PHYSICS OF THE SOLID STATE
VOLUME 41, NUMBER 2
FEBRUARY 1999
SEMICONDUCTORS, DIELECTRICS Differences in the atomic environment of nonequivalent sites in SiC-polytype structures A. E. Madison St. Petersburg State Electrical Engineering University, 197376 St. Petersburg, Russia
~Submitted February 18, 1998; accepted for publication July 10, 1998! Fiz. Tverd. Tela ~St. Petersburg! 41, 183–186 ~February 1999!
A general algorithm is proposed for calculating the Q series of SiC polytypes. The obtained Q series of the main SiC polytypes can be useful in calculating lattice sums, in particular when using the Mellin transform of the Q series. By expanding the Q series in the Jacobi parameter, one obtains sequences of coordination numbers for crystallographically nonequivalent atomic sites in the main SiC polytypes. A nontrivial interrelationship is demonstrated between these numerical sequences and the local symmetry of the nonequivalent sites. © 1999 American Institute of Physics. @S1063-7834~99!00202-6#
of the stability of atomic clusters in the initial stages of crystal growth,16 the study of fine structures of impurity states,13 and in some other problems. Of all the SiC and ZnS polytypes, the Q series are known only for the simplest structures—sphalerite and wu¨rzite.16,17 With this fact in mind, the main task of the present work was to determine the Q series of the basic polytypes of silicon carbide, and also to determine the coordination numbers for crystallographically nonequivalent atomic sites in the structures of the most important polytypes. To obtain the Q series of polytypes of SiC and ZnS we used a general approach rigorously developed in Refs. 17 and 18. First it is necessary to note the following fact. We assume that the Q series is already known for a case in which the origin coincides with one of the sites of one of the sublattices. If we then translate the origin to one of the atoms of the other sublattice, the fundamental picture of the atomic environment does not change and it is sufficient to replace all the Si atoms by C atoms and conversely. Thus, it suffices to consider the Q series by successively locating the origin at nonequivalent sites of just one of the sublattices. For the case in which the origin coincides with one of the sites of the other sublattice, the Q series can be obtained by a formal substitution of atoms, and it is necessary just to bear in mind the strict 1–1 correspondence of the nonequivalent sites of the different sublattices. This conclusion is valid for almost all the main polytypes of SiC and ZnS. As an exception we may cite the example of the quite rarely encountered polytype 21R-ZnS or (2311) 3 in Zhdanov symbols.7 For it, the Q series of the zinc sublattice in the (2311) 3 structure are equivalent to the series of the sulfur sublattice in the structure (3211) 3 , and the converse. All Q series can be expressed in terms of Jacobi Q functions.16,19 After substantial simplifications thanks to the use of some of their properties, and introducing the a , b and w functions derived in the Appendix, we succeeded in showing that the Q series of any nonequivalent site in the struc-
DIFFERENCES IN THE ATOMIC ENVIRONMENT OF NONEQUIVALENT SITES IN SIC-POLYTYPE STRUCTURES
SiC polytypes are of great practical significance as a class of promising semiconductor materials having a range of band gap widths.1–6 In addition, they have definite academic interest as natural superlattices and as a models for a proper study of polytypism.7–9 ZnS also crystallizes in analogous structures. A characteristic of polytypes is the presence in their structures of crystallographically nonequivalent sites differing in the local symmetry of their immediate environment, and also the number and mutual arrangement of the atoms in the distant coordination spheres. Their existence has a serious effect, in particular, on the kinetics of impurity capture during crystal growth,10,11 the multiplet structure of the impurity states,12,13 and the optical, electrical, and some other properties of SiC polytypes. All polytype structures differ from one another by the stacking sequence of double Si–C layers along some preferred direction ~the c axis!. Structural characteristics of the main polytypes of SiC and our notation for nonequivalent sites are depicted in Fig. 1. We find it necessary to draw attention to the strict 1–1 correspondence of nonequivalent sites of atoms of different sublattices. In particular, for the polytype 6H the ordering of nonequivalent sites (h,c 1 ,c 2 , . . . ) along the c axis in the silicon sublattice corresponds to the ordering of nonequivalent sites (h,c 2 ,c 1 , . . . ) in the carbon sublattice. We investigated differences in the atomic environment of nonequivalent sites using the method of Q series. Besides the fact that use of Q series can substantially expedite the task of calculating lattice sums, they are of independent interest in crystallography. As a result of expanding them in a series in the Jacobi parameter, the coordination numbers for successive coordination spheres can be obtained without especial effort.14–17 Such information is useful, for example, in the analysis 1063-7834/99/41(2)/4/$15.00
160
© 1999 American Institute of Physics
Phys. Solid State 41 (2), February 1999
A. E. Madison
161
¯ 0) planes for the main FIG. 1. Arrangement of atoms in the (112 SiC polytypes using a hexagonal unit cell. For rhombohedral polytypes, only one-third of the unit cell is shown. In view of the absence in the literature of common notation for crystallographically nonequivalent sites, here we use our own notation. The rhombohedral polytype 9R-ZnS is hardly ever encountered for silicon carbide, but it is one of the common polytypes of zinc sulfide.
ture of any polytype can be obtained using relations of general form. In particular, if we place the origin at one of the nonequivalent sites in the silicon sublattice, then the relative positions of all the remaining silicon atoms will be determined by the Q series Q Si–Si~ z ! 5 a ~ z !
(w kPA
1b~ z !
S
2 2 k n z, 3 n
(w kPB
S
D
D
2 2 k n z, , 3 n
~1!
and the Q series defining the relative positions of the carbon atoms can be obtained using
Q Si–C~ z ! 5 a ~ z !
(w kPA
1b~ z !
(
kPB
S
2 2 n z, 3
S
3 4
D
k1
3 4
k1
2 2 n z, w 3
n
n
D
.
~2!
In these equations, n is the number of layers per unit cell of given polytype and the argument k successively takes all values of some set A, for the first sum, and from some set B, for the second sum. Placing the origin now at one of the nonequivalent sites, we consider atoms in successive layers 0
nation tetrahedra were assumed to be ideal, and the ratio of dimensions of the unit cell was also taken to be equal to its ideal value c/a5n( A6/3) for all polytypes. Utilizing the hexagonal arrangement of axes and the above-described method for analyzing the cubic polytype 3C ~sphalerite!, we obtained the Q series for the Si sublattice which, to within a scale factor, coincides with the series of the PZK lattice published in Ref. 16, where a cubic arrangement of the coordinate axes was used. The carbon sublattice corresponds to an analogous series found by translating the origin to one of the tetrhedral voids.16 After expanding the Q series of the 2H –SiC structure ~wu¨rzite! in powers of the Jacobi parameter it is clear that our results also agree with the results of Ref. 17. The coordination numbers of the most important SiC polytypes for the first 43 coordination spheres are listed in Table I. The first column gives the square of the radius of the coordination sphere, expressed in units of the lattice constant a. This number can be considered as an index of the coordination sphere m, which can be fractional. The following columns give the number of atoms in it S m for the most important polytypes, including the structures 2H and 3C. The last column indicate the kind of atom ~Si or C! of which this spheres is composed. The most widespread polytype of SiC is 6H. Like 2H, it belongs to the space group P6 3 mc, but this structure has three crystallographically nonequivalent sites—one hexagonal and two cubic ~see Fig. 1!. The Q series for them can be obtained from Eqs. ~1! and ~2! if we set n56 in them. An analysis of the coordination numbers given in the table for 6H-SiC points up their nontrivial interrelationship with the site ~hexagonal or cubic! at which the atom is found. In particular, an atom at a hexagonal site has the same local
162
Phys. Solid State 41 (2), February 1999
A. E. Madison
TABLE I. Sequences of coordination numbers for nonequivalent sites of SiC polytypes. 3C 2H
4H
6H
15R
r2
c
h
h
c
h
c1
c2
h1
c1
h2
c2
c3
0 3/8 1
1 4 12 –
1 4 12 1
1 4 12 –
1 4 12 1
1 4 12 –
1 4 12 1
1 4 12 –
1 4 12 –
1 4 12 1
1 4 12 –
1 4 12 1
1 4 12 –
Si C Si C
12
9
12
9
12
9
12
12
9
12
9
12
C
1
6 –
6 6
6 –
6 6
6 –
6 6
6 –
6 –
6 6
6 –
6 6
6 –
Si C
3
12
9
12
9
12
9
12
12
9
12
9
12
C
1
1 24 3 18
2 2 24 28 2 23
–
2
–
2
–
1
1
–
2
–
1
1
Si
3
24 16
18 9
24 15
18 9
24 16
21 9
21 16
24 16
18 9
24 15
21 9
21 15
Si C
2
–
12
–
12
–
6
6
–
12
–
6
6
Si
3 38 33 17 3 24
–
3
3
3
–
3
–
–
3
3
3
3
C
12 –
6 6
12 –
6 6
12 –
9 6
9 –
12 –
6 6
12 –
9 6
9 –
Si C
3 48 17 4 24
24
18
18
18
24
18
24
24
18
18
18
18
C
–
3
3
3
–
3
–
–
3
3
3
3
C
5
24 –
12 7
24 –
12 7
24 –
18 6
18 –
24 –
12 7
24 –
18 6
18 1
Si C
12
3
12
3
12
6
12
12
3
12
6
9
C
–
12
–
12
–
6
6
–
12
–
6
6
Si
–
6
6
6
–
6
–
–
6
6
6
6
C
8 –
6 6
6 –
6 6
8 –
6 –
6 –
7 –
6 6
7 –
6 –
6 6
Si C Si
4 1
4 24
1
5 24 3 58 2 53 17 5 24
6 1
6 24 1
63
–
6
6
6
–
6
6
3
6
3
6
6
3 68 2 63
24
12
18
12
24
15
24
24
12
18
15
15
C
–
12
–
12
–
6
6
–
12
–
6
6
Si
7
48 –
24 1
36 1
24 1
48 –
30 –
30 –
42 –
24 –
42 1
30 1
30 1
Si C
–
6
6
6
–
6
6
3
6
3
6
6
Si
36
15
27
15
36
24
36
36
18
27
21
21
C
–
6
6
6
–
6
–
–
6
6
6
6
C
6 –
– 24
6 6
– 24
6 –
3 12
3 –
6 –
– 18
6 6
3 18
3 12
Si C
–
12
12
12
–
12
12
6
12
6
12
12
Si
12
9
9
9
12
12
12
12
12
9
9
9
C
1
7 24 1 73 3 78 17 7 24
8 1
8 24 1 83 3 88 17 8 24
–
3
3
3
–
3
–
–
3
3
3
3
C
1
36 –
12 6
24 –
12 6
36 –
18 –
18 –
30 –
12 6
30 –
18 –
18 6
Si C
98
3
28
12
21
12
28
24
28
27
18
22
18
15
C
2 93 17 9 24
–
24
–
24
–
12
12
–
24
–
12
12
Si
–
9
9
9
–
9
–
3
9
6
9
9
C
24
12
12
12
24
12
12
18
12
18
12
12
Si
9 9 24
10
symmetry of its first two coordination spheres as in wu¨rzite, and the number of atoms in the far coordination spheres coincides exactly with the corresponding numbers for sphalerite ~i.e., the cubic modification!. The identity of the corresponding Q series graphically illustrates the fact that fundamentally different structures can have identical Q series. For one of the two cubic sites, the sequence of coordination numbers differs not too greatly from the correspond-
ing sequence for sphalerite and, for the second cubic site, it differs not too greatly from the corresponding sequence for wu¨rzite. The first differences from the ideal sequences show up starting at the eighth coordination sphere, which corresponds to '267 nearest interatomic distances. Such results are entirely unexpected at first glance. The 4H-SiC structure has two nonequivalent sites: hexagonal and cubic ~see Fig. 1!. The sequence of coordination numbers for the hexagonal site differs only insignificantly from the corresponding sequence for sphalerite, and the differences appear starting from the tenth coordination sphere ~3 nearest interatomic distances!. The analogous sequence for the cubic site coincides with the sequence of coordination numbers of wu¨rzite ~i.e., the hexagonal modification!. The 15R-SiC structure has five nonequivalent sites: two hexagonal and three cubic ~see Fig. 1!. The coordination numbers for them are also listed in the table. The nontrivial interrelationship between the coordination numbers and the local symmetry of the site noted by us for the 6H and 4H-SiC structures also holds for the other two polytypes. In conclusion we note the following salient points. A general algorithm for calculating the Q series of SiC and ZnS polytypes has been proposed. The Q series have been analyzed in detail for the most important SiC polytypes. The sequences of coordination numbers so obtained for all the crystallographically nonequivalent atomic sites in the main SiC polytypes allow us to conclude that there is a nontrivial interrelationship between these sequences and the local symmetry of their immediate environment.
APPENDIX
In general, a Q series is defined as a holomorphic function of a complex variable Q L~ z ! 5
(
rPL
q N ~ r! 5
(m S m q m ,
where q5e i p z is the Jacobi parameter, N(r)5r 2 is the lattice vector norm, and S m is the number of vectors in the lattice L with norm equal to m. The expansion coefficients of the Q series in powers of the parameter q give the coordination numbers in the successively considered coordination spheres with radius Am. The structures of all SiC polytypes can be considered as a union of several primitive hexagonal sublattices L 0 , shifted relative to one other by the vector u. In this case, for the kth nonequivalent site, the Q series is defined as the sum of the series of sublattices18 n
Q L~ z ! 5
( (
q N ~ r1u j 2uk ! .
j51 rPL 0
All Q series can be expressed in terms of the Jacobi Q functions.16,19 To avoid the alternative interpretations encountered in the literature, we are forced to give specific equations
Phys. Solid State 41 (2), February 1999
A. E. Madison
1`
q 3 ~ j ,z ! 5
( e i~ 2m j 1 p m z !, m52` 2
b~ z !5
Im~ z ! .0;
F
SD
S DG
1 z z q ~ z !q2 1 q 3~ z ! q 3 2 2 3 3
163
1 2 a~ z !. 2
1`
q 3 ~ z ! 5 q 3 ~ 0, z ! 5 p q 2 ~ z ! 5e i 4 z
q3
(
2
qm ;
m52`
S D
p z,z 5 qS 2 m52`
(
D
2
1 2 m1 2 .
Developing the approach in Ref. 16, we deem it useful to introduce the function 1`
w ~ z, t ! 5e
ipzt2
q 3 ~ p t z,z ! 5
( q ~ m1 t ! . m52` 2
It is easy to show that the parameter t can always be selected from the interval 0< t <1/2. It is also clear that the standard Jacobi functions q 2 (z) and q 3 (z) are particular cases of the function w introduced above, specifically q 3 (z)5 w (z,0) and q 2 (z)5 w (z, 21 ). For the function w it is possible to prove the identity k21
(w j51
S
z, t 1
Yu. M. Tairov, Mater. Sci. Eng., B 29, 83 ~1995!. R. F. Davis, G. Kelner, M. Shur, J. W. Palmour, and J. A. Edmond, Proc. IEEE 79, 677 ~1991!. 3 R. F. Davis, J. W. Palmour, and J. A. Edmond, Diamond Relat. Mater. 1, 109 ~1992!. 4 J. H. Edgar, J. Mater. Res. 7, 1, 235 ~1992!. 5 R. F. Davis, J. Vac. Sci. Technol. A 11, 4, 829 ~1993!. 6 C. M. Venvliet, G. Bosman, and L. L. Hench, Ann. Res. Mater. 18, 381 ~1988!. 7 A. R. Verma, P. Krishna, Polymorphism and polytypism in crystals, ~John Wiley & Sons, N.Y. 1966; Mir, Moscow, 1969!. 8 Yu. M. Tairov and V. F. Tsvetkov, Prog. Cryst. Growth Charact. 7, 111 ~1983!. 9 M. Dubey, U. Shanker Ram, K. Nath Rai, and G. Singh, Phys. Status Solidi A 18, 2, 689 ~1973!. 10 G. B. Dubrovskii and E. I. Radovanova, Phys. Status Solidi 48, 875 ~1971!. 11 D. J. Larkin, P. G. Neudeck, J. A. Powell, and L. G. Matus, Appl. Phys. Lett. 65, 1659 ~1994!. 12 G. Pensl and W. J. Choyke, Physica B 185, 264 ~1993!. 13 Y. A. Vodakov, G. A. Lomakina, E. N. Mokhov, E. I. Radovanova, V. I. Sokolov, M. M. Usmanova, G. F. Yuldashev, and B. S. Machmudov, Phys. Status Solidi A 35, 37 ~1976!. 14 I. J. Zucker, J. Math. Phys. 16, 11, 2189 ~1975!. 15 M. L. Glasser and I. J. Zucker, Theor. Chem.: Adv. Persp. 5, 67 ~1980!. 16 N. J. A. Sloane and B. K. Teo, J. Chem. Phys. 83, 6520 ~1985!. 17 N. J. A. Sloane, J. Math. Phys. 28, 1653 ~1987!. 18 A. M. Odlyzko and N. J. A. Sloane, Stud. Sci. Math. Hung. 15, 4, 461 ~1980!. 19 Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, edited by M. Abramowitz and I. A. Stegun, National Bureau of Standards, Appl. Math. Series-55 ~1964!. 1
1`
D S D
j z 5w 2 ,tk , k k
where k is an arbitrary integer. The Jacobi functions are interrelated by an entire labyrinth of useful relations,19 many of which are special cases of the last equation. Making use of it, it is possible to show that the Q series obtained by analysis of the polytype structures can be expressed in terms of a combination of Jacobi functions, for which it is convenient to introduce separate notations
a ~ z ! 5 q 2 ~ z ! q 2 ~ 3z ! 1 q 3 ~ z ! q 3 ~ 3z ! ,
Translated by Paul F. Schippnick
