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Short-range order in amorphous germanium
JOURNALOFNON-CRYSTALLINESOLIDS1 (1969) 371-387© North-Holland Publishing Co., Amsterdam
S H O R T - R A N G E O R D E R IN A M O R P H O U S
GERMANIUM
R. GRIGOROVICI and R. MA, N/'~IL/'~
Institute of Physics of the Academy, Calea Victoriei, 114, Bucharest, Romania Received 11 February 1969 The structure of amorphous germanium is described by an array of atoms, linked together in distorted tetrahedra. There is a limited number of short-range orders with only two admitted configurations (staggered and eclipsed) present if all bonds are to be satisfied (ideal structure). Analysis of the ideal structures shows that in small clusters groups with five-fold symmetry, containing preferentially eclipsed bonds are energetically favoured, thus explaining the formation of an amorphous phase when germanium is built up atom by atom at sufficiently low temperature. Defects can be introduced into the ideal structures by cumulation of misfits, by blocking of atoms in wrong configurations, when neighbouring clusters grow together, etc. The model agrees with many experimental facts and data.
1. Introduction
Amorphous germanium, like other amorphous semiconducting elements or mllIBV compounds whose structure is based on essentially covalent bonds with tetrahedral symmetry, can be obtained only in the form of thin films built up atom by atom (or, eventually, from very small clusters of 2-8 atoms already linked together) by vacuum evaporation1), by cathodic sputtering2), by electrolysis 3), or by rf discharge 4). As in the case of crystalline solids the understanding of the physical properties of amorphous semiconductors must be based ultimately on the knowledge of their structure, i.e. of the arrangement in space of their atoms. But unlike crystalline solids, the periodic structure of the lattice is lost in amorphous solids and the whole formalism which links the structure of the reciprocal space with the energy band structure in k-space is no longer applicable. What can be inferred about the structure of amorphous solids from the isotropic X-ray or electron diffraction patterns is an atomic or electronic radial distribution function. The corresponding radial distribution curve (r.d.c.) consists of a series of rather broad maxima and minima which display the change in atom or electron density versus the radial distance from an arbitrary chosen central atom. The structure is implicitly supposed to be spherically symmetrical. This may be for some purposes a fair description of the short-range order of liquid metals, where the first coordination 371
372
R. GRIGOROVICI AND R. MANAIL,~
number is relatively high (10-11). It is certainly a very poor description of the short-range order of amorphous semiconductors with tetrahedral symmetry, where this number is four. If diffraction experiments could be performed on volumes of a few ~mgstr/Sms radius, i.e. much smaller than it is actually done and possible to do, both the diffraction pattern and the atom or electron density would be far from isotropic. It must also be recalled that the r.d.c, is obtained by the Fourier transformation of the scattered intensity versus scattering angle function. Instead of being extended to infinity, the Fourier integrals have unavoidably a finite upper limit. As a consequence the maxima of the r.d.c, are broadened and ripples without any correspondence to physical reality appear on both sides of these maxima (end-of-series effect). The present knowledge about the short-range order of amorphous germanium (a-Ge) can be summarized as follows: l) The area under the first maximum of the r.d.c, yields the first coordination number fourS,~). The a v e r a g e symmetry of the nearest neighbours of the central atom is therefore tetrahedral and the bonds are due to the overlapping of one of the sp a orbitals of these neighbours with those of the central atom. Hence the first coordination sphere is similar to that in the Ge crystal. 2) The analysis of the position and the shape of the first maximum of the r.d.c, shows s, 7) that the first interatomic distance is increased by about 3% as compared with the first interatomic distance in the Ge crystal (2.50 versus 2.43 A) and that, apart from the Fourier broadening due only to the mathematical procedure, there is also an intrinsic broadening due to fluctuations in this distance which correspond in non-annealed a-Ge films deposited at 300°K to a standard deviation A r l of +0.19 A or ___7.6% of this distance, respectively. 3) In order to duplicate the whole experimental r.d.c, of a-Ge by that of a structural model in which all Ge atoms are linked with one another tetrahedrally by s p 3 covalent bonds (ideal amorphous structure), one is forced to admit that: a) The fluctuations of the successive interatomic distances are increasing like r n, where 0.5 < n < 1 (ref. 7); as the second interatomic distance in both the Ge crystal and in a-Ge corresponds to the length of the tetrahedron edge, it follows that each of the real tetrahedra formed by the four nearest neighbours of any of the atoms in a-Ge is in general not only radially but also angularly distorted. In non-annealed a-Ge films deposited at 300°K the fluctuation in the lengh of the edge of these tetrahedra lies around A r 2 = ( r 2 / r l ) °'Ts x A r 1 =(4.09/2.50) 0.75 x 0.19 = +0.28 A
or + 6.8%.
SHORT-RANGE ORDER IN AMORPHOUS GERMANIUM
373
b) Neighbouring Ge atoms are linked together in two different configurations called "staggered" and "eclipsed", respectively, after the relative orientation of the two groups of three orbitals not involved in the binding of these atoms (fig. 1). The Ge lattice contains the staggered configuration only. In the wurtzite lattice a quarter of the bonds are eclipsed, the rest are staggered; Ge atoms linked together consistently only by eclipsed bonds form five-fold rings with only small angular misfits; these rings can merge into a regular dodecahedron (a so-called "amorphon") having 20 Ge atoms at its corners; these dodecahedra can again merge into one another along a five-fold ring and so on. It was shown by us a) that the experimental r.d.c. of a-Ge is matched by the r.d.c, of a 50/50 per cent mixture of small, slightly expanded, heavily distorted Ge crystallites and of amorphonic groupings. Independently Coleman and Thomas s) proposed the same model, with a slightly different ratio (60/40) of crystals and amorphons to describe the short-range structure of a-Si films. Obviously the five-fold symmetry of the rings of Ge atoms linked together in eclipsed configuration only explains the lack of translation symmetry in a-Ge and a-Si and hence there seems to be some truth in this model. But in many other respects the model suggested by us and by Coleman and Thomas is rather rough and unsatisfactory. There are still many questions to be answered. The following is a list of such questions: 1) Are the two types of configurations intimately intermixed in a-Ge or are they forming a mixture of two different "phases", crystalline and amorphonic, separated by a sort of grain boundaries. Is an intimate mixture of staggered and eclipsed configurations geometrically possible without leaving many covalent bonds unsatisfied. Could one therefore describe a real amorphous solid by introducing into an ideal amorphous semiconductor, having all bonds satisfied, some local structural defects, where this condition is not satisfied. 2) If an ideal amorphous semiconductor containing only s p 3 bonds can be formed, is the presence of radial and angular fluctuations within its basic unit, the tetrahedron, an unavoidable consequence of the angular misfit introduced by the eclipsed configurations and are these fluctuations of the right order of magnitude.
S tagge~cl
E~ed
Fig. 1. Staggeredand eclipsed configurations.
374
R. GRIGOROVICI AND R. MANAIL.~.
3) Could one find an energetical justification for the formation of eclipsed bonds when solid amorphous germanium is built up atom by atom on a substrate. 4) I f so, what must the influence be of the substrate temperature or of subsequent annealing on the detailed structure of a-Ge films and how do these changes fit experimental observations, especially the experimental r.d.c, and the electrical properties. The final aim of the present paper consists in giving a much more comprehensive and detailed description than usual of the short-range order in a-Ge, which could form the basis for a calculation of the energy band structure of this amorphous element, e.g. on the lines proposed recently by Fletcher 9).
2. The ideal structure of amorphous germanium The basic unit used in modelling the ideal structure of a-Ge, i.e. having all covalent bonds satisfied, is a modified Voronoi polyhedron of the diamond-like lattice used previously6, s). The exact Voronoi polyhedron of the diamond-like lattice is shown in fig. 2a and represents the volume occupied by a Ge atom within the lattice. Covalent sp 3 bonding between neighbouring atoms corresponds to perfect contact between the big hexago-
a
b
c
d
Fig. 2. (a) Voronoi polyhedron in diamond-like lattice. (b) Simplified Voronoi polyhedron. (c) Fragment of diamond lattice built up of simplified Voronoi polyhedra. (d) Amorphon. (e) Fragment of wurtzite lattice. 0-1, 0-2, 2-3 and 4-5 are staggered bonds ; bond 2-4 is eclipsed. 0 and 5 touch along triangular face.
375
SHORT-RANGE ORDER IN AMORPHOUS GERMANIUM
nal faces of two contiguous polyhedra. Contact between Voronoi polyhedra along two small triangular surfaces corresponds to the much weaker link with second order neighbours. These links are neglected in our considerations, as nearly free rotation about the c o m m o n bond is assumed (see below). Therefore the four small triangular pyramids have been completely flattened to form each a single triangular face (fig. 2b). These modified Voronoi polyhedra can be used to build up not only the diamond lattice (fig. 2c), but also the a m o r p h o n (fig. 2d), which takes the form of a regular icosahedron* with five-edged pits at its corners, and the wurtzite lattice (fig. 2e). In this latter case some of the second-order neighbours touch along the small triangular faces, while the bonds between nearest neighbours within the atomic layers perpendicular to the c-axis are staggered and those between nearest neighbours lying in adjacent atomic layers are eclipsed. As mentioned earlier, a certain misfit in covalent bonding must be tolerated as a consequence of eclipsed bonding in the amorphonic structure. This is due to the fact that there is a difference of 1 ° 28' between the angle of 70 ° 32' formed by two hexagonal faces of the basic unit and the central angle of 72 o of the five triangles which form a regular pentagon. This is illustrated in fig. 3a where a five-fold ring of basic units linked in eclipsed configuration is viewed along the c o m m o n edge C. The consequences of a small misfit are: a) a certain increase in the interatomic distance a' as compared with the corresponding distance a in the diamond-like lattice (fig. 3b); within a five-
\
I
\-.i/
/
./
b
a
Fig. 3. (a) Five-fold ring of simplified Voronoi polyhedra; mean angular misfit 0 = 1o 28" is exaggerated for the sake of clarity, a ' = mean interatomic distance. (b) Interatomic distance a between two exactly fitting Voronoi polyhedra. * Incidentally the regular icosahedron was the form attributed by Platon (Timaion, 55b-56b) to the solid basic unit of water, the third element.
376
R. GRIGOROVICI AND R. MAN,~ILA
fold ring or a single amorphon (icosahedron), the mean relative increase (a'-a)/a due to this misfit amounts to 1.8~; b) a smaller overlapping between the s p 3 orbitals of neighbouring atoms due both to the increase in the interatomic distance and to an increase in the angle between the symmetry axes of these orbitals from zero to 0 (fig. 3a). Within a five-fold ring or a single amorphon the mean value of 0 is 1 ° 28'. The possible ideal structural models of a-Ge were obtained by adding to a central atom successive correlation layers, i.e. groups of basic units touching (or nearly touching) one another along their hexagonal faces and containing all neighbours of a given order (first, second, third, etc.)*. Only those configurations which permit the addition of the next correlation layer, i.e. without letting any free hexagonal face uncovered by the basic units of the next layer, were taken into consideration. This latter condition greatly restricted the number of admissible structures and gave us the possibility of solving the problem without using a computer, by simply manipulating a moderate number of basic units made of polystyrene. The units could be linked together by a bunch of strained rubber strings converging into the central atom and crossing the basic units along channels drilled perpendicularly to the centres of the hexagonal faces of the truncated tetrahedra. In this way both the directional character of the s p 3 covalent bond and the free rotation about this bond was succesfully simulated. This technique was used only to the second correlation layer. Structures extending to the third and fourth correlation layer were obtained by gluing the basic units together. While in the first correlation layer formed by the four nearest neighbours, the above-mentioned condition did not restrict in any way the relative rotation of the basic units, purely geometrical conditions restricted these rotations from the second correlation layer to only the staggered and the eclipsed configurations. (For energetical arguments concerning free rotation and the stability of the two configurations within a system of only two atoms, see ref. 6.) This greatly simplifies the problem and eliminates at once the structural model of a-Ge proposed by Richter and BreitlingS), who admit these rotations to be aleatory. Their great merit remains that they have drawn the attention to the importance of this rotation. By restricting ourselves only to these two configurations, the number of different structures containing the central atom and the first two correlation layers nevertheless amounts to 4096. However, when applying the condition * Obviously there is no identity between the so-defined correlation layer and the usual coordination sphere. A correlation layer generally contains atoms situated at different distances from the central atom; hence they are members of different coordination spheres (see table 1).
S H O R T - R A N G E ORDER I N A M O R P H O U S G E R M A N I U M
377
of ideality, this number drops to 33 structures of which only 10 are distinct. Table 1 displays the results of this investigation. The first three correlation layers of the different distinct ideal structures are characterized as follows: a) the ratio s/e between the number of staggered and eclipsed configurations; b) the statistical weight of the structure, if it is supposed that the deposition of a given correlation layer takes place on an already frozen structure containing one less correlation layer (see however section 4) and that there is an equal probability of a newly arriving atom to be bound in any of the two configurations, staggered or eclipsed; c) the total number of atoms within the correlation layer; d) the number of perfect and imperfect bonds (because of angular misfit and increased interatomic distance); e) the number of free hexagonal faces (unsatisfied bonds); f) the number of wurtzite-type bonds (contact between triangular faces); g) the energy of the structure. The data for the fourth correlation layer are less complete: only the total number of atoms, of satisfied and non-satisfied bonds, of wurtzite-type bonds, and the energy are recorded. Indeed changes from staggered to eclipsed configurations do not change the energy of the structure or the distribution of atoms over the coordination spheres below about 7 A, i.e. within the region which is decisive for checking the model by comparison between the calculated and the experimental r.d.c. (see section 4). Inspection of table 1 as well as a series of attempts to go farther with the building up of succesive correlation layers showed that no structure which proved to be continuable at a given order of correlation became noncontinuable at a higher order. On the contrary, the number of continuable structures increased with increasing correlation order. We can therefore conclude quite safely that there are many ways in which an ideal structure can be realized in a-Ge. Defects in the ideal structure must appear wherever the ideality condition is not fulfilled and, as a consequence, overlapping between orbitals becomes very poor or, on the contrary, more than two orbitals overlap significantly. Both types of defects may occur and there might even be a high probability to find two defects of opposite type close to one another. It is however to be expected that the first type of defect will prevail in a-Ge. What are the arguments now against or for a-Ge being simply a mixture of small Ge crystallites and amorphonic gloupings of atoms in nearly equal parts. It was pointed out earlier that to duplicate the experimental r.d.c, of a-Ge both the crystallites and the amorphonic structures must present relatively big fluctuations in interatomic distances. There is no reason why this should
378
R. GRIGOROVICI AND R. M,~NAILA
Tt I st layer
2nd layer
Bonds
Bonds
Imperfect
-~
4/0
1/16
4
4
-
-
12
-
--1.307
3/1
4/16
4
4
-
-
12
-
-1.307
Imperfect
12/0 9/3 6/6 3/9 0/12
1/16 4/16 6/16 4/16 1/16
12 12 12 12 12
12 12 12 12 12
-
-
36 36 36 36 36
12/0
1/5
12
12
-
-
36
9/3
4/5
12
12
-
-
-1 --1 --1 --1 --I
-I
36
-
-1
2/2
6/16
4
4
-
-
12
-
--1.307
10/2
1
12
12
1
-
34
-
- 1
1/3
4/16
4
4
-
-
12
-
--1.307
6/6
1
12
12
3
-
30
-
-1
0/4
1/16
4
4
-
-
12
-
-1.307
0/12
1
12
12
6
-
24
-
-2
happen
to such
an extent
r e a s o n w h y it m u s t h a p p e n
in crystallites.
The
question
in the ideal amorphous
is i f t h e r e
is a n y
structure.
3. Radial and angular fluctuations in ideal amorphous structures Let us look first at a five-fold ring or even a whole amorphon only
eclipsed
links.
Both
structures
possess
a centre
outside the basic units and therefore the angular be equally
distributed
between
If we now first one, but amorphon, bonds
try to construct having
lying
i n fig. 3 a c a n
of the structure.
distances couldbut must not appear so far.
a second
its centre
amorphon
of symmetry
the angular misfit cannot
b e c a u s e it is o f o p p o s i t e
misfit shown
all the five (or 60) bonds
Thus fluctuations in the interatomic
containing
of symmetry
merged
partly with the
at the periphery
be equally distributed
sign in the two
partly
of the first
over all eclipsed
merged
structures.
379
SHORT-RANGE ORDER IN AMORPHOUS GERMANIUM
~tructures of amorphous germanium (structural characteristics and binding energy) 4th layer
3nd layer Bonds
Bonds
& >,
Imperfect
•~,
~
1
24 25 25 26
36 36 36 36
1
27
1
28
4/64 16/64 24/64 16/64 4/64 1/2 1/2
24 24 24 24 24 25 25
36 36 36 36 36 36 36
60 60 60 60 60 64 64
7 7
6/24 12/24 6/24
24 24 24
34 34 34
62 62 62
I/'2 I/2
24 24
30 30
3 3
-
1
24
24
12
-
1
i
.-~
~
1/2 1,/2
E .~
&
Imperfect
o
42 43 44 43 42 40
60 64 64 64 60 52
6 12
-
--108 10S 112 108 96 84
7 6 12 12
2,070 2.070 --2.070 -- 2.070 2.070 2.022 2.022
42 43 44 45 46 43 44
60 60 60 60 60 64 64
2 4 6 6 -
-
-
108 108 108 108 112 108 112
6 9 12 6 25 7
--2.202 2.215 --2.224 - - 2. 2 4 0 --2.215 --2.227 2.202
4 4 4
2.032 2.032 --2.032
43 44 45
62 62 62
2 6 10
4 4 4
102 98 9J~
8 7 6
--2.302 2.352 2.400
60 60
6 6
--2.068 -- 2.068
42 43
60 60
3 9
3 3
96 88
12 19
--2.314 2.400
48
-
--2.298
36
36
12
6
72
-
-
2
-
60 64 64 64
36
6
-
60
36
12
-
52
7 7 I0 6
--2.0%0 --2.020 --2.020 --2.048 --2.150 --2.317
[ I ] i ] I
I
-
-
-2.201 2.227 --2.201 --2.240 2,348 --2,452
This fact can easily be visualized within a two-dimensional ideal amorphous structure. A two-dimensional " a n a l o g o n " of Ge is formed by a plane array of trivalent atoms linked together by covalent directional bonds (see Fletcher 0)). The corresponding crystalline lattice is to be seen in fig. 4a; the Voronoi polygon is an equilateral triangle. The two-dimensional "analogon" of a-Ge would consist of a mixture of five-fold, six-fold, and eventually seven-fold rings. The rather high angular misfit within an isolated five-fold ring can be distributed equally over all five bonds and would amount in this case to 12 ° (fig. 4b). Within a second five-fold ring, which has two atoms in common with the first one but is centred on its periphery, homogenous distribution of misfits is no longer possible because of the opposite sign of the same misfit angle c~ relative to the two symmetry centres C and C' (fig. 4b). Therefore, within an ideal amorphous structure with tetrahedral symmetry
2.350
380
R. GRIGOROVIC1 AND R. IVlAN,~ILA
angular and radial fluctuations about some average values must be present. Essentially, this is a consequence of the lack of statistical homogeneity of the amorphonic structure in which the misfits cumulate and the density diminishes 8) with increasing distance from the central atom. Finally, staggered bonds must alternate with eclipsed ones to limit and even reduce the misfit, till new eclipsed bonds appear again. A nice example is to be seen in fig. 5 where the amorphon which contains only eclipsed bonds is compared
£/
b
Fig. 4. (a) Model of two-dimensional lattice of trivalent atoms with symmetrically oriented covalent bonds. The Voronoi polygon is an equilateral triangle. (b) Fragment of the short-range order in a two-dimensionalamorphous solid based on the same Voronoi polygon. Uniform distribution of misfits is possible around centre C, but not around centre C'.
t
!
Fig. 5. (a) Model of an amorphon with minimum misfits concentrated towards initial atom 0 and maximum misfits concentrated around keystone 20. (b) Model of a structure containing 15 atoms with 35 eclipsed and 10 staggered bonds. with a structure of 15 atoms in which 10 intermediate bonds are staggered. As a consequence there is a much smaller cumulated misfit in the second structure which no longer has 12 but only a single axis of five-fold symmetry. Obviously the fluctuations will be the bigger, the more extended the domains with prevailing eclipsed bonds will be. As mentioned earlier the relative experimental increase in the first interatomic distance in a-Ge films deposited at room temperature is of about 3~, while the average increase within a single amorphon amounts to 1.89/o. We must therefore suppose that the size of the domains in which eclipsed bonds are strongly prevailing exceeds in general the size of an amorphon and hence includes at least parts of the fourth correlation layer.
SHORT-RANGE ORDER IN AMORPHOUS GERMANIUM
381
In order to evaluate the order of magnitude of the fluctuations in our model let us first consider the highest possible relative deviation from the interatomic distance within the Ge lattice of the last atom closing the shell of the amorphon like a keystone (see fig. 5a). It is the twentieth atom within the amorphon and the only one of the very numerous members of the fifth correlation layer which is a component of the amorphon. If the misfits are reduced to a minimum towards the opposite pole of the structure, the distances between this last atom and its three neighbours are increased by 24~o. The mean fluctuation within the amorphon can be evaluated therefore to lie somewhat below 6700. The experimental standard deviation within the same a-Ge films as before is higher (7.6~) and we must again conclude that the size of the domains with prevailing eclipsed bonds exceeds that of an amorphon. 4. Analysis of the model The next step in our investigation will be to find arguments in favour of our model in an energetical analysis which must explain the preference for the formation of domains with prevailing eclipsed bonds of the abovementioned size. The procedure we used was the following: all admissible models of ideal structures with complete successive correlation layers (table 1) were build up in such a way as to concentrate the misfits on the smallest possible number of pairs of contiguous hexagonal faces. It was supposed that in view of the smallness of the misfits the decrease in binding energy within a structure containing many small misfits will be about the same as within a structure in which these small misfits have been concentrated into few big ones. This procedure was continued to the fourth correlation layer inclusively. Four types of hexagonal faces emerged: i) in perfect contact (binding energy - 1.633 eV per bond, i.e. per pair of touching faces); ii) in imperfect contact, with a simple angular misfit of 1 ° 28' (binding energy - 1.600 eV per bond); iii) in imperfect contact, with a double angular misfit of 2 ° 56' (binding energy - 1.500 eV per bond); iv) free faces (binding energy zero). The values of the binding energy per bond in imperfect contact were calculated in the following manner. We allowed for the angular misfit by calculating the bond strength F of tWO s p 3 orbitals whose symmetry axes form an angle 0 of 1 ° 28' and 2 ° 56', respectively. The bond strength is given after Pauling n) by F = 2 1- + 2 cos0,
382
R. GRIGOROVICI AND R. M,~N.~IL,~
and the binding energy by e(0) =e(0) × ½F.
(1)
We allowed for the relative increase in the interatomic distance by using the following relation
e(r) =½K(r -a) z,
(2)
which holds for small deviations from the equilibrium distance a. Here K = = 1.565 x 104 dyne cm-1, a value which results from the vibration frequency of the atoms within the Ge lattice12). Other evaluations of K f r o m the characteristic temperature of crystalline Ge, from the vibration frequency of the atoms within the Ge2 molecule, and from the elastic constants of the Ge single crystal provide values lying between 1.16 and 3. ! x 104 dyne cm-1. It is finally supposed that the changes in energy due to both sorts of distortions are additive. The results of this calculation are to be found in table 1 and are also shown in fig. 6, where the binding energy per atom is plotted for 15 of the 19 different ideal structures containing a complete third correlation layer versus the order of correlation. It was shown earlier in this paper (see section 2) that there is no need to differentiate between numerous ideal structures containing a complete fourth correlation layer if only the binding energy is to be considered. Fig. 6 shows that including the second correlation layer the structures with prevailing eclipsed bonds are strongly favoured energetically. When one includes the third correlation layer, the continuation of structures with prevailing staggered bonds by prevailing eclipsed bonds is also energetically very favourable. Between the third and the fourth correlation order there is a tendency in favouring the structures which contain staggered links. This is in good agreement with the conclusions drawn in the preceding paragraph about the size of the structures with prevailing eclipsed bonds. For an infinite number of correlation layers, the curve corresponding to exclusively staggered bonds must become the lowest of all and reach the lattice value of - 3.266 eV. Essentially, this behaviour is due to the fact that the configurational energy per atom of small clusters is determined chiefly by the ratio between the number of free and contiguous hexagonal faces and that this ratio is much lower in clusters in which most bonds are eclipsed. With increasing cluster size the decrease in binding energy due to the increasing misfit and the ever decreasing ratio between surface and volume tend to favour progressively the structure lacking any misfit, i.e. the crystalline structure where all bonds are staggered.
383
SHORT-RANGE ORDER IN AMORPHOUS GERMANIUM
\
\ \
~/o
\
-3/I 9/3 4/0 e/e -
.\\..
\
-20
/o 12/o 3/I IZ/O 4/0 3/9 ~- ~/o o / n
\ \
k
\
9/3 4/0 e/6 r~/o ; 12/oI"~I° ~,-3/~ 1///-;/1 t./.s/l
\
o/4 o / ~ \ \ \
--4/0 9/3 go/a
\
le/o ez/9 ~ - - - e / e lo/z ee/e "+--l/e 8/8 18/Is a/¢ om o/s~
--~/o
~/o
~--,~ -25~
gI
12/o 3e/o g/3 2?/9 ze/o 3e/o g/s 3s/3 9/s ~s/o
3/g
z~/le
e/e 15/le O/lZ le/38
SI 4
Order o f tt~e correlahon {ager Fig. 6. Binding energy per atom versus order of correlation for 15 of the 19 structures contained and described in table 1. Four structures have not been represented for the sake of clarity (3/1 12/0 33/3; 3/1 12/0 30/6; 3/1 12/0 24/12; 2/2 10/2 23/11).
It must also be emphasized that a structure of higher energy can be transformed into one of lower energy by rotating neighbouring atoms in such a way about their common valence bond as to diminish the number of free hexagonal faces. These rearrangements are hampered to some extent by the existence of a certain activation energy of the rotation. (In organic
384
R. GRIGOROVICI AND R. IvlAN~tILA
molecules this activation energy amounts to about 3 kcal/mol=0.13 eV.) Even more hindering is the often intervening necessity to break some bonds already formed with other atoms of the same or the next correlation layer in order to achieve this rotation. Due to this possibility of rearrangement we must expect to find in general, and especially in annealed samples, more structures with prevailing eclipsed bonds than would correspond to the statistical weights given in table I. To check this point the number N 3 of atoms within the third coordination sphere ( r = 4 . 8 0 A ) was calculated from the data of table 1. The third coordination sphere was selected to perform this check, because it is very sensitive to the type of structure. Indeed this coordination sphere is occupied by 12 atoms in the Ge crystal, by 9 atoms in a hypothetical wurtzite crystal and by none in the amorphon. The calculation yielded N 3 = 8.02, while the experimental r.d.c, of a-Ge films deposited at room temperature fits well a calculated r.d.c, with N3 = 6 (ref. 6). In a-Si films deposited at room temperature, too, the best fit is obtained with N 3 =7.2 (ref. 8); indeed taking into account the higher crystallization temperature of a-Si, these later films were less "annealed" than the Ge films deposited under identical conditions.
5. The real structure of amorphous germanium The real structure of a-Ge films is thus not at all well defined. Even if we neglect the structural imperfections, there is an incommensurate number of possible ideal structures compatible with a real film. Only a statistical description could be given under certain well-defined deposition and annealing conditions. Structural imperfections can also be introduced in a number of different ways. A Ge atom bound on the surface of a cluster in a wrong configuration and blocked in this position by another neighbouring atom so as to be unable to rotate into the right configuration which satisfies the condition of ideality must give rise to a point defect. So can a foreign atom introduced into the film during the deposition, though such an atom can also saturate a non-satisfied valence bond. Cumulation of angular misfits may also give rise to a point defect similar to a vacancy in a crystalline lattice. Another structural imperfection similar to a high angle grain boundary or a crystallite interface must arise when two independent clusters grow together. Here, as previously, the number of defects with incomplete overlapping of orbitals (pseudo-vacancies) will undoubtedly exceed the number of defects with too many overlapping orbitals (pseudo-interstitials). The first ones wiU act as acceptors and are known to occur in polyerystalline Ge; the last ones will act as donors and might compensate some of the acceptors.
S H O R T - R A N G E ORDER I N A M O R P H O U S G E R M A N I U M
385
Obviously there is no sharp distinction between a point defect of the vacancy type and a local angular misfit. Both lead to poor overlapping of orbitals which differ only in degree. It may be conjectured that only vacancies act as relatively deep acceptors, while angular misfits give rise to the often postulated tail of states protruding into the energy gap. If so, the density of states function should not drop to zero, but only perhaps go through a minimum between the band edge and the acceptor levels maximum. Transport phenomena agree qualitatively with this picture: a-Ge is a p-type, apparently partly compensated semiconductor13). The acceptor density in a-Ge films deposited at room temperature is of the order of 1018 cm -3. Let us now try to predict on the basis of our model what the real structure of an a-Ge film should be under different conditions and compare these predictions with what is known from experiment. If deposited on a substrate held at low temperature, the film should consists of many small domains in which the statistical proportion of staggered and eclipsed bonds should hold because neither rotation about the bonds nor surface or volume migration can take place to any significant degree. Where these domains meet, grain boundaries as described above should provide an important number of predominantly acceptor-like defects. Indeed such films show an electrical resistivity which exceeds that of polycrystalline Ge films only by a little more than one order of magnitudela). Also, due to the lack of surface migration, a-Ge films have a very smooth surfacelS). Annealing should increase the misfit in the nearly ideal regions to allow for a high degree of binding within the grain boundaries. This binding can be achieved either by the "recombination" of a pseudo-vacancy with a pseudo-interstitial or by the migration of a pseudo-vacancy to the surface of the film. This process must lead to a reduction both in the acceptor density and in the carrier mobility, which happens indeed in a-Ge films16). An increasing degree of misfit and of fluctuations has also been observed in a-Ge as a consequence of annealing between room temperature and 200 ° C ~, 6). At even higher temperatures not only defects but also Ge atoms start migrating, first on the surface, later within the volume of the film. In the first process fewer and looser bonds have to be broken and the atom when reaching a domain with predominantly staggered bonds is integrated into a crystallite, where the binding is stronger. The growing of the crystallite will stop when the surface of the partly emptied amorphous regions round the crystallite gets so rough, that surface migration is suppressed again. The process is therefore very similar to a reconstructive
386
R. GRIGOROVIC1 AND R. M/kNAILA
phase transformation 17) favoured by the existence of an intermediate phase - the two-dimensional "gas" of atoms moving on the surface of the film - which is however exhausted in time. Electron microscope pictures taken during the annealing of a-Ge films show indeed the formation between 200 and 430 °C of crystalline islands within an amorphous matrix (fig. 7)*. These islands cover only a few per cent of the surface, their size reaches a certain limit at a given temperature, and the amorphous matrix surrounding them until a certain distance is thinner and has a rougher surface than the rest. Finally at 430 °C volume migration contributes essentially to the growing of the crystalline islands. This process is analogous to a reconstructive transformation without a catalyst17), i.e. at this temperature crystallization should proceed slowly in time, because bonds must be broken and formed again in a new geometry, until the whole volume of the film has crystallized. Indeed it takes 20 to 30min to crystallize an a-Ge film completely at 430°C la) and the onset of progressive crystallization is quite sharp. An estimate of the heat of crystallization of a-Ge based on our model can also be made. Let us assume that the additional energy of the amorphous phase results from the mean increase of the first interatomic distance r 1 - a =7.35 × 10 - 2 / ~ (i.e. 3~o of the crystalline distance), as well as from the corresponding average misfit 0 = 0 . 0 4 3 = 2 ° 26' of the interatomic bonding. By means of eqs. (1) and (2) a total supplementary energy of 6 . 4 0 x 10 - 3 eV per atom or 2.0 cal/g can be inferred. No attempt has yet been made to measure the heat of crystallization of a-Ge. A comparison with the figure given above would constitute another check of our model.
i
Fig. 7. Electron microscope picture of an amorphous Ge film annealed at 280°C. In the right-hand side some crystallites (dark spots) embedded in the amorphous matrix can be seen. They are surrounded by an amorphous region which is both thinner and rougher. * We are indebted to Dr. E. F. P6cza, Dr. P. B. Barna and A. Barna from the Institute of Technical Physics in Budapest for taking this picture.
SHORT-RANGE ORDER IN AMORPHOUS GERMANIUM
387
6. Final considerations The way in which o u r m o d e l o f a m o r p h o u s G e was o b t a i n e d is o p e n to m a n y criticism. W e c o n s t r u c t e d o u r clusters in space; in fact they are g r o w i n g on a substrate whatever the process o f b u i l d i n g up the a m o r p h o u s film. N o t only isolated atoms, b u t also g r o u p s o f 2-8 a t o m s might be the real b u i l d i n g stones o f the films o b t a i n e d by v a c u u m deposition. N o a t t e m p t was m a d e to t a k e into a c c o u n t the r e p u l s i o n which m u s t a p p e a r when the basic units t o u c h one a n o t h e r a l o n g the small t r i a n g u l a r faces, i.e. the shortrange o r d e r is similar to t h a t in a n o t existing wurtzite-type G e crystal. The estimates o f the a n g u l a r misfit, o f the m e a n increase in i n t e r a t o m i c distances a n d their fluctuations, a n d o f the energy o f the different structures are very r o u g h a n d b a s e d on r a t h e r naive geometrical c o n s i d e r a t i o n s r e m i n d i n g o f Demokritos. Nevertheless the a u t h o r s feel that a certain progress has been achieved by this d e s c r i p t i o n o f the s h o r t - r a n g e o r d e r in a n a m o r p h o u s solid a n d t h a t further i m p r o v e m e n t in the quantitative c h a r a c t e r i z a t i o n o f the statistical c o r r e l a t i o n s within this structure c o u l d f o r m the basis for a c a l c u l a t i o n o f the energy b a n d structure o f a n actual t h r e e - d i m e n s i o n a l s e m i c o n d u c t o r .
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17)
H. K6nig, Reichsber. Phys. 1 (1944) 4. E. Kirkorian and R. J. Sneed, J. Appl. Phys. 37 (1966) 3665. G. Szekely, J. Electrochem. Soc. 98 (1951) 318. H. F. Sterling and R. C. G. Swan, Solid-State Electron. 8 (1965) 653. H. Richter and G. Breitling, Z. Naturforsch. 13a (1958) 988. R. Grigorovici and R. M~n~.ilh, Thin Solid Films 1 (1968) 343. R. Grigorovici, R. M~n~.ilftand A. A. Vaipolin, Acta Cryst. B24 (1968) 535. M. V. Coleman and D. J. D. Thomas, Phys. Status Solidi 24 (1967) K 111. N. H. Fletcher, Proc. Phys. Soc. (London) 91 (1967) 724; 92 (1967) 265. G. F. Voronoi, Reine Angew. Math. 134 (1908) 198. L. Pauling, The Nature of the Chemical Bond (Cornell Univ. Press, Ithaca, New York, 1960) p. 116. K. M. Guggenheimer, Proc. Phys. Soc. (London) 58 (1946) 456. R. Grigorovici, Mater. Res. Bull. 3 (1968) 13. N. Croitoru and L. Vescan, private communication. J. Tauc, A. Abrahfim, L. Pajasov~i, R. Grigorovici and A. Vancu, in: Proc. Intern. Conf. Non-Crystalline Solids, Delft, 1964 (North-Holland, Amsterdam, 1965) p. 606. R. Grigorovici, N. Croitoru, A. D6v6nyi and E. Teleman, in: Proc. Intern. Conf. Phys. Semiconductors, Paris, 1964, p. 423. M. J. Buerger, Crystallographic Aspects of Phase Transformations, in: Symposion on Phase Transformations in Solids, Cornell Univ., 1948, p. 196.
S H O R T - R A N G E O R D E R IN A M O R P H O U S
GERMANIUM
R. GRIGOROVICI and R. MA, N/'~IL/'~
Institute of Physics of the Academy, Calea Victoriei, 114, Bucharest, Romania Received 11 February 1969 The structure of amorphous germanium is described by an array of atoms, linked together in distorted tetrahedra. There is a limited number of short-range orders with only two admitted configurations (staggered and eclipsed) present if all bonds are to be satisfied (ideal structure). Analysis of the ideal structures shows that in small clusters groups with five-fold symmetry, containing preferentially eclipsed bonds are energetically favoured, thus explaining the formation of an amorphous phase when germanium is built up atom by atom at sufficiently low temperature. Defects can be introduced into the ideal structures by cumulation of misfits, by blocking of atoms in wrong configurations, when neighbouring clusters grow together, etc. The model agrees with many experimental facts and data.
1. Introduction
Amorphous germanium, like other amorphous semiconducting elements or mllIBV compounds whose structure is based on essentially covalent bonds with tetrahedral symmetry, can be obtained only in the form of thin films built up atom by atom (or, eventually, from very small clusters of 2-8 atoms already linked together) by vacuum evaporation1), by cathodic sputtering2), by electrolysis 3), or by rf discharge 4). As in the case of crystalline solids the understanding of the physical properties of amorphous semiconductors must be based ultimately on the knowledge of their structure, i.e. of the arrangement in space of their atoms. But unlike crystalline solids, the periodic structure of the lattice is lost in amorphous solids and the whole formalism which links the structure of the reciprocal space with the energy band structure in k-space is no longer applicable. What can be inferred about the structure of amorphous solids from the isotropic X-ray or electron diffraction patterns is an atomic or electronic radial distribution function. The corresponding radial distribution curve (r.d.c.) consists of a series of rather broad maxima and minima which display the change in atom or electron density versus the radial distance from an arbitrary chosen central atom. The structure is implicitly supposed to be spherically symmetrical. This may be for some purposes a fair description of the short-range order of liquid metals, where the first coordination 371
372
R. GRIGOROVICI AND R. MANAIL,~
number is relatively high (10-11). It is certainly a very poor description of the short-range order of amorphous semiconductors with tetrahedral symmetry, where this number is four. If diffraction experiments could be performed on volumes of a few ~mgstr/Sms radius, i.e. much smaller than it is actually done and possible to do, both the diffraction pattern and the atom or electron density would be far from isotropic. It must also be recalled that the r.d.c, is obtained by the Fourier transformation of the scattered intensity versus scattering angle function. Instead of being extended to infinity, the Fourier integrals have unavoidably a finite upper limit. As a consequence the maxima of the r.d.c, are broadened and ripples without any correspondence to physical reality appear on both sides of these maxima (end-of-series effect). The present knowledge about the short-range order of amorphous germanium (a-Ge) can be summarized as follows: l) The area under the first maximum of the r.d.c, yields the first coordination number fourS,~). The a v e r a g e symmetry of the nearest neighbours of the central atom is therefore tetrahedral and the bonds are due to the overlapping of one of the sp a orbitals of these neighbours with those of the central atom. Hence the first coordination sphere is similar to that in the Ge crystal. 2) The analysis of the position and the shape of the first maximum of the r.d.c, shows s, 7) that the first interatomic distance is increased by about 3% as compared with the first interatomic distance in the Ge crystal (2.50 versus 2.43 A) and that, apart from the Fourier broadening due only to the mathematical procedure, there is also an intrinsic broadening due to fluctuations in this distance which correspond in non-annealed a-Ge films deposited at 300°K to a standard deviation A r l of +0.19 A or ___7.6% of this distance, respectively. 3) In order to duplicate the whole experimental r.d.c, of a-Ge by that of a structural model in which all Ge atoms are linked with one another tetrahedrally by s p 3 covalent bonds (ideal amorphous structure), one is forced to admit that: a) The fluctuations of the successive interatomic distances are increasing like r n, where 0.5 < n < 1 (ref. 7); as the second interatomic distance in both the Ge crystal and in a-Ge corresponds to the length of the tetrahedron edge, it follows that each of the real tetrahedra formed by the four nearest neighbours of any of the atoms in a-Ge is in general not only radially but also angularly distorted. In non-annealed a-Ge films deposited at 300°K the fluctuation in the lengh of the edge of these tetrahedra lies around A r 2 = ( r 2 / r l ) °'Ts x A r 1 =(4.09/2.50) 0.75 x 0.19 = +0.28 A
or + 6.8%.
SHORT-RANGE ORDER IN AMORPHOUS GERMANIUM
373
b) Neighbouring Ge atoms are linked together in two different configurations called "staggered" and "eclipsed", respectively, after the relative orientation of the two groups of three orbitals not involved in the binding of these atoms (fig. 1). The Ge lattice contains the staggered configuration only. In the wurtzite lattice a quarter of the bonds are eclipsed, the rest are staggered; Ge atoms linked together consistently only by eclipsed bonds form five-fold rings with only small angular misfits; these rings can merge into a regular dodecahedron (a so-called "amorphon") having 20 Ge atoms at its corners; these dodecahedra can again merge into one another along a five-fold ring and so on. It was shown by us a) that the experimental r.d.c. of a-Ge is matched by the r.d.c, of a 50/50 per cent mixture of small, slightly expanded, heavily distorted Ge crystallites and of amorphonic groupings. Independently Coleman and Thomas s) proposed the same model, with a slightly different ratio (60/40) of crystals and amorphons to describe the short-range structure of a-Si films. Obviously the five-fold symmetry of the rings of Ge atoms linked together in eclipsed configuration only explains the lack of translation symmetry in a-Ge and a-Si and hence there seems to be some truth in this model. But in many other respects the model suggested by us and by Coleman and Thomas is rather rough and unsatisfactory. There are still many questions to be answered. The following is a list of such questions: 1) Are the two types of configurations intimately intermixed in a-Ge or are they forming a mixture of two different "phases", crystalline and amorphonic, separated by a sort of grain boundaries. Is an intimate mixture of staggered and eclipsed configurations geometrically possible without leaving many covalent bonds unsatisfied. Could one therefore describe a real amorphous solid by introducing into an ideal amorphous semiconductor, having all bonds satisfied, some local structural defects, where this condition is not satisfied. 2) If an ideal amorphous semiconductor containing only s p 3 bonds can be formed, is the presence of radial and angular fluctuations within its basic unit, the tetrahedron, an unavoidable consequence of the angular misfit introduced by the eclipsed configurations and are these fluctuations of the right order of magnitude.
S tagge~cl
E~ed
Fig. 1. Staggeredand eclipsed configurations.
374
R. GRIGOROVICI AND R. MANAIL.~.
3) Could one find an energetical justification for the formation of eclipsed bonds when solid amorphous germanium is built up atom by atom on a substrate. 4) I f so, what must the influence be of the substrate temperature or of subsequent annealing on the detailed structure of a-Ge films and how do these changes fit experimental observations, especially the experimental r.d.c, and the electrical properties. The final aim of the present paper consists in giving a much more comprehensive and detailed description than usual of the short-range order in a-Ge, which could form the basis for a calculation of the energy band structure of this amorphous element, e.g. on the lines proposed recently by Fletcher 9).
2. The ideal structure of amorphous germanium The basic unit used in modelling the ideal structure of a-Ge, i.e. having all covalent bonds satisfied, is a modified Voronoi polyhedron of the diamond-like lattice used previously6, s). The exact Voronoi polyhedron of the diamond-like lattice is shown in fig. 2a and represents the volume occupied by a Ge atom within the lattice. Covalent sp 3 bonding between neighbouring atoms corresponds to perfect contact between the big hexago-
a
b
c
d
Fig. 2. (a) Voronoi polyhedron in diamond-like lattice. (b) Simplified Voronoi polyhedron. (c) Fragment of diamond lattice built up of simplified Voronoi polyhedra. (d) Amorphon. (e) Fragment of wurtzite lattice. 0-1, 0-2, 2-3 and 4-5 are staggered bonds ; bond 2-4 is eclipsed. 0 and 5 touch along triangular face.
375
SHORT-RANGE ORDER IN AMORPHOUS GERMANIUM
nal faces of two contiguous polyhedra. Contact between Voronoi polyhedra along two small triangular surfaces corresponds to the much weaker link with second order neighbours. These links are neglected in our considerations, as nearly free rotation about the c o m m o n bond is assumed (see below). Therefore the four small triangular pyramids have been completely flattened to form each a single triangular face (fig. 2b). These modified Voronoi polyhedra can be used to build up not only the diamond lattice (fig. 2c), but also the a m o r p h o n (fig. 2d), which takes the form of a regular icosahedron* with five-edged pits at its corners, and the wurtzite lattice (fig. 2e). In this latter case some of the second-order neighbours touch along the small triangular faces, while the bonds between nearest neighbours within the atomic layers perpendicular to the c-axis are staggered and those between nearest neighbours lying in adjacent atomic layers are eclipsed. As mentioned earlier, a certain misfit in covalent bonding must be tolerated as a consequence of eclipsed bonding in the amorphonic structure. This is due to the fact that there is a difference of 1 ° 28' between the angle of 70 ° 32' formed by two hexagonal faces of the basic unit and the central angle of 72 o of the five triangles which form a regular pentagon. This is illustrated in fig. 3a where a five-fold ring of basic units linked in eclipsed configuration is viewed along the c o m m o n edge C. The consequences of a small misfit are: a) a certain increase in the interatomic distance a' as compared with the corresponding distance a in the diamond-like lattice (fig. 3b); within a five-
\
I
\-.i/
/
./
b
a
Fig. 3. (a) Five-fold ring of simplified Voronoi polyhedra; mean angular misfit 0 = 1o 28" is exaggerated for the sake of clarity, a ' = mean interatomic distance. (b) Interatomic distance a between two exactly fitting Voronoi polyhedra. * Incidentally the regular icosahedron was the form attributed by Platon (Timaion, 55b-56b) to the solid basic unit of water, the third element.
376
R. GRIGOROVICI AND R. MAN,~ILA
fold ring or a single amorphon (icosahedron), the mean relative increase (a'-a)/a due to this misfit amounts to 1.8~; b) a smaller overlapping between the s p 3 orbitals of neighbouring atoms due both to the increase in the interatomic distance and to an increase in the angle between the symmetry axes of these orbitals from zero to 0 (fig. 3a). Within a five-fold ring or a single amorphon the mean value of 0 is 1 ° 28'. The possible ideal structural models of a-Ge were obtained by adding to a central atom successive correlation layers, i.e. groups of basic units touching (or nearly touching) one another along their hexagonal faces and containing all neighbours of a given order (first, second, third, etc.)*. Only those configurations which permit the addition of the next correlation layer, i.e. without letting any free hexagonal face uncovered by the basic units of the next layer, were taken into consideration. This latter condition greatly restricted the number of admissible structures and gave us the possibility of solving the problem without using a computer, by simply manipulating a moderate number of basic units made of polystyrene. The units could be linked together by a bunch of strained rubber strings converging into the central atom and crossing the basic units along channels drilled perpendicularly to the centres of the hexagonal faces of the truncated tetrahedra. In this way both the directional character of the s p 3 covalent bond and the free rotation about this bond was succesfully simulated. This technique was used only to the second correlation layer. Structures extending to the third and fourth correlation layer were obtained by gluing the basic units together. While in the first correlation layer formed by the four nearest neighbours, the above-mentioned condition did not restrict in any way the relative rotation of the basic units, purely geometrical conditions restricted these rotations from the second correlation layer to only the staggered and the eclipsed configurations. (For energetical arguments concerning free rotation and the stability of the two configurations within a system of only two atoms, see ref. 6.) This greatly simplifies the problem and eliminates at once the structural model of a-Ge proposed by Richter and BreitlingS), who admit these rotations to be aleatory. Their great merit remains that they have drawn the attention to the importance of this rotation. By restricting ourselves only to these two configurations, the number of different structures containing the central atom and the first two correlation layers nevertheless amounts to 4096. However, when applying the condition * Obviously there is no identity between the so-defined correlation layer and the usual coordination sphere. A correlation layer generally contains atoms situated at different distances from the central atom; hence they are members of different coordination spheres (see table 1).
S H O R T - R A N G E ORDER I N A M O R P H O U S G E R M A N I U M
377
of ideality, this number drops to 33 structures of which only 10 are distinct. Table 1 displays the results of this investigation. The first three correlation layers of the different distinct ideal structures are characterized as follows: a) the ratio s/e between the number of staggered and eclipsed configurations; b) the statistical weight of the structure, if it is supposed that the deposition of a given correlation layer takes place on an already frozen structure containing one less correlation layer (see however section 4) and that there is an equal probability of a newly arriving atom to be bound in any of the two configurations, staggered or eclipsed; c) the total number of atoms within the correlation layer; d) the number of perfect and imperfect bonds (because of angular misfit and increased interatomic distance); e) the number of free hexagonal faces (unsatisfied bonds); f) the number of wurtzite-type bonds (contact between triangular faces); g) the energy of the structure. The data for the fourth correlation layer are less complete: only the total number of atoms, of satisfied and non-satisfied bonds, of wurtzite-type bonds, and the energy are recorded. Indeed changes from staggered to eclipsed configurations do not change the energy of the structure or the distribution of atoms over the coordination spheres below about 7 A, i.e. within the region which is decisive for checking the model by comparison between the calculated and the experimental r.d.c. (see section 4). Inspection of table 1 as well as a series of attempts to go farther with the building up of succesive correlation layers showed that no structure which proved to be continuable at a given order of correlation became noncontinuable at a higher order. On the contrary, the number of continuable structures increased with increasing correlation order. We can therefore conclude quite safely that there are many ways in which an ideal structure can be realized in a-Ge. Defects in the ideal structure must appear wherever the ideality condition is not fulfilled and, as a consequence, overlapping between orbitals becomes very poor or, on the contrary, more than two orbitals overlap significantly. Both types of defects may occur and there might even be a high probability to find two defects of opposite type close to one another. It is however to be expected that the first type of defect will prevail in a-Ge. What are the arguments now against or for a-Ge being simply a mixture of small Ge crystallites and amorphonic gloupings of atoms in nearly equal parts. It was pointed out earlier that to duplicate the experimental r.d.c, of a-Ge both the crystallites and the amorphonic structures must present relatively big fluctuations in interatomic distances. There is no reason why this should
378
R. GRIGOROVICI AND R. M,~NAILA
Tt I st layer
2nd layer
Bonds
Bonds
Imperfect
-~
4/0
1/16
4
4
-
-
12
-
--1.307
3/1
4/16
4
4
-
-
12
-
-1.307
Imperfect
12/0 9/3 6/6 3/9 0/12
1/16 4/16 6/16 4/16 1/16
12 12 12 12 12
12 12 12 12 12
-
-
36 36 36 36 36
12/0
1/5
12
12
-
-
36
9/3
4/5
12
12
-
-
-1 --1 --1 --1 --I
-I
36
-
-1
2/2
6/16
4
4
-
-
12
-
--1.307
10/2
1
12
12
1
-
34
-
- 1
1/3
4/16
4
4
-
-
12
-
--1.307
6/6
1
12
12
3
-
30
-
-1
0/4
1/16
4
4
-
-
12
-
-1.307
0/12
1
12
12
6
-
24
-
-2
happen
to such
an extent
r e a s o n w h y it m u s t h a p p e n
in crystallites.
The
question
in the ideal amorphous
is i f t h e r e
is a n y
structure.
3. Radial and angular fluctuations in ideal amorphous structures Let us look first at a five-fold ring or even a whole amorphon only
eclipsed
links.
Both
structures
possess
a centre
outside the basic units and therefore the angular be equally
distributed
between
If we now first one, but amorphon, bonds
try to construct having
lying
i n fig. 3 a c a n
of the structure.
distances couldbut must not appear so far.
a second
its centre
amorphon
of symmetry
the angular misfit cannot
b e c a u s e it is o f o p p o s i t e
misfit shown
all the five (or 60) bonds
Thus fluctuations in the interatomic
containing
of symmetry
merged
partly with the
at the periphery
be equally distributed
sign in the two
partly
of the first
over all eclipsed
merged
structures.
379
SHORT-RANGE ORDER IN AMORPHOUS GERMANIUM
~tructures of amorphous germanium (structural characteristics and binding energy) 4th layer
3nd layer Bonds
Bonds
& >,
Imperfect
•~,
~
1
24 25 25 26
36 36 36 36
1
27
1
28
4/64 16/64 24/64 16/64 4/64 1/2 1/2
24 24 24 24 24 25 25
36 36 36 36 36 36 36
60 60 60 60 60 64 64
7 7
6/24 12/24 6/24
24 24 24
34 34 34
62 62 62
I/'2 I/2
24 24
30 30
3 3
-
1
24
24
12
-
1
i
.-~
~
1/2 1,/2
E .~
&
Imperfect
o
42 43 44 43 42 40
60 64 64 64 60 52
6 12
-
--108 10S 112 108 96 84
7 6 12 12
2,070 2.070 --2.070 -- 2.070 2.070 2.022 2.022
42 43 44 45 46 43 44
60 60 60 60 60 64 64
2 4 6 6 -
-
-
108 108 108 108 112 108 112
6 9 12 6 25 7
--2.202 2.215 --2.224 - - 2. 2 4 0 --2.215 --2.227 2.202
4 4 4
2.032 2.032 --2.032
43 44 45
62 62 62
2 6 10
4 4 4
102 98 9J~
8 7 6
--2.302 2.352 2.400
60 60
6 6
--2.068 -- 2.068
42 43
60 60
3 9
3 3
96 88
12 19
--2.314 2.400
48
-
--2.298
36
36
12
6
72
-
-
2
-
60 64 64 64
36
6
-
60
36
12
-
52
7 7 I0 6
--2.0%0 --2.020 --2.020 --2.048 --2.150 --2.317
[ I ] i ] I
I
-
-
-2.201 2.227 --2.201 --2.240 2,348 --2,452
This fact can easily be visualized within a two-dimensional ideal amorphous structure. A two-dimensional " a n a l o g o n " of Ge is formed by a plane array of trivalent atoms linked together by covalent directional bonds (see Fletcher 0)). The corresponding crystalline lattice is to be seen in fig. 4a; the Voronoi polygon is an equilateral triangle. The two-dimensional "analogon" of a-Ge would consist of a mixture of five-fold, six-fold, and eventually seven-fold rings. The rather high angular misfit within an isolated five-fold ring can be distributed equally over all five bonds and would amount in this case to 12 ° (fig. 4b). Within a second five-fold ring, which has two atoms in common with the first one but is centred on its periphery, homogenous distribution of misfits is no longer possible because of the opposite sign of the same misfit angle c~ relative to the two symmetry centres C and C' (fig. 4b). Therefore, within an ideal amorphous structure with tetrahedral symmetry
2.350
380
R. GRIGOROVIC1 AND R. IVlAN,~ILA
angular and radial fluctuations about some average values must be present. Essentially, this is a consequence of the lack of statistical homogeneity of the amorphonic structure in which the misfits cumulate and the density diminishes 8) with increasing distance from the central atom. Finally, staggered bonds must alternate with eclipsed ones to limit and even reduce the misfit, till new eclipsed bonds appear again. A nice example is to be seen in fig. 5 where the amorphon which contains only eclipsed bonds is compared
£/
b
Fig. 4. (a) Model of two-dimensional lattice of trivalent atoms with symmetrically oriented covalent bonds. The Voronoi polygon is an equilateral triangle. (b) Fragment of the short-range order in a two-dimensionalamorphous solid based on the same Voronoi polygon. Uniform distribution of misfits is possible around centre C, but not around centre C'.
t
!
Fig. 5. (a) Model of an amorphon with minimum misfits concentrated towards initial atom 0 and maximum misfits concentrated around keystone 20. (b) Model of a structure containing 15 atoms with 35 eclipsed and 10 staggered bonds. with a structure of 15 atoms in which 10 intermediate bonds are staggered. As a consequence there is a much smaller cumulated misfit in the second structure which no longer has 12 but only a single axis of five-fold symmetry. Obviously the fluctuations will be the bigger, the more extended the domains with prevailing eclipsed bonds will be. As mentioned earlier the relative experimental increase in the first interatomic distance in a-Ge films deposited at room temperature is of about 3~, while the average increase within a single amorphon amounts to 1.89/o. We must therefore suppose that the size of the domains in which eclipsed bonds are strongly prevailing exceeds in general the size of an amorphon and hence includes at least parts of the fourth correlation layer.
SHORT-RANGE ORDER IN AMORPHOUS GERMANIUM
381
In order to evaluate the order of magnitude of the fluctuations in our model let us first consider the highest possible relative deviation from the interatomic distance within the Ge lattice of the last atom closing the shell of the amorphon like a keystone (see fig. 5a). It is the twentieth atom within the amorphon and the only one of the very numerous members of the fifth correlation layer which is a component of the amorphon. If the misfits are reduced to a minimum towards the opposite pole of the structure, the distances between this last atom and its three neighbours are increased by 24~o. The mean fluctuation within the amorphon can be evaluated therefore to lie somewhat below 6700. The experimental standard deviation within the same a-Ge films as before is higher (7.6~) and we must again conclude that the size of the domains with prevailing eclipsed bonds exceeds that of an amorphon. 4. Analysis of the model The next step in our investigation will be to find arguments in favour of our model in an energetical analysis which must explain the preference for the formation of domains with prevailing eclipsed bonds of the abovementioned size. The procedure we used was the following: all admissible models of ideal structures with complete successive correlation layers (table 1) were build up in such a way as to concentrate the misfits on the smallest possible number of pairs of contiguous hexagonal faces. It was supposed that in view of the smallness of the misfits the decrease in binding energy within a structure containing many small misfits will be about the same as within a structure in which these small misfits have been concentrated into few big ones. This procedure was continued to the fourth correlation layer inclusively. Four types of hexagonal faces emerged: i) in perfect contact (binding energy - 1.633 eV per bond, i.e. per pair of touching faces); ii) in imperfect contact, with a simple angular misfit of 1 ° 28' (binding energy - 1.600 eV per bond); iii) in imperfect contact, with a double angular misfit of 2 ° 56' (binding energy - 1.500 eV per bond); iv) free faces (binding energy zero). The values of the binding energy per bond in imperfect contact were calculated in the following manner. We allowed for the angular misfit by calculating the bond strength F of tWO s p 3 orbitals whose symmetry axes form an angle 0 of 1 ° 28' and 2 ° 56', respectively. The bond strength is given after Pauling n) by F = 2 1- + 2 cos0,
382
R. GRIGOROVICI AND R. M,~N.~IL,~
and the binding energy by e(0) =e(0) × ½F.
(1)
We allowed for the relative increase in the interatomic distance by using the following relation
e(r) =½K(r -a) z,
(2)
which holds for small deviations from the equilibrium distance a. Here K = = 1.565 x 104 dyne cm-1, a value which results from the vibration frequency of the atoms within the Ge lattice12). Other evaluations of K f r o m the characteristic temperature of crystalline Ge, from the vibration frequency of the atoms within the Ge2 molecule, and from the elastic constants of the Ge single crystal provide values lying between 1.16 and 3. ! x 104 dyne cm-1. It is finally supposed that the changes in energy due to both sorts of distortions are additive. The results of this calculation are to be found in table 1 and are also shown in fig. 6, where the binding energy per atom is plotted for 15 of the 19 different ideal structures containing a complete third correlation layer versus the order of correlation. It was shown earlier in this paper (see section 2) that there is no need to differentiate between numerous ideal structures containing a complete fourth correlation layer if only the binding energy is to be considered. Fig. 6 shows that including the second correlation layer the structures with prevailing eclipsed bonds are strongly favoured energetically. When one includes the third correlation layer, the continuation of structures with prevailing staggered bonds by prevailing eclipsed bonds is also energetically very favourable. Between the third and the fourth correlation order there is a tendency in favouring the structures which contain staggered links. This is in good agreement with the conclusions drawn in the preceding paragraph about the size of the structures with prevailing eclipsed bonds. For an infinite number of correlation layers, the curve corresponding to exclusively staggered bonds must become the lowest of all and reach the lattice value of - 3.266 eV. Essentially, this behaviour is due to the fact that the configurational energy per atom of small clusters is determined chiefly by the ratio between the number of free and contiguous hexagonal faces and that this ratio is much lower in clusters in which most bonds are eclipsed. With increasing cluster size the decrease in binding energy due to the increasing misfit and the ever decreasing ratio between surface and volume tend to favour progressively the structure lacking any misfit, i.e. the crystalline structure where all bonds are staggered.
383
SHORT-RANGE ORDER IN AMORPHOUS GERMANIUM
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Order o f tt~e correlahon {ager Fig. 6. Binding energy per atom versus order of correlation for 15 of the 19 structures contained and described in table 1. Four structures have not been represented for the sake of clarity (3/1 12/0 33/3; 3/1 12/0 30/6; 3/1 12/0 24/12; 2/2 10/2 23/11).
It must also be emphasized that a structure of higher energy can be transformed into one of lower energy by rotating neighbouring atoms in such a way about their common valence bond as to diminish the number of free hexagonal faces. These rearrangements are hampered to some extent by the existence of a certain activation energy of the rotation. (In organic
384
R. GRIGOROVICI AND R. IvlAN~tILA
molecules this activation energy amounts to about 3 kcal/mol=0.13 eV.) Even more hindering is the often intervening necessity to break some bonds already formed with other atoms of the same or the next correlation layer in order to achieve this rotation. Due to this possibility of rearrangement we must expect to find in general, and especially in annealed samples, more structures with prevailing eclipsed bonds than would correspond to the statistical weights given in table I. To check this point the number N 3 of atoms within the third coordination sphere ( r = 4 . 8 0 A ) was calculated from the data of table 1. The third coordination sphere was selected to perform this check, because it is very sensitive to the type of structure. Indeed this coordination sphere is occupied by 12 atoms in the Ge crystal, by 9 atoms in a hypothetical wurtzite crystal and by none in the amorphon. The calculation yielded N 3 = 8.02, while the experimental r.d.c, of a-Ge films deposited at room temperature fits well a calculated r.d.c, with N3 = 6 (ref. 6). In a-Si films deposited at room temperature, too, the best fit is obtained with N 3 =7.2 (ref. 8); indeed taking into account the higher crystallization temperature of a-Si, these later films were less "annealed" than the Ge films deposited under identical conditions.
5. The real structure of amorphous germanium The real structure of a-Ge films is thus not at all well defined. Even if we neglect the structural imperfections, there is an incommensurate number of possible ideal structures compatible with a real film. Only a statistical description could be given under certain well-defined deposition and annealing conditions. Structural imperfections can also be introduced in a number of different ways. A Ge atom bound on the surface of a cluster in a wrong configuration and blocked in this position by another neighbouring atom so as to be unable to rotate into the right configuration which satisfies the condition of ideality must give rise to a point defect. So can a foreign atom introduced into the film during the deposition, though such an atom can also saturate a non-satisfied valence bond. Cumulation of angular misfits may also give rise to a point defect similar to a vacancy in a crystalline lattice. Another structural imperfection similar to a high angle grain boundary or a crystallite interface must arise when two independent clusters grow together. Here, as previously, the number of defects with incomplete overlapping of orbitals (pseudo-vacancies) will undoubtedly exceed the number of defects with too many overlapping orbitals (pseudo-interstitials). The first ones wiU act as acceptors and are known to occur in polyerystalline Ge; the last ones will act as donors and might compensate some of the acceptors.
S H O R T - R A N G E ORDER I N A M O R P H O U S G E R M A N I U M
385
Obviously there is no sharp distinction between a point defect of the vacancy type and a local angular misfit. Both lead to poor overlapping of orbitals which differ only in degree. It may be conjectured that only vacancies act as relatively deep acceptors, while angular misfits give rise to the often postulated tail of states protruding into the energy gap. If so, the density of states function should not drop to zero, but only perhaps go through a minimum between the band edge and the acceptor levels maximum. Transport phenomena agree qualitatively with this picture: a-Ge is a p-type, apparently partly compensated semiconductor13). The acceptor density in a-Ge films deposited at room temperature is of the order of 1018 cm -3. Let us now try to predict on the basis of our model what the real structure of an a-Ge film should be under different conditions and compare these predictions with what is known from experiment. If deposited on a substrate held at low temperature, the film should consists of many small domains in which the statistical proportion of staggered and eclipsed bonds should hold because neither rotation about the bonds nor surface or volume migration can take place to any significant degree. Where these domains meet, grain boundaries as described above should provide an important number of predominantly acceptor-like defects. Indeed such films show an electrical resistivity which exceeds that of polycrystalline Ge films only by a little more than one order of magnitudela). Also, due to the lack of surface migration, a-Ge films have a very smooth surfacelS). Annealing should increase the misfit in the nearly ideal regions to allow for a high degree of binding within the grain boundaries. This binding can be achieved either by the "recombination" of a pseudo-vacancy with a pseudo-interstitial or by the migration of a pseudo-vacancy to the surface of the film. This process must lead to a reduction both in the acceptor density and in the carrier mobility, which happens indeed in a-Ge films16). An increasing degree of misfit and of fluctuations has also been observed in a-Ge as a consequence of annealing between room temperature and 200 ° C ~, 6). At even higher temperatures not only defects but also Ge atoms start migrating, first on the surface, later within the volume of the film. In the first process fewer and looser bonds have to be broken and the atom when reaching a domain with predominantly staggered bonds is integrated into a crystallite, where the binding is stronger. The growing of the crystallite will stop when the surface of the partly emptied amorphous regions round the crystallite gets so rough, that surface migration is suppressed again. The process is therefore very similar to a reconstructive
386
R. GRIGOROVIC1 AND R. M/kNAILA
phase transformation 17) favoured by the existence of an intermediate phase - the two-dimensional "gas" of atoms moving on the surface of the film - which is however exhausted in time. Electron microscope pictures taken during the annealing of a-Ge films show indeed the formation between 200 and 430 °C of crystalline islands within an amorphous matrix (fig. 7)*. These islands cover only a few per cent of the surface, their size reaches a certain limit at a given temperature, and the amorphous matrix surrounding them until a certain distance is thinner and has a rougher surface than the rest. Finally at 430 °C volume migration contributes essentially to the growing of the crystalline islands. This process is analogous to a reconstructive transformation without a catalyst17), i.e. at this temperature crystallization should proceed slowly in time, because bonds must be broken and formed again in a new geometry, until the whole volume of the film has crystallized. Indeed it takes 20 to 30min to crystallize an a-Ge film completely at 430°C la) and the onset of progressive crystallization is quite sharp. An estimate of the heat of crystallization of a-Ge based on our model can also be made. Let us assume that the additional energy of the amorphous phase results from the mean increase of the first interatomic distance r 1 - a =7.35 × 10 - 2 / ~ (i.e. 3~o of the crystalline distance), as well as from the corresponding average misfit 0 = 0 . 0 4 3 = 2 ° 26' of the interatomic bonding. By means of eqs. (1) and (2) a total supplementary energy of 6 . 4 0 x 10 - 3 eV per atom or 2.0 cal/g can be inferred. No attempt has yet been made to measure the heat of crystallization of a-Ge. A comparison with the figure given above would constitute another check of our model.
i
Fig. 7. Electron microscope picture of an amorphous Ge film annealed at 280°C. In the right-hand side some crystallites (dark spots) embedded in the amorphous matrix can be seen. They are surrounded by an amorphous region which is both thinner and rougher. * We are indebted to Dr. E. F. P6cza, Dr. P. B. Barna and A. Barna from the Institute of Technical Physics in Budapest for taking this picture.
SHORT-RANGE ORDER IN AMORPHOUS GERMANIUM
387
6. Final considerations The way in which o u r m o d e l o f a m o r p h o u s G e was o b t a i n e d is o p e n to m a n y criticism. W e c o n s t r u c t e d o u r clusters in space; in fact they are g r o w i n g on a substrate whatever the process o f b u i l d i n g up the a m o r p h o u s film. N o t only isolated atoms, b u t also g r o u p s o f 2-8 a t o m s might be the real b u i l d i n g stones o f the films o b t a i n e d by v a c u u m deposition. N o a t t e m p t was m a d e to t a k e into a c c o u n t the r e p u l s i o n which m u s t a p p e a r when the basic units t o u c h one a n o t h e r a l o n g the small t r i a n g u l a r faces, i.e. the shortrange o r d e r is similar to t h a t in a n o t existing wurtzite-type G e crystal. The estimates o f the a n g u l a r misfit, o f the m e a n increase in i n t e r a t o m i c distances a n d their fluctuations, a n d o f the energy o f the different structures are very r o u g h a n d b a s e d on r a t h e r naive geometrical c o n s i d e r a t i o n s r e m i n d i n g o f Demokritos. Nevertheless the a u t h o r s feel that a certain progress has been achieved by this d e s c r i p t i o n o f the s h o r t - r a n g e o r d e r in a n a m o r p h o u s solid a n d t h a t further i m p r o v e m e n t in the quantitative c h a r a c t e r i z a t i o n o f the statistical c o r r e l a t i o n s within this structure c o u l d f o r m the basis for a c a l c u l a t i o n o f the energy b a n d structure o f a n actual t h r e e - d i m e n s i o n a l s e m i c o n d u c t o r .
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17)
H. K6nig, Reichsber. Phys. 1 (1944) 4. E. Kirkorian and R. J. Sneed, J. Appl. Phys. 37 (1966) 3665. G. Szekely, J. Electrochem. Soc. 98 (1951) 318. H. F. Sterling and R. C. G. Swan, Solid-State Electron. 8 (1965) 653. H. Richter and G. Breitling, Z. Naturforsch. 13a (1958) 988. R. Grigorovici and R. M~n~.ilh, Thin Solid Films 1 (1968) 343. R. Grigorovici, R. M~n~.ilftand A. A. Vaipolin, Acta Cryst. B24 (1968) 535. M. V. Coleman and D. J. D. Thomas, Phys. Status Solidi 24 (1967) K 111. N. H. Fletcher, Proc. Phys. Soc. (London) 91 (1967) 724; 92 (1967) 265. G. F. Voronoi, Reine Angew. Math. 134 (1908) 198. L. Pauling, The Nature of the Chemical Bond (Cornell Univ. Press, Ithaca, New York, 1960) p. 116. K. M. Guggenheimer, Proc. Phys. Soc. (London) 58 (1946) 456. R. Grigorovici, Mater. Res. Bull. 3 (1968) 13. N. Croitoru and L. Vescan, private communication. J. Tauc, A. Abrahfim, L. Pajasov~i, R. Grigorovici and A. Vancu, in: Proc. Intern. Conf. Non-Crystalline Solids, Delft, 1964 (North-Holland, Amsterdam, 1965) p. 606. R. Grigorovici, N. Croitoru, A. D6v6nyi and E. Teleman, in: Proc. Intern. Conf. Phys. Semiconductors, Paris, 1964, p. 423. M. J. Buerger, Crystallographic Aspects of Phase Transformations, in: Symposion on Phase Transformations in Solids, Cornell Univ., 1948, p. 196.