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Density of electronic states in amorphous carbons
Diamond and Related Materials 12 (2003) 891–899
Density of electronic states in amorphous carbons G. Fanchini, S.C. Ray, A. Tagliaferro* Dip. Fisica & Unita` INFM, Corso Duca Abruzzi 24, Politecnico, Turin, Italy
Abstract In this paper we deal with the origin and features of the density of electronic states in the various types of a-C and a-C:H films. In particular we discuss the effects of disorder, distortions and composition (i.e. hydrogen and sp2 carbon fractions) on the density of states and the main optoelectronic properties. We show that the wide differences among the various amorphous carbons can be understood by introducing three characteristic lengths of the structure (i.e. distance between neighbouring clusters, decay length of p states and ‘accommodation length’ of sp2 clusters in the sp3 network). Finally, we discuss additional features crucial in determining film properties, such as the correlation energy. 䊚 2003 Elsevier Science B.V. All rights reserved. Keywords: Amorphous carbon; Amorphous hydrogenated carbon; Band structure; Defects
1. Introduction Amorphous (hydrogenated or not) carbon films have been widely studied over the past years. Despite the limited number of elements (one or two) by which they are composed (neglecting contamination like oxygen), the structure and the physical properties of such films can vary in striking ways w1x. Hence, the existence of a common basis for the interpretation of such different films seems to be rather questionable. One of the aims of this paper is to show that such a basis exists. The origin and features of the density of electronic p and p* states will be discussed in this light. The structural characteristics such as local disorder, bond distortions, « affecting the energy distribution of states are considered and their effect on the density-of-states (DOS) is discussed. As the DOS close to Fermi level determines the optoelectronic properties of the films, such properties will be discussed. As films having quite different physical properties have sometimes very close optical gap values w1x, such a point will be addressed and the improper use of terms such as ‘Urbach-tail’ and ‘Taucgap’ in amorphous carbons is discussed. The last item *Corresponding author. Tel.: q39-011-564-7347; fax: q39-011564-7399. E-mail address: [email protected] (A. Tagliaferro).
will be a discussion about the nature of ‘defects’ and the difference between ‘defects’ and traps. 2. Amorphous hydrogenated silicon 2.1. Disorder and density of states In a-Si:H the local environment of Si ions from one site to another varies. Still, the vast majority of Si atoms are four-fold co-ordinated w3x, bond angle and bond lengths having values close to those of c-Si, although the loss of long range order leads to modification in the DOS. The Tauc model w2x, considering a ‘virtual crystal’ and relaxing momentum conservation, calculates the valence and conduction bands (described, at least close to the band edges, by power laws) and properly describes the optical transitions in a-Si:H w2x. In order to later discuss amorphous carbon it is important to focus on the consequences related to the loss of long range order w3x: 1. extended states still exist, but their phase coherence is lost, as the mean free path between scattering events of carriers is close to the interionic distance w4 x ;
0925-9635/03/$ - see front matter 䊚 2003 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 9 6 3 5 Ž 0 2 . 0 0 3 7 6 - X
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Fig. 1. Optical absorption coefficient of some of the amorphous carbons types described in the text. DLHC and PLHC data are taken from Teo et al. w14x. A typical curve for a-Si:H is reported for comparison.
2. co-ordination defects such as dangling bonds arise; 3. bands arising from defect states are Gaussian-like w3x because of bond distortions; and 4. the presence of weak Si–Si bonds gives rise to (Urbach) exponential tails. The DOS of a-Si:H is composed by extended bands, Urbach tails and Gaussian (localised) bands. The twofold role of hydrogen in a-Si:H is to reduce the density of co-ordination defects (improving electronic properties) and the local structural disorder (leading to steeper Urbach tails). 2.2. JDOS and optical absorption coefficient In the one-electron approximation, the DOS of a material is used to describe its optical properties through the so-called joint ‘density-of-states’ (JDOS):
In Fig. 1 a typical absorption coefficient curve for aSi:H is reported. The change in slope at approximately 2 eV separates the so-called Urbach region (at lower energies) from the Tauc region (at higher energies). The change in the nature of the states involved in the optical transition from localised (belonging to exponential tails, Urbach region) to extended (belonging to parabolic bands, Tauc region) is observed. By analysing such data with the Tauc model the (optical) Tauc gap is obtained w2x. Both the Tauc gap and E04 (the photon energy at which the absorption coefficient equals 104 cmy1) are close to the mobility gap (the energy spacing between the onset of the valence and conduction bands), a quantity crucial in determining the electronic properties of a-Si:H films. Moreover, the steepness of Urbach tails scales with the photoconductivity w3x. For this reason the analysis of optical data became a very popular tool for a rough and simple investigation of the electronic quality of a-Si:H w3x.
Eph
|
JDOSŽEph.s
NoŽZyEph.NuŽZ. dZ
(1)
0
where No(x) and Nu(x) represent the density of occupied and unoccupied states, respectively, and Eph is the energy of the impinging photon. The optical absorption coefficient is given by w2x: aŽEph.sKwyQ2ŽEph. yEphz~JDOSŽEph. ynŽEph. x
|
(2)
y1 where K is a suitable constant, Q(Eph)AEph is the matrix element describing the probability of transition between the electronic states and n(Eph) is the refractive index. Hence, the energy trend of a can be used to investigate the shape of DOS (and JDOS).
3. Amorphous carbons: structure and DOS 3.1. Structural considerations Amorphous carbon films are composed of phases in each of which the co-ordination number of carbon ions is the same w1x. Amorphous carbon films differ in many ways from a-Si:H as the hybridisation of atomic orbitals can lead to the formation of both s and p states, having different characteristics w1x. Three typical local structures are formed: sp3 (diamond-like, each carbon atom participating to 4 s bonds), sp2 (graphite-like, 3Øs and 1Øp bonds) and sp1 co-ordination (2Øs and 2Øp bonds). The undistorted structures formed by sp3, sp2 and sp1 coordinated sites are, respectively, tridimensional, bidimen-
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Fig. 2. Sketch of the ‘typical lengths’ introduced in Section 3.1. Clusters are represented as round-shaped for simplicity.
sional (i.e. planar) and monodimensional (i.e. linear). The tridimensional backbone of the films is often w1x formed by the sp3 co-ordinated phase. The matching of the sp2 andyor sp1 co-ordinated clusters to the sp3 network (which occurs through s states) can lead to the distortion of the sp2 clusters (example: ‘boat’ and ‘chair’ structures w5x). The amount and relevance of the distortions are closely related to: 1. the rigidity of the sp3 phase (roughly described by the local co-ordination number xloc, defined as the average number of bonds per atom) nearby a given sp2 ysp1 cluster; 2. the dimension Lc of the clusters; and 3. the distance Lcyc between neighbouring clusters. For instance, a highly overconstrained w5x network will force clusters to distort (i.e. to vary bond length and bond angles) if the cluster is small, this effect will propagate to all sp2 bonds in the cluster. A reciprocal effect is, however, present. By introducing a ‘non-sp3 co-ordinated’ cluster, the local structure of the sp3 network will be distorted and the ‘original’ structure will be recovered only at a certain distance from the cluster. Finally, the presence of hydrogen in the sp3 phase reduces xloc w6x, reducing distortions induced by clusters. In order to classify the various amorphous carbons and understand the DOS features, we define three characteristic lengths (Fig. 2): 1. Lp (the decay length of p states). Lp is the distance at which the wavefunction becomes negligible, not the decay length Ly1 w4x of a spatially decaying wavefunction (i.e. Lp;4–5 Ly1). Lp can vary in a given sample at different energies but, being interested in gross features only, we assume Lp to be similar for all states of a given sample. The presence of H reduces the Lp value w7x. 2. Lcyc (the distance between neighbouring clusters): although this value varies from one pair of neighbouring clusters to the other, we are interested in its average value, correlated to the sp3 content and the cluster size.
3. Lac (the ‘accommodation’ distance of clusters in the sp3 network). An ideal totally sp3 co-ordinated amorphous carbon network will have local distortions and disorder. By introducing in such a network a cluster, the structure of the sp3 network in its surroundings will be modified. Such modification will affect the sp3 network up to a distance Lac from the incorporated cluster. Lac is defined as the maximum distance at which the incorporation of a sp2 cluster will modify the local structure of the sp3 network. Generally speaking Lac increases as the sp3 network becomes more rigid (i.e. at increasing xloc). 3.2. The density-of-states 3.2.1. Fermi and ‘non-bonding’ levels When p (p*) states do not percolate, a single common Fermi level for all clusters does not exist since the Fermi level is a statistically based quantity (reflecting the charge neutrality of the system) that loses physical meaning when the number of electrons involved are not sufficiently large (i.e. largely exceeding 106). As the number of electrons belonging to a single cluster are order of magnitudes lower, we can only focus on atomic-like quantities, such as the non-bonding level (NbL, i.e. the energy level of the p-atomic states prior to hybridisation). The NbL of a cluster is determined by the condition that charge neutrality in the cluster and its surrounding is fulfilled w8x. Hence, any effect that leads to a transfer or spatial redistribution of electronic charge will lead to a shift in the NbL. The sp3 network is continuous, and a Fermi level can be defined for it. Such level represents a reference energy level for describing the DOS (in the following by the term Fermi level we will indicate such level). 3.2.2. The DOS: gross features The formation of molecular bonds gives rise to bonding and antibonding levels w8x. If the matching with the sp3 network does not lead to relevant bond distortions, the relative position of the bonding and antibonding levels with respect to the local NbL remains almost fixed. This will eventually lead to the formation
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of bands for bonding (p) and antibonding (p*) levels. These bands (when no percolation occurs) are formed by localised (i.e. belonging to a single cluster) states, as shown by the participation ratio of p and p* states w1x and it is crucial for the opto-electronic properties of the films. The energy spacing between a p state and its corresponding p* one is generally much lower than for a s–s* pair w1x. Hence, optoelectronic properties are mostly (although not completely) determined by p states (i.e. by non sp3-co-ordinated atoms). We will then focus our attention on the sp2 and sp1 co-ordinated regions at which p states belong. As clusters are formed (if percolation does not occur) by a limited number of atoms, each cluster will have a finite number of electronic states. The DOS and, consequently wEqs. (1)–(3)x, the absorption coefficient should show a number of well defined features, not observed (Fig. 1). Since clusters having different sizes andyor shapes exist, each of them in a given amount, the DOS (and the absorption coefficient curve) should be more structured, but still featuring non observed bumps and valleys. We must then consider a further effect: the role of the sp3 phase. In fact, the matching between the sp3 phase and the clusters has strong consequences on both sides. Let us consider an ‘ideal’ material in which all clusters have the same structure and the DOS of each cluster is represented by a limited number of sharp features. Each cluster has to be ‘accommodated’ in the sp3 phase and this will cause distortions in bond angle and bond lengths in the cluster, as well as a local redistribution of charge. This leads w8x to a shift of the p and p* levels energies from the ‘undistorted’ values. If, as a reasonable approximation, we assume the distributions of bond lengths and bond angles to be Gaussianlike we end up with Gaussian-shaped p and p* bands w8 x : w 1 B EyEp E2z F| NpŽE.sNpMAXexpxy C 2 D sp G ~ y
w 1 B EyEp* E2z F| Np*ŽE.sNp*MAXexpxy C 2 D sp* G ~ y
(3)
in which the zero of energy is set at the Fermi level of the sp3 phase, –Ep (Ep*) represents the energy of the peak of the p (p*) band and sp (sp*) the width of the p (p*) band. The two bands are not necessarily symmetric, neither in peak position nor in width, with respect to Fermi level w8x. In a real film clusters of different sizes andyor shapes exist w1x. Hence, DOS bands will not strictly be Gaussian-like, as the contribution of the DOS of the various clusters should be summed:
NpŽE.s8NpiŽE.
(4)
i
Anyhow, the JDOS calculated from Gaussian-shaped p and p* bands often describes the energy trend of the absorption coefficient over a wide energy range w9x. Restricting our analysis of the absorption coefficient curves to a limited energy range, we can reasonably approximate the curves with straight lines (Fig. 1). An exponential behaviour can then be attributed to a, sometimes on a quite large energy range (see the ta-C curve in Fig. 1). However, it has neither been shown nor proved that such behaviour is due to the same physical processes leading to the Urbach behaviour in a-Si:H. Hence, although the ‘Urbach’ slope represents an appealing parameter to characterise an amorphous carbon w1x, its meaning is unclear. 3.3. Local structure and DOS A first order structural classification can be made on the basis that p states do or do not percolate through the film. Let us notice that, since p states decay outside clusters, they can percolate even when the sp2 (or sp1) phase does not. 3.3.1. Films with non-percolating p states (2ØLpLcyc; case 2ØLac-Lcyc) Fig. 3a shows that, since 2ØLp-Lcyc, p states do not overlap to form extended states. There is also no overlap between the distortions to the sp3 network caused by neighbouring clusters. Fluctuations in the NbL energy will lead (Section 3.2) to the formation of Gaussianlike p and p* bands (Fig. 4a) made by localised states. When the sp3 network becomes more rigid, the disorder in bond lengths and bond angles of the sp2 phase increases, leading to wider fluctuations in the NbL energy. This will lead to larger values of sp as Ep increases w9,10x. Indeed the dimensionality of clusters will also be increased, for instance through ‘boat’ and ‘chair’ distortions, and states closer to Fermi level created w5x. Both effects will lead to wider p and p* bands and to an increase of DOS close to Fermi level. 3.3.2. Films with non-percolating p states (2ØLpLcyc; case 2ØLac)Lcyc) Fig. 3b shows that the distortions of the sp3 network due to neighbouring clusters ‘overlap’. Consequently, the cluster distortions that cause the shift of the NbLs are not independent. Such distortions cannot be considered as randomly fluctuating and the p DOS will depart from the Gaussian shape (Fig. 4b). 3.3.3. Films with percolating sp2 phase (Lcycs0) The sp2 phase is percolating when a non-interrupted path along sp2 carbon sites exist on a macroscopic scale (Fig. 3c). Since sp2 (sp1) structures are planar (linear)
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Fig. 3. Sketch of the various cases detailed in Section 3.3. (a) Non-percolating p states, (b) non-percolating p states and ‘interacting distortions’, (c) percolating sp2 phase, and (d) percolating p states.
and p states do not have spherical symmetry, the percolation threshold is reached w11x at higher sp2 contents (;0.8) than expected (;0.3) from a simple isotropic model. The p states extend over the whole sp2 phase, so that a Fermi level for the sp2 phase can be defined. The percolation of p states lead to the formation of extended bands, similar to the case of graphite (Fig. 4c). However, the lack of an infinite bidimensional sp2 co-ordinated structure leaves (in absence of distortions) a gap between extended p and p* states. Extended states conduction takes place and the activation energy (typically in the range 0.04–0.10 eV w7x) can be regarded as the spacing between the
Fermi level and the threshold of extended p* (or p) states. 3.3.4. The ‘partial percolation’ (2ØLp)Lcyc) case As described in Fig. 3d, although the sp2 phase does not percolate, p states do w12x, since they decay outside clusters. This will, however, occur only if, at a given energy, the density of p states at a given energy is high enough and this will happen near the peak energies. Hence, thresholds energies separating extended from localised states exist w9x. The density of states at such threshold energies, because of the non-3d nature of the states, will be higher than the typical threshold energy
Fig. 4. Sketch of the density-of-states corresponding to the structure of (a) Fig. 3a, (b) Fig. 3b, (c) Fig. 3c, (d) Fig. 3d. Shaded areas correspond to extended states.
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Table 1 Characteristics of the various a-C and a-C:H types Type (§ 3.4)
H
sp3C
Case
xloc
Lcl
2ØLacyLcyc (§ 3.2)
Local disorder
E04 (eV)
Ns (cmy3)
GC a-C) ta-C GLHC DLHC ta-C:H PLHC
– – – Low Low Low High
;0 Low High Low Average High High
§ § § § § § §
;3.2 )3.2 )3.8 -3 ;3.2 ;3.5 -2.5
Perca Large Small Large Average Small Small
– ;1 41 -1 41 41 <1
High Average High Low High High Low
;0 ;0.8 ;2.5 ;1.8 ;1.5 ;2.4 )3
– ;1=1020 )1=1020 ;1=1019 )5=1020 ;5=1019 F1018
3.3.3 3.3.2 3.3.2 3.3.1 3.3.2 3.3.2 3.3.1
GC, glassy carbon. a p states percolation.
in a-Si:H (5=1021 states eVy1 cmy3). Over the threshold energy a departure of the DOS from the Gaussianshape is found (w14x, Fig. 4d). 3.4. Classification of the various a-C and a-C:H types We analyse the DOS and the properties of the various types of a-C and a-C:H on the basis of their local structure, sp3 content and hydrogen content. Glassy carbon and ‘sp2 phase percolating films’ (Section 3.3.3 case) (sp2 fraction exceeding ;0.5): percolation of the sp2 phase occurs. The DOS have no Gaussian-like character and a relevant density of states near the midgap is present. As a consequence, the films are black and conductive w1x. ta-C (tetrahedral amorphous carbon) films (Section 3.3.2 case): here we group all non-hydrogenated sp3rich films (sp3 fraction )0.6) having mechanical properties approaching those of diamond. sp2 sites are mainly paired up or diluted in the highly constrained sp3 matrix w1x. Relevant distortions and the high level of local disorder lead to broad bands w14x having a non-Gaussian shape. This reflects onto the shape of the absorption coefficient spectrum (Fig. 1). Relaxation of the structure by annealing lead to an increase of the Gaussian character of the DOS bands witnessed by the changes in the absorption coefficient w15x. Hydrogen poor (-30 at.%), sp2 rich (sp3 ysp2 below 0.5) films: some films, at energies exceeding a threshold value, can fit to Section 3.3.4 category (i.e. 2ØLp) Lcyc). Most, however, fit in the Section 3.3.1 case. Their DOS is well described by Gaussian bands on a quite large energy range (from 1 to 3 eV on each side of the Fermi level) w9x. Although the structural relaxation due to hydrogen is limited, most of the carbon is sp2-coordinated. This reduces xloc, the spread in energy of local no bonding levels and the width of the Gaussian bands w9x. Hydrogen poor (-30 at.%), sp3 rich (sp3 ysp2 over 0.8) diamond-like films (Section 3.3.2 case): although the sp2 phase and the p states might not percolate, the increased value of xloc, due to the low hydrogen content
and the high amount of sp3 sites, introduces an indirect correlation between neighbouring clusters embedded in the sp3 network. Relevant distortions affect both sp3 network and sp2 clusters. The DOS bands show departures from the Gaussian shape and are wider than in the previous case w13x. ta-C:H (hydrogenated tetrahedral amorphous carbon) (Section 3.3.2 case): this material is similar to the previous one. However, sp2 content is lower and sp2 inclusions are smaller in size andyor further apart w1x: lower distortions are caused to the sp3 network and Lac is reduced. DOS bands are more Gaussian-like w16x. Hydrogen rich ()40 at.%) polymer-like films (Section 3.3.1 case): Lac and Lp w8x are strongly reduced. p bands are Gaussian-like and narrow w10x. Electrons are confined in clusters and films are highly photoluminescent w1x. A given film can readily be classified in one of the above classes (i.e. a hint on its DOS can be obtained) on the basis of its H and sp3 contents (see Table 1). A more sophisticated procedure will require the evaluation of the lengths introduced in Section 3.1. Lp can be estimated to ;4 nm for non-hydrogenated films and ;1 nm for hydrogenated ones w7x. The determination of Lac and Lcyc, however, is a difficult task and can be achieved only through crude approximations. For instance, to evaluate Lcyc in a non hydrogenated film we can assume that only a single cluster type exist. The size Lc of such ‘average’ cluster can be for instance estimated from the Raman spectra w17x and the number Nc of atom in the cluster consequently evaluated. Taking into account the density r and the sp2 fraction, as well as the mass mC of a carbon atom, we can easily write: B
E1y3 mc F yLc 2 D rŽsp yNc. G
LcycsC
(5)
As an example, for a ta-C film (sp2;0.2, r;2.4 gy cm3 w1x) assuming that all sp2 sites are paired up (i.e. ˚ we obtain Lcycf3 A. ˚ We stress, however, Lc;1.4 A), that Eq. (5) (and its twin for a-C:H films) leads only
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Fig. 5. Trends of U vs. E04. Data are taken from Fanchini and Tagliaferro w8x.
to very crude estimates of Lcyc, so that its usefulness is limited to clear-cut cases. 3.5. About s bands Detailed experimental information about s bands are lacking. Surface investigation have just given a very rough picture of such band near the surface w18x, but the bulk has not been investigated. All carbon atoms, irrespective of their hybridisation, contribute states to s bands, but those belonging to sp3 sites are closer to the Fermi level, as bonds are longer w1x. Hence, if we are interested in the edge of s bands, we can look at aSi:H for hints. For instance, we will expect s bands to be described by power laws and the tails of localised states to be exponential. The steepness of such tails will be affected by the local disorder and the distortions in the sp3 phase, a lower amount of them leading to steeper tails. Hence, we expect steeper tails in films for which xloc in the sp3 phase is lower. This will be, for instance, the case for hydrogen-rich polymer-like films. 4. Additional effects The above outlined picture is of course far from complete, as the above discussion refers to overall features, that can be somewhat modified by the spread in local configurations. Moreover, a few more effects deserve a short discussion as they lead to important consequences on the film properties: 1. Band asymmetry. The calculation of p and p* energy levels is made from a ‘molecular’ point of view w1x.
One effect to be considered is the asymmetry between p and p* states with respect to the cluster NbL w8x. As a consequence, the p band peak is narrower (sp-sp*) and closer (NEpN-NEp*N) to the Fermi level w8x. This can lead to p-type conduction in undoped films w7x. 2. Correlation energy. p and p* states are localised in or nearby a cluster: the ‘isolated’ system is formed by a limited number of electrons. Consequently, if a state is doubly occupied, the one-electron approximation fails and a proper description of the p* state energy requires the introduction of a correlation energy U. The value of U is related to the local structure and the ability of the sp3 network to relax w7x. Roughly speaking, U represents the additional energy needed to host two electrons (with opposite spins) in the same state. Hence, the states belonging to an energy region across Fermi level of width U will remain singly occupied. Such states will contribute to the paramagnetic signal detected by electron spin resonance. The values of U can be obtained from the analysis of optical data w8x. Fig. 5 shows that a clear increasing trend of U vs. E04 (i.e. vs. sp3 content) can be identified for each set of films. Since a crucial contribution to reduce U comes from the local reconstruction of the network w7x the trend of U can be attributed to the increase in xloc and in the rigidity of the network that render local reconstruction more and more costly from an energetic point of view. This is also confirmed by the fact that (as already reported in Ristein et al. w18x) at a given E04, softer films (open squares in Fig. 5) have lower U values.
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3. s–p mixing: when clusters are heavily distorted, the orthogonality between s and p states is removed and mixing of these states occur w19x. Transitions to and from mixed states are possible and the DOS bands as well as the absorption coefficient trend are deformed. This might be the case for ta-C. 5. Optical properties of amorphous carbons The optical absorption coefficient in the photon energy range 1–6 eV (Fig. 1) is mainly determined by interband transitions and is therefore strictly related to the p and p* DOS wEqs. (1)–(4)x, although some modifications, not crucial to our qualitative discussion, occur when the effects of correlation energy and of states belonging to odd-fold rings are considered w8x. In many amorphous carbons (Section 3.4) the states relevant to the optical absorption are localised into clusters w1x, so that the relevant change in slope observed in a-Si:H at approximately 2 eV (witnessing a change from extended to localised states, Section 2) is not observed. Even when percolation occurs over a given threshold energy, the change in DOS shape is not abrupt (DLHC and a-C curves in Fig. 1). Hence, a proper Urbach tail is not present in most amorphous carbon plots. Nevertheless, local disorder is crucial in determining the DOS and the optical properties of amorphous carbons w8x and can sometimes lead to quasi-linear behaviours over wide energy ranges (ta-C sample, Fig. 1) improperly termed (Section 3.2.2) Urbach-like. Although deprived of the physical meanings they have in a-Si:H (Section 2.2) Tauc gap and E04 are useful parameters to estimate the optical transparency of the system. Anyhow an empirical correlation between E04 and sp2 content has been shown w1x, E04 increasing as sp2 content decreases. Ep, on its turn, does not show a clear trend vs. sp2 content, most literature values being in the range 2.3–2.7 eV. sp, usually having values in the range 0.8–1.2 eV, is sensitive to the amount of local disorder. sp increases when local disorder increases w9,10x, although large spreads in values are indirectly caused by variations in the H content. 6. ‘Defect states’ and electronic transport in amorphous carbons The term ‘defect’ is not necessarily a synonym for trap. The presence of a large correlation energy U for states close to the Fermi level w8x forbid such states from becoming viable recombination paths. These states remain singly occupied and contribute to the spin density detected, for instance, by electron spin resonance. In fact, up to three orders of magnitudes difference between the density of traps estimated by C-v-T measurements w20x and the spin density measured by ESR w6x are
reported. Despite their large spin density, photoconductivity has been reported for some amorphous carbons w7x, supporting this point of view. The correlation energy is crucial in determining the spin density and, on its turn, is affected by network rigidity, distortions and disorder w6x. We expect higher spin densities in sp3-rich, hydrogen-poor films (see Table 1). In low spin density films, additional ESR inactive recombination paths are present, which are detectable by optical absorption measurements w1x. 7. Conclusion We have qualitatively discussed the genesis and nature of the p and p* states and their collection into bands. We have detailed the effect of the local structure on the DOS shape and distribution and we have shown that a common point of view can be taken in the analysis of all amorphous carbons by introducing three characteristic lengths. Finally, we have addressed the role of correlation energy in determining the difference between trap states and defect states, as well as the spin density. An important role, not considered in the present paper, can be played by layering effects or the presence of large scale inhomogeneities. A more thorough and quantitative analysis about specific subjects such as optical w8,9x and transport related w7,21x properties can be found elsewhere. Additional references can be found in Roberston w1x. References w1x J. Robertson, Mater. Sci. Eng. Rev. 37 (2002) 129. w2x G.D. Cody, in: J.J. Pankove (Ed.), Hydrogenated Amorphous Silicon (Semiconductor and Semimetals), 21B, Academic Press, New York, 1984, p. 11. w3x R.A. Street, Hydrogenated Amorphous Silicon, Cambridge University Press, UK, 1991. w4x N.F. Mott, E.A. Davis, Electronic Processes in Non Crystalline Materials, Clarendon Press, Oxford, 1989. w5x J. Robertson, Diamond Relat. Mater. 4 (1995) 298. w6x F. Giorgis, A. Tagliaferro, M. Fanciulli, in: R. Silva (Ed.), Amorphous Carbon: State of the Art, World Scientific, Singapore, 1998, p. 143. w7x A. Ilie, Diamond Relat. Mater. 10 (2001) 208. w8x G. Fanchini, A. Tagliaferro, Diamond Relat. Mater. 10 (2001) 191. w9x G. Fanchini, A. Tagliaferro, D.P. Dowling, et al., Phys. Rev. B 61 (2000) 5002. w10x C. Oppedisano, A. Tagliaferro, Appl. Phys. Lett. 75 (3650). w11x J. Robertson, Phil. Mag. B 76 (1997) 335. w12x C.W. Chen, J. Robertson, J. Non-Cryst. Solids 227 (1998) 602. w13x V. Paret, PhD Thesis, Universite´ Paris VII, Paris, 1999. w14x K.B.K. Teo, A.C. Ferrari, G. Fanchini, et al., Diamond Relat. Mater. 11 (2002) 1086.
G. Fanchini et al. / Diamond and Related Materials 12 (2003) 891–899 w15x A.C. Ferrari, B. Kleinsorge, N.A. Morrison, et al., J. Appl. Phys. 85 (1999) 7191. w16x N.M.J. Conway, A.C. Ferrari, A.J. Flewitt, et al., Diamond Relat. Mater. 9 (2000) 765. w17x A.C. Ferrari, J. Robertson, Phys. Rev. B 61 (2000) 14095. w18x J. Ristein, J. Schafer, L. Ley in ref. 6, p. 163.
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w19x G. Fanchini, V. Paret, A. Tagliaferro, M.L. Theye, J. NonCryst. Sol. 299y302 (2002) 858. w20x B. Racine, M. Benlahsen, K. Zellama, et al., Diamond Relat. Mater. 10 (2001) 200. w21x R.C. Barklie, Diamond Relat. Mater. 10 (2001) 184.
Density of electronic states in amorphous carbons G. Fanchini, S.C. Ray, A. Tagliaferro* Dip. Fisica & Unita` INFM, Corso Duca Abruzzi 24, Politecnico, Turin, Italy
Abstract In this paper we deal with the origin and features of the density of electronic states in the various types of a-C and a-C:H films. In particular we discuss the effects of disorder, distortions and composition (i.e. hydrogen and sp2 carbon fractions) on the density of states and the main optoelectronic properties. We show that the wide differences among the various amorphous carbons can be understood by introducing three characteristic lengths of the structure (i.e. distance between neighbouring clusters, decay length of p states and ‘accommodation length’ of sp2 clusters in the sp3 network). Finally, we discuss additional features crucial in determining film properties, such as the correlation energy. 䊚 2003 Elsevier Science B.V. All rights reserved. Keywords: Amorphous carbon; Amorphous hydrogenated carbon; Band structure; Defects
1. Introduction Amorphous (hydrogenated or not) carbon films have been widely studied over the past years. Despite the limited number of elements (one or two) by which they are composed (neglecting contamination like oxygen), the structure and the physical properties of such films can vary in striking ways w1x. Hence, the existence of a common basis for the interpretation of such different films seems to be rather questionable. One of the aims of this paper is to show that such a basis exists. The origin and features of the density of electronic p and p* states will be discussed in this light. The structural characteristics such as local disorder, bond distortions, « affecting the energy distribution of states are considered and their effect on the density-of-states (DOS) is discussed. As the DOS close to Fermi level determines the optoelectronic properties of the films, such properties will be discussed. As films having quite different physical properties have sometimes very close optical gap values w1x, such a point will be addressed and the improper use of terms such as ‘Urbach-tail’ and ‘Taucgap’ in amorphous carbons is discussed. The last item *Corresponding author. Tel.: q39-011-564-7347; fax: q39-011564-7399. E-mail address: [email protected] (A. Tagliaferro).
will be a discussion about the nature of ‘defects’ and the difference between ‘defects’ and traps. 2. Amorphous hydrogenated silicon 2.1. Disorder and density of states In a-Si:H the local environment of Si ions from one site to another varies. Still, the vast majority of Si atoms are four-fold co-ordinated w3x, bond angle and bond lengths having values close to those of c-Si, although the loss of long range order leads to modification in the DOS. The Tauc model w2x, considering a ‘virtual crystal’ and relaxing momentum conservation, calculates the valence and conduction bands (described, at least close to the band edges, by power laws) and properly describes the optical transitions in a-Si:H w2x. In order to later discuss amorphous carbon it is important to focus on the consequences related to the loss of long range order w3x: 1. extended states still exist, but their phase coherence is lost, as the mean free path between scattering events of carriers is close to the interionic distance w4 x ;
0925-9635/03/$ - see front matter 䊚 2003 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 9 6 3 5 Ž 0 2 . 0 0 3 7 6 - X
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Fig. 1. Optical absorption coefficient of some of the amorphous carbons types described in the text. DLHC and PLHC data are taken from Teo et al. w14x. A typical curve for a-Si:H is reported for comparison.
2. co-ordination defects such as dangling bonds arise; 3. bands arising from defect states are Gaussian-like w3x because of bond distortions; and 4. the presence of weak Si–Si bonds gives rise to (Urbach) exponential tails. The DOS of a-Si:H is composed by extended bands, Urbach tails and Gaussian (localised) bands. The twofold role of hydrogen in a-Si:H is to reduce the density of co-ordination defects (improving electronic properties) and the local structural disorder (leading to steeper Urbach tails). 2.2. JDOS and optical absorption coefficient In the one-electron approximation, the DOS of a material is used to describe its optical properties through the so-called joint ‘density-of-states’ (JDOS):
In Fig. 1 a typical absorption coefficient curve for aSi:H is reported. The change in slope at approximately 2 eV separates the so-called Urbach region (at lower energies) from the Tauc region (at higher energies). The change in the nature of the states involved in the optical transition from localised (belonging to exponential tails, Urbach region) to extended (belonging to parabolic bands, Tauc region) is observed. By analysing such data with the Tauc model the (optical) Tauc gap is obtained w2x. Both the Tauc gap and E04 (the photon energy at which the absorption coefficient equals 104 cmy1) are close to the mobility gap (the energy spacing between the onset of the valence and conduction bands), a quantity crucial in determining the electronic properties of a-Si:H films. Moreover, the steepness of Urbach tails scales with the photoconductivity w3x. For this reason the analysis of optical data became a very popular tool for a rough and simple investigation of the electronic quality of a-Si:H w3x.
Eph
|
JDOSŽEph.s
NoŽZyEph.NuŽZ. dZ
(1)
0
where No(x) and Nu(x) represent the density of occupied and unoccupied states, respectively, and Eph is the energy of the impinging photon. The optical absorption coefficient is given by w2x: aŽEph.sKwyQ2ŽEph. yEphz~JDOSŽEph. ynŽEph. x
|
(2)
y1 where K is a suitable constant, Q(Eph)AEph is the matrix element describing the probability of transition between the electronic states and n(Eph) is the refractive index. Hence, the energy trend of a can be used to investigate the shape of DOS (and JDOS).
3. Amorphous carbons: structure and DOS 3.1. Structural considerations Amorphous carbon films are composed of phases in each of which the co-ordination number of carbon ions is the same w1x. Amorphous carbon films differ in many ways from a-Si:H as the hybridisation of atomic orbitals can lead to the formation of both s and p states, having different characteristics w1x. Three typical local structures are formed: sp3 (diamond-like, each carbon atom participating to 4 s bonds), sp2 (graphite-like, 3Øs and 1Øp bonds) and sp1 co-ordination (2Øs and 2Øp bonds). The undistorted structures formed by sp3, sp2 and sp1 coordinated sites are, respectively, tridimensional, bidimen-
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Fig. 2. Sketch of the ‘typical lengths’ introduced in Section 3.1. Clusters are represented as round-shaped for simplicity.
sional (i.e. planar) and monodimensional (i.e. linear). The tridimensional backbone of the films is often w1x formed by the sp3 co-ordinated phase. The matching of the sp2 andyor sp1 co-ordinated clusters to the sp3 network (which occurs through s states) can lead to the distortion of the sp2 clusters (example: ‘boat’ and ‘chair’ structures w5x). The amount and relevance of the distortions are closely related to: 1. the rigidity of the sp3 phase (roughly described by the local co-ordination number xloc, defined as the average number of bonds per atom) nearby a given sp2 ysp1 cluster; 2. the dimension Lc of the clusters; and 3. the distance Lcyc between neighbouring clusters. For instance, a highly overconstrained w5x network will force clusters to distort (i.e. to vary bond length and bond angles) if the cluster is small, this effect will propagate to all sp2 bonds in the cluster. A reciprocal effect is, however, present. By introducing a ‘non-sp3 co-ordinated’ cluster, the local structure of the sp3 network will be distorted and the ‘original’ structure will be recovered only at a certain distance from the cluster. Finally, the presence of hydrogen in the sp3 phase reduces xloc w6x, reducing distortions induced by clusters. In order to classify the various amorphous carbons and understand the DOS features, we define three characteristic lengths (Fig. 2): 1. Lp (the decay length of p states). Lp is the distance at which the wavefunction becomes negligible, not the decay length Ly1 w4x of a spatially decaying wavefunction (i.e. Lp;4–5 Ly1). Lp can vary in a given sample at different energies but, being interested in gross features only, we assume Lp to be similar for all states of a given sample. The presence of H reduces the Lp value w7x. 2. Lcyc (the distance between neighbouring clusters): although this value varies from one pair of neighbouring clusters to the other, we are interested in its average value, correlated to the sp3 content and the cluster size.
3. Lac (the ‘accommodation’ distance of clusters in the sp3 network). An ideal totally sp3 co-ordinated amorphous carbon network will have local distortions and disorder. By introducing in such a network a cluster, the structure of the sp3 network in its surroundings will be modified. Such modification will affect the sp3 network up to a distance Lac from the incorporated cluster. Lac is defined as the maximum distance at which the incorporation of a sp2 cluster will modify the local structure of the sp3 network. Generally speaking Lac increases as the sp3 network becomes more rigid (i.e. at increasing xloc). 3.2. The density-of-states 3.2.1. Fermi and ‘non-bonding’ levels When p (p*) states do not percolate, a single common Fermi level for all clusters does not exist since the Fermi level is a statistically based quantity (reflecting the charge neutrality of the system) that loses physical meaning when the number of electrons involved are not sufficiently large (i.e. largely exceeding 106). As the number of electrons belonging to a single cluster are order of magnitudes lower, we can only focus on atomic-like quantities, such as the non-bonding level (NbL, i.e. the energy level of the p-atomic states prior to hybridisation). The NbL of a cluster is determined by the condition that charge neutrality in the cluster and its surrounding is fulfilled w8x. Hence, any effect that leads to a transfer or spatial redistribution of electronic charge will lead to a shift in the NbL. The sp3 network is continuous, and a Fermi level can be defined for it. Such level represents a reference energy level for describing the DOS (in the following by the term Fermi level we will indicate such level). 3.2.2. The DOS: gross features The formation of molecular bonds gives rise to bonding and antibonding levels w8x. If the matching with the sp3 network does not lead to relevant bond distortions, the relative position of the bonding and antibonding levels with respect to the local NbL remains almost fixed. This will eventually lead to the formation
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of bands for bonding (p) and antibonding (p*) levels. These bands (when no percolation occurs) are formed by localised (i.e. belonging to a single cluster) states, as shown by the participation ratio of p and p* states w1x and it is crucial for the opto-electronic properties of the films. The energy spacing between a p state and its corresponding p* one is generally much lower than for a s–s* pair w1x. Hence, optoelectronic properties are mostly (although not completely) determined by p states (i.e. by non sp3-co-ordinated atoms). We will then focus our attention on the sp2 and sp1 co-ordinated regions at which p states belong. As clusters are formed (if percolation does not occur) by a limited number of atoms, each cluster will have a finite number of electronic states. The DOS and, consequently wEqs. (1)–(3)x, the absorption coefficient should show a number of well defined features, not observed (Fig. 1). Since clusters having different sizes andyor shapes exist, each of them in a given amount, the DOS (and the absorption coefficient curve) should be more structured, but still featuring non observed bumps and valleys. We must then consider a further effect: the role of the sp3 phase. In fact, the matching between the sp3 phase and the clusters has strong consequences on both sides. Let us consider an ‘ideal’ material in which all clusters have the same structure and the DOS of each cluster is represented by a limited number of sharp features. Each cluster has to be ‘accommodated’ in the sp3 phase and this will cause distortions in bond angle and bond lengths in the cluster, as well as a local redistribution of charge. This leads w8x to a shift of the p and p* levels energies from the ‘undistorted’ values. If, as a reasonable approximation, we assume the distributions of bond lengths and bond angles to be Gaussianlike we end up with Gaussian-shaped p and p* bands w8 x : w 1 B EyEp E2z F| NpŽE.sNpMAXexpxy C 2 D sp G ~ y
w 1 B EyEp* E2z F| Np*ŽE.sNp*MAXexpxy C 2 D sp* G ~ y
(3)
in which the zero of energy is set at the Fermi level of the sp3 phase, –Ep (Ep*) represents the energy of the peak of the p (p*) band and sp (sp*) the width of the p (p*) band. The two bands are not necessarily symmetric, neither in peak position nor in width, with respect to Fermi level w8x. In a real film clusters of different sizes andyor shapes exist w1x. Hence, DOS bands will not strictly be Gaussian-like, as the contribution of the DOS of the various clusters should be summed:
NpŽE.s8NpiŽE.
(4)
i
Anyhow, the JDOS calculated from Gaussian-shaped p and p* bands often describes the energy trend of the absorption coefficient over a wide energy range w9x. Restricting our analysis of the absorption coefficient curves to a limited energy range, we can reasonably approximate the curves with straight lines (Fig. 1). An exponential behaviour can then be attributed to a, sometimes on a quite large energy range (see the ta-C curve in Fig. 1). However, it has neither been shown nor proved that such behaviour is due to the same physical processes leading to the Urbach behaviour in a-Si:H. Hence, although the ‘Urbach’ slope represents an appealing parameter to characterise an amorphous carbon w1x, its meaning is unclear. 3.3. Local structure and DOS A first order structural classification can be made on the basis that p states do or do not percolate through the film. Let us notice that, since p states decay outside clusters, they can percolate even when the sp2 (or sp1) phase does not. 3.3.1. Films with non-percolating p states (2ØLpLcyc; case 2ØLac-Lcyc) Fig. 3a shows that, since 2ØLp-Lcyc, p states do not overlap to form extended states. There is also no overlap between the distortions to the sp3 network caused by neighbouring clusters. Fluctuations in the NbL energy will lead (Section 3.2) to the formation of Gaussianlike p and p* bands (Fig. 4a) made by localised states. When the sp3 network becomes more rigid, the disorder in bond lengths and bond angles of the sp2 phase increases, leading to wider fluctuations in the NbL energy. This will lead to larger values of sp as Ep increases w9,10x. Indeed the dimensionality of clusters will also be increased, for instance through ‘boat’ and ‘chair’ distortions, and states closer to Fermi level created w5x. Both effects will lead to wider p and p* bands and to an increase of DOS close to Fermi level. 3.3.2. Films with non-percolating p states (2ØLpLcyc; case 2ØLac)Lcyc) Fig. 3b shows that the distortions of the sp3 network due to neighbouring clusters ‘overlap’. Consequently, the cluster distortions that cause the shift of the NbLs are not independent. Such distortions cannot be considered as randomly fluctuating and the p DOS will depart from the Gaussian shape (Fig. 4b). 3.3.3. Films with percolating sp2 phase (Lcycs0) The sp2 phase is percolating when a non-interrupted path along sp2 carbon sites exist on a macroscopic scale (Fig. 3c). Since sp2 (sp1) structures are planar (linear)
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Fig. 3. Sketch of the various cases detailed in Section 3.3. (a) Non-percolating p states, (b) non-percolating p states and ‘interacting distortions’, (c) percolating sp2 phase, and (d) percolating p states.
and p states do not have spherical symmetry, the percolation threshold is reached w11x at higher sp2 contents (;0.8) than expected (;0.3) from a simple isotropic model. The p states extend over the whole sp2 phase, so that a Fermi level for the sp2 phase can be defined. The percolation of p states lead to the formation of extended bands, similar to the case of graphite (Fig. 4c). However, the lack of an infinite bidimensional sp2 co-ordinated structure leaves (in absence of distortions) a gap between extended p and p* states. Extended states conduction takes place and the activation energy (typically in the range 0.04–0.10 eV w7x) can be regarded as the spacing between the
Fermi level and the threshold of extended p* (or p) states. 3.3.4. The ‘partial percolation’ (2ØLp)Lcyc) case As described in Fig. 3d, although the sp2 phase does not percolate, p states do w12x, since they decay outside clusters. This will, however, occur only if, at a given energy, the density of p states at a given energy is high enough and this will happen near the peak energies. Hence, thresholds energies separating extended from localised states exist w9x. The density of states at such threshold energies, because of the non-3d nature of the states, will be higher than the typical threshold energy
Fig. 4. Sketch of the density-of-states corresponding to the structure of (a) Fig. 3a, (b) Fig. 3b, (c) Fig. 3c, (d) Fig. 3d. Shaded areas correspond to extended states.
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Table 1 Characteristics of the various a-C and a-C:H types Type (§ 3.4)
H
sp3C
Case
xloc
Lcl
2ØLacyLcyc (§ 3.2)
Local disorder
E04 (eV)
Ns (cmy3)
GC a-C) ta-C GLHC DLHC ta-C:H PLHC
– – – Low Low Low High
;0 Low High Low Average High High
§ § § § § § §
;3.2 )3.2 )3.8 -3 ;3.2 ;3.5 -2.5
Perca Large Small Large Average Small Small
– ;1 41 -1 41 41 <1
High Average High Low High High Low
;0 ;0.8 ;2.5 ;1.8 ;1.5 ;2.4 )3
– ;1=1020 )1=1020 ;1=1019 )5=1020 ;5=1019 F1018
3.3.3 3.3.2 3.3.2 3.3.1 3.3.2 3.3.2 3.3.1
GC, glassy carbon. a p states percolation.
in a-Si:H (5=1021 states eVy1 cmy3). Over the threshold energy a departure of the DOS from the Gaussianshape is found (w14x, Fig. 4d). 3.4. Classification of the various a-C and a-C:H types We analyse the DOS and the properties of the various types of a-C and a-C:H on the basis of their local structure, sp3 content and hydrogen content. Glassy carbon and ‘sp2 phase percolating films’ (Section 3.3.3 case) (sp2 fraction exceeding ;0.5): percolation of the sp2 phase occurs. The DOS have no Gaussian-like character and a relevant density of states near the midgap is present. As a consequence, the films are black and conductive w1x. ta-C (tetrahedral amorphous carbon) films (Section 3.3.2 case): here we group all non-hydrogenated sp3rich films (sp3 fraction )0.6) having mechanical properties approaching those of diamond. sp2 sites are mainly paired up or diluted in the highly constrained sp3 matrix w1x. Relevant distortions and the high level of local disorder lead to broad bands w14x having a non-Gaussian shape. This reflects onto the shape of the absorption coefficient spectrum (Fig. 1). Relaxation of the structure by annealing lead to an increase of the Gaussian character of the DOS bands witnessed by the changes in the absorption coefficient w15x. Hydrogen poor (-30 at.%), sp2 rich (sp3 ysp2 below 0.5) films: some films, at energies exceeding a threshold value, can fit to Section 3.3.4 category (i.e. 2ØLp) Lcyc). Most, however, fit in the Section 3.3.1 case. Their DOS is well described by Gaussian bands on a quite large energy range (from 1 to 3 eV on each side of the Fermi level) w9x. Although the structural relaxation due to hydrogen is limited, most of the carbon is sp2-coordinated. This reduces xloc, the spread in energy of local no bonding levels and the width of the Gaussian bands w9x. Hydrogen poor (-30 at.%), sp3 rich (sp3 ysp2 over 0.8) diamond-like films (Section 3.3.2 case): although the sp2 phase and the p states might not percolate, the increased value of xloc, due to the low hydrogen content
and the high amount of sp3 sites, introduces an indirect correlation between neighbouring clusters embedded in the sp3 network. Relevant distortions affect both sp3 network and sp2 clusters. The DOS bands show departures from the Gaussian shape and are wider than in the previous case w13x. ta-C:H (hydrogenated tetrahedral amorphous carbon) (Section 3.3.2 case): this material is similar to the previous one. However, sp2 content is lower and sp2 inclusions are smaller in size andyor further apart w1x: lower distortions are caused to the sp3 network and Lac is reduced. DOS bands are more Gaussian-like w16x. Hydrogen rich ()40 at.%) polymer-like films (Section 3.3.1 case): Lac and Lp w8x are strongly reduced. p bands are Gaussian-like and narrow w10x. Electrons are confined in clusters and films are highly photoluminescent w1x. A given film can readily be classified in one of the above classes (i.e. a hint on its DOS can be obtained) on the basis of its H and sp3 contents (see Table 1). A more sophisticated procedure will require the evaluation of the lengths introduced in Section 3.1. Lp can be estimated to ;4 nm for non-hydrogenated films and ;1 nm for hydrogenated ones w7x. The determination of Lac and Lcyc, however, is a difficult task and can be achieved only through crude approximations. For instance, to evaluate Lcyc in a non hydrogenated film we can assume that only a single cluster type exist. The size Lc of such ‘average’ cluster can be for instance estimated from the Raman spectra w17x and the number Nc of atom in the cluster consequently evaluated. Taking into account the density r and the sp2 fraction, as well as the mass mC of a carbon atom, we can easily write: B
E1y3 mc F yLc 2 D rŽsp yNc. G
LcycsC
(5)
As an example, for a ta-C film (sp2;0.2, r;2.4 gy cm3 w1x) assuming that all sp2 sites are paired up (i.e. ˚ we obtain Lcycf3 A. ˚ We stress, however, Lc;1.4 A), that Eq. (5) (and its twin for a-C:H films) leads only
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Fig. 5. Trends of U vs. E04. Data are taken from Fanchini and Tagliaferro w8x.
to very crude estimates of Lcyc, so that its usefulness is limited to clear-cut cases. 3.5. About s bands Detailed experimental information about s bands are lacking. Surface investigation have just given a very rough picture of such band near the surface w18x, but the bulk has not been investigated. All carbon atoms, irrespective of their hybridisation, contribute states to s bands, but those belonging to sp3 sites are closer to the Fermi level, as bonds are longer w1x. Hence, if we are interested in the edge of s bands, we can look at aSi:H for hints. For instance, we will expect s bands to be described by power laws and the tails of localised states to be exponential. The steepness of such tails will be affected by the local disorder and the distortions in the sp3 phase, a lower amount of them leading to steeper tails. Hence, we expect steeper tails in films for which xloc in the sp3 phase is lower. This will be, for instance, the case for hydrogen-rich polymer-like films. 4. Additional effects The above outlined picture is of course far from complete, as the above discussion refers to overall features, that can be somewhat modified by the spread in local configurations. Moreover, a few more effects deserve a short discussion as they lead to important consequences on the film properties: 1. Band asymmetry. The calculation of p and p* energy levels is made from a ‘molecular’ point of view w1x.
One effect to be considered is the asymmetry between p and p* states with respect to the cluster NbL w8x. As a consequence, the p band peak is narrower (sp-sp*) and closer (NEpN-NEp*N) to the Fermi level w8x. This can lead to p-type conduction in undoped films w7x. 2. Correlation energy. p and p* states are localised in or nearby a cluster: the ‘isolated’ system is formed by a limited number of electrons. Consequently, if a state is doubly occupied, the one-electron approximation fails and a proper description of the p* state energy requires the introduction of a correlation energy U. The value of U is related to the local structure and the ability of the sp3 network to relax w7x. Roughly speaking, U represents the additional energy needed to host two electrons (with opposite spins) in the same state. Hence, the states belonging to an energy region across Fermi level of width U will remain singly occupied. Such states will contribute to the paramagnetic signal detected by electron spin resonance. The values of U can be obtained from the analysis of optical data w8x. Fig. 5 shows that a clear increasing trend of U vs. E04 (i.e. vs. sp3 content) can be identified for each set of films. Since a crucial contribution to reduce U comes from the local reconstruction of the network w7x the trend of U can be attributed to the increase in xloc and in the rigidity of the network that render local reconstruction more and more costly from an energetic point of view. This is also confirmed by the fact that (as already reported in Ristein et al. w18x) at a given E04, softer films (open squares in Fig. 5) have lower U values.
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3. s–p mixing: when clusters are heavily distorted, the orthogonality between s and p states is removed and mixing of these states occur w19x. Transitions to and from mixed states are possible and the DOS bands as well as the absorption coefficient trend are deformed. This might be the case for ta-C. 5. Optical properties of amorphous carbons The optical absorption coefficient in the photon energy range 1–6 eV (Fig. 1) is mainly determined by interband transitions and is therefore strictly related to the p and p* DOS wEqs. (1)–(4)x, although some modifications, not crucial to our qualitative discussion, occur when the effects of correlation energy and of states belonging to odd-fold rings are considered w8x. In many amorphous carbons (Section 3.4) the states relevant to the optical absorption are localised into clusters w1x, so that the relevant change in slope observed in a-Si:H at approximately 2 eV (witnessing a change from extended to localised states, Section 2) is not observed. Even when percolation occurs over a given threshold energy, the change in DOS shape is not abrupt (DLHC and a-C curves in Fig. 1). Hence, a proper Urbach tail is not present in most amorphous carbon plots. Nevertheless, local disorder is crucial in determining the DOS and the optical properties of amorphous carbons w8x and can sometimes lead to quasi-linear behaviours over wide energy ranges (ta-C sample, Fig. 1) improperly termed (Section 3.2.2) Urbach-like. Although deprived of the physical meanings they have in a-Si:H (Section 2.2) Tauc gap and E04 are useful parameters to estimate the optical transparency of the system. Anyhow an empirical correlation between E04 and sp2 content has been shown w1x, E04 increasing as sp2 content decreases. Ep, on its turn, does not show a clear trend vs. sp2 content, most literature values being in the range 2.3–2.7 eV. sp, usually having values in the range 0.8–1.2 eV, is sensitive to the amount of local disorder. sp increases when local disorder increases w9,10x, although large spreads in values are indirectly caused by variations in the H content. 6. ‘Defect states’ and electronic transport in amorphous carbons The term ‘defect’ is not necessarily a synonym for trap. The presence of a large correlation energy U for states close to the Fermi level w8x forbid such states from becoming viable recombination paths. These states remain singly occupied and contribute to the spin density detected, for instance, by electron spin resonance. In fact, up to three orders of magnitudes difference between the density of traps estimated by C-v-T measurements w20x and the spin density measured by ESR w6x are
reported. Despite their large spin density, photoconductivity has been reported for some amorphous carbons w7x, supporting this point of view. The correlation energy is crucial in determining the spin density and, on its turn, is affected by network rigidity, distortions and disorder w6x. We expect higher spin densities in sp3-rich, hydrogen-poor films (see Table 1). In low spin density films, additional ESR inactive recombination paths are present, which are detectable by optical absorption measurements w1x. 7. Conclusion We have qualitatively discussed the genesis and nature of the p and p* states and their collection into bands. We have detailed the effect of the local structure on the DOS shape and distribution and we have shown that a common point of view can be taken in the analysis of all amorphous carbons by introducing three characteristic lengths. Finally, we have addressed the role of correlation energy in determining the difference between trap states and defect states, as well as the spin density. An important role, not considered in the present paper, can be played by layering effects or the presence of large scale inhomogeneities. A more thorough and quantitative analysis about specific subjects such as optical w8,9x and transport related w7,21x properties can be found elsewhere. Additional references can be found in Roberston w1x. References w1x J. Robertson, Mater. Sci. Eng. Rev. 37 (2002) 129. w2x G.D. Cody, in: J.J. Pankove (Ed.), Hydrogenated Amorphous Silicon (Semiconductor and Semimetals), 21B, Academic Press, New York, 1984, p. 11. w3x R.A. Street, Hydrogenated Amorphous Silicon, Cambridge University Press, UK, 1991. w4x N.F. Mott, E.A. Davis, Electronic Processes in Non Crystalline Materials, Clarendon Press, Oxford, 1989. w5x J. Robertson, Diamond Relat. Mater. 4 (1995) 298. w6x F. Giorgis, A. Tagliaferro, M. Fanciulli, in: R. Silva (Ed.), Amorphous Carbon: State of the Art, World Scientific, Singapore, 1998, p. 143. w7x A. Ilie, Diamond Relat. Mater. 10 (2001) 208. w8x G. Fanchini, A. Tagliaferro, Diamond Relat. Mater. 10 (2001) 191. w9x G. Fanchini, A. Tagliaferro, D.P. Dowling, et al., Phys. Rev. B 61 (2000) 5002. w10x C. Oppedisano, A. Tagliaferro, Appl. Phys. Lett. 75 (3650). w11x J. Robertson, Phil. Mag. B 76 (1997) 335. w12x C.W. Chen, J. Robertson, J. Non-Cryst. Solids 227 (1998) 602. w13x V. Paret, PhD Thesis, Universite´ Paris VII, Paris, 1999. w14x K.B.K. Teo, A.C. Ferrari, G. Fanchini, et al., Diamond Relat. Mater. 11 (2002) 1086.
G. Fanchini et al. / Diamond and Related Materials 12 (2003) 891–899 w15x A.C. Ferrari, B. Kleinsorge, N.A. Morrison, et al., J. Appl. Phys. 85 (1999) 7191. w16x N.M.J. Conway, A.C. Ferrari, A.J. Flewitt, et al., Diamond Relat. Mater. 9 (2000) 765. w17x A.C. Ferrari, J. Robertson, Phys. Rev. B 61 (2000) 14095. w18x J. Ristein, J. Schafer, L. Ley in ref. 6, p. 163.
899
w19x G. Fanchini, V. Paret, A. Tagliaferro, M.L. Theye, J. NonCryst. Sol. 299y302 (2002) 858. w20x B. Racine, M. Benlahsen, K. Zellama, et al., Diamond Relat. Mater. 10 (2001) 200. w21x R.C. Barklie, Diamond Relat. Mater. 10 (2001) 184.