Amorphous carbon

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Amorphous carbon
J. Robertson
a
a
Central Electricity Research Laboratories, Leatherhead, Surrey,
KT22 7SE, England
Version of record first published: 02 Jun 2006.
To cite this article: J. Robertson (1986): Amorphous carbon, Advances in Physics, 35:4, 317-374
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ADVANCES IN PHYSICS, 1986, VOL. 35, NO. 4, 317-374
Amorphous carbon
By J. ROBERTSON
Central Electricity Research Laboratories,
Leatherhead, Surrey K T 2 2 7SE, England
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[Received 3 October 1986]
Abstract
The properties of various types of amorphous carbon and hydrogenated
amorphous carbon are reviewed with particular emphasis on the effect of atomic
structure on the electronic structure. It is shown how the proportion of sp 3 and sp 2
sites not only defines the short-range order but also a substantial medium-range
order. Medium-range order is particularly important in amorphous carbon because
it is the source of its optical gap, whereas short-range order is usually sufficient to
guarantee a gap in other amorphous semiconductors. The review discusses the
following properties: short-range order and the radial distribution function, the
infrared and Raman spectra, mechanical strength, the electronic structure,
photoemission spectra, optical properties, electron energy-loss spectra, core-level
excitation spectra, electrical conductivity, electronic defects and the electronic
doping of hydrogenated amorphous carbon.
Contents
1. Introduction
PAGE
318
2. Atomic structure of amorphous carbon
2.1. The hierarchy of carbons
2.2. Structural determinations of amorphous carbon
2.3. Structural modelling
2.4. Structure of a-C :H
2.5. Extended X-ray absorption fine structure
2.6. Vibrational properties
2.7. Strength and medium-range order
320
320
323
326
329
330
331
336
3. Electronic structure of amorphous carbon
3.1. Structural stability in the n electron systems
3.2. Tight binding Hamiltonian
3.3. Results for graphite, diamond and the random network models
3.4. n Bonded clusters
3.5. Hydrogen configurations
3.6. Mobility edges
339
339
342
344
347
349
350
4. Electronic structure: comparison with experiment
4.1. Photoemission spectra
4.2. X-ray near-edge spectra
4.3. The optical absorption edge
4.4. The wide-band optical spectra
4.5. Electron energy-loss spectra
4.6. Heat treatment of a-C : H
4.7. Ion-beam deposited carbon
351
351
352
352
355
356
358
359
5. Localized states
5.1. Origins of localized states
5.2. Conductivity
361
361
363
318
J. Rohertson
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5.3. Electron spin resonance and photo-luminescence
5.4. Doping
365
367
6. Conclusions
369
Acknowledgments
370
References
370
1. Introduction
The field of disordered carbon covers a wide range of materials and properties:
carbon fibres, of importance for their strength in composites (Reynolds 1973); chars and
cokes involved as intermediate species in the pyrolysis of carbonaceous materials into
graphite (Clar 1964); glassy carbon, formed by heating certain organic polymers (Noda
et al. 1969); microcrystallinc (pc) carbon, produced by irradiating graphite (Kelly 1981);
amorphous carbon (a-C) produced by evaporation in an electron beam or carbon arc
or by sputtering (McLintock and Orr 1973); and hydrogenated amorphous carbon
(a-C:H) films, produced by plasma deposition or ion-beam deposition of gaseous
hydrocarbons (Anderson 1977, Weissmantel et al. 1982) and used as hard, transparent,
coating materials. Of this range, the present review concentrates on amorphous carbon
and in particular on the inter-relation of their local atomic structure and electronic
properties.
Let us first summarize the bonding possibilities of a carbon atom. In principle, a
carbon atom can adopt three different bonding configurations, sp 3, sp 2 and sp a
(figure 1). In the sp 3 configuration, each of the carbon's four valence electrons is
assigned to a tetrahedrally directed sp 3 hybrid orbital, which then forms a strong a
bond with an adjacent atom. At a carbon sp z site, three of the four electrons are
assigned to the trigonally directed sp z hybrids which form o- bonds; the fourth electron
lies in a p~ (pn) orbital lying normal to the o-bonding plane. The pn orbital forms weaker
n bonds with adjacent pn orbitals. At sp a sites, only two of the electrons form o- bonds,
along _ 0x, and the two other electrons are left in orthogonal py and p= orbitals to form
n bonds. A a bond between two sites is called a single bond, and is represented by a
single line, while a a - n bond pair is called a double bond and is represented by two lines.
A hydrocarbon containing only single bonds is called 'saturated'. Unsaturated systems
can take the form of a system of separate double bonds in 'olefinic' systems such as
ethylene, HzC = CH2, or as delocalized or 'conjugated' n bonded systems such as the
'aromatic' six-membered rings in benzene ( C 6 H 6 ) and graphite.
Z
X
X
Z
sP 3
sp 2
sp
Figure 1. Schematic representation of sp a, sp 2 and sp I hybridized atoms.
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A m o r p h o u s carbon
319
Table 1 gives values of some of the key properties for diamond, graphite and four
forms of disordered carbon. Diamond consists of sp 3 sites. The saturated bonding
produces the wide 5.5 eV band gap and low conductivity, and the isotropy of the
bonding gives it its strength. Graphite consists of hexagonal layers of sp 2 sites, weakly
bonded together by van der Waals forces into a ABAB stacking sequence along the c
axis. Conductivity and strength are high along the basal plane but are low along the c
axis. As graphite is the stable allotrope of carbon, many disordered forms of carbon
have structures based on its lattice. The structures of/~c-carbon and glassy carbon are
frequently classified in terms of a basal plane correlation length L, and a c axis
correlation length Lc, and table 1 shows that both materials are essentially metallic.
Evaporated a-C and a-C : H differ from glassy carbon in being truly amorphous and
semiconducting. The presence of a semiconducting band gap is a crucial difference and
their structures are not so easily classified. It is now wise to derive the structure of both
materials from first principles; first defining the proportion of sp z and sp 3 sites, then
their local arrangement and finally, in a-C : H, the proportion and arrangement of the
hydrogen atoms. Summarizing the data discussed in detail in later sections, it is
generally believed that glassy carbon contains approximately 100~ s p 2 sites,
evaporated a-C 1-10~ sp 3 sites, while a - C : H may comprise 30-60~o hydrogen with
perhaps 30~o of the carbon sites having a n s p 2 configuration, but this is strongly
dependent on heat-treatment. Ion-beam deposition methods are able to raise the
proportion ofsp 3 sites in both a-C and a - C : H and produce harder films. There is little
evidence for sp 1 sites in unhydrogenated carbons, but there is some evidence for minor
amounts of - C - C H
groups being present in a - C : H .
These two parameters, the carbon bonding and the hydrogen content, define the
short-range order in amorphous carbon. However, they do not entirely define its
structure. This is because there exists a substantial degree of medium-range order on
the ~ 10/~ scale; the sp 2 sites ofa-C tend to occur in warped graphite layer clusters and
the sp 2 and sp 3 sites in a - C : H are somewhat segregated and clustered.
The structure of amorphous carbon is of fundamental importance for a variety of
reasons, of particular interest here is the effect of disorder in arc electron system. Since
the ~ states are weakly bound, they lie closer to the Fermi level EF than the o- states
(figure 2). Consequently, the filled rr states will form the valence band and the empty ~*
states will form the conduction band and so determine the size of the gap.
Table 1. Room-temperature conductivity (aR~), optical gap, density and hardness of forms of
diamond, graphite glassy carbon, evaporated a-C, ion-beam deposited a-C, and
plasma/ion-beam deposited a-C:H. References: 1 Dischler and Brandt (1985), 2 Moore
(t973), 3 Jenkins and Kawamura (1976), 4 Noda et al. (1969), 5 Hauser (1975), 6 Fink et al.
(1983), 7 Savvides (1986), 8 Zelez (1983) 9 Kaplan et al. (1985), and 10 Weissmantel et al.
(1982).
Diamond
Graphite
Glassy carbon
Evaporated a-C
Ion-beam a-C
a-C:H
O'RT
(f~- 1cm- 1)
Eg
(eV)
10-18
2"5 × 10 4 ( I c )
102 103
~ 10 -~
~ 10 2
10 7 10-16
5'5
-0-04
10 -2
0-44).7
0-4-3"0
1-54
Density
(g cm- 3)
3.515
2"267
1-3-1.55
~2.0
1.8-2'7
1-4-1-8
Hardness
(kg mm z)
10 4
800-1200
20-50
1250 6000
Reference
1
2
3, 4
1, 5, 6
7, 8
1, 6, 9, 10
J. Robertson
320
filled
valence
states
empty
conduction
states
Energy
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Figure 2. Schematic representation of the electronic spectrum of an unsaturated hydrocarbon.
The effects of disorder in ~-bonded systems such as a-Si are well known (Weaire
1971, Mott and Davis 1979). It is understood that the absence of long-range order
causes a tailing of localized states into the gap from the two band edges, while the states
in the bands remain extended. A 'pseudogap' remains between the extended states if the
atoms can retain their short-range order, i.e. their tetrahedral coordination in the case
of a-Si.
Disorder effects in ~ electron systems have been studied only recently from a
theoretical viewpoint (O'Reilly et al. 1985, Robertson and O'Reilly 1986). The situation
is now more complex; even with defined short-range order, it is found that the ~ * gap
can vary between 0 and --~5 eV entirely as a function of medium-range correlations
between ~ states. Thus, in principle, the absence of long-range order could have two
effects, it could produce a band tailing as in ~ systems, or it could allow new types of
medium-range order which could either create or close up the gap. This dependence of
the gap on medium-range order is a unique feature of ~ systems.
The review is organized as follows. In §2 we describe how the techniques of
diffraction, and infra-red and Raman spectroscopy are used to analyse the structure of
amorphous carbon. Section 3 then describes their electronic structure, particularly of
their ~ states. These results are then used to interpret the various electronic spectra-photoemission, core absorption, optical, energy loss, etc.--which also turn out to
provide very valuable information on the atomic structure. Variations of properties
with deposition and annealing conditions are discussed in §§ 4.6 and 4.7 while gap states
and electrical conductivity are discussed in § 5.
2. Atomic structure of amorphous carbon
In this section we first describe the structures of related disordered forms of carbon
in order to place a-C and a - C : H in context. The radial distribution function and
scattering function of amorphous carbon are then analysed, first in terms of the
graphite lattice and then in terms of continuous random network models. The
vibration spectrum is dependant on the structure and in the case of amorphous carbon
it is found to provide information on the size of graphite layers and the bonding of the
hydrogen atoms.
2.1. The hierarchy of carbons
Since graphite is the stable crystalline allotrope of carbon and because synthetic
graphite can be produced by the pyrolysis of many organic materials, it is natural to use
the graphite lattice as the first reference for the structure of disordered forms of carbon.
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Amorphous carbon
321
Perfect graphite consists of hexagonal layers of atoms in an ABAB stacking
sequence with unit cell dimensions of a = 2-461 A and ¢ = 6"708/~, corresponding to a
bond length of 1.421A, an inter-planar spacing of d=3.354/~ and a density of
P0 = 2.267 g cm- 3 (table 1).
The most ordered form of synthetic graphite is called highly oriented pyrolytic
graphite (h.o.p.g.), and is produced by heat and pressure treatments up to 3500°C. Its d
spacing is 3-354-3.359/~, its density exceeds 2-26 g c m - 3 and its crystallite sizes are of
the order of 1 #m (Moore 1973).
Warren (1941) used the concept of a turbostratic graphite lattice to analyse the
structure of pyrolytic carbons. The turbostratic lattice is defined as a series of graphite
layers with random orientation and random stacking. The diffraction pattern of this
lattice possesses the (hkO) and (00/) spots of graphite, but the (hkl) l ¢ 0 spots denoting
the three-dimensional ordering have become broadened out into a faint background.
This reference form of disorder is believed to increase the inter-planar spacing to 3-44/~
and reduce the density Po to 2.22 g cm- 3. Warren (1941) further modelled the disorder
in terms of cylindrical crystallites of mean diameter L, and mean height Lc, which were
estimated from the sharpness of the (110) and (002) peaks, respectively.
The pyrolysis of organic material into graphite occurs in three stages (Fischbach
1971, Jenkins and Kawamura 1976), the loss of volatile matter containing H, N and O
at ~ 400-700°C (carbonization), the formation of the graphitic sheets of pyrolytic
carbon at 60(~1200°C (polymerization) and the gradual evolution of the three
dimensional graphite lattice (graphitization) at 1200-3000°C. However, the structure
of pyrolytic carbon depends not only on the heat-treatment temperature T~ but also on
the starting material. Franklin (1951) noted that pyrolytic carbon could be classified as
being either graphitizing or non-graphitizing, according to whether the graphitization
occurred at T , - 1200-2000°C or only with difficulty above Ta ~- 3000°C. She noted that
the structure of graphitizing or soft carbons consisted of turbostratic crystallites whose
dimensions L, and L c increased steadily with T,. In contrast, Franklin (1950, 1951)
found that non-graphitizing carbon, later called glassy carbon, consisted of an ordered
and a disordered component, and that parallel layers developed with much more
difficulty--only the (002) interlayer peak was observed and Lc tended to remain below
30/~ until Ta reached ~ 3000°C. Franklin (1951) focused on the movement of planes
into parallelism as the key step in the graphitization process and so she distinguished
the structures of the two carbons in terms of whether or not they opposed this
movement.
When pyrolytic carbon is examined using electron microscopy it is found that the
sheet size tends to exceed L, as determined by X-ray diffraction by a factor of ~ 10
(Jenkins et al. 1972). Thus, L, was re-interpreted as being a 'correlation length' over
which a layer was flat. L~ needed no change, however, and corresponds to the distance
over which layers are parallel.
Electron microscopy and other results led Jenkins et al. (1972) to develop a new
model for glassy carbon, different from that of Franklin (1951). It is shown in figure 3 (a)
and consists of entangled ribbons of graphitic polymeric molecules. Lc denotes the
thickness of the ribbon and L a the width. It is also perhaps locally flat over a length La.
The difficulty of graphitization and its strength is attributed to the entanglement of the
ribbons. Its chemical inertness is regarded as being due to the ribbons having no ends,
but rather merging and separating at 'confluences', as shown. The entanglements also
give glassy carbon its porosity and its low density (1.2-1.6, table 1), although figure 3 (a)
over-emphasizes the voidage somewhat (Jenkins and Kawamura 1976).
Typicalstrongconfluence
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:~::::::ii !!ii:................
:
.
Weakconfluence
~L~
-i ,-~
(a)
I
200
(b)
i-'r
I
Figure 3. Schematic illustration of the structure of (a) glassy carbon and (b) carbon fibres, after
Jenkins and Kawamura (1976).
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Amorphous carbon
323
Carbon fibres are produced by the carbonization of, for example, polyacrylonitrile
fibres. Their structure resembles that of glassy carbon in that they are non-graphitizing
and their diffraction patterns show only the (hkO)and (002) peaks of the graphite lattice,
from which it is deduced that L, = 61~120 • and L c _~40 ~ (Jenkins and Kamawura
1976). Electron microscopy reveals that carbon fibres are anisotropic and consist of an
array of carbon fibril units of length at least 1000/~ with a preferential parallel
alignment. These ideas are summarized schematically in figure 3(b) and by the
description of carbon fibres as 'textured glassy carbon' (Jenkins and Kawamura 1976).
These concepts do however leave some questions regarding the structure of glassy
carbon--the nature of the disordered component observed by Franklin (1950), and the
possibility ofsp 3 bonded sites suggested by Noda and Inagaki (1964) which will now be
discussed.
2.2. Structural determinations of amorphous carbon
In common with other vapour-quenehed systems, evaporated a-C is a highly
disordered form of carbon. Table 1 shows that although a-C is denser than glassy
carbon, it has a larger gap and a much lower conductivity. In the next sections therefore
we consider a wider range of structures, not just variations on the graphite lattice.
The first parameter to be determined is the sp2:sp 3 site concentration ratio.
The most direct techniques to probe this ratio are the spectroscopic methods
nuclear magnetic resonance (NMR) and X-ray near-edge structure (XANES). The
XANES technique gives unequivocal evidence of the existence of sp 2 sites in a sample; it
depends on the electronic structure and is described in § 4.2. The N M R method gives
quantitative data on the concentration ofsp 2 and sp 3 carbon sites, as discussed in § 2.4,
but unfortunately screening prevents its use to date in unhydrogenated material. Sadly,
there is as yet no technique that gives direct evidence ofsp 3 sites in a-C. The analysis of
wide-band optical spectra is perhaps the best means of quantifying the sp 2 : sp 3 ratio in
a-C and this gives an upper limit of sp 3 sites of ~ 12~ as discussed in § 4.4. Sputtered
a-C is generally similar to evaporated a-C in its structure and electronic properties.
Recently, ion-beam sputtering techniques have produced a-C with minimal hydrogen
content and there is evidence that some of these have a substantial proportion of sp 3
sites (see, for example, Mori and Namba 1983), but their structure has yet to be
measured and analysed in any detail; investigators have concentrated on their hardness
and optical properties (§ 4.7).
Electron, X-ray and neutron diffraction techniques have been employed to
investigate the structure of a-C. Diffraction studies provide a one-dimensional
representation of the scattering properties of an amorphous system, the scattering
intensity, F(k) where k is the scattering wave-vector. F(k) is an oscillatory function
whose peaks correspond to the Bragg spots of a crystal. F(k) is related by a Fourier
transform to the radial distribution function (r.d.f.), J(r):
F(k)=
f ] {J(r)/r-4npor} sin(kr)dr.
(2.1)
The r.d.f. (figure 4) consists of a series of peaks, which eventually merge into a smooth
parabolic curve
J(r) = 4nrZpo,
(2.2)
where Po is the bulk density. Its first peak lies at the average first-neighbour separation
or bond length, rl, and it contains an area equal to the first-neighbour coordination
324
J. Robertson
I
F
I
(I
4ot
30-
t~
E
/
o
2O
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Z
10
!
0
1
~ i,,'
2
3
J
4
5
Figure 4. Experimental radial distribution functions of glassy carbon from Mildner and
Carpenter (1982) (solid curve) and of evaporated carbon from Kakinoki et al. (1960a)
(dashed curve) and Boiko et al. (1968) (dot-dashed curve).
number n 1. The second peak lies at the second-neighbour distance rz from which the
bond angle 0 may be extracted using
r 2 = 2r 1 sin (0/2).
(2.3)
Figure 5 shows the scattering intensity of glassy carbon measured by Mildner and
Carpenter (1974) using neutron diffraction, and of evaporated a-C by Kakinoki et al.
(1960 a) and Boiko et al. (1968) using electron diffraction. The F(k) of glassy carbon
measured by Egrun (1976) using X-ray diffraction is very similar to that found by
Mildner and Carpenter (1974), except that the first (002) peak is sharper.
Table 2 gives the values of some important parameters of the r.d.f, and of F(k), as
measured by various authors, and compares them to values for graphite and diamond.
The four parameters rl, nx, 0 and Po are often sufficient to establish the structure of
other amorphous semiconductors, ri should be particularly useful in the case of carbon
because the bond length of graphite (1.42/~) is much shorter than that of diamond
(1.54/~). In contrast, porosity can severely affect the value of the density Po and thereby
the coordination numbers, so these parameters must be treated with caution. An
additional important parameter is the first peak in the F(k), if this lies around k = 1.88
it relates to the (002) peak of graphite and provides evidence of layering.
Mildner and Carpenter (1974, 1982) measured the F(k) of glassy carbon out to a
cutoff of kmax= 25/~- 1 and so produced a well-resolved r.d.f. The peaks in F(k) can be
indexed as the (hkO) and (002) peaks of turbostatic graphite and their positions are quite
close to their ideal values (table 2). Mildner and Carpenter (1982) estimated La"~ 49
and L c - 3 1 ~ from the widths of the (100) and (002) peaks. No higher order (00l) and
(hkl) l ~ 0 peaks were apparent. The calculated r.d.f, in figure 4 has a sharp first peak at
r i = 1.425/~ with an area n 1 = 2"99 atoms, and a second peak at r 2 = 2-45 ~, all close to
325
A m o r p h o u s carbon
r
I
'
I
I
I
'
I
~
b
i
I
I
I
~
I
I
I
k
I
i
L
i
I
i
I
i
I
I
(a) -~
E
IN
(b) :
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tl.
(c)
I
0
2
4
6
8
10
12
~4
16
k (~,-~)
Figure 5. Experimental interference functions of (a) glassy carbon measured by neutron
diffraction by Mitdner and Carpenter (1974), and evaporated a-c measured by electron
diffraction by (b) Boiko et al. (1968) and (c) by Kakinoki et al. (1960a).
Table 2. Peak positions, in/~- 1, in the interference functions of Mildner and Carpenter (1974).
Kakinoki et al. (1960 a), the 1200°C annealed sample of Kalanoki et al. (1960 b), and Boiko
et al. (1968), compared to the indexed reflections of graphite and diamond, and compared
to the peak positions in the continuous random network models of Beeman et al. (1984).
Peak position
(~- 1)
Sample
Mildner
Kakinoki
Kakinoki (1200°C)
Boiko
Graphite(hkl)
Diamond (hkl)
Models Cl120
C340
C256
C519
1.8
2.98
1.2
2.96
1-84 2.96
1-0
2.9
(002) (100)
1.88 2.95
(100)
3-06
1-8
3.1
1-1
3-2
1.1
2.9
-3-1
5"11 5"96 7"8
8"8
10.2
5"5
--8-6
10.4
5.12 5.91 7.87
8.86 10-2
5.1
(110) (200) (210) (300) (220)
5"11 5-90 7"82
8.86 10.2
(200) (220) (311) (331)
3.53
5.00
5"86 7-71
5-7
9-0
5-7
9.2
5-5
8.9
5.5
8.7
their graphite values. Their high cutoff (kmax)gives confidence in the peak positions in
their r.d.f., although the coordination numbers required precise measurement of the
microscopic density (table 3). These observations led Mildner and Carpenter (1982) to
conclude that their data provided no evidence for any sp 3 sites in glassy carbon.
Franklin (1950) measured F(k) for glassy carbon by X-ray diffraction with
kmaX= 14 ~ - 1. She concluded that 65 per cent of the atoms were in turbostatic graphitic
domains with L a - ~ 1 6 ~ and Lc_~5/~, with the remainder in disordered regions of
Downloaded by [Duke University Libraries] at 15:26 12 November 2012
326
J. Robertson
unspecified character. The (002) peak corresponded to a greater than ideal inter-layer
spacing d = 3.70/~. The results differ considerably from those of Mildner and Carpenter
(1982), but this is due in part to the less peaky character of Franklin's value of F(k),
which in turn is due to the much lower heat-treatment temperature of her sample:
T, = 1000°C compared to T, = 2000°C.
Noda and Inagaki (1964) and N oda, Inagaki and Yamada (1969) also studied glassy
carbon by X-ray diffraction, but their results were restricted by low resolution and a
low km,x. They concluded that glassy carbon may contain a proportion of sp 3 sites.
However, we would question whether this could be supported in view of the low kmax.
The scattering intensity of evaporated a-C was measured by Kakinoki et al.
(1960a, b) out to kmax=27/k -1. This F(k) differs substantially from that of glassy
carbon in that the first peak is weaker and has moved to a much lower value of k, the
following peaks are broader out to 8/~-1 and the subsequent peaks decay rapidly
(figure 5). This decay causes the peaks of the r.d.f. (figure 4) to be much broader. Most
importantly, the first peak of the r.d.f, has moved out to 1-50/~, and so Kakinoki et al.
(1960 a) conclude that their sample consisted of a mixed random network, containing
an equal proportion of sp 2 and sp 3 sites. Indeed, Kakinoki et al. (1960 a) and Kakinoki
(1965) proposed a microcrystallite model of a-C, consisting of domains of graphitic and
random sp3-bonded regions.
The scattering intensity obtained by Boiko et al. (1968) only extends to
km~x= 8.5/~ - 1 (figure 5) and it is this low limit which broadens the peaks in their r.d.f, in
figure 4. In general, the Boiko r.d.f, may be described as being a broadened version of
the Mildner r.d.f, and indeed Boiko et al. (1968) concluded from their r.d.f, that a-C had
graphitic short-range order.
It is clear from their F(k) in figure 5 that the structures of evaporated and glassy
carbon are different. Evaporated a-C is clearly a much more disordered form of carbon.
The much longer bond length is the most significant feature of this study and it would
be interesting to find the corresponding change in the F(k). Again, however, some of the
difference is due to the heat treatment. Kakinoki et al. (1960 b) and Oberlein et al. (1975)
found that the F(k) for evaporated carbon sharpens up with heat treatment. In
particular, the (002) peak is either barely visible or is at a displaced position in samples
prepared at low temperature, but with heat treatment its intensity and sharpness
increases dramatically.
2.3. Structural modelling
We now consider three types of structural modelling that have been applied to a-C;
the strained layer model of Egrun (1973, 1976), the domain model of Stenhouse and
Grout (1978) and the random network models of Beeman et al. (1984).
Egrun (1976) noted that the F(k) value of glassy carbon has peaks corresponding to
the (hkO) and (002) positions of graphite but the other (hkl) peaks are not visible. He
suggested that glassy carbon consists of strained graphite layers stacked in a disordered
manner. He argued that the strain-induced broadening would exceed that due to finite
crystallite sizes in the turbostratic model. This model produced a small broadening of
the (hkO) peaks and a larger broadening of (001) peaks which, in practice, washed out all
but the (002) peak. In this way Egrun (1976) reproduced the observed F(k) of his glassy
carbon sample. Clearly, no sp 3 sites exist in this model. It would be interesting to see
this model applied to more disordered carbons.
Stenhouse and Grout (1978) tried to develop a theory which would handle a greater
degree of disorder. They postulated that a-C could be modelled by considering it as
327
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A m o r p h o u s carbon
microcrystallites of graphite inter-linked by an sp3-bonded r a n d o m network. They
calculated the r.d.f, as a function of microcrystallite size and the proportion of sp 3 sites.
They also included a correction to the calculated electron or X-ray scattering
intensities, which allowed for the covalent bonding producing a slightly non-spherical
electron distribution around each atom. They compared calculated scattering
intensities with those measured by Kakinoki et al. (1960 a) and Franklin (1950). They
concluded that the Kakinoki sample contained ~ 75 per cent sp 3 sites and had ~ 12
graphitic domains, while the Franklin (1950) sample has ~ 50 per cent sp 3 sites and
20/~ graphitic domains. However, Mildner and Carpenter (1982) and Summerfield
et al. (1984) found an error in the analysis of Stenhouse and G r o u t which caused it to
over-estimate the importance of sp 3 regions in the case of small crystallites.
Beeman et al. (1984) considered even more disordered structures. They constructed
four model random networks containing different proportions of sp 2 and s p 3 sites,
whose characteristics are summarized in table 3. Their C1120 model consisted of four
warped spZ-bonded layers (C280). The C340 model contained 9-1 per c e n t s p 3 sites and
the C356 model contained 51.4 per cent sp 3 sites. The C519 model contains only sp 3
sites and is the Polk (1971) model rescaled from Ge to sp 3 carbon. The bond lengths are
set to those appropriate to the bonding. All the models contain a sizeable fraction of
odd-membered rings. The C340 and C356 models differ from the domain model of
Stenhouse and G r o u t (1978) in that the sp 2 and sp 3 sites are intimately mixed.
The r.d.f.s of the four models are compared in figure 6 with representative
experimental r.d.f.s, of Kakinoki et al. (1960 a) and of Mildner and Carpenter (1982).
Beeman introduced an extra broadening into all r.d.f.s, as he was interested in the
overall disposition of the peaks. It is clear that the density of models, as constructed,
decreases with the fraction of sp 2 sites because of their wide non-bonded spacing.
Beeman et al. (1984) noted that the density of the C1120 model, containing only sp 2
sites and a rather large void space, was closest to the experimental value of
1"8~-0 g cm - 3 and that the density of the C340 and C356 models could be reduced by a
greater clustering of their sp 2 sites.
Table 3. Distances and coordinations in the radial distance functions of Mildner and Carpenter
(t974, 1982), Kakinoki et al. (1960 a), and Boiko et al. (1968) compared to the continuous
random network models of Beeman et al. (1984). The coordinations depend crucially on
the value of the density, P0; the value of Mildner and Carpenter (1974) is a microscopic
density after allowing for porosity, the Kakinoki value is considered to be a gross
overestimate compared to that found by other workers (1-7-2'0) and that used by Boiko
was imposed, not measured.
Sample
Graphite
Diamond
a-C: Mildner
Kakinoki
Boiko
Models Cl120
C340
C356
C519
rl
1.42
1"55
1.425
1"5
1-43
1.42
1-42
1-51
1.55
nl
r2
n2
Po (g cm-3)
3
4
2.99
3"45
3"3
3
3-28
3.53
4.0
2.45
2"52
2.45
2-53
2"53
2-44
2-43
2.55
2.52
6
12
6.1
2-25
3.51
1.49
2-4
2-1
2.11
2-69
3-21
3.39
8-8
6
12
328
J. Robertson
C l 1 2 0 100%sp 2
40
(.)
20
~ o
o 4O - C340 91% sp 2
-6
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(b)
(c)
g
....,-..;/XJ"',.,.
~ 20
g 0
~ 40 _ C356 4
E
-g
i:5 20
9
~5 0
f
C51 9 0% sp 2
/
_
.
,
~
/
.
40
(d)
20
0
1
2
5
4
5
R (~)
Figure 6. Radial distribution functions of the four random network models of Beeman et al.
(1984) (solid curves) compared with experimental radial distribution functions of
Kakinoki et al. (1960 a) for evaporated carbon (dashed curves) and Mildner and Carpenter
(1974) for glassy carbon (dotted curves).
Because the r.d.f.s have been broadened, the first peak was not used as a key
comparison parameter. The second peak of the Kakinoki r.d.f, occurs at ~2.52/~,
which is closer to that in C356 and C519 models. Overall, subsequent peaks fit more
closely to the C340 model than to the Cl120 or C356 models.
The scattering intensities of the four models are shown in figure 7. All four models
produce a peak in F(k) at around 3 ~ - 1 and a main peak at around 5.5/~ - 1. The C1120
model and, to a lesser extent, the C340 model have the pre-peak at k ~ 1-8/~- x, the
analogue of the (002) graphite peak. It is very interesting that this peak survives the
introduction of sp 3 sites into the C340 model, in which the more obvious layer
characteristics have been lost. In fact, it is typical of layer materials to retain a pre-peak
in their amorphous phases, as in, for example, a-As and a-AszSe 3 (Apling et al. 1977).
Beeman et al. (1984) noted that the interference function of Boiko et al. (1968) was
best fitted by the C340 model containing 9 per cent sp 3 sites and that of Kakinoki et al.
(1960 a) by the C356 with ~ 50 per cent sp 3 sites. They also felt that the C1120 model
was not a good model because its pre-peak was too large. However, we do not consider
this a problem as this peak is a function of the heat treatment and is certainly not too
large for the data of Mildner and Carpenter (1974) or Egrun (1976).
The random network models also have some incorrect features. Firstly, all
experimental F(k) curves except for those of Kakinoki et al. (1960 a) possess a shoulder
or peak at ~ 6 ~ 1, the (200) graphite peak, but this is absent in all the model F(k)
curves, even that of the C1120 model. This suggests that a-C possesses more ordering in
329
Amorphous carbon
/\
A
C1120 100% sp2
C340 91% sp 2
(b) .~;
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c3 6 49%,s,2
//~
0
q
I
2
I
I
4
t
C51 9 0% sp ~
I
6
I
I
8
k (~-1)
i
I
t0
~
I
12
I
I
14
i
16
Figure 7. Interference functions of the four random network models of Beeman
et al. (1984).
its s p 2 sites than is present in the Beeman models. In particular, electronic structure
considerations discussed in § 3 indicate that the s p 2 sites should form few oddmembered rings, unlike in the C1120 and C340 models.
We therefore conclude that the short-range order of glassy carbon is reasonably
well modelled by the strained graphite layer model of Egrun (1976) and that its longerrange order is represented by entangled fibrils as shown in figure 3 (a). The random
network models of Beeman et al. are valuable in pointing out the types of disorder
present in evaporated a-C, but they probably possess insufficient correlations between
their sp z sites.
2.4. Structure o f a-C : H
Direct measurement of the sp a and sp 3 site fractions in a - C : H is given by C 13
magic-angle-spinning NMR. Kaplan et al. (1985) directly verified the presence of s p 2
sites in a-C : H and showed that their concentration varied with deposition conditions
and tended to decrease with hydrogen content. A typical sample with an optical gap of
1-7 eV was found to possess 50 per c e n t s p 2 and 50 per cent sp a sites. Further support for
the presence of sp a sites in a-C : H is provided by the XANES spectra discussed in § 4.2
and further quantitative estimates of the s p 2 : s p 3 ratio are provided by the analysis of
wide-band optical spectra in § 4.4.
In contrast to a-C, there have been few diffraction studies ofa-C : H. McKenzie et al.
(1983 a) and Sproul et al. (1986) have studied a-C : H by electron diffraction, but only to
a cut-off of kma x = 10/~- 1. This is really too low for a good analysis of a system which
optical data (§4.3) suggest is highly clustered. Nevertheless, McKenzie et al. (1983 a)
interpreted their scattering function, shown in figure 8, in terms of both graphitic and
hydrocarbon polymer domains. Hydrogen does not contribute to the F(k). The
330
J. Robertson
I
P
~o
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la.
0
I
1
2
i
l
4
i
I
6
I
~
I
I
t
10
12
k (~,-~)
Figure 8. Interference function of a-C: H (upper curve) and after annealing at 500°C for 1 hour
(lower curve), after McKenzie et al. (1983 a).
measured F(k) tends to peak at wave-vectors characteristic of graphite. Interestingly,
there is also a pre-peak at ~ 2/~- 1 which McKenzie et al. (1983 a) thought was unlikely
to correspond to the graphite interlayer separation and instead interpreted it as being a
pre-peak from the hydrocarbon domains.
2.5. Extended X - r a y absorption fine structure
Extended X-ray absorption fine structure (EXAFS) measurements for amorphous
carbon have been taken by Kincaid et al. (1978), Batson and Craven (1979), Wesner
et al. (1983) and by Fink et al. (1983), as shown in figure 9. The EXAFS correspond to
the small oscillations in absorption intensity which occur at energies above an X-ray
absorption edge. These are caused by interference between the outgoing wave of the
photoelectron and its back-reflections from the surrounding atoms altering the
absorption cross-section for the X-ray photon. The EXAFS intensities are given
approximately by
= ~, s 0 sin (2kr i + rl) exp ( - 2alZk2) exp ( - 2ri/2e(k)),
(2.4)
where t/is a phase shift and k the wave-vector of the photoelectron: ~.e is the electron
mean-free path and this factor damps out the effects of neighbours with rl ~ 5/~. In
disordered solids, the effects of second- and higher-order neighbours tend to be damped
out by their static broadening, a i. Thus, the EXAFS of graphite and diamond extend to
,-~400 eV above the carbon ls edge (figure 9), but those of a-C are strongly damped.
Fink et al. (1983) noted that this shows that the first-neighbour distance of a-C has a
large static broadening of o-~ =0-1-0.2/~, presumably due to the difference in the sp 2
and sp 3 bond lengths. The oscillations are damped slightly less strongly in the a-C : H
spectra. This is unusual because optical measurements suggested that their a-C sample
contained > 88 per c e n t s p 2 sites, i.e. nearly all spZ--while the a - C : H samples has
32 per cent sp 2 sites--and so the latter could be expected to have the greater static
broadening.
331
Amorphous carbon
I
I
t
,
I
I
i
~
i
~
i
I
I
I
graphite
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II 2--,
~/ ~
Ta=20°C
~ d l a ~ o n d
500
350
Energy(eV)
400
Figure 9. Extended X-ray absorption fine structure spectra of various forms of carbon, after
Fink et al. (1983).
2.6. Vibrational properties
The vibrational (phonon) spectra provide valuable evidence to the structure of a-C
and a-C : H. Infrared and Raman activity in crystals is determined by their symmetry.
In diamond, there is one Raman active mode (at 1332 cm-1) and no infrared active F
modes. There are two Raman active modes in graphite, both of Eag symmetry, the main
1580 cm t mode and the 50 cm 1 rigid-layer mode, and there are two infrared active
nodes, the main Elu 1585cm -~ mode and the 868cm -1 out-of-plane Azu mode
(Nemanich et al. 1977). In amorphous carbon, there is no k-conservation because of the
loss of long-range order, and consequently all modes are allowed; we therefore expect
that the infrared spectrum should resemble the phonon density of states (DOS) and that
the Raman spectrum resemble the phonon DOS weighted by a matrix element. The
matrix may peak for certain modes and tends to vary a s ( . o 2 at low frequency (co)(Alben
et al. 1975). The infrared and Raman spectra may also reveal disorder-induced changes
in the phonon DOS.
The Raman spectrum of sputtered and evaporated a-C was measured by Wada
et al. (1980), Solin et al. (1978), Lannin (1977) Nathan et al. (1974) and Solin and
Kobliska (1974), and of glow-discharge deposited a-C : H by Dillon et al. (1984). The
spectrum of a-C is dominated by a large peak at ~ 1550 cm- 1 labelled G in figure 10
with a small shoulder at ~ 1350 cm- 1 labelled D. The infrared spectrum of evaporated
a-C measured by Knoll and Geiger (1984) has less strong features (figure 10); it has a
peak at 703 cm-1 and a hump with two features at 1233 and 1465 cm-1.
The Raman spectrum is interpreted as providing strong evidence in favour of
graphite bonding. The 1550cm-1 G peak is close to the mode in graphite and well
332
J. R o b e r t s o n
i
I
I
~
Infrared
absorption
]
I
I
~
F
r
[
I
J
I
I
]
I
I
I
I
~
Raman scattering
~--~-~
Theory
g
J~D
CH20
~ ~
"6
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~G
- __.~'-,
C 340
3 ~'~ 356
2
~
ca.
C 519
diamond
0
~
i
500
r i i i i
1000
1500
Wave number (cm-I)
2000
Figure 10. Infrared absorption (Knoll and Geiger 1984),and Raman scattering intensity (Wada
et al. 1980) of evaporated a-C, compared to the theoretical phonon density of states of
graphite, diamond and four random network models (Beeman et al. 1984).
above any mode frequency possible in an sp3-bonded lattice. The lack of features at the
diamond mode frequency (1332 cm-i) cannot itself be used as evidence against s p 3
bonding because diamond has a much lower Raman scattering efficiency (Wada et al.
1980). The shoulder and peak D around 1350 cm- 1 are not attributed to sp 3 bonds, but
to a disorder mode of graphite microcrystals, as we now explain.
Figure 11 compares the Raman spectra of a-C and a-C: H with those of other
graphitic carbons. The spectrum of highly.oriented pyrolytic graphite (h.o.p.g.) exhibits
just the single Ezg mode of the ideal graphite lattice. The microcrystalline graphite
sample shows an additional mode at 1355cm-1, which is also found in the glassy
carbon and coal samples. The 1355cm -1 mode is also found in graphite made
microcrystalline by irradiation (Elman et al. 1981). This 'D' mode is associated with
disorder in the graphite lattice. 1355 cm- 1 corresponds to a peak in the phonon DOS of
graphite due to the Alg mode at the K point of the Brillouin zone. The eigenvector of the
Alg mode for a single graphite layer is compared in figure 12 with that of the E2g
1550 cm- 1 mode. The Alg mode is inactive for an infinite layer, but develops a strong
Raman activity when k is no longer conserved (Tuinstra and Koenig 1970). The
intensity of the mode in irradiated graphite has been found to vary inversely with the
crystallite diameter L a with the latter measured by X-ray diffraction (Tuinstra and
Koenig 1970, Nemanich and Solin 1979). Indeed, the D mode can dominate the G
mode in small crystallites, as can be seen in figure 11. Thus, the D peak in figures 11 and
12 is evidence not of the development of sp 3 bonding with annealing, but rather the
development of graphitic medium-range order within the sp2-bonded layers.
333
A m o r p h o u s carbon
S
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r~
E
t~
or"
"o
o
iF
0
500
1000
1500
aO00
Wflvenumber (cm-I)
Figure 11. First-order Raman scattering intensities (in arbitrary units) for highly-oriented
pyrolytic graphite (h.o.p.g.),a mierocrystallinegraphite with a crystallite size of La-~25/~,
glassy carbon (all after Solin and Nemanich (1979)), coal (Tsu et al. 1977), a-C:H glow
discharge-depositedat 25°C(Dillon et al. 1984),sputtered and evaporated a-C (afterWada
et al. (1980)).
The sharpness of the Raman spectrum of glassy carbon in figure 11 is the clearest
evidence obtained so far that it possesses much more ordering than does a-C. It is also
interesting that the sp 2 component of a-C : H is evident in producing the D shoulder in
its spectrum.
The Raman spectra of a-C and a-C:H change significantly with annealing
(figure 13). Wada et al. (1980) and Dillon et al. (1984) find that the D mode grows from a
shoulder into a peak and only at their highest temperature, 1000°C, did Dillon et al.
find the peak began to decrease. We interpret these intensity changes thus: the decrease
above 1000°C corresponds to the growth ofmicroerystallite size in samples which have
now become graphitized, the initial increase at 200-800°C presumably corresponds to
an increase in short-range order (e.g. reduction in bond-angle disorder) which allows
the Raman matrix element of the mode to increase. Wada et al. (1980) used the strength
of the D mode as an argument for proposing that a-C consisted of graphitic layers,
L, = 15-20 ~ in diameter. A similar L, was estimated from the optical gap of a-C by
Robertson and O'Reilly (1986), as reviewed in §4. Dillon et al. (1984) also found that
their G peak shifted to higher frequencies as the proportion of sp 3 sites decreased with
carbonization.
These interpretations are generally confirmed by the lattice dynamics calculations
of Beeman et al. (1984) who calculated the phonon DOS for graphite, diamond and for
their four random-network models (figure 10) using the equation of motion method
J. Robertson
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334
Alg
E2g Rctmon m o d e o t 1~
disorder
mode
Figure 12. Atomic eigenvectors of the Raman active E2g zone centre mode and the Alg
'disorder mode', after Tuinstra and Koenig (1970).
25°c
-~ 2oo°c
.,
400°C
g 5oo°c
g 6oooc
(1!
700°C
8 00°C
k i i
1000
i
J I L i
1500
i
i
2000
Wave n u m b e r (cm -I)
Figure 13. Annealing temperature dependence of the Raman scattering intensity of a-C : H,
after Dillon et al. (1984).
(Beeman and Alben 1977) and a simple valence force field (VFF) approximation to the
atomic interactions. The V F F takes the form
AE = ~1k r A r 2 +~korlAO
1 2 2 + ~1k ~ r2l A # .2
(2.5)
For sp 3 sites the two parameter V F F has a bond-stretching force constant k, of
270 N m - 1 and a bond-bending force constant k o of 25 N m - 1 in order to fit the
1332cm-1 R a m a n peak of diamond. For sp 2 sites, there is an additional four-body
force constant k, which opposes the puckering of the layers (Young and Toppel 1965).
The constants k s, k o and k u take the values 363, 36 and 134 N m - 1, respectively for sp 2
sites (Beeman et al. 1984, Nemanich et al. 1977), in order to fit the R a m a n modes of
graphite.
The phonon D O S for crystalline graphite and diamond shown in figure 10 do not
show the expected Van Hove singularities because the equation-of-motion method
causes a smoothing. Nevertheless, the graphite spectrum shows a large peak at 13501550cm -a due to bond-stretching vibrations and two lower peaks at ~ 3 0 0 and
600 cm 1 due to mixed bond-stretching/bending modes, seen in more accurate D O S
(Nicklow et al. 1972), together with a large peak near v = 0 due to rigid-layer
translational modes. The diamond spectra have a large peak at 1100-1280 c m - 1 due to
-
335
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A m o r p h o u s carbon
bond-stretching modes and again two lower peaks at 900 and 600 c m - 1. The phonon
spectrum of the random s p 3 model (C519) shows the expected features (compare Alben
et al. 1975); a slight peak broadening and a merging together of the 600 and 900 cm-1
peaks due to odd-membered rings. The DOS of the C1120 random sp 2 model resemble
a broadened version of the graphite DOS, and its computed Raman intensity shows
that the D peak at 1310 c m - 1 is developing Raman activity. As a whole, the four models
show a gradual transition of features from a graphite-like to a diamond-like spectrum.
In particular, the main Raman peak shifts downwards in rough proportionality to the
percentage of sp 3 sites, being at 1528, 1489, 1374 and 1265 cm-1 in the C1120, C340,
C356 and C519 models, respectively. Beeman et al. (1984) suggested that a DOS
intermediate between the Cl120 and C340 models most resembled the experimental
DOS of Wada et al. (1980). It is possible however, as mentioned in § 2.3, that these two
models under-represent the ordering of their sp 2 sites; an increased ordering would
shift the G peak upwards and increase the intensity of the D peak, as is needed for better
agreement. It would be interesting if the Raman spectrum of a model containing a more
clustered arrangement of sp 2 and sp 3 sites could be calculated, to see at what point the
effects of sp 3 sites become evident.
The Raman spectrum ofa-C : H shown in figure 11 is very similar to that of a-C. The
a - C : H sample was prepared by ion-beam sputtering (Dillon et al. 1984) but its
hydrogen content and sp 2 content were not quoted. If 25°C deposited a-C: H contains
30 per cent sp 2 sites, well clustered as argued later, and if these dominate the Raman
scattering, this would account for the similarity between the two spectra. Also, the
a - C : H loses hydrogen above 400°C and will anneal similarly to a-C above this
temperature.
On the other hand, the infrared spectrum of a-C : H measured by Dischler et al.
(1983 a) shown in figure 14 has more features in the 700-1600 c m - 1 band than the a-C
infrared spectrum of Knoll and Geiger (1984). A large 1300cm -1 peak due to C - C
stretching can now be distinguished from the two higher peaks at 1430 and 1570 cm 1
due to graphitic bond stretching. Dischler et al. (1983 b) assign a 1620 c m - t shoulder to
C = C (olefinic) bond stretching (table 4).
X (/~m)
5
I
4
5
6
8
I
I
I
I
10 12 t4
I
T
E
o
~
N
~)
~
[
• --
254
III
I
4000
5000
I
2000
Wove
number,
1500
t000
700
u (cm -1)
Figure 14. Infrared absorption, its double derivative and band indices for a-C: H, after Dischler
et al. (1983 a).
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336
J. Robertson
The infrared spectrum of a-C : H around 3000 cm 1 provides detailed information
about the C - H bonding configurations. Figure 14 shows the infrared absorption, its
second derivative and the indices of the bands. The assignments in table 4 follow
Dischler et al. (1983a, b) and McKenzie et al. (1983b) who used the standard
frequencies found in molecules (Herzberg 1950) for comparison. The concentrations of
each species were taken from the line intensities, normalized using the molecular
analogue. Dischler et al. (1983b) conclude that monohydride species dominate
dihydride groups, and that hydrogen bonds to sp 3 and olefinic sp / sites with similar
probability (see also Fink et al. 1983). However, m a n y other workers find that hydrogen
bonds preferentially to sp 3 sites (e.g. Nadler et al. 1984). The weakish 3300 c m - 1 line
due to - C H groups is the only firm evidence of any sp 1 hybridized carbon in a-C : H or
a-C.
Dischler et al. (1983b) also followed the changes in hydrogen bonding with
annealing from the deposition temperature of 50°C up to 600°C. The weakly bound
hydrogen and - C H groups quickly disappear. Major hydrogen loss occurs in the
range 300-600°C, and parallels the closing of the optical gap. H y d r o g e n associated
with = C H z and aliphatic - C H groups is lost sooner, by 500°C. The carbon skeleton
also changes from sp 3 and olefinic sp / towards aromatic sp 2 over a similar temperature
range, so the net result is that most hydrogen is bonded to aromatic (graphitic) carbon
at 600°C (see also § 4.6).
2.7. Strength and medium-range order
It is generally recognized that the strength of any carbon based structure must
depend on how its primary (a and re) bonds are arranged in the structure, but
developing a complete theory for this has been difficult.
The strength of glassy carbon and carbon fibres was discussed by Jenkins (1973) and
Jenkins and K a w a m u r a (1976) in terms of the schematic structures shown in figure 3. In
Table 4.
Infrared absorption bands observed in a-C:H and their assignments according to
Dischler et al. (1983 a, b) and McKenzie et al. (1983 b).
Observed frequency
(cm 1)
Predicted frequency
(cm -t)
2'
3300
3045
-3000
3
3'
4
5
6
7
8
9
10
14
15
-2920
2920
2850
1620
1570
1509
1430
1367
1300
880
780-820
3305
3050
3020
3000
2960
2950
2925
2915
2855
1620
1575
Band
1
2
-
-
1355
1330
Assignment
sp I C H
sp 2 CH (aromatic)
sp 2 CH 2 (olefinic)
sp 2 CH (olefinic)
sp 3
sp 2
sp 3
sp a
sp 3
CH
CH
CH
CH
CH
a
2 (olefinic)
2
2
sp 2 C = C (otefinic)
sp2 ~ C (aromatic)
sp 2 C - C (aromatic)
Disorder mode
sp 3 C - C s t r e t c h
sp 2 C - H (aromatic)
sp 2 C - H (aromatic)
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Amorphous carbon
337
graphite, all the strong bonds are arranged in parallel layers, and the weak interlayer
bonds cannot prevent the easy sliding of one layer over another. In contrast, a strong
glassy carbon (one heat treated at 1200-1800°C) consists of an entanglement of
graphitic ribbons, perhaps 40/~ x 40/~ in cross-section,, which acquires its strength
from a degree of cross-linking and the large degree of knotting.
This relatively macroscopic model is entirely appropriate to the more ordered
carbons, but we now describe a model based on coordination numbers for the more
microscopically disordered a-C.
Following Phillips (1979 a, 1981), Thorpe (1983) considered a glass to consist of
'rigid' and 'floppy' regions, and that a phase transition occurs as the mean coordination
increases so that the rigid portion percolates entirely through the network, making it
'rigid' or 'overconstrained' overall.
Following Thorpe, consider a random network of N atoms whose energetics obey a
valence force field ofnearest-neighbour bond-stretching and bond-bending forces. Van
der Waals forces and dihedral angle constraints are second order and are omitted.
When the network coordination n 1 is low, there are many ways in which it can be
deformed at no cost in energy. Deformations are possible in which bond lengths and
bond angles are unchanged. The number of these deformations or zero-frequency
vibration modes is given by the number of degrees of freedom (3N) minus the number of
constraints. There is one constraint associated with each bond and 2 n i - 3 associated
with the angles of each hi-coordinated atom. Thus, the fraction of zero frequency modes
f is given by
=2-5/6n,
where
(2.7)
( nl)/N
is the mean coordination. For n = 2 , corresponding to isolated polymeric chains,
f = 1/3. As n increases by, say, forming crosslinks, f decreases and passes through zero
at
np = 2-4
(2.8)
For n < np we have an 'underconstrained', polymeric glass with rigid and floppy regions
in which the rigid regions do not percolate; for n > np the rigid regions percolate and we
have an 'overconstrained', rigid solid. The idea that rigidity can depend on an average
coordination has been confirmed by calculations (He and Thorpe 1985) and by
experiments on Ge-Se glasses (Bresner et al. 1986).
We now apply the Phillips-Thorpe constraints model to a-C. First we determine
the number of constraints at a n s p 2 site. For a pyramidal site, the number of bond-angle
constraints is three, corresponding to the number of bond angles. For an sp z site
obeying the valence force field (equation (2.5)), there are only two constraints associated
with bond-angle variations within the planes, and another is needed for the out-ofplane mode, #, again giving a total of three. Thus, equations (2.6-2.8) still apply to
carbon, and an array of sp z sites should have
f = -½
(2.9)
or f=O, as negative values of f are not strictly allowed. Thus the array should be
338
J. R o b e r t s o n
overconstrained. However, the graphite structure is a special case, as its ordered layer
structure permits zero-frequency modes. The two rigid-layer translational modes of a
layer are unopposed by valence forces and so have zero frequency. As there are six
modes in all and two atoms per unit cell in a layer, we have
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f = 1/3
(2.10)
and graphite is underconstrained, purely because it has parallel layers. So, comparing
equations (2.9) and (2.10) we can conclude that if an a-C network of sp 2 sites is
disordered so that its cross-linking connects the whole sample, it will be rigid, whilst if it
is highly layered, it will not be rigid.
This constraints theory can also delineate glassy and polymeric a-C : H. Assuming
first that all carbon sites are sp 3, then the mean coordination of a-Cl_xHx is
n=4-3x.
(2.11)
Substituting into equation (2.7) requires that a-C : H be a weak polymer for a hydrogen
content over
Xp= 53 per cent,
(2.12)
but rigid for x < xp. Taking the more accurate result (see later) that the sp 3 : s p 2 ratio is
~2:1,
n = 11/3 - 8x/3
(2.13)
and xp is lowered:
Xp= 47-5 per cent.
(2.14)
This is roughly in accord with experiment. Jenkins e t al. (1972) found that the strength
of glassy carbons dropped to zero for a hydrogen content of ~ 50 per cent.
We now consider why a-C possesses medium-range order, and what limits it.
Sections 2.3 and 2.6 conclude that a-C contains aromatic domains of size L a -~ 15-20/~.
The presence of an optical gap requires that the aromatic character does not percolate
entirely through the sample, thus the sp 2 bonded domains are true islands, in spite of
the fact that >95 per cent of all sites might be sp 2. Robertson and O'Reilly (1986)
suggested that the islands form to prevent the accumulation of strain that would occur
in larger domains, by analogy to the Phillips (1979 b) model of medium range order in
a-Si.
Phillips (1979 b) noted that random networks with an average coordination of over
2.4 were overconstrained. He suggested that such an infinite ~-bonded network would
not be able to relieve strain throughout the network entirely by bond-angle distortions
but would prefer instead to form islands with intrinsically generated dangling bonds
around their surfaces. He then proposed that the majority of these dangling bonds
would reconstruct into weak bonds, leaving islands surrounded by a weakly bonded,
defective surface.
Robertson and O'Reilly (1986) proposed a similar model for a-C, modified to
include the presence of rc and o-bonding. The rc bonding places a very strong constraint
on the network of sp a sites. The interaction energy between adjacent rc orbitals varies as
cos ~b,where q5is the dihedral angle along the bond. The rc bond energy is maximized by
aligning the rc orbitals into planes--noting that the out-of-plane force constant k u is of
similar magnitude to kr (equation (2.5)). Following Phillips (1979 b), the ~ bond energy
is maximized by having almost all the r~ orbitals aligned in islands with ~b~- 0, and then
having a complete break in the rc bonding at the island edges, rather than having a
Amorphous carbon
339
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broad distribution of q~. The re bonding can be broken at the island edges either by the
presence of sp 3 sites or by aligning the sp 2 sites with ~b,-~90 °. The n bonding is then
contained within the island. As ku,~ k r this mechanism will provide significant strain
relief. The model does not predict the number of peripheral sites expected, but it is
interesting that the probable concentration of s p 3 sites in a-C ( ,-~5 per cent) is similar
to the concentration of intrinsic broken bonds in Phillips' (1979 b) model of a-Si
(--~3"5 per cent).
3. Electronic structure of amorphous carbon
This section considers the bonding configurations that are likely to be found in
types of amorphous carbon from the quantum chemical viewpoint, and then describes
their electronic band structures.
3.1. Structural stability in the n electron systems
Let us first consider the coordination number. Group IV elements with four valence
electrons are expected to form four bonds. As a bonds are more stable than n bonds,
one might expect the diamond lattice with its four a bonds per site to be the most stable.
Recently, realistic total-energy calculations for group IV elements in a variety of
crystalline lattices have been carried out by Yin and Cohen (1981, 1982, 1983, 1984). Si
and Ge are found to be typical group IV elements; they are more stable in the diamond
lattice and less stable in the three-fold coordinated graphite structure or the six-fold
simple-cubic lattice. Carbon is atypical It is slightly more stable in the graphite lattice,
and increasingly unstable in lattices of higher coordination. The fundamental reason
for this behaviour is that all first row elements are atypical in having no p-like core
electrons (Yin and Cohen 1981, 1983). This causes their p orbitals to be more compact
and more tightly bound, compared to the s states and compared to Si. This effect
strengthens carbon's s p 2 O" bonds compared to its sp 3 bonds, and favours the graphite
structure.
There is also the possibility of a divalent, sp~-bonded carbon, known as polyyne or
carbyne (figure 15), in a long straight chain of alternating single and triple bonds. Its
bond lengths alternate to prevent the two sets of n bonds from delocalizing along the
chain (Hoffmann 1966, Kertesz et al. 1978). Polyyne is not found'in nature but finite
chains of polyyne are a possible configuration of carbon found in outer space.
Polyynes are less stable than graphite because the presence of a second weak n bond per
site is now just too destabilizing.
We now consider ways of arranging the o- and ~zbonds in amorphous carbons. As
usual, the o- states form the backbone of the random network. This is because their
energetics are local and follow a valence force field model of bond-stretching and bondbending forces (equation (2.5)). Any structure minimizes the total a energy if its bond
lengths and bond angles lie close to their ideal values (bond angles of 109"5° for sp 3 sites
and 120 ° for sp 2 sites etc.).
The rc states have a more subtle, often non-local effect, which cannot in general be
treated so simply. The rc states as a whole form a half-filled band. Therefore, any
structural change which opens up a gap at EF is likely to be stabilizing. In molecular
orbital terminology this opens up a gap between the highest occupied molecular orbital
and the lowest unoccupied molecular orbital, (the H O M O - L U M O gap).
J. Robertson
340
group
(~'~)n
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(=-)n
Figure 15.
name
polyacetylene
Etot/~
1
carbynes
2
benzerie
1.333
[~
napthalene
t -368
<~
azulene
t "336
[~
quinoid
t.240
(ii[~:i)n
polybenzoid
1.403
(ii~ii)n
polyquinoid
1'216
IIQII
gr(lphite
1'616
bonding configurations and their total n energy per site, in units of ft.
We analyse the energetics of n states using a series of approximations. First, we
separate the total Hamiltonian into its o- and n parts
H=H.+H~+H.~.
(3.1)
The H . term is minimized within the valence force field model. We can then decouple
the n states entirely, by noting that
H.~---0
(3.2)
because local symmetry minimizes this interaction (the orbitals lie in perpendicular
planes as shown in figure 1) and secondly because the a and n states tend to be at
different energies (figure 2). Therefore the problem is reduced to minimizing the total n
electron energy per site, which we shall call Eto t.
The second approximation is to treat n electrons as in the Huckel approximation.
This is a one-electron, atomic orbital model, which retains only the nearest-neighbour
interaction,//= V(ppn). F o r convenience we also set the n orbital energy Ep = 0. The
Huckel approximation has the effect of mapping the original structure of sp 2 and s p 3
sites into a sparser network of n states, which only interact if they are adjacent. In
practice, this separates the whole network into a series of independent clusters. The
possible configurations of such clusters are just those of the analogous organic
molecules, whose properties are well known (Streitwieser and Heathcock 1976,
Albright et al. 1985, Salem 1966, Pitzer and Clementi 1959).
Some of the possible configurations of n states are shown in figure 15 with their Eto t
values, and in figure 16 with their electronic spectra. The most stable configurations
Amorphous carbon
341
ethylene
I
I
N-fold rings
N=~
N=6
N=7
N=8
fused rings
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Bill
II
I
II
i
Ir
II I
I II
tl
It
J II
tl
II I
tll
I II IIIIJ
JII II I
I IIfl II I
g r o p h i ~
~3
2
t
0 -I -2 - 3
Energy (,8)
Figure 16. Energy levels of ethylene, planar rings with N=5, 6, 7, 8, three and four aromatic
rings, a single graphite layer and a graphite layer containing two 5- and two 7-fold rings.
follow certain rules, whose basis is described in more detail as follows (Robertson and
O'Reilly (1986)).
(a) The rc orbitals on adjacent sites try to orient themselves in a parallel
arrangement, so as to maximize their interaction. This is because/~ varies with
dihedral angle ~b as:
/~ = V(pp~z)cos ~b.
(3.3)
(b) Clusters with an odd number of z states are unfavoured because they possess at
least one half-filled level near E = 0, and so have not minimized Etot, i.e. they
have no gap at Ev (see for example the five- and seven-fold rings in figure 16).
(c) The isolated six-fold benzoid ring is strongly favoured. It is planar, with equal
bond lengths, i.e. it has D6h symmetry. This configuration is favoured by two
effects, the 120 ° bond angle is optimum for sp 2 hybrids, and the rc bonds are
stabilized by conjugation or, as in this case, aromaticity--the delocalization of
the three zcbonds over all the six available positions increases the rc bond energy
from 6/3 to 8/L giving an increase in stability of/3/3 per site (figure 15).
Conversely, planar octagons are not favoured; they have a double degenerate
level at E = 0 and so suffer from a Jahn-Teller instability causing them to
distort into a tub structure.
(d) A quinoid ring formed by adding further z states at the 1 and 4 positions of a
benzene ring is marginally less stable than a separate benzene ring and ~r bond.
Thus, if the network allows, a quinoid ring is likely to dissociate into its
components. This also applies to a fused row of quinoid rings, the polyquinoid
according to figure 15.
342
d. Robertson
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(e) We can build up benzoid rings in a variety of ways, into rows or into more
compact clusters, eventually giving a graphite layer. Many of the nonlinear
clusters have an odd number of sites and so are unstable by rule (b). Five- and
seven-membered rings are permitted if they occur together as pairs, as this then
gives an even number of sites. However they still have a marginally lower
stability (compare naphthalene versus azulene in figure 15).
(f) Graphitic sheets of fused benzoid rings are the most stable configuration for
large n cluster. Compact clusters are more stable than other combinations such
as rows (figure 17).
The Huckel model used here to predict structures and later to predict optical
spectra is reasonably successful. Its one major failing is that it is a one parameter theory,
so a value offl = - 1-4 eV is used for total energies, but a value offl = - 2.9 eV is used for
band energies. The former value is found by fitting the n bond energies of ethylene and
benzene given by Pauling (1960); the latter value is found by fitting the band structure of
graphite (§ 3.2).
3.2. Tight-binding Hamiltonian
Electronic structure calculations on random networks are most easily performed
using a basis of localized orbitals, as in the tight-binding (molecular orbital) method.
Therefore in this section the band structures of graphite and diamond are used to
calibrate the parameters of a tight-binding Hamiltonian.
Figure 18 shows the three-dimensional band structure of a graphite crystal
calculated by a mixed-basis set pseudopotential method by Holzwarth et al. (1982),
together with the band structure of diamond calculated by a first principles linear
combination of atomic orbitals method by Painter et al. (1971).
The band structure of diamond can be fitted adequately for our purposes using a
spas * basis and including only nearest-neighbour interactions (Vogl et al. 1983). The
interactions are chosen to fit the theoretical band structure together with a number of
experimental band energies, in particular the valence band width of 21 eV, as seen by
photoemission (Pate et al. 1980), the 5"5 eV width of the indirect gap (Clark et al. 1964),
and the position of the F~ and F 15 conduction band states as seen by photoemission
compQct
clusters
1.5
1-4
"
even ~
J °
~dd
5
?oF
1.~
1.2
i
i
i
i
i
i
i
j
I.
i
10
Number of rings(M)
i
i
i
i
i
i
i
100
Figure 17. 7zenergy per atom in units of fl for clusters of 6-fold rings, both compact and linear
rows. For compact clusters, those with even and odd numbers of sites are distinguished.
343
Amorphous carbon
~
10
Evoc.
fo
-Evac
t
0
///"
"....
.;,'1"-
Ld
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-10
graphite
/
-10
-2C
t_20
A
F
E
MT
KHL
A
(a)
F
x
A
(b)
Figure 18. Three-dimensional band-structure of graphite, after Holzwarth et al. (1982), and a
first-principles band structure of diamond (Painter et aL 1971).
(Himpsel et al. 1980). The resulting parameters are given in table 5 and the tight-binding
band structure is given in figure 19 (b).
A tight-binding band structure for a single layer of graphite, shown in figure 19 (a),
was found by fitting to the single-layer band structure of Painter and Ellis (1970), which
is closely related to the full band structure shown in figure 18 (a). In the case of graphite,
special attention was paid to fitting the ~ states (Robertson 1984); the V(pprc)
interaction between the two rc states was allowed to differ from that between two ostates, and a second neighbour V(ppr0 interaction was also included. The resulting
bands are shown in figure 19 (a) and reproduce most of the expected features of the
valence bands, but they under-estimate the width of the a* conduction bands.
It is interesting to compare our parameters for the graphite rc bands with those
found by Tatar and Rabii (1982) for the Slonczewski-Weiss (1958)-McCture (SWC)
model, which is a frequently used parameterization of the ~ bands of graphite near E F,
based on the tight-binding method. Our value of/~ = V(pzrc) = - 2.9 eV compares with
Tatar and Rabii's value of the SWC parameter Yl = - 2"9 eV and a best fit to experiment
of 71 = - 3.16 eV. The second neighbour parameter Vz(pzrt)= 0.2 eV here, compared
Table 5. Energies and interaction parameters in eV for C, H, C-C and C-H bonds. Energies are
referred to EF of graphite, lying 4.7 eV below Evac.
Energy or interaction parameter (eV)
Atom or
bond
C
H
C-C
C-H
E(s)
E(p)
E(s*)
- 5"35 0"45
-2-3
14-0
V(ss) V(sp) V(p~r) V(p~) V(pz~) Vz(Pz~) V(s*p)
-4'55
- 7-5
5-2
8"9
5"45
-1"6
-2-9
0-15
4-5
344
J. Robertson
t0
5
C
-5
Ld
-10
-15
/
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-20
diamond
-25
K
A
F
>2
LL
A
F
(a)
n
X
Z
F
(b)
Figure 19. Tight-binding fitted band structure of (a) a single layer of graphite and (b) diamond,
using the parameters of table 5.
to 72 =0.27eV in Tatar and Rabii (1982) and 72 =0-39eV from fitting experiments.
Higher-order terms in the SWC model are not included in our Hamiltonian.
It has been possible to use a common set of parameters for both sp 2 and sp 3 sites,
although strictly they are slightly different. Indeed, using the vacuum level, Evac, as a
common reference, it was also possible to use a common set of orbital energies, E(s),
E(p) and E(s*). In graphite, Ev, c lies 4-7 eV above E F (Willis et al. 1974) while in diamond
Evac lies at ~ 7"0 eV above the valence-band maximum (Pate et al. 1981, Himpsel et al.
1979).
The C - H interactions are parameterized on the energy levels of the methane
molecule. This is a saturated molecule with four strong C - H bonds. In fact the large
~ a * splitting causes the ~* states to be above E .... which in turn is ~ 13 eV above the
highest occupied level (Herzberg 1966, Cavell et al. 1973). Thus the first optical
transitions are to Rydberg states rather than a* states. As recent calculations use
extended basis sets and concentrate on these Rydberg states, we instead fit our
parameters (table 5) to the older minimal basis set results of Nesbet (1960).
3.3. Results for graphite, diamond and the random network models
Figure 19 shows the tight-binding bands of graphite and diamond, and figure 20
shows the associated density of states (DOS). The band structure of diamond consists of
four valence bands, followed by an indirect F ~ A gap, and the conduction bands. The
upper two valence bands are p-like and the lower two have mixed s p character.
The band structure of graphite consists of a and ~ states. The dispersion of the
occupied n states is consistent with that found by angle-resolved photoemission
(Eberhardt et al. 1980) and the unoccupied ~* states are consistent with the inverse
photoemission spectra of Fauster et al. (1983). The widths of the a and a* bands are
consistent with those found by photoemission (Willis et al. 1974, McFreely et al. 1974).
The band energies are referenced both to EF of graphite and to the vacuum level.
It is instructive to consider the analytic form of the 7cband energies for a single layer,
which is:
E = + flA,
(3.4)
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Amorphous carbon
345
'3
-z5-zo-15-to-5
o 5
Energy (eV)
Figure 20.
1o 15
Local density of states for single layer of graphite, diamond, and of the four random
network models of Beeman et al. (1984).
with
A=lexp{i(2kx+ky)a/3}+exp{i(-kx+ky)a/3}+exp{i(-kx-2ky)a/3}],
(3.5)
where a is the cell dimension, and kx, k r the wave-vectors of the x, y axes of the
hexagonal unit cell. At F, where k~ = ky = 0,
E = + 3//.
(3.6)
At the K point where k:, = k r = 4rc/(3a) we have
E=0.
(3.7)
At the M point where k~ = 2n/a and k r = 0 we have
= +/~.
(3.8)
Thus, we find that the overall g/rt* b a n d w i d t h is 6/3 and that this is symmetrically
disposed a b o u t EF until a second-neighbour interaction is included. Secondly, EF
occurs at the K point in the two-dimensional b a n d structure, where the rc and n* b a n d s
touch. This causes a single layer of graphite to act as a zero b a n d gap semiconductor.
The interlayer interactions cause a 0-04 eV b a n d overlap in the three-dimensional b a n d
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346
J. R o b e r t s o n
structure and make graphite act like a metal. Finally, the two peaks in the ~ band D O S
(figure 20) arise from the large area of the zone around the M point and lie at 4-ft.
The key feature of the electronic structure of a-C and a-C: H is that it possesses an
optical gap, 0-4-0.7 eV in the case o f a - C and usually ~ 1.2-2.5 eV in the case o f a - C : H.
This is to be contrasted with the slight band overlap in graphite and glassy carbon and
the 5.5 eV band gap of diamond. The presence of a gap places strong constraints on
possible local bonding configurations.
Beeman et al. (1984) constructed four r a n d o m network models of a m o r p h o u s
carbon containing various fractions of sp z and sp 3 sites. O'Reilly et al. (1985) calculated
their electronic D O S (figure 20) using the tight-binding parameters of table 5 and the
recursion method (Haydock 1980). It is immediately apparent that there is no gap at E F
in those random networks containing n states. Their disorder has filled in the strong dip
at E F that is present in the n D O S of graphite. It is particularly interesting that the D O S
at E v, N(EF) , is largest in the network containing 50 per cent sp 3 sites, C356, and this
immediately refutes one notion concerning the electronic structure of a-C, that the
mere presence of sp 3 sites is sufficient to open up a gap. Clearly, the existence of a gap
requires that the sp z sites be spatially correlated in a different manner to that found in
these networks.
It was noted in § 3.1 that a network ofsp z and sp 3 sites could be broken down into a
series of independent ~ orbital clusters for the purpose of studying the gap. It was also
noted that structures with a gap were energetically favoured because the existence of a
gap usually lowers the total rc electron energy. A series of rules were given for
minimizing the rc energy; these rules also tend to maximize the gap.
The most probable cause of the large D O S at E F in the C1120/280 100 per cent sp 2
model is the high proportion of five- and seven-fold rings. To emphasize the effect of
ring statistics, we have calculated the s-band D O S of the C280 model, a graphite layer,
and of a group of two five- and two seven-fold rings embedded in a graphite layer, as
shown in figure 21. If the dihedral angle dependence of the zc interaction between
orbitals is omitted, it is equivalent to replacing the ~ orbitals with s orbitals, whose
interactions are isotropic. The s band D O S now contains only the effects of topological
(b)
g
tm
3
2
I
0
-I
-2
-~
Energy (/9)
Figure 21. S band density of states for (a) the C280 model, (b) a pair of 5- and 7-fold rings
embedded in a graphite layer and (c) a single layer of graphite.
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Amorphous carbon
347
disorder. Comparing figure 20 (c) and 21 (a), it is clear that the N(EF) is similar. A similar
but smaller effect is seen for the five- and seven-fold ring groups in figure 20(b). This
shows that the finite N(Ev) of the C280 model is principally due to its topological
disorder. Similar arguments apply to the density of states of the C340 network. In the
case of the C356 network, various reasons may have caused its high value of N(Ev), but
the peak in N(E) at E v suggests that it is principally due to the presence of clusters of
odd numbers of n orbitals, which automatically produce states at Ev, as in figure 16. We
therefore conclude from these electronic structure results that a-C possesses a greater
degree of ordering of its sp 2 sites than in the Beeman models. A similar conclusion
followed from the discussion of the scattering intensity and the Raman scattering.
The behaviour of the a states in figure 20 is much more conventional. The a a* gap
is maintained at 5 eV or more. Disorder in the C519 (Polk) model causes a smoothing of
features and a slight steepening of the leading edge of the valence band, while its oddmembered tings cause a merging of the two s band peaks around - 20 eV, just as in the
case of a-Si (Thorpe et al. 1973, Joannopoulos and Cohen 1973).
All the bands of graphite and diamond discussed so far are derived from the
interaction of 2s and 2p valence orbitals and are two-dimensional in character.
However, a new conduction band of F + symmetry had recently been discovered in
graphite (Posternak et al. 1983, Holzwarth et al. 1982) and is labelled s* in figure 19. It is
derived from carbon 3s orbitals and its charge density is concentrated between the
layers. It is unusual in that it lies at such a low energy, ~ 2 eV above Ev, where it has
recently been detected by inverse photoemission (Fauster et al. 1983). Although this
band cannot be accurately included in our tight-binding calculations, it would be
interesting to follow its dependence on disorder experimentally.
3.4. n Bonded clusters
We now consider the electronic DOS and the size of the gap in some typical nbonded clusters. The n states determine the gap in these systems, so, as in § 3.1, we
neglect the a states and use the Huckel approximation with/~ = - 2 . 9 eV.
An interesting general property of aromatic and olefinie clusters is that their energy
spectrum is symmetric about E = 0 (in the first neighbour approximation). Therefore,
clusters with an odd number of sites must have one state at E = 0. Such clusters produce
defects, as discussed in § 5, The bulk bands are therefore due to clusters with an even
number of sites. The symmetry about E = 0 implies that we expect the band edges of any
amorphous carbon to be roughly symmetric about midgap, whatever its precise
structure.
The levels of various small n-bonded clusters are shown in figure 16. The levels of an
isolated double bond lie at _+]/~1,so its gap is 2mile.The levels of an isolated benzoid ring
are seen to lie at _+2]fll and _+I//6,so its gap is also 2 B. The gap of a quinoid ring is 0.61 ibm,
much less than that of benzene.
The DOS of finite layers of fused benzene rings is shown in figure 22, as a function of
the number of rings, M. We note that already by M = 18 the DOS resembles that of a
graphite layer ( M ~ ~), and by M = 43 it is also quite smooth. Also of interest is the high
density of states at _+0.4/~ in the finite clusters. These shoulders are still present in rather
large clusters, but not in graphite. The shoulder in the occupied states is responsible for
the lower Etot of finite clusters.
The variation of the gap itself for finite clusters of fused six-fold rings is shown in
figure 23. As in the discussion of total energies we distinguish between compact and
J. Robertson
348
Number of rinI
I
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1
:5
Figure 22.
2
1
1
0 -1
Energy(/~)
-2
-5
band density of states for compact clusters of fused 6-fold rings of increasing size.
non-compact clusters. It is found that the gap for compact clusters declines rather
unevenly (because of the changing symmetry) but eventually follows the trend.
Eg= 2lfllM -°'5.
(3.9)
Less compact clusters have smaller gaps. Figure 23 also shows the variation of Eg for
the case of a linear row of fused rings. Now it is found that Eg ultimately decreases quite
rapidly according to
Eg~M 2
(3.10)
at large M.
The infrared spectra (Dischler et al. 1983 b) suggest that olefinic (ethylene-like)
double bonds are present in a-C: H. If such double bonds are arranged in chains, as in
polyacetylene, and if their rc orbitals are aligned parallel, then the n eigenvalues are
given by
E = 2fl cos
n --- 1,..., N
(3.11)
and for N even, the two closest states to E F lie at E'-" +__flrc/N for large N, giving,
Eg',, 2fln/N
(3.12)
(to be compared with Eg = 2fl for N = 1). Dihedral angle disorder will reduce this gap. In
the unlikely event that such a polymeric chain with N > 2 0 is present, a Peierls
distortion will occur and increase the gap by causing a bond alternation, as in
polyacetylene (Salem 1966).
Amorphous carbon
349
10
o.o,
o-c~
i
1-0
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o.~
0.t
0"01
10
100
Number of rings {M)
Figure 23. Minimum gap for clusters of fused 6-fold rings in units of I/~land eV, for/3 = - 2"9 eV.
Inset shows probable cluster sizes found in a-C and a-C:H.
This simple nearest-neighbour Huckel theory slightly over-estimates the energies of
optical transitions in n systems. The principle peak in the optical spectrum of graphite
corresponds to zorn* transitions at the Q point in the two-dimensional Brillouin zone
(corresponding to the M L direction in three dimensions). The transition energy is
21fl[= 5"8 eV in our Huckel model compared to the experimental value of 4.5 eV (Doni
and Pastori-Parravicini 1969). Similarly, the transition energies in simple unsaturated
molecules are also slightly overestimated. For example, the optical gaps of ethylene
CH 2 = CH 2 and benzene are both 21//I in Huckel theory (figure 16). Of course, optical
transitions in molecules differ from those in solids because of the greater importance of
spin; triplet excited states lying below singlet excited states. Thus, in ethylene, the lowest
triplet transition occurs at 4.2 eV and the singlet at 7-6 eV (Herzberg 1966, Bender et al.
1972) and in benzene they lie at 3.9 and 4.9 eV respectively (Hay and Shavitt 1973, 1974),
so their average lies somewhat below our Huckel estimate (5.8 eV).
3.5. Hydrogen configurations
The local hydrogen electronic DOS of various hydrogen configurations calculated
by tight-binding are shown in figure 24; an s p 3 f - H bond in a diamond lattice (formed
by interposing two hydrogens along a C-C bond), an sp2C-H bond in the graphite
lattice (formed by interposing two hydrogens along a C-C bond within the plane), an
sp3C-H bond in the graphite lattice (formed by placing two hydrogen atoms, one
above and one below, two adjacent carbon sites in a graphite lattice, thereby saturating
a C-C bond), sp3C-H bonds in a layer structure (analogous to the A7 layer structure of
c-As, but with 109.5 ° bond angles) and finally a trans polyethylene chain (CH2),. It is
clear that the hydrogen-like states lie at least 2 eV away from E v and therefore well
away from the gap. The H-like states form rather broad band resonances in a-C : H,
much broader than the H states in a-Si:H (Ching et al. 1980).
J. Robertson
350
[
i
i
i
I
i
I
--=C-H in diamond
= C - H in graphite
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4~
- - C - H in graphite
s
(CH) n layer
-25 -20
-t5 -10 -5
0
Energy (eV)
5
10
15
Figure 24. Calculated local density of states at hydrogen sites, for a tetrahedral - C-H site in a
diamond lattice, a planar = C-H site in a graphite lattice, a tetrahedral - C - H site in a
graphite lattice, a (CH)~ layer of A7 symmetry and a ( C H 2 ) o ~ chain.
3.6. Mobility edges
This section considers some general aspects of localization in amorphous carbon.
The mobility edge is an energy which separates localized and extended states.
Considering a single band, if the disorder is strong, the whole band can be localized. For
weaker disorder, the centre of the band consists of extended states and only the band
edge states are localized. Mott developed a convenient rule of thumb that mobility
edges occur at energies where N(E) is roughly one third of its free electron value (Mott
and Davis 1979). Experiments on a-Si:H confirm this estimate (Jackson et al. 1985).
In amorphous carbon, both valence and conduction band edges are due to ~ states.
The ~ band DOS in a-C is clearly high enough to support extended states on this
criterion. However, in a-C : H, as the proportion of sp 2 sites falls, the 7r DOS falls and
the complete ~ and ~* bands will become localized. This transition should be
experimentally accessible in high H-content films.
The second unique aspect of amorphous carbon is its medium range order. Mott
and Davis (1979) stress that disorder is short-ranged in the usual amorphous
semiconductor and that mobility edges are determined by quantum-mechanical
tunnelling rather than such classical concepts as percolation. In a-C and particularly in
a-C: H, the medium-range order is of greater importance than the short-range order in
determining the band gap. We shall describe in detail in § 43 that a-C and a - C : H
contain a series of ~-bonded clusters of varying size and consequently of varying gap
width. Therefore, an electron near a band edge may be forced to tunnel to a distantneighbour cluster to find an allowed state. Thus, when the scale of medium-range order
is large, the mobility edges will depend on the smallest sized, largest gap clusters and
there will be a long tail of localized states due to states from the larger clusters.
351
A m o r p h o u s carbon
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4. Electronic structure: comparison with experiment
4.1. P h o t o e m i s s i o n spectra
The valence band DOS is measured by photoemission. Figure 25 compares the
spectra of various forms of carbon as measured by X-ray photoemission (XPS) by
McFreely et aL (1974), and figure 26 shows the 120 eV photon photoemission spectra
(labelled UPS) of Wcsner et al. (1983). Similar UPS results were also found by Oelhafen
I/1
+6
"5
¢--
n ~ G r a p h i ,
30
20
10
eV
0
Figure 25. X-ta+~ pholoemission spectra of diamond, graphite and glassy carbon, after
McFrccly et al. (1974).
UPS
~
~
XANES
+r+t0
O.,X-
S
77-~
-25 -20 -t5 -t0 -5 0
Energy(eV)
IT '¢
5
t0
15
Figure 26. 120eV photoemission spectra, 'UPS', and XANES spectra of different forms of
carbon, after Wesner et al. (1983) and Fink et al. (t983) respectively.
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352
J. Robertson
et al. (1984). The differences in peak intensities between the XPS and UPS spectra are
due to the strong photon-energy dependence of the matrix elements.
There are three features of note. Firstly that the zc states are apparent as a shoulder
on the leading edge of the graphite spectrum as shown in figure 26. The rt states are not
apparent in the 25C deposited a-C : H sample, even though N M R proves that s p 2 sites
are present in such samples. The rc states only appear after annealing to 350°C and over,
when hydrogen has been expelled and the sample is in the process of changing into a-C.
A second point is that hydrogen-related features are not seen in the UPS spectra.
This is mainly due to the very low photoemission matrix element of H relative to C, and
also to the broadness of the hydrogen-like features shown, in figure 24.
Thirdly, the s band at ~ - 1 6 eV has two peaks in the case of graphite, diamond,
glassy carbon (figure 25) and a-C (figure 26) but only one peak for a-C : H (figure 26).
This two-peak feature is well known to be washed out by the presence of oddmembered rings in the random network (Joannopoulos and Cohen 1973, Thorpe et al.
1973). An sp 3 network allows odd-membered rings, and consequently no splitting is
found for a-C : H which is largely sp 3. An sp 2 network strongly favours six-fold rings as
discussed in § 3.1, and consequently there is a splitting for glassy carbon and a-C.
4.2. X - r a y near-edge spectra
The XANES show the first ,~ 10 eV of oscillations in the excitation spectra of the
carbon ls core electrons. Figure 26 shows the XANES spectra of various carbons, as
measured by Fink et al. (1983). Similar spectra have been found by Wesner et al. (1983),
Batson and Craven (1979) and Denley et al. (1980). The XANES intensity is
proportional to the conduction band DOS, but distorted by a weighting factor and
shifted in energy due to the interaction of the core hole and excited electron (Mele and
Risko 1979).
The XANES spectra are particularly useful in the case of amorphous carbon
because they unequivocably indicate the presence of sp 2 sites. The s~Tc* transitions
produce a peak at ~ 285 eV, below the onset of s ~ o * transitions at ~289 eV. This
effect is also apparent in unsaturated molecules (Eberhardt et al. 1976). Thus, even
small concentrations ofsp 2 sites can be identified in a-C: H, as here, or in a-SixC 1_x : H
(McKenzie et al. 1986). Unfortunately the matrix element effects prevent the use of
XANES to give a quantitative estimate of the s p 2 concentration. (NMR and wide-band
optical spectra are best for this.)
The XANES technique can be used to probe for any structural anisotropy in a-C
and a - C : H on an atomic scale; the s--,~r* transition is polarization dependent and its
intensity would vary with angle of incidence if anisotropy was present, as it does in
graphite (Rosenberg et al. 1986).
4.3. The optical absorption edge
The optical spectra provide some of the most valuable data on the structure of
amorphous carbon. For instance, we shall find that the energy loss spectrum can be
used to evaluate the ratio of rc to rr electrons, and thus the ratio of sp 2 to sp 3 sites, while
the optical absorption edge gives information on the degree of clustering of the sp 2
sites--the band edges are g-like in the both a-C and a-C:H.
The complex dielectric function is given by
e = e 1 q-ie 2.
(4.1)
Amorphous carbon
353
e 1 and e2 are related by the Kramers-Kronig relationship
~l(E) 1 + l (c° __g2(E') dE'.
rc J o E ' - - E
(4.2)
e2 is given by
e2(E) = [(2rce2)Z/NA]R2(E)J(E),
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where e is the electronic charge, R(E) the position matrix element between valence and
conduction band states separated by energy e, NA the atomic density and J(E) the joint
density of valence and conduction band states, given by
J(E) =
Nv(E')Nc(E' + E) dE'.
(4.3)
o
The position matrix element R is related to the dipole matrix element P by (Jackson
et al. 1985)
P(E) = [mE/h 2] 2R(E).
(4.4)
The optical absorption coefficient 7 is related to e2 by
t; 2 "~-
Cmoch/ E,
(4.5)
where n o is the refractive index and c is the speed of light.
The optical absorption around the absorption edge has been measured by many
workers, by Adkins et al. (1970), Jungk and Lange (1972), Grigorovichi et al. (1972),
Arakawa et al. (1977) and Hauser (1977) for evaporated a-C, by Savvides (1985, 1986)
and Zelez (1983) for magnetron-sputtered a-C, and by Anderson (1977), Jones and
Stewart (1982), McKenzie et al. (1983b, c) Bubenzer et al. (1983), Dischler et al.
(1983 a,b) Weissmantel (1979) and Smith (1984) for a-C: H. Disorder causes the
absorption edge to be broad. In many amorphous semiconductors the edge can be
divided into two regions, above and below a - 1 0 4 cm-1. In the upper region, with
transitions between extended states the energy dependence of ~ often gives a linear
dependence on the 'Tauc plot' (Tauc 1973, Mort and Davis 1979), written as
(c~E)1/2 = B1/2(E _ ET).
(4.6)
This linearity requires the DOS to be parabolic at each band edge and P(E) be energy
independent, as discussed by Jackson et al. (1985). Thus there are two popular
definitions of the optical gap: the Tauc gap ET defined by the intercept in equation (4.6),
and the Eo4 gap, the energy at which ~ = 104 cm- 1. Absorption below ~ 104 c m - 1 often
declines exponentially with E and is due to localized states.
Figure 27 shows the optical absorption energy for typical samples of a-C and
a-C: H. We find that Eo4-~0.5 eV for a-C and Eo4-~ 1-8 eV for a-C :H. Furthermore
the Tauc slopes B are rather low; B " ~ 3 " 7 x l 0 4 e V - l c m -1 for a-C and
B " 6 x 104 eV- 1 cm- 1 for a-C: H, compared to B ~-,6 x 105 eV- 1 cm- 1 for a-Si : H.
The values of Eg and B for forms of amorphous carbon indicates the ordering of the
sp z sites--the gap itself gives the size of the largest significant cluster and the slope gives
the range of duster sizes. From figure 23, we see that a gap of 0.5 eV needs clusters of
M - 120 rings, equivalent to La = 18 ~: This upper estimate of the cluster size in a-C is
compatible with values L, = 15-20/~ found previously in § 2.5 from Raman scattering.
d. Robertson
354
106
f E105
a--C
k,=
104
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,~ 10 ~
102
0
i
1"0
2-0
Energy (eV)
3"0
4"0
Figure 27. Typical optical absorption edge of a-C (Hauser 1977) and a-C : H (Smith 1984).
The optical gap of a-C : H is remarkably small--not only does a 1-8 eV gap imply
rc~ re* transitions, it also requires clusters of 8-10 rings in compact clusters, or 3 4 rings
in row clusters (figure 23) or eight atom polyene chains. Recalling that only ~ 35 per
cent of carbon sites are sp 2 in a-C : H, this is a substantial degree of clustering.
The broad slope of the a-C and a-C : H absorption edge is due to the wide range of
cluster sizes. Noting that s 2 is proportional to J(E)/E if the matrix element P is
independent of energy, we plot in figure 28 J(E)/E for various finite compact clusters of
I
I
I
I
I
Number of Rings
o
LLI
CO
"6
o
0
1
2
3
4
Energy (,Q)
5
{5
Figure 28. Calculated joint density of states, weighted by the factor 1/E, for aromatic clusters of
various sizes.
Amorphous carbon
355
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increasing size. We see that J(E) rapidly grows towards its maximum, for energies
immediately above the gap. This is because of the shoulders in the one band DOS at
E-~ + 0-4//seen in figure 22. Therefore, once E exceeds the Eg of a particular cluster, it
absorbs strongly. Thus, the broad slopes in figure 27 correspond to the successive
absorption edges of a range of clusters. The Eo4 gap is the gap of the largest clusters
present at a concentration of ~ 10 per cent.
4.4. The wide-band optical spectra
In the wide-band optical spectra, the excitations of a and n electrons show up as two
separate peaks and this can be used to provide a fairly accurate estimate of the relative
concentration of sp 3 and sp z sites. Figure 29 compares e2 for diamond (Roberts and
Walker 1967), graphite (Taft and Phillip 1965, see also Klucker et al. 1974 and
Venghaus 1974) and a - C : H (Fink et al. 1984). The 540°C annealed sample of a - C : H
will largely resemble a-C and so this curve is used to represent a-C, in lieu of any other
data. (Taft and Phillip (1965) did measure the reflectivity of glassy carbon but did not
quote a ~2 curve.)
The c2 curve of graphite (for E±e) shows three features; the singularity at E = 0
expected for a metal, a peak at 4.5 eV due-to n ~ n * transitions along the M L direction
(see figure 16, and Doni and Pastori-Parravicini 1969), and a peak around 15 eV due to
a~o-* transitions. Diamond has a band gap of 5.5 eV (Clark et al. 1964) followed by a
single broad band culminating in the E2 peak at 12.2 eV. Little absorption occurs below
8eV.
The e2 curve of a-C : H shows, as expected, two peaks due to n--+n* transitions
around 5 eV and due to a ~ a * transitions around 13 eV. In annealed a - C : H the 5 eV
peak has grown slightly at the expense of the 13 eV peak. In glassy carbon (Taft and
Phillip 1965), the n and a peaks occur at slightly lowerenergy than in graphite.
8
4
2
0
!
~
d
~e
0~ . ~ . . _ ~ ~
I/,.
a-C:H
oV v
0
--
4o°c
10
20
eV
30
40
Figure 29. Optical constants of diamond (Roberts and Walker 1967), graphite (Taft and Phi]lip
1965) and a-C:H (Fink et aL 1984).
356
J. Robertson
The relative intensities of the n and a peaks are best compared by calculating neff by
the sum rule,
(2n2NAe2h2/m)neff =
Ee2(E) dE,
(4.7)
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1
where neff is the effective number of electrons per atom which have contributed to the
excitations between the two cutoffenergies E1 and E2. neff is plotted for graphite by Taft
and Phillip (1965) and is seen to reach a plateau of one electron at ~ 9 eV and a second
plateau of four electrons by 30 eV. A similar plot of neff for diamond shows only a very
small neff below 8eV and a plateau o f four electrons by 30eV. We may therefore
approximate by stating that all transitions at 0-8 eV are due to n excitations and all
transitions above 8 eV are due to a excitations and apply equation (4.7) accordingly. In
this manner Fink et al. (1983, 1984) calculated the ncff(n) and neff(a) for a variety of
amorphous carbons. Fink et al. (1983) quote an 88 per cent sp 2 concentration, but Fink
et al. (1984) quote a ~ 100 per cent sp 2 concentration for evaporated a-C. Fink et al.
(1983) also found that a-C: H deposited by glow discharge from benzene at 20°C had
~ 32 per cent sp 2 sites. This is consistent with the presence of predominantly sp a
bonding, as seen in the infra-red spectra, while some sp 2 bonding is needed to give the
optical gap, the XANES spectra and the 1355 c m - 1 Raman peak.
There are three possible sources of error in this method. Firstly the n-g* cutoff
energy of 8-9 eV is considerably less than the total n-n* bandwidth of 17 eV and it may
occur in some disordered systems that the g electron oscillator strength is not
exhausted by this energy. Secondly, although diamond does not absorb significantly
below 8 eV, this energy is certainly above the minimum a - a * gap and such transitions
could become significant when k is no longer conserved. Thirdly, the microscopic
density and hence N g is often not known with great reliability. It is therefore preferable
to measure e 2 up to 25 eV to find the ratio of neff(g) and neff(a) as in Fink et al. (1983,
1984), rather than to only know e 2 up to 9 eV and estimate neff(n) by comparison with
graphite, as several authors do.
It is tempting to use the shape of the n peak around 5 eV to identify the possible sp 2
configurations present in a-C : H, as in McKenzie et al. (1983 c), and Fink et al. (1984).
However, we believe that this could be risky. Figure 28 shows that in the Huckel
approximation all sizes of aromatic clusters have a peak in J(E) and hence in ~2 at
~ 21/~1= 5.8 eV, so we do not expect the n peak position to depend strongly on ordering.
Of course, the Huckel theory is only approximate, and experimentally this peak occurs
at 4.5 eV in graphite and at ,,, 5-5 eV in molecular benzene. Fink et al. (1984) noticed
that the rc peak consists of two features, a lower peak at ~ 4 eV and an upper peak at
6"5 eV whose intensity decreased relative to the 4 eV peak as the sample is annealed.
They argue that the upper peak is larger in small benzene-like clusters, and that such
clusters become more prevalent during annealing. This is a valid interpretation, and
requires matrix element effects to emphasise higher energy n transitions in very small
clusters, as is likely.
4.5. Electron energy-loss spectra
A further use of the optical spectra is for calculating the energy loss function:
-- Im (l/e) = ~2/(~12-[- e22),
(4.8)
where Im stands for 'imaginary part of'. Figure 30 shows the loss function for graphite
and diamond, calculated from their optical constants e 1 and e2 (Taft and Phillip 1965,
357
A m o r p h o u s carbon
O
o-C
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0
I...----~
0
10
t
t
20
r
50
T
i
40
Energy (eV)
Figure 30. Energy-loss function of graphite (Taft and Phillip 1965), various amorphous
carbons (Fink et al. 1984), and diamond (Roberts and Walker 1967).
Roberts and Walker 1967). Similar loss functions have been measured directly using the
inelastic electron scattering technique by Liang and Cundy (1969), Klucker et al. (1974)
and Buchner (1977) for graphite and by Egerton and Whelan (1974) for diamond. The
loss function also appears as satellites on the C ls core-level photoemission spectra
(McFreely et al. 1974). The loss functions for a-C and a-C : H were measured by inelastic
electron scattering by Fink et al. (1983, 1984) and McKenzie et al. (1983 c). Loss
functions for a-C were also measured by Wada et al. (1980) and for glassy carbon by
McFreely et al. (1974).
The loss function can show features due to one-electron inter-band transitions and
due to plasma oscillations of the valence electrons. Plasma oscillations produce a peak
in the loss function if the plasmon energy, given by
Ep = [(47ze2h2 /m)N Aneff] 112,
(4.9)
occurs where e 1 is small, as is the case for carbon. Thus, the loss function of carbon
generally consists of two peaks, a lower peak assigned to rc oscillations and an upper
peak due to the oscillations of all the valence electrons (re + tr). Epevaluated for the ideal
neff values is compared with the measured peak positions in table 6. The upper peak Ep
is seen to compare very well with its experimental position, in graphite and diamond.
However, the rc plasmon in graphite occurs at a lower energy than expected
because the rc oscillation is strongly screened by the a electrons-- the unscreened rc
plasmon lies at 12.5 eV, in the middle of the a excitation band where ~1 would be large,
and is forced down to lower energy where el changes sign (Taft and Phillip 1965). The
total plasmon energies shown in table 6 for a-C and a-C : H are consistent with their
atomic densities, but again the ~ plasmon is depressed in energy. Because of this
depression, the neff values calculated from ~2 are preferred to the experimental ~z and
(a + re) plasmon energies as a means of estimating the proportion of sp 2 and sp 3 sites.
358
J. Robertson
Table 6. Positions of the peaks in - I m ( l / e ) of various forms of carbon compared to the
expected plasma frequencies of free electrons (marked theory). Method of measurement:
KK = Kramers Kronig analysis of optical absorption data, EELS = electron-energy-loss
spectroscopy. References: 1 Liang and Cundy (1969) (EELS), 2 Taft and Phillip (1965)
(KK), 3 Egerton and Whelan (1974) (EELS), 4 Phillip and Taft (1964), Roberts and Walker
(1967) (KK), 5 Fink et al. (1983) and 6 Fink et al. (1984).
gp
(eV)
Form of carbon
Ta
(°C)
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Graphite
Diamond
Glassy carbon
a-C
a-C:H
25
600
~
tr + z
7.2
7"2
12-5
----
26-8
25-2
25-1
33-3
30-32
31
21
24-9
20-8-24.0
21
6
7
6
Density
(g cm 3)
2.25
3"5
~1-5
~ 2-0
1"5- t-8
Reference
1
2
theory
3
4
theory
2
5
5
6
Inelastic electron scattering can be used to measure the wave-vector dependence of
Ep (Buchner 1977). A dispersion relation of the form
Ep = E°p + (hEe/2m)k 2
(4.10)
is expected, with c~= 1 for free electrons. A value of ~ = 0.5 is found for the rc plasmon in
graphite but Fink et al. (1983, 1984) find minimal dispersion for both a-C and
unannealed a - C : H . They suggest that this indicates strong localization of the 7z
electrons, but we noted in § 3.6 that the centre of the rc band of a-C must consist of
extended states as N(E) is large there. Thus, the cause of this absence of dispersion is not
understood and deserves further attention.
4.6. H e a t treatment o f a-C : H
The variation of the structure of a-C with heat-treatment can be determined
adequately by diffraction and by R a m a n spectroscopy, as discussed previously in § 2.
Some changes in the structure of a-C: H were apparent in the diffraction and R a m a n
spectra, but the effects in the optical spectra are more dramatic. Figure 31 summarizes
results from various workers, whose samples are broadly similar but not identical.
The annealing of a-C: H above ~ 300°C causes the loss of hydrogen, which can be
followed by measuring the strength of the C - H vibrational bands (Dischler et al.
(1983 b), summarized by Fink et al. (1984)). However, unlike in a-Si: H, a-C : H loses its
hydrogen not just as H 2 molecules, but also as hydrocarbon molecules, as emphasized
by Smith (1984) because there is a ~ 35 per cent decrease in the sample mass. The
difficulty in apportioning gas evolution to hydrogen itself may account for the initial
slight increase in H content in figure 31 at 300°C, seen by Dischler et al. (1983 b). The
loss of polymeric material causes the density of the films to increase. The loss of
hydrogen causes the bonding to become gradually more unsaturated, and the
proportion ofsp 2 rises from ~ 30 per cent towards 75 per cent at 700°C. Note however,
that the fraction of sp a sites is only ~ 70 per cent when the hydrogen evolution is
essentially complete at 600°C and will not approach 100 per cent until above 800°C.
359
Amorphous carbon
8C
04
J
I
i
]
200
I
4-0
o
1
o
Z
~__0.5
o
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~ol.4
2
j
1
0
~E105
L~
"7
[13
o
4
400
I
I
600
To (°C)
Figure 31. Effect of annealing temperature T, on various properties ofa-C: H; percentage ofsp 2
sites (Fink et al. 1983), bonded hydrogen content (Fink et al. 1984), density, optical gap Eg
and Tauc slope B for films deposited at Ta = 250°C, all after Smith (1984). Also shown is the
variation of Eg with deposition temperature.
The increase in the sp 2 site fraction causes a reduction in the gap, Eg, as expected.
The decline is more rapid as a function of deposition temperature than with annealing
temperature. The variation in size of the gap is from Smith (1984). Other workers found
broadly similar results for films deposited under widely differing conditions, although
the absolute value of the gap would alter. Conductivity results discussed in the next
section suggest a similar variation of Eg with T, and Td.
Interestingly, Natarajan et al. (1985) found that the gap of their films, prepared by
the decomposition of methane or propylene, did not close up even at 800°C.
4.7. Ion-beam deposited carbon
As diamond is the hardest solid known, amorphous carbon films have naturally
been studied with a view to using them as a hard, transparent coating material. Sungren
and Hentzell (1986) have recently reviewed various hard coating materials grown from
the vapour phase. In the case of carbon, the hardest films are prepared by some form of
ion-assisted deposition technique, such as sputtering, magnetron sputtering, biased
plasma deposition, ion beam deposition or ion plating. Since the first experiments of
Aisenberg and Chabot (1971) these techniques have been intensively studied, and the
films were initially called 'diamond-like carbon' to emphasize their hardness. However,
when their sp 2 c o n t e n t became apparent, Weissmantel (1979) proposed instead the
term i-C (for ion beam).
Weissmantel (1979, 1982) and Weissmantel et al. (1979, 1980, 1982) prepared
carbon films from benzene using ion plating with a d.c. bias voltage of up to 3 kV. Their
films attained a peak Vickers hardness of ~ 6000 kg m m - 2 (equivalent to ~ 9.4 on the
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360
d. Robertson
Mohs scale (see, for example, Tabor 1961)) at a bias of 1 kV (Weissmantel et al. 1982).
Microcrystallites were sometimes observed in their films. A similar peak hardness was
also achieved by Mori and Namba (1983) at a bias of 400eV and over, with ionized
deposition from methane. The diffraction pattern of their samples also suggests the
presence of microcrystals.
Dischler and Brandt (1985) produced uniformly amorphous films from benzene
using a bias plasma deposition system. These films had a lower hardness of 1250
1650kgmm -2 as measured by the Knoop test, equivalent to 6 7 on the Mohs scale.
This group has, however, thoroughly studied the dependence of the optical gap,
refractive index, infrared absorption, and density on the deposition parameters (bias
voltage, gas pressure and deposition temperature) (Bubenzer et al. 1983, Dischler et al.
1983 a, b, Fink et al. 1983, 1984). The electronic and optical properties of these a-C:H
films were discussed earlier. The problems involved in measuring the hardness of
a-C : H films were discussed by Dischler and Brandt (1985) and Sundgren and Hentzell
(1986).
Films from various other laboratories quote hardnesses in the range equivalent to
5-8 on the Mohs scale (Zarowin et al. 1986, Natarajan et al. 1985, Angus et al. 1984,
Vora and Moravec 1981).
An inverse correlation between hydrogen content and hardness over extreme
changes in deposition parameters has been remarked on by Nadler et al. (1983, 1984)
and Kaplan et al. (1985). As hydrogenation tends to increase the fraction of sp 3 sites;
this cannot be attributed to 'diamond-like' behaviour. We attribute the decrease in
hardness with hydrogenation to the growth of soft polymeric regions containing
= CH2 units being faster than the growth of crosslinking associated with the higher
carbon coordination. This result is in accord with the arguments of § 2.7.
Other workers have produced minimally hydrogenated carbon films by ionassisted methods. Zelez (1983) prepared a-C:H films by a hybrid gias sputtering
plasma deposition technique from butane. The hydrogen content of these films was
very low, < 1 per cent. The density of the films was remarkably high, 2.8 g c m - 3, which
is only possible if they contain substantial sp 3 bonding (c.f. table 1). Zelez (1983) also
measured a Tauc optical gap of Eg= 3"05 eV, which is the highest known for an
essentially unhydrogenated a-C. Clearly this is an important result and implies that the
remaining sp 2 sites are much less clustered than in evaporated or sputtered a-C.
Savvides (1985, 1986) has also produced a-C by an ion assisted technique-magnetron sputtering, but this time the gap is more like than in conventional a-C
(0.4q).8 eV). However, he used the optical absorption spectrum up to 9 eV to calculate
neff(~), as in § 4.4, to show that the films now had a sizeable fraction of sp 3 sites, of order
40-75 per cent depending on deposition conditions. Following our analysis in §§ 3.4
and 4.3, we must therefore assume the remaining sp 2 sites still to be arranged in clusters
as large as in evaporated a-C. The results of Zelez and Savvides demonstrate that ionassisted deposition greatly increases the fraction of sp 3 sites in a-C. Equivalent results
are not available for a-C: H, where plasma deposited a-C:H already contains
considerable sp 3 sites.
How does ion-assisted deposition increase the fraction of sp 3 sites? In evaporation,
the incident particles have thermal energies. In plasma deposition, the major species in
the plasma are believed to be radicals (Kampas and Griffith 1980, Longeway et al.
1984). In the plasma deposition of electronic quality a-Si: H, deposition was shown to
be rate-limited by a surface-plasma radical reaction--in effect a chemical vapour
deposition regime (Tsai et al. 1986). Higher plasma power levels produce poorer a-Si : H
Amorphous carbon
361
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with a columnar morphology and m a n y defects. A physical deposition regime now
exists with radicals arriving directly from the bulk plasma at the substrate. The
introduction of a d.c. bias voltage of order 1 kV, or the use of magnetron sputtering
increases the importance of ionic species in the plasma. The bias voltage also gives the
positive ions incident on the substrate a higher kinetic energy. According to the model
of Weissmantel (1982), this higher kinetic energy is crucial for a-C: H. The incident
particles cause an intense local thermal and pressure,spike. Recalling that diamond is
stable in the high temperature/high pressure zone of carbon's phase diagram,
Weissmantel (1982) asserts that the spikes favour the formation of sp 3 sites. This
results in a harder film because increasing the fraction of sp 3 sites at a constant
hydrogen fraction increases the cross-linking of the network.
5.
Localized states
5.1. Origins of localized states
The review has so far emphasized the presence of a gap in a m o r p h o u s carbon and
how it arose. We now consider the nature of states within the gap, and the electronic
properties which they control--the conductivity, electron spin resonance, sub-band
gap luminescence and doping.
Electronic states in a typical a m o r p h o u s semiconductor are usually classified
according to their type; extended states, the tail states and the deep gap states (Mott and
Davis 1979, Robertson 1983). The extended states are the states well within the bands.
The gap of an a m o r p h o u s semiconductor should strictly be called a pseudogap or
mobility gap. Localized states cannot conduct electricity at 0 K. Tails of localized states
exist above the valence band mobility edge E v and below the conduction band mobility
edge E c. The width of the optical absorption edges (figure 27) suggests that amorphous
carbon has broad tails. In a-C itself, the pseudogap is quite narrow (0.4-0-7 eV) and the
two tails will overlap at the gap centre. In wider gap amorphous semiconductors,
however, a set of deep gap states can usually be distinguished. These are usually more
localized than the tail states and are usually associated with 'defect' sites whose bonding
differs from that in the bulk. The best known example are the midgap states of the
'dangling bond' defect in a-Si:H, which is a trivalent Si site. One expects a - C : H to
belong to this latter class, because of its wider gap.
The chemistry of a m o r p h o u s carbon immediately suggests that its defects will be
more complicated that those of a m o r p h o u s silicon. The dangling bond in a-Si is formed
by breaking a a bond; in amorphous carbon we can form a defect by breaking a a bond
or a n bond. Naturally, as n bonding is weaker, we expect n defects to have a lower
creation energy and to predominate. A second difference is that n defects can delocalize
very easily if associated with a conjugated n electron system.
F r o m the Huckel model of § 3, we can define a defect as a state with energy near
E = 0. We noted there that the ~ electron spectrum tends to be symmetric about E = 0.
Hence any cluster with an odd number o f n orbitals will produce a state near E = 0, and
it will be half-filled--i.e, it will have a paramagnetic defect level. The two simplest such
defects are the three-site chain, analogous to the allyl radical C H 2 . C H . C H 2, and the
cluster of three aromatic rings, analogous to the perinaphthene radical:
(5.1)
362
J. R o b e r t s o n
As noted in § 3.1, the presence of half-filled n states at E-- 0 makes a cluster less stable.
Therefore, two identical defect clusters with an odd number of sites (N) could, in
principle, disproportionate into two even-membered clusters, with more and less sites
respectively. It is therefore possible, in principle, to define a defect creation energy E d as
the energy evolved in this process,
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Ed =(EN-- ff,N)N.
(5.2)
Here EN is the total n energy per site of the defect cluster and EN the total n energy per
site of the products; the total n energy a defect cluster of its size would have had if it had
an even number of sites. This quantity can be calculated for clusters of fused aromatic
rings from figure 17. Etot for odd clusters lies consistently below the trend line of Eta t for
even clusters; so defining E N from the trend line, figure 32 shows E a plotted in units offl
against N. Figure 32 also includes the variation of Ea for linear conjugated chains. Ea is
seen to decrease rapidly with N for linear chains but it decreases only slowly and rather
unevenly for ring clusters.
Figure 32 illustrates two important differences between n hefects and a defects.
Firstly, there is no single E d for n defects, only a spectrum of values which could
decrease down to Ed = 0, if sufficiently large clusters were present. In contrast the
dangling bond has a reasonably well defined energy, about half the o- bond energy. The
second difference is that the average Ed of n defects is quite low, ~0-4 eV for the
aromatic centres. This compares with Ed ~ 1"8 eV for a dangling bond in a-C, estimated
from the a bond energy. This islargely due to the relative weakness of n bonding. This
lower creation energy causes defect concentrations to be larger than in a-Si:H.
The effective correlation energy U is an important electronic parameter of defects. It
is defined as the extra energy needed to place a second electron in the defect. Thus, the
singly occupied defect has energy E and the doubly occupied defect has energy 2E + U.
The usual case of Coulombic electron-electron repulsion corresponds to U >0.
However, if the electronic levels couple strongly to lattice modes, it is possible for this
coupling to create a net negative U. In this case the defect electrons have a net attraction
(Matt and Davis 1979, Robertson 1983). In the case of U >0, all three defect states,
1
1
compact
graphitic
~- o-5
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i
i
f i i i i1
5
i
i
10
" ~
i
i
50
i i i i[
100
Number of sifes,Nlodd)
Figure 32. Calculated dependence of defect creation energy E a plotted against defect cluster
size N, for both linear chains of~ states and compact graphitic clusters. In units of/~ and eV
(//~- - 1-4 e V ) .
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Amorphous carbon
363
empty D +, half-filled D O and filled D - are found as Ev rises through E. In the case
U < 0, raising E v causes D + to change immediately to D - and the paramagnetic D o
configuration is only metastable. The dangling bond in a-Si : H is the best example of
U > 0 (Street 1984), while the valence alternation defects in a-As2Se3 are the best
examples of U < 0.
We expect the U value of 7t defects in amorphous carbon to be small and positive.
Firstly, the rc defect state is quite delocalized; for instance, the E = 0 state of the allyl
radical extends over all three sites, the 'soliton' radical of a long polyacetylene chain
extends over about 20 sites (Su et al. 1979), while the E = 0 state of the aromatic radical
(equation (5.1)) is delocalized over all its external sites. As delocalization reduces the
repulsion between two electrons in the defect state, we expect the coulombic
contribution to U to be small. There is also no strong electron-coupling mechanism to
produce a negative value of U. Thus, a small positive U value is expected.
In a mixed ~r, rc bonded system such as amorphous carbon, the dangling bond must
be defined as an isolated three-fold coordinated carbon site. Its creation energy is
expected to be E a ~ 1.8 eV, as noted earlier. Its electronic character is however expected
to be ~-like. This is because its analogous molecular radical, the methyl radical CH 3, is
found to be p l a n a r rather than pyramidal (Streitwieser and Heathcock 1976), and
consequently the unpaired electron will occupy arc orbital normal to the plane of the
hydrogens. Thus, the 'dangling bond' is expected to have ~ character in a-C : H and not
sp 3 character as in a-Si:H.
The tail states of a m o r p h o u s carbon are expected to be the n states of larger than
average rc clusters. The correlation energy of their states is also expected to be low and
positive. Therefore, in this model, the orbital character of tail states and defect states is
qualitatively the same.
5.2. Conductivity
dependence
of the conductivity a in amorphous
Generally, the temperature
semiconductors varies as
a = a o exp { - (Tn/T)"}.
(5.3)
A regime with n = 1 indicates that conduction is predominantly by thermal activation
to states of energy E=kBTn away from EF, where kB is Boltzmann's constant. If
% > 103 f~-1 c m - 1 this suggests that conduction is by activation to extended states
beyond E~ or E v, while if a 0 < 103f2 -1 cm -1 suggests that conduction occurs by
hopping between localized states on near-neighbour sites (Mott and Davis 1979). If
n < l , this suggests that conduction is occurring by a variable range hopping in
localized states around E F. This occurs if N(EF) is low so that as the temperature falls,
an electron must hop (tunnel) to more distant sites (a distance Ropt away) to find a level
within ~ kBT of its energy, n =¼ is the classic power law for variable range hopping at
E F (Mott and Davis 1979). n>¼ can suggest a rapidly changing N(E) near E v or
dimensionality effects. If the thickness of the film decreases below Root, a twodimensional regime with n = 1/3 is expected. In the three-dimensional T 1/4 regime
16~ 3
T4 = kBN(EF)
(5.4)
364
J. Robertson
and in the two-dimensional T 1/3 regime
8ct2
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T 3 = kBdN(EF) ,
(5.5)
where e-1 is the decay length of the localized state wavefunction and d is the film
thickness. If only a T 1/4 regime is found it is conventional to guess e - 1 _ 10 ~ and
estimate N(EF). If a transition between T 1/4 and T 113 occurs with decreasing d, this is
preferable, as N(EF) and 0~-1 can then be uniquely determined.
The conductivity of a-C has been studied by Adkins et aL (1970), Adkins and
Hamilton (1971), Hamilton et al. (1974), Morgan (1971), Grigorovichi et al. (1972),
McLintock and Orr (1972), Orzeszko et al. (1984) and Hauser (1975, 1977).
Hamilton et al. (1974) found that o-ofa-C obeyed various laws, n = 0.254).5, over an
increasing temperature interval. They also found that a. was anisotropic, being lower
along the film than through it. (Kakinoki et al. (1960b) also found a structural
anisotropy in annealed a-C films.) Hauser (1977) showed that this resulted from the use
of an imperfect evaporation source and the presence of very conductive graphite grains
in their films. Hauser and Patel (1976) created an isotropically conducting a-C by
carbon-ion bombardment of a diamond crystal and then, using this conductivity as a
reference, Hauser (1977) produced sputtered a-C with similar properties. Using films of
varying thickness, Hauser (1977) evaluated
o~- 1 ,.~ 1 2 / ~
(5.6)
and
N ( E F ) ~ 1018 eV- 1 cm-3,
(5.7)
for films sputtered at 95 K and annealed up to 573 K, similar values to those proposed
by other workers, including Hill (1971). Hauser (1977) estimated the mobility gap to be
0-8 eV, slightl) larger than his Eo4 optical gap of ~ 0.6 eV.
The conductivity of a-C depends strongly on heat treatment. Hauser (1975, 1977)
found that annealing to 300 K initially decreased a, but then further annealing at 450 K
and over increased a from 10 3 to 10 ~~ ~cm 1. The latter increase was found to be
largely duc to an increase in N(E~), as the optical gap was largely unchanged (Hauser
1977).
Summarizing, there is a continuous distribution of localized states across the
pseudogap in a-C, their extent is ~12/k and their density is ~ 1 0 1 a e V - l c m -3,
increasing with heat treatment.
The conductivity of a - C : H has been measured by Anderson (1977), Jones and
Stewart (1982), Meyerson and Smith (1980 a) and Orzeszko et al. (1984). All authors
found that the temperature dependence of the conductivity (figure 33) was curved and
did not obey any law particularly well. Anderson (1977) found o- to follow a simple
thermally activated regime at high temperatures:
a = a 0 exp (
AE/kT),
(5.8)
with AE = 1.1 - 1"46eV. Meyerson and Smith (1980 a) found that o- in figure 33 loosely
followed equation (5.8) above ~ 4 0 0 K . The resulting values of a 0 were low,
10-4_10-2 ~-'~-1cm-1, suggesting conduction by hopping in the band tails. This is
consistent with an extensive localization of the band edges (§ 3.6). Additionally,
Meyerson and Smith (1982) measured the thermopower and found undoped a-C : H to
365
Amorphous carbon
i
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i
[
i
I
/
I0-4
10-6
'IE
u 10-8 ~ ~ - - ~ 3 0 0
5~
C"
L
10-lo
2ootI
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o i0-12
10-14
-
~
~15°°c
k t,,
1.5
2-0
I""T~75~ c
2.5
5-0
3"5
IO00/T (K -l)
Figure 33. Temperature dependence of conductivity cr for a-C : H films of different deposition
temperature Ta, from Meyerson and Smith (1980a).
be p-type below 100°C and n-type above 100°C. By comparing AE with the optical gap,
Jones and Stewart (1982) suggested that E v lay just above midgap, giving n-type films.
As the thermopower is low this suggests that transport always occurs close to EF and
that the defect states again form a continuous distribution across the gap.
Figure 33 also shows how the conductivity depends on the deposition temperature.
Deposition (or annealing) at temperatures above 300 K produces an increase in a and a
decline in AE, in line with the decline in the optical gap discussed in § 4.6.
5.3. Electron spin resonance and photo-luminescence
Electron spin resonance (ESR) is used to detect unpaired spins associated with halffilled defect states around E F, or in chemical language, unpaired spins associated with
radical species. Mzorowski (1971, 1980, 1982), Singer and Lewis (1978) and Lewis
(1982) have used ESR to study the electroffic and chemical processes involved in
graphitization. Orzeszko et al. (1984) found a spin density of ~101Scm -3 in
evaporated a-C, which was relatively independent of annealing temperature. The
density is consistent with conductivity data, but its temperature dependence is not.
Jansen et al. (1985) also found a spin density of ~ 10 is cm -3 in ion-beam sputtered a-C.
Orzeszko et al. (1984), Gambino and Thompson (1970), Miller and McKenzie
(1983), Jansen et al. (1985) and Watanabe and Okumura (1985) have studied ESR in
a-C : H. Spin densities of 1016 cm-3 were quoted by Jansen et al. (1985) and 101 s cm- 3
by Gambino and Thompson (1970) and Miller and McKenzie (1983). Figure 34 shows
the dependence of spin density on deposition temperature Td found by Watanabe et al.
(I 982) in a-C : H films deposited from ethylene. The spin density increases at Ta where
the optical gap is found to decrease. Both Miller and McKenzie (1983) and Watanabe
and Okumura (1985) were able to resolve two spin signals in a-C : H, a narrow line with
g = 2.003 and a wider line with g = 2.011. A photo-induced spin density was also found
by Watanabe.
366
J. Robertson
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350
Downloaded by [Duke University Libraries] at 15:26 12 November 2012
(°C)
Figure 34. Dependence of spin density N= ol a-C : H films on deposition temperature Td, from
Watanabe et aL (1982).
Sub-bandgap photo-induced luminescence can give valuable information on the
nature of gap states, for example, their trapping characteristics and their coupling to
lattice modes. Watanabe et al. (1982), Lin and Feldman (1983) and Wagner and
Lautenschlager (1986) have found a very broad (white) luminescence (figure 35)
associated with an excitation energy of 2.5~.7 eV. The luminescence peak decreases in
energy with Td (figure 35), due to the decrease in Eg (which was rather large in those
samples, Eg = 2-3-6 eV). The luminescence is quite efficient and only decreases slowly
with temperature. Thus, a-C:H still luminesces well at room temperature, unlike
a-Si : H.
The correlation between defect creation energy and cluster size (figure 32), and
between cluster size and band gap (figure 17) implies that the defect density in a-C : H
will correlate inversely with the optical gap. Thus a-C:H deposited at higher
temperatures has more gap states because of its smaller gap. Hydrogenation therefore
lowers the defect density not by saturating dangling bonds, as in a-Si:H, but by
reducing cluster sizes and thereby increasing the defect creation energy and decreasing
their thermodynamic probability.
I
b
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Ta (C)
200
.~
\
,20]~
r
I r
°
~
2
=
3
i
1,2
1.6
2"0
Energy(eV)
Figure 35. Photoluminescence spectrum of a-C: H for various deposition temperatures.
Excitation energy is 2-54eV (after Watanabe et al. 1982).
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Amorphous carbon
367
5.4. Doping
Mayerson and Smith (1980b) discovered that a - C : H could be doped by adding
B2H 6 or P H 3 to the hydrocarbon gas stream arriving at the plasma deposition reactor.
Figure 36 shows the typical changes in the room-temperature conductivity activation
energy AERT and compares this with the analogous changes for a-Si : H found by Spear
and Le Comber (1975). It is clear that a substantial change in conductivity has
occurred, increasing the room-temperature conductivity 0"RTfrom ~ 10- a2 ~ - 1 cm 1
to ~ 10- 7 ~ - 1 cm - t for gas phase doping concentrations of 10 per cent. However, it is
also clear from figure 36 that the doping is less efficient than in a-Si : H; a greater dopant
concentration is required to achieve a large reduction in AERT, the conductivity aRT can
only be increased to ~ 10 - 7 f~ - 1 cm - ~ rather than 10 - 2 f~ - a cm - t as in a-Si : H, and
AERT can only be reduced to ~ 0-3 eV compared to ~ 0.18 eV in the best a-Si :H. Jones
and Stewart (1982) also studied doped a-C: H prepared from a variety of hydrocarbons
and found lower doping efficiencies than Meyerson and Smith (1980 b), in particular
they found much smaller changes in AE. This led Jones and Stewart (1982) to wonder if
true substitutional doping was occurring, or whether a change in the defect D O S was
responsible for an upward shift in E F. Meyerson and Smith (1982) confirmed that
substitutional doping was indeed occurring when they found the expected changes in
thermopower; phosphorus doped films becoming n-type and b o r o n doped films
becoming p-type. Interestingly, Jones and Stewart (1982) also found that a-Ci H could
be doped by nitrogen, whereas nitrogen is only able to dope a - S i : H very weakly
(Dunnett et al. 1986).
The mechanism of substitutional doping in amorphous group IV elements has a
number of unique features. Phosphorus, for example, will dope a-Si : H because it enters
a four-fold coordinated site and has a fifth electron which is not bound in bonds to its
neighbours. In crystalline (c-) Si, this fifth electron occupies a shallow d o n o r state and is
easily excited into the conduction band. In a-Si : H, the donor electrons must first fill all
the gap states before E F c a n rise into the band tails (Spear and Le C o m b e r 1975, Spear
1976). The doping efficiency therefore depends on the number of donors and the
number of gap states. A major difference between c-Si and a-Si : H is that in a-Si : H a
I
i
I
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I J l l l l
1"0
(1- Si.: H
Ik
IX
0'8
i1)
o.-C:H
0-6
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klA
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\
0"4
\\k
0-2
i
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I
10 -6
I
0 10-6
i
I
I0 -4
I
]
10- 2
I
[PH 5]/~2H2]
Figure 36. Variation of the apparent room-temperature activation energy of the conductivity,
AERT, with gas-phase dopant concentration for a-C:H and for a-Si:H for comparison,
after Meyerson and Smith (1980b).
d. Robertson
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368
group III or group V atom need not enter a four-fold substitutional site. The absence of
the topological constraints associated with long-range order permits the dopant atom
to exert its normal chemical valence which is three in both cases. Before doping was
discovered, it was believed that this possibility would completely rule out doping. It is
now realized that dopants can form four-fold sites, but with a reduced probability
because they are energetically less favourable. This factor causes the first decrease in
doping efficiency.
The second factor is that doping in a-Si:H is accompanied by the formation of
additional dangling-bond defects. The defect levels are initially half-filled so that they
will accept the electron from a donor and E v will not now rise. Using subscripts to
denote coordination, this reaction can be denoted as an equilibrium between the nondoping po sites and the donor-plus defect configuration (Street 1982):
poce;
+ D-.
(5.9)
If it is assumed that this equilibrium holds during deposition, the law of mass action can
be applied to equation (5.9) giving K 1 = n2/(N - n), where N is the total phosphorus
concentration and n is the concentration of P~- sites. If a doping efficiency I/1 is defined
as the proportion of four-fold sites then
rlt ~-n/N
(5.10)
~-(KdU) 1/2
(5.11)
for N >>n, as is usual. This decline in doping efficiency with dopant concentration was
indeed found in a-Si: H (Street 1982). The magnitude of~h now depends on K~ and the
relative stability of p0, p~- and D - .
A third factor is the number of carriers created by doping. Equation (5.9)
exaggerates the case where all the donor electrons are absorbed by defect states.
Referring to figure 37, we can identify a classical case of compensation. The donor
electron gains energy A if it drops into a defect level. The creation of a new defect to
accommodate this carrier is dependant on whether or not A exceeds the creation
enthalpy of the defect E d. Thus, there is a second equilibrium between free and trapped
carriers:
e-~D-
(5.12)
K 2= [e-]/[D-]
(5.13)
with an equilibrium constant
- exp { - (A - Ea)/(kTo) },
I
--IP--
(5.14)
donor
defect
Ev
Figure 37. Schematic representation of energy levels in a compensating semiconductor.
369
A m o r p h o u s carbon
where both A and E a are positive quantities. If A > E d the creation of defects is
energetically favourable (Robertson 1986). This is the case in a-Si : H where measurements of the carrier concentration (Stutzmann and Street 1986) give
Downloaded by [Duke University Libraries] at 15:26 12 November 2012
K2 ~0"1,
(5.15)
and so E v never moves closer than ~ 0.2 eV to the impurity level or the mobility edges.
To in equation (5.14) is a fictive temperature which characterizes the energy spectrum of
its excited configurations (see Bar-Yam et al. 1986).
This model can be applied to a-C : H. Firstly it should be noted that the presence of
a large proportion of carbon sp 2 sites does not prevent some of the dopants forming
substitutional sites, and in fact K 1 could have a value similar to that in a-Si : H. The
major change is that E a is much lower in a-C : H because of the possibility of forming rc
defects. Figure 33 showed that the mean value of E a ~_0.4 eV and this will substantially
reduce K 2 and thereby the true doping efficiency ~, defined as
q = [e-]/N.
(5:16)
These effects are illustrated by means of a schematic density of states for undoped
and n-type doped a-C : H (figure 38). Figure 38 (a) for undoped a-C : H shows the a and
a* bands, the rc and re* bands and the uniform distribution of gap states across the
pseudogap, with E v lying close to midgap. Figure 38 (b) for doped a - C : H shows the
donor states associated with P~ bound below the o-* band and an increase in the defect
DOS around midgap due to the D - levels, which will tend to merge into the valence
band tail, as they do in a-Si : H. E F has moved abovemidgap but is still well below E cin
the conduction band tail. The clearest evidence of compensation causing defect
creation is a decrease in photoluminescence efficiency with doping level. This is seen in
a-Si:H (Street et al. 1981) but has not been looked for in a - C : H .
6. Conclusions
The diffraction patterns, Raman spectra and conductivities emphasize that glassy
carbon, evaporated amorphous carbon and ion-beam deposited a-C have different
(a) ~ to2O
'~
EF
10t8
1o 2
(b)
o
.~"
Ev
t020
10t8
-6
-4
-2
0
2
Energy (eV)
4
6
Figure 38. Schematic density of states for (a) undoped and (b) n-type doped a-C : H.
370
J. Robertson
structures. Glassy carbon is metallic. It appears to be quasi-crystalline on a 5/~ scale
and disordered on a 30 & scale. It comprises entangled ribbons of graphitic molecules.
Evaporated a-C is amorphous on a 5/~ scale. It contains largely sp 2 sites and perhaps
some sp 3 sites. The sp 2 sites have some order on a 15 ~ scale; various properties (optical
gap, Raman spectra) suggest that they form islands bounded by sp 3 sites. The size of the
islands determines the band gap. It is possible to increase the hardness and the
proportion of sp 3 sites in a-C by ion-assisted deposition processes, a-Carbon is a
semiconductor and electrical conduction is by variable range hopping in states near
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E F.
Hydrogenated a-C contains both sp 2 and sp 3 sites in similar numbers. The relative
proportion can be altered by a factor of ,-~3 by changing the deposition conditions. The
role of hydrogen is to stabilize sp 3 sites, but it does not eliminate them. Deposition
above 300°C or annealing above 400°C strongly reduces the proportion of sp 3 sites.
Ion-assisted deposition techniques produce extremely hard a - C : H films with a
hardness of 1200-6000 kg m m - 2 or ~ 6.4-9.4 Mohs, which are very valuable coating
materials. To date, the hardness is found to vary inversely with hydrogen content
because hydrogen seems to increase the soft polymeric component of the structure
more than it enhances the cross-linking sp 3 site fraction. The band edge states in a-C : H
are n-like. The optical gap in a-C: H is surprisingly low and indicates that the
remaining sp 2 sites are clustered, confirming that the gap is a function of medium-range
order in all amorphous carbons. The gap increases with hydrogenation because this
reduces the clustering of the remaining sp 2 sites. The fraction of sp 2 and sp 3 sites can be
measured directly by N M R in a-C : H but not in a-C. The initial peak due to n* states in
the XANES spectrum is an unambiguous signature ofsp 2 sites in both a-C and a-C : H.
However, there is as yet no direct unambiguous signature ofsp 3 sites in a-C. a-C : H can
be doped n- and p-type by group V and III elements. A true substitutional mechanism
is involved.
Acknowledgments
The author would like to express his gratitude to his co-workers E. P. O'Reilly and
D. Beeman. This paper is published with the permission of the Central Electricity
Generating Board.
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