Philosophical Magazine, 11 October 2003 Vol. 83, No. 29, 3351–3365
Field-enhanced electrical transport mechanisms in amorphous carbon films C. Godety, Sushil Kumarz Laboratoire de Physique des Interfaces et des Couches Minces, Unite´ Mixte de Recherche CNRS 7647, Ecole Polytechnique, 91128 Palaiseau-Cedex, France
V. Chu Instituto de Engenharia de Systemas e Computadores, Microsistemas e Nanotecnologias, Rua Alves Redol, 9, 1000-029 Lisboa, Portugal [Received 28 January 2003 and accepted in revised form 26 June 2003]
Abstract In order to investigate the localized electronic states in hydrogenated amorphous carbon (a-C : H) films, the temperature and electric field dependences of the current density have been measured in low-field coplanar (Al/a-C : H/Al) and in high-field transverse (TiW/a-C : H/TiW) geometries. The very-low-field conductivity ¼ 00 exp[(T0/T)1/4] reveals a band-tail hopping transport mechanism in an Ohmic regime (at least for films above 20 nm thickness) while, for electric field values F > 4 103 V cm1, a different behaviour is evidenced of the form ¼ n0 exp½ðF=Fn Þn with a transition in the exponent value from n ¼ 2 at low fields (F 3 105 V cm1). This field enhancement of electrical transport can be interpreted as arising from either a three-dimensional Poole–Frenkel effect for charged empty defects, or field-assisted hopping out of neutral empty defects (the Apsley–Hughes hopping model). Although a clear discrimination between both models would require very high electric fields (F > 8kT/e, where 1 is the localized wavefunction radius), the Apsley–Hughes model describes accurately the experimental temperature-dependent and field-enhanced transport within localized states (scaling as eF/2kT with 1 ¼ 2.8 0.4 nm) and is also consistent with the variable-range hopping mechanism observed in the Ohmic regime.
} 1. Introduction During the last three decades, much effort has been devoted to high-electric-field transport characteristics of amorphous semiconductors or insulating materials (Mott and Davis 1979, Kao and Hwang 1981, Bottger and Wegener 1990, Shklovskii et al. 1990, Baranovskii and Thomas 1996, Hofsa¨ss 1998). Amorphous carbon (a-C) films
y Author for correspondence. Email:
[email protected]. z Present address: Thin Film Technology Group, National Physical Laboratory, New Delhi-110 012, India. Philosophical Magazine ISSN 1478–6435 print/ISSN 1478–6443 online # 2003 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/14786430310001605010
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represent an interesting model material to understand high-field electronic transport for two reasons. (i) The hydrogenated amorphous carbon (a-C : H) bandgap between p and p* bands can be easily changed over a very wide range (0.1–4.0 eV, i.e. from a semimetal to an insulating material) by controlling the growth parameters and the hydrogen content (Robertson 1986). (ii) Variations in the dihedral angle localizes all p and p* electronic states within the s–s* gap (Chen and Robertson 1998), leading to very small time-of-flight or field-effect carrier mobilities (109–105 cm2 V1 s1) (Godet 2003). As a consequence of the fully localized p and p* state distributions, any mechanism based on extended state transport (e.g. in states or * states) is irrelevant for most a-C : H films. The study of hopping transport in a-C films is also important for electronic devices such as electrochemical electrodes, cold cathodes, organic light-emitting diodes, active matrix displays and thin-film transistors. In a-C films, several mechanisms have been proposed to describe the field enhancement of carrier transport through localized states distributed in energy: thermionic field emission (Chan et al. 1992), space-charge-limited conduction (Sarangi et al. 2000), thermal activation modified by the Poole–Frenkel (PF) effect (Adkins et al. 1970, Egret et al. 1997, Hofsa¨ss 1998, Huang et al. 2000, Khan and Silva 2000), thermally activated transitions between localized defect band states around the Fermi level, tunnelling between neighbouring sp2 clusters (Dasgupta et al. 1991), multistep tunnelling (Hastas et al. 2001), band-to-band tunnelling (Hastas et al. 2002), and variable-range hopping (VRH) in tail states approaching the Fermi level (Hofsa¨ss 1998, Godet 2001). Electric-field enhancement of the spacecharge-limited current (SCLC) due to phonon-assisted tunnelling through a reduced potential barrier of traps has also been proposed, in the absence of a manifestation of the PF mechanism (Bozhko et al. 2002). In a-C films, the observed temperature dependence of the Ohmic conductivity, given by " # T 1=4 ¼ exp 0 , T 00
ð1Þ
has previously been ascribed to a three-dimensional (3D) hopping transport mechanism within band-tail states which gives access to the localization parameter N(EF) 3, where N(EF) is the density of states at the Fermi level and 1 is the localization radius (Godet 2001, 2002a, b, 2003). Our hopping transport model is based on a single-phonon hypothesis, unlike the earlier multiphonon interpretation of hopping transport in a-C (Shimakawa and Miyaka 1988). The electric-field dependence of the current density J(F) (measured up to Fmax ¼5 105 V cm1) is expected to supply additional information on the localized states (Hill 1971b), provided that the transport is limited by bulk mechanisms. From a comparison with existing models, the relationship ¼ n0 exp
F Fn
n ð2Þ
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is followed by our experimental results (where is defined as J/F). This is consistent with 3D transport mechanisms. The J–V characteristics can be interpreted either as detrapping assisted by the electric field (PF effect for charged empty defects) or as field-enhanced hopping out of neutral empty defects (the Apsley–Hughes (AH) (1974, 1975) hopping model). This paper reports the temperature and electric-field dependences of the current density in a-C films with different thicknesses (21 and 42 nm) grown using an electron cyclotron resonance technique. In order to cross-check the experimental results, the current density has been characterized both in low-field coplanar (Al/a-C : H/Al) structures and in high-field transverse (TiW/a-C : H/TiW) geometries. } 2. Experimental details The a-C : H films used for this investigation contain about 30 at.% H. They have been codeposited near room temperature on glass, crystalline silicon (c-Si) and TiW(N) electrodes by using the decomposition of acetylene in an electron cyclotron resonance (ECR) plasma (Godet et al. 2002). The energy of the positively charged ions has been set at 160 eV by polarizing the substrate holder with a radio-frequency bias. The typical optical bandgap E04 ¼ 2.1 eV is the energy corresponding to an absorption coefficient ¼ 1 104 cm1 (Godet et al. 2002). In transverse devices, electric fields (up to 5 105 V cm1) have been applied, as deduced from the applied voltages V and the film thicknesses d ( 5%) determined by ellipsometry (a-C : H/c-Si). The current density has been measured in 0.1 cm gap-cell coplanar (Al/a-C : H/Al) structures on glass substrates and in symmetrical (Al(150 nm)/TiW(N)(15 nm)/ a-C : H/TiW(N)(15 nm)/Al(150 nm)) square-shaped devices obtained at the intersection of perpendicular metal lines (top and bottom electrodes). The width of the intersecting lines defines the active area of the device. The Al/TiW(N) electrodes have been deposited using dc magnetron sputtering and the a-C has been etched using reactive ion etching in a CF4, O2 and argon plasma in order to minimize current leakage in the devices. The TiW(N) layer (Ti, 10 wt%; W, 88 wt%; N, 2 wt%) is important as a diffusion barrier to minimize the migration of metal (aluminium) into the carbon films (Paul and Clough 2002). The current density has been measured as a function of the device area (in the range from 5 mm 5 mm to 200 mm 200 mm), and the symmetry of the J–V characteristics with respect to the voltage polarity has been checked. The advantage of using variable-area devices is twofold: (i) detection of possible pinhole effects; (ii) avoidance of very high currents through the electrometer by using smaller areas. } 3. Field-enhanced transport mechanisms The effects of high fields on the J–V characteristics are generally caused by a change in the (non-equilibrium) distribution function and mobility of carriers, or by a change in the rate of carrier generation or injection, or by both (Kao and Hwang 1981). To obtain a physical interpretation of the dominant transport mechanisms under high electric fields, the first difficulty arises from the usual assumption of uniform electric fields and thermal equilibrium of the carriers. When the energy gained in the field is only partially released to the lattice, the latter hypothesis is
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not fulfilled and a hot-electron theory should then by applied (Fro¨lich 1947, Marianer and Shklovskii 1992, Cleve et al. 1995). In principle, the field-enhanced conductivity can provide valuable information on the charge state of the empty electron states left by the dominant charge carriers, because neutral empty traps can be described by a short-range Dirac potential whereas a long-range potential arises from the mutual Coulombic interaction of carriers leaving charged empty traps. In the following, we examine two classes of carrier detrapping models, corresponding either to charged empty defects (the PF effect) or to neutral empty defects (a model proposed by AH for field-assisted hopping within localized states distributed in energy, and a model for tunnelling through a collection of independent triangular barriers). Both models predict equation (2), which describes our experimental results.
3.1. Poole–Frenkel effect It is important to separate field-enhancement effects related to barrier lowering at the film–contact interface (Schottky (1914)-type thermionic emission and tunnelling near the injecting contact) and at bulk traps (the PF effect (Frenkel 1938)). The Schottky effect is an attenuation of a metal–insulator barrier arising from the electrode image force with electric field, while the PF effect is a lowering of the Coulombic potential barrier in the bulk of the materials. Both effects are due to the Coulombic interaction between the escaping electron and a positive charge, but they differ in that the positive charge is fixed for the PF trapping barrier, while the positive charge is a mobile image for the Schottky barrier. For a distribution of trapped carriers, the trap depth EC ED below the mobility edge EC (or the transport energy Et in the case of fully localized states) is effectively lowered in the field direction to (EC ED) PFF1/2 in an electric field F, giving the onedimensional (1D) PF equation (Frenkel 1938) ! PF F 1=2 ¼ ðTÞ exp : kT 0
ð3Þ
The PF constant is PF ¼ (e3/p"0"r)1/2, where e and "0"r are the elementary charge and the high-frequency dielectric constant of the solid. It has been pointed out (Hartke 1968, Hill 1971a, Ieda et al. 1971) that this calculation corresponds to a 1D approach which overestimates the effective reduction in the trap depth, because only in the direction of the applied field is the edge of the potential well lowered by as much as that given by equation (3). Indeed, a trapped electron can be thermally released not only in the forward direction of the applied field but also in other directions, although the probability of escape is smaller. The argument is as follows. The Coulombic potential F ¼ e2/4p"0"rR eFR is lowered by F ¼ PF(F cos )1/2 which occurs at RPF ¼ (e/4p"0"rF cos )1/2 where is the angle between F and R. The potential barrier is lowered only in the forward direction for 044p/2. In the reverse direction, Ieda et al. (1971) have assumed that there is a state energy denoted by (of the order of kT) in which, by interaction with phonons, the transition probability of an electron at a distance R to become a free or mobile carrier is much larger than that to the trap ground state.
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They have derived an expression for R as R ¼
e2 4"0 "r
and for the increase of the potential barrier in the reverse direction as 2PF F cos : 4 In the forward direction, the effective barrier lowering for RPF4R (high field) is E ¼
EPF ¼ PF ðF cos Þ1=2 and that for RPF 5 R (low field) is given by 2PF F cos : 4 Equation (3) based on a 1D treatment is thus replaced by the following equations based on a 3D treatment: 8 ! 2 > 4 F > 0 > 2 sinh ð4 aÞ for F 1=2 4 2, > > > 4 F > > > < 0 1 1=2 1=2 ¼ 2 ð F 1Þ exp ð F Þ F > > > ! > for F 1=2 5 2, ð4 bÞ > 2 > F > > 2 exp þ 2 þ exp ðÞ > : 4 EPF ¼
where ¼ PF/kT and ¼ /kT. At low fields, equation (4 b) is slightly different from the original paper by Ieda et al. It can be seen that at very low fields the conductivity follows a simple Ohm’s law and that at high fields it approaches asymptotically equation (3). In addition, an F2 dependence of ln (/ 0) is predicted at low and intermediate fields. 3.2. Field-assisted variable-range hopping For localized states distributed in energy, as occurs in molecular solids and disordered materials, the carrier mobility is described by the concepts of wavefunction localization and quantum tunnelling (hopping) electronic transport (Mott and Davis 1979). Hopping stands for phonon-assisted tunnelling transitions from occupied to unoccupied state, the state energy difference being bridged by absorption or emission of a phonon. The transition rates are chosen to be consistent with the detailed equilibrium principle (Ambegaokar et al. 1971). At very low fields, Mott (1969) has predicted an Ohmic conductivity given by equation (1) for VRH among a uniform distribution of neutral empty traps within a 3D space. The relationship between the slope T01=4 and the prefactor 00, as a function of variations in the parameter N(EF) 3, is a very sensitive signature to discriminate hopping within a uniform density of states (DOS) at the Fermi level and band-tail hopping within an exponential distribution of localized states (Godet 2002a). Interestingly, the filling rate approach provides a correct order of magnitude of the conductivity prefactor using physical DOS parameters (Godet 2001, 2003). Within the single-phonon hopping transition theory, the field dependence of conductivity has been considered along two main directions. The percolation theory
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has been used with the critical path approach including directional constraints created by the spatial correlations (Pollak and Riess 1976, Van der Meer et al. 1982). Alternatively, a modelling based on the weighted integration of the partial carrier mobility for each energy level has been developed by AH (1974, 1975). Both calculations were performed in three dimensions for a uniform density of states distribution. In both models, at very high fields, the conductivity is described by a temperature-independent equation: ln (F/Fn)1/4. It is emphasized that this analytic dependence is not predicted by the PF equation which describes merely thermally activated jumps over a barrier. However, at low and intermediate fields (eF/kT
