

5J ltM ul•i§^i-§i

I'?5i-l||a « 0 8 •? "5. »

^1 « ^* «t "^ *^•fi .•* .= S3 — — to

• llTr*-s2 .2 -5 § * •£ =iilitsi"" > .5 g '5 =3JS J'E.2 "4 P? - £ o '^? S= - 5 •> '3 g i

•Pi-liif§i i,s|- s -c T -S S _S

.« W « • • » < • "

S I | ¥=:•*-=

1.2 e ? S -s i:' *s -3 a § •=i. t. . M -i 5E S

Sd

5111JH IfllltIIH| ii llliii ? = N™* w o —

?"~ a S tfe'K'.i5 = ^ J 5 —r-s r 3 5 "8 s-is-

='«|

^ - = -s ><>"4r- C — 5 O

~ i2 — — .2. 5

e

i

-- 8.

i-l*5!.*:s s — •• •*• •'y •> ~= <* "c =u -5 * .= T

v5 » i 3 Is«• * « _ 3 .«-^ .0 .0 o 5 r-

-|-| a s s-s.5 5 - -3 § s:J5.i"i3-S. ;- * s 5 S S313 I 4! M-»f

• • -» - , ttl > O k. —

••£ " 5 Se -r.y .- = •» - .,I 1.8 ^ Jl S

-= ~

- <s a ^ o - ~* ~ S -5 ° =

O! 5 -r fl » s'-"r" i"«s <• « •• s .•

•r -«:.= S £ £•=

SH^r«l .-jwr

c> r» n

a o CM

s-'-SS a

H

s -a i?— a n-a 3 3H - s - I

0 ^ '= fc

tj *'

M

i!h •=!*'

-I "!T "!ei b

e«-r<ee

^

«lz 2-f -=

— * ' 5 f c s . o a r — . ~i t i.s »•£ £ r - 2 34 **^-5^ .£ «, s. a. « r v--S n ^ J-- — — s = «>j — il — — ' '~

OF POOR

*--

*!

-as

sori

alnl

wil

li

Zf J

• "S ~ 5

J

US '•£

- 5 - ^ -a s 87 s— C e - S 5 S - "% fci < S i ^ S -

. (C

OfWQINAl.OF POOR

«

i

.

:H!i»HUii*"= a 2 5 i ^ IL 2? ± <£

*s o ii ** 3 3 •£ g J-$jo i Sf 5 S 1 ~* •** t* 4i ?

w "5P *» ^* .i a w JT J C£•= •3 fc -3 .22 s-S-S.esW p ! 2 5 , ! * w 7 j r t 5 ' * * * « ' »— rtjsrtoys. h. .—= S = r 3 < = i . 2 > J g *** •" U S _• w i ^* V*• S5 — ^ VJ *" w 2 a^. JJi _ J

° t c x * ^ Q -s ta "" '* .2 ^

iSl^^Jj^li1.s g 3 r i < S l : i s £ s «< 2 ^ - § 3 O g -o *3 .S 4J ^.-* • j iSv . i ^T 'g *

tfi*HHi«IHiitljiiiS3 * " "" S^T^ 'Sa ;iS's'sl '5-"?!! '*ill s^ r* — ? • - « • • »3 O <

J S ". £ 1 ts I ? S-o* I— 5» -fi ~* .'s ^; J5 » C j*

| ' S 2 s ' S ' * £ x . I ^ " e.:! 1 3 ?'H Illl

m 4: 'IS J> .c — ai si '.S v ~ ™

5 j j lsS-sl-Hsn3 £ £ st i .H -^ » —"— - • C. If C <j « .- •; ~ C — •»

- 5 2 3 * f* * >. ji S. S =

^ -2 J £' | .§ i ! I" f 1 .1' Z > - ? « 2 i ~ " ' i f . T - : : 5" —

" ' i; ! * B S g ' i ; . 2 i H

PAG€ fSQUALITY

->*=.5

. £ — • ^^ ^ ^^ *^ ^^ ^f ^^ -^*^ _ ' l» ^ OO ^ ^ ^ ^ GC— • ~ J ' " V v . O iff ^

rt (

** "^ uTI^toe^ toStn's.-_ -=j r; • ^ — ^ ,2 .2

"-; S *<J L? tf M to n CM_ ^_ ^) r? — cc oa>

"~ C '-• *— >,i

«s ^ 1 rt rt rt ^rt « • • «

£ E rt rt rt ^« w «• • «

> 5 '?- — rC t* ?. rt rt « «•« — s ' C3 w < ^

^5 T ^cSr*-1- -^ d J? ^

• J S ^ = ~ 5 £ = ~ 5 3•si 1 *- c ce = —;

*3 o i ... ^» -^ ^*•1 •* £ ff ~ ~ <7 JT -5 J2

•-5 ^ ' * * * * * ,i /2 ** ** ^ '

S ** *^ -**^"?^? '?-*^-?'«'5*3 i n n it tt < « M I « I «rt w rt rt r* *« rt (• r^ r<5

%• ^2 I »• »<* W ^* r* •• W Wh. ^ IM« ^ <w w W W ^ <^V £ ~ — • -^ C.

. . V.> i

-S 1-3 5r* •-

-•J 2 -5 u S ss"S fe c» . r . «> _- « to= a. "i-^ "- • 1^U.T9 _>* .13 - ^ * S * —

Si -r _ =. "^ -if ~i 5 c» i* "C 5 * - • - • - S i— • -

. « 1 ?" 1 :~ 1 ^ - * ~^.t/: <T^ " « t T a ^ c S• ff- ^ ~ ~ ^3 ^^ rt ^ J^ "•

• ^* "^ ^ S — 3i — • ^S — " S1-

p>i i— "•"• 5 " « ^ — • —

™ = 4 ••" = » ° S s" § =^ s . - - ^ . „• > I <N •=* z *~* - •- "Z • S 5 -3 -2is* of M * 3 -a> ':t; i J« . - ~* • " > « ! > s x " " SS- SI s" J 6 1 ^ 0 ! ? ' -~ >, 2s 2 l r - « ^ a , c . w >of -| " ? £ !>§• si

="•§ gl: * 4 i sl^ I

il sl £ * 1 i'l--? 1'" **• - • s = £ — -H ? -

T ™ .^ O *•• "™ S£ ^

ri j ~ 1 •- 1 2 "i 5 3 -^ ",: ^ 1 '* ^. .Z -z. -5 '-S S- - 2 -

• 42 s"! " 2 - 2 « i^| iIS -f.f SS " n ^ g - J i ^ ^

2 .= ~ — •, j n — •- T! ~5 r= j5 ; = = = — : ^ . ^ . £ s ^

= S -. '-^ ^ '; = o ~ - ~ "&- ~^ - ^ "?s O1 - !— C- — S = Suir-: S

« -- « -?T •* « «

8 S c ! S i : - g ^ g ) S ' S 5 -1^ OJ C h. -C ty *Q * .« ^ -S

s | l l * * " S S S r j "1* ^ S & S j S ^ w J < a — r r> . v n . . » « < « « * . . AJ t2.h •= Sis'S-Y J £ s •:l £ ^ a ™ - ' ° i " - B > i .is"o S B ' ^ - g - S ^ " , , ; ? ? "i

tlifll^I i \;i2i!,Hi4il 1:|lji!!l!lf Jg - ^ S ' s S f c . ^ jS-B,B -•= -c S e 0 T5 •£ -S ; - •; S- j: «2 * T 5 - - 2 S > 3 . e

S E § < .g T

^Ii|!<i^!l- 1 <=" 2 S 5 * ~ « ' B ' « w ' S —fl lf Irsi- ie .; ^» 5 i g ? S g ^ - g g . 2 - - - ..,:?" T S o O - j l S l S S '= "^^« 8 - s s . 2 ^ : i a . s a | ^ c3s•3 = H I i | = ji 1 s 1 .1.s • - c " £ 5 - = S ' 3 - _ . — .,» - * " ? . i s ! ?^ ' 3 ^^ ~ ; r t2 * ' ? 8 8 o H 2 S « . t — "2C3- 5 ^ | S £ . « - ° ^ < a ^ — Si

"li^lli^lils; -* J°: r=&=?3ga^I t l "

— t » 5 * ' * S r t - 3 c y 4 ' w - J * t a •S £ 5 C 3 5 B M g 5 2 - r ^ - 5

= *•» — ! 2 . - w J i e ^ ~ •i ' S > e 3 C : r 5 > ' S « e £ = 2 ~i j = t ; ? s f j s » ' g si- - < • ? « -a » „ "8 - o js .2 =£ It c |?=2|1 i7 -sj ^ ^ r l ^ S x l s i - s ? ^ '1

"Id ililli iiIsS'lf isl lJi- 's 3 §

M — -rh «

*! 1?^ ^ **^ c. «s ^— < "•- £i « *; =

- tS -3 13e . 3 ™ 5 *2 -2 ? 5 ~— s •£ o cCT .^ - a. K

'V -I's -|^— •« " *9 "2 *— ' * —

S ft 5 — « .iO . — -^ ^ f ^*

g-^** 2-2 •S'ssS & efe"3 « -5.

S-3 5S -asa ^S § 2 "g c?« o 5 j« | _*• • "" S --J1 -£5 --• *» | ^"?

*". 1 1^ " " >"= <* • f u "3 * *'r =" "• '= 8 £' En •= or e ._ _s s •'«••= "-••

• — C? " S x"^S *""* -•« < K -= t

•fl — — ~ ^^^« _• "".- 5 u 2>, 5 "S 3 j; — =- A .0-— . o -- or — s_ _. « x _=

^ » •• J i s! i «ao ~ e» ' c '"-^

«T ^ ** .S ia ?•3f ** O ^ * a

"= "^ — '£ .? ?.^ t. ? T !I

« jj B •* *•

"~ «— ** •" *5 *2

••pjiM •« S — " ftC **•

*U 2 yi .s e £i -1 ^ - j" 5 ^ '— — «

•f Z * •=• 5 S 4- •« n s ~ - 5 - 3 5 j

J: s T « 2- J

^5 Uk " "*" fc(f BO O •— ^ 2*

•§ 5 -w ^ £ 20 ^3 ff ^ {£ OO V " ^ >M l#

» £.£•"•• 2 •= .

lilfiT2 a *— T r~- « c .= r- -

js 2 = — - = —-•== -.i