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Microstructural relaxation of hydrogenated amorphous carbon thin films
Thin Solid Films 482 (2005) 90 – 93 www.elsevier.com/locate/tsf
Microstructural relaxation of hydrogenated amorphous carbon thin films R. Bouzerar*, M. Benlahsen, J.C. Picot Laboratoire de Physique de la Matie`re Condense´e, Universite´ de Picardie, UFR Sciences, 80039 Amiens, France Available online 15 December 2004
Abstract Capacitive spectroscopy measurements carried out on small and wide gaps as-deposited hydrogenated amorphous carbon thin films evidence a long-time evolution of the capacitive response of the films. The reported phenomenon analyzed within the classical Cohen–Lang model evidences a slow variation with time of the density of states at the Fermi level indicating a likely defect creation within the films. Based on this conclusion, we built up a model describing the influence of the viscous stress field on the electronic properties. D 2004 Elsevier B.V. All rights reserved. Keywords: Relaxation; Carbon thin films; Capacitive response
1. Introduction Many plasma-based deposition techniques such as radiofrequency magnetron sputtering or ECR-RF glow discharge plasma [1] lead to the bgrowthQ of strongly disordered solid films. Two disordering processes contribute to the residual stresses. The impinging species penetrate the film subsurface inducing strong atomic displacements. This first process results in strong frozen-in spatial fluctuations of the short-range order [2]. The second process consists of defect creation such as dangling bonds [3] and paramagnetic centers in a-C:H films of all types [4]. These two types of disorders might be distinguished by the way they influence the electronic spectrum: Structural disorder leads to a broadening of the band-edges generating band-tail states [5] while defect offers additional states lying around the Fermi level [3]. We intend here to show that strongly disordered a:C:H thin films exhibit ageing effects such as in glassy systems. We prove that the reported long-time evolution of the films is compatible with a stress field relaxation likely in keeping with some defect production within our films. We built up a
* Corresponding author. Tel.: +33 322 82 76 24; fax: +33 322 82 78 91. E-mail address: [email protected] (R. Bouzerar). 0040-6090/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2004.11.124
phenomenological model accounting for the influence of the residual stress field on the electronic density of states.
2. Experimental details and results The hydrogenated amorphous carbon thin films were prepared by plasma-enhanced chemical vapor deposition (PECVD) of methane at low pressure (0.35 Pa) in a dual electron cyclotron resonance–radiofrequency glow discharge system. The full characterization of the samples by IR spectroscopy, TDMS spectroscopy and optical absoption was reported in a previous paper [6]. The reader will also find in this paper some details regarding the analysis of the capacitance curves within the Cohen and Lang model [7]. The typical thermal evolution of the capacitance C(T )— imaginary part of the admittance—of the films is presented in Fig. 1 (small gaps samples) and Fig. 2 (wide gaps samples). Both curves exhibit a low temperature plateau stretching here over the temperature range 20–150 K followed at higher temperature by an increase of the capacitive response up to one order of magnitude (thermal emission processes). The ageing effect is clearly present in both series consisting of a capacitance increasing with time. Nevertheless, these likely relaxational processes are characterized by two very different time scales: the evolution is
R. Bouzerar et al. / Thin Solid Films 482 (2005) 90–93
91
defect production–annihilation kinetics. Oppositely, this kinetics might influence the relaxational dynamics. We propose in this section a simplified phenomenological model accounting for the dependence of the stress relaxation upon the defect content of the films. 4.1. The simplified electron Hamiltonian
Fig. 1. Thermal evolution of the capacitive response (in F) of the small gaps samples at an excitation frequency f=130 Hz. The lower curve was picked up after the deposition. The other ones were recorded 6 and 8 weeks later.
clearly faster in the wide gaps series (typically a few hours) and far much longer (up to several weeks) in the small gaps samples.
3. Ageing effect and structural relaxation In this section, we focus on the ageing effects. These effects suggest a coupling between the electronic properties of the films and the mechanical ones. More precisely, the ageing effect might reflect the relaxation of the built-in stress field. We will show in this section how the overall density of states of the films depend on their stress state and how the electronic properties influence the relaxational dynamics. In amorphous systems, the stress relaxation reflects a local ordering of the microstructure by releasing the bonds’ length and orientation distorsions (this is one of the main contributions to the stress field). If it was the only process (ordering) to occur, we should have observed a substantial decrease of the density of states at the Fermi level. We thus hypothesize that the reported increase of the capacitance with time reflects defect creation accompanying the stress field relaxation. Nevertheless, the nature of these defects (paramagnetic, five- or sevenfold rings, etc.) cannot be deduced from the present experiments. Indeed, a study of the effect of annealing on the films’ properties coupled to ESR measurements could be a good test for this hypothesis since annealing is a good way to achieve the stress relaxation and determine the interplay between ordering and disordering processes.
In a dynamical regime, the stress tensor Aij (including the residual stress) of any solid medium acquires a timedependent part, the so-called P Y viscousYstress component [8] r ijV so that it reads ij ðr ; t Þ ¼ rij ðr Þ þ rijVðt Þ where the first term is the static residual stress field distribution. In the weak amplitude stress limit, the viscous stress depends Bvj i linearly on the strain rate tensor e˙ ij u 12 Bv þ where the Bxj Bxi Y vj ðr ; t Þ are the components of the velocity field at any point and at time t. This reads rijV ¼ gijkl e˙ kl where the Einstein’s convention of summation over repeated indices was used. In an amorphous and homogeneous medium (that is the case of our films), the four-rank viscosity tensor g ijkl possessing only two independent components g and n simplifies to: 1 ð1Þ rijV ¼ 2g e˙ ij e˙ ðt Þdij þ n˙e ðt Þdij 3 P where e˙ ðt Þ ¼ l e˙ ll is the trace of the strain rate tensor corresponding to a volume variation rate. We now assume of the system occupies a wavefunction / that any electron Y Y rY þ Rl localized around an atom at position Rl . Because of the structural relaxation process, the atomic positions vary slightly with time. The total interaction HamiltonianP of the electrons with the atoms reads then H ¼ int l;k P Y Y Y where the fRk g1VkVN are the atomic rY þ R R l
k
positions where indices l and k run over neighbouring atomic positions due to the strongly localized nature of the electronic states. The atomic relative positions can be Y Y Y decomposed into two terms, namely Rkl ¼ Rl Y0 Rk ¼ Y0 Rkl þ uY RY0 ; t uY RY0 ; t where the terms Rkl refer l k to the realization of the ideal local short-range order
4. A model for the coupling between electronic and mechanical properties The ordering process is thus limited by these defects arising from the stress field relaxation. According to this point of view, the stress relaxation process controls the
Fig. 2. Thermal evolution of the capacitive response (in F) of the wide gaps samples at an excitation frequency f=130 Hz taken after the deposition and a few hours later.
92
R. Bouzerar et al. / Thin Solid Films 482 (2005) 90–93
(local relaxation of the distorted bonds). The relative Y Y displacement field uY ðRl0 ; tÞ uY ðRk0 ; tÞ ¼ d uYl ðt Þ d uYk Y Y0 Y Y0 ðt Þ þ U ðRl ; tÞ U ðRk ; tÞ; contains a fast varying component (phonon contribution) generating small amplitude random fluctuations of the short-range order. We will give up this noise term by averaging the displacement field over a long enough time. The second term is a slowly varying displacement responsible for the relaxation of the structure which might be rewritten in first approximation, X Y0 Y Y Y0 ðU ðt Þ U ðt ÞÞa c ð Rkl Þb eab ð Rk ; tÞ ð2Þ l
k
b
Making use of the elastic constants, this expression can be equivalently expressed with the stress tensor (Aab ) components leading to the P final form of theP approximate interaction Y Y Hamiltonian Hint c l;k PðrY ; Rkl0 ; ab ðRk0 ; tÞÞuK. We will address now the problem of the calculation of the density of states, essential for the interpretation of the capacitive data. The small amplitude of the viscous stresses (the relaxation is very low) allows a power expansion of this Hamiltonian up to first order, X Y Y0 X Y Y0 Hint c Pð r ; Rkl ; rab ðR 0k ÞÞ þ bklij rijVðRk ; tÞ ð3Þ l;k l;k;i;j ! Y P BP Y r ;Rkl0 ; P ab where the symbols bklij ¼ are in fact B
ij
rV¼0
the components of a tensor operator. The extra term in the Hamiltonian acquires a time dependence induced by the viscous stress tensor. This term corresponds in fact to the above-mentioned slow evolution of the environment of the electrons. The b operators depend on the static part of the stress tensor, and consequently on the structural realization of disorder within our system. 4.2. A simplified treatment of the coupling The treatment of this slowly varying perturbation being difficult, we will adopt a phenomenological point of view which allows us to bypass this necessity of modelling of the structural disorder. With this aim in view, we assume that the viscous stresses vary slowly in space. Within this approximation, the local electronic density of states reads, Y Y G E; r cG 0 E U r ; t ð4Þ where G 0 is the density of states of the system in the absence of viscous stresses and the function X Y Y U r ;t ¼ h/E jbˆ klij r j/E irijVðt Þ ð5Þ l;k;i;j
is expressed as a linear combination of the components of the viscous stress whose coefficients are the expected values of the b operators in the electron eigenstates / E of energy E. In fact, this result is obtained through a perturbative expansion of the viscous term truncated to first order.
R The mean electronic density is n¯ ðt Þ ¼ VNðtÞ u VI film n Y R r ; t Þd3 rY with a local density n ðrY ; t Þ ¼ Spect f ð E ÞG 0 ðE UðrY ; t ÞÞdE . Using the usualR properties of the Fermi– l P Dirac function f, we obtain n¯ c l G0 ð E UðrY ; t ÞÞ dE, the bar indicating an averaging over the film volume. Because of the low relaxation rate, the function U is small (that is |U|Vl) allowing an expansion in powers of U truncated to first order in such a way that N N c G0 ðlÞU¯ ðt Þ ð6Þ V V0 This might be rewritten in the form N V 1ðtÞ V10 c DV ðt Þ U¯ ðt ÞG0 ðlÞ or, more explicitly, noticing that N V2 ¼ P 0 D V ðt Þ=V0 ¼ l ell ðt Þueðt Þis the global relative deformation, XP U¯ ðt Þ ¼ bklij rijVðt Þ ¼ n¯ 0 eðt Þ=G0 ðlÞ ð7Þ n¯ ¼
l;k;i;j
where the tensor b is evaluated around the Fermi level Ecl. This relationship is interesting since it establishes a kind of linear variation of the viscous stress with the global deformation analogous to the Hooke’s law but with a mechanical modulus depending on the density of states at the Fermi level l. In fact, Eq. (7) is a relaxation equation as can be seen by inserting Eq. (1) into it: XP XP 2g bklij e˙ ij þ ðn 2g=3Þ˙e ðt Þ bklii ¼ n¯ 0 eðt Þ=G0 ðlÞ l;k;i;j
l;k;i
ð8Þ The first term in the left-hand side can be decomposed into two terms by distinguishing the i=j and ipj contributions: X P XP 2g bklij e˙ ij þ n˙e ðt Þ bklii ¼ n¯ 0 eðt Þ=G0 ðlÞ ð9Þ l;k;i;j;ipj
l;k;i
where we noticed that for a quasi-homogeneous film 1 e˙P ii ¼ 3 e˙ ðt Þ for each index i. This requires the condition P l;k;i bklii b0 (since nN0 as expected for a viscosity) to hold for any relaxation process to be present (this is in fact a kind of mechanical stability P condition), characP terized by a time scale sfnj l;k;i blkii jG0 ðlÞ=¯n 0 proportional to the viscosity and the density of states at the Fermi level. The connection with defect configurations is made natural by this expression, and more especially by the dependence of the time scale s upon the density of states G 0(l). This connection with defects is reinforced by the dependence upon viscosity which is a signature of the defect creation/annihilation kinetics.
5. Connection with the defect kinetics Following Robertson [3], we will define a defect configuration of an amorphous system as an atomic configuration creating states deep in the bandgap, which is around the Fermi energy l. It is usually believed that the two essential contributions to the overall density of localized
R. Bouzerar et al. / Thin Solid Films 482 (2005) 90–93
states in the bandgap arise from structural distorsions (that is frozen-in spatial fluctuations of the bond lengths and orientations) and from these defects. Structural distorsions lead to a localisation of the electronic states close to the band-edges which tail into the pseudogap. These states are characterized by a DOS G1 which is predominant near the band-edges and which overlaps only weakly near the Fermi level. Conversely, the deep states arising from defect configurations overlap strongly near the Fermi level (deep traps) where it gives birth to a defect band. The DOS G2 associated with these defects is the dominating contribution around l. It is reasonable to assume that the two types of bands do not overlap in such a way that the overall DOS is the sum of the above-mentioned contributions. In spite of this simplification, it is clear that the characteristic time s is dominated by the defect content. A high enough defect density, or equivalently G 0(l), induces a slow relaxational dynamics (time s long) since the atomic configurations responsible for the defect structure require a high energy to be relaxed. We intend now to derive from Eq. (9) a phenomenological equation controlling the defect kinetics. Eq. (9) shows that only states around the Fermi level l are involved in the relaxational dynamics. As was previously noticed, these states dominate the defect band. The defect density at any point ofR the film should evolve slowly with time as ND ðrY ; t Þ ¼ Defband G0 ð E UðrY ; t ÞÞdE since for deep states G 0(E)cG 2(E). As these defect states are lying around E=l, we assume this band to be dnarrowT enough of small width U for theP average defect density to be simplified into N¯ D ðt ÞcU G0 ðl UðrY ; t ÞÞ . U is in fact a characteristic energy whose nature depends on the type of defect. The evolution of the density is thus dN¯ D =dt ¼ U
P 0 Y
BU BG ðl Uðr ; t ÞÞ : Bt Bl
ð10Þ
After a straightforward expansion, the last expression reads,
93
property in Eq. (10), we obtain the kinetic equation governing the defect creation/annihilation rate: XP bklij r˙ ij V þ U G0W ðlÞ dN¯ D =dt ¼ U G0V ðlÞ klij
XP b2klij rijVr˙ ijV
ð12Þ
klij
6. Conclusion There are mainly three interesting aspects in our model. The first consists of the prediction of the relaxation time PoneP sfnj l;k;i blkii jG0 ðlÞ=¯n 0 of the viscous stress, which appears to increase with the DOS at the Fermi level. This point agrees with the observation of a slower relaxation of the denser films, characterized by a highest DOS. The second point regards the dependence of the defect creation/ annihilation rate upon the shape of the density of states around the Fermi energy l, that is, the first and second derivatives of the DOS. The behaviour of the solutions is actually in progress. A final consequence of our model is the direct connection between the capacitance of the films as revealed by admittance spectroscopy and the defect creation kinetics. Indeed, the squared capacitance per unit area reads at low enough temperature (low temperature plateau) P e e q2 P Y 0 r C 2 ¼ e 0 e r q2 G l / r ; t c ND ðt Þ U
ð13Þ
It is thus clear that an increase of the capacitance with time reflects a defect creation and oppositely annihilation kinetics will manifest as a decrease.
References
P
dU¯ BU þ U G0W ðlÞ U ð11Þ dN¯ D =dt ¼ U G0V dt Bt P P P P Y Y ˙Vi Vj V in which U BU ikjll iVjVkVlV bklij ðr ÞbkVlViVjV ðr Þ rijVðt Þr Bt ¼ and G8V and G 0W are the first and second derivatives of the DOS. The kinetic Eq. (11) is rather complex because of the second term of the r.h.s. If the b klij are independent components (we just have to remember that these terms acquire a random part due to their dependence upon the static stress influenced by disorder), we can write P P bklij ðrY ÞbkVlViVjV ðrY Þ ¼ b2klij dkkV dllV diiV djjV . Making use of this
[1] R.J. Schul, S.J. Pearton, Handbook of Advanced Plasma Processing Techniques, Springer, 2000. [2] D. Adler, in: D. Adler, H. Fritsche, S.R. Ovshinsky (Eds.), Physics of Disordered Materials, Plenum Press, New York, 1985. [3] J. Robertson, Philos. Mag., B 76 (1997) 335. [4] R. Kleber, K. Jung, H. Erhardt, J. Muhling, K. Breuer, H. Metz, F. Engelke, Thin Solid Films 205 (1991) 274. [5] P. Van Mieghem, Rev. Mod. Phys. 64 (3) (1992) 755. [6] R. Bouzerar, M. Benlahsen, B. Racine, J.C. Picot, J. Appl. Phys. 93 (2003) 11. [7] J.D. Cohen, D.V. Lang, J.P. Harbison, Phys. Rev., B 25 (1982) 5285. [8] L. Landau, E. Lifschitz, Theory of Elasticity, MIR, Moscou, 1967.
Microstructural relaxation of hydrogenated amorphous carbon thin films R. Bouzerar*, M. Benlahsen, J.C. Picot Laboratoire de Physique de la Matie`re Condense´e, Universite´ de Picardie, UFR Sciences, 80039 Amiens, France Available online 15 December 2004
Abstract Capacitive spectroscopy measurements carried out on small and wide gaps as-deposited hydrogenated amorphous carbon thin films evidence a long-time evolution of the capacitive response of the films. The reported phenomenon analyzed within the classical Cohen–Lang model evidences a slow variation with time of the density of states at the Fermi level indicating a likely defect creation within the films. Based on this conclusion, we built up a model describing the influence of the viscous stress field on the electronic properties. D 2004 Elsevier B.V. All rights reserved. Keywords: Relaxation; Carbon thin films; Capacitive response
1. Introduction Many plasma-based deposition techniques such as radiofrequency magnetron sputtering or ECR-RF glow discharge plasma [1] lead to the bgrowthQ of strongly disordered solid films. Two disordering processes contribute to the residual stresses. The impinging species penetrate the film subsurface inducing strong atomic displacements. This first process results in strong frozen-in spatial fluctuations of the short-range order [2]. The second process consists of defect creation such as dangling bonds [3] and paramagnetic centers in a-C:H films of all types [4]. These two types of disorders might be distinguished by the way they influence the electronic spectrum: Structural disorder leads to a broadening of the band-edges generating band-tail states [5] while defect offers additional states lying around the Fermi level [3]. We intend here to show that strongly disordered a:C:H thin films exhibit ageing effects such as in glassy systems. We prove that the reported long-time evolution of the films is compatible with a stress field relaxation likely in keeping with some defect production within our films. We built up a
* Corresponding author. Tel.: +33 322 82 76 24; fax: +33 322 82 78 91. E-mail address: [email protected] (R. Bouzerar). 0040-6090/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2004.11.124
phenomenological model accounting for the influence of the residual stress field on the electronic density of states.
2. Experimental details and results The hydrogenated amorphous carbon thin films were prepared by plasma-enhanced chemical vapor deposition (PECVD) of methane at low pressure (0.35 Pa) in a dual electron cyclotron resonance–radiofrequency glow discharge system. The full characterization of the samples by IR spectroscopy, TDMS spectroscopy and optical absoption was reported in a previous paper [6]. The reader will also find in this paper some details regarding the analysis of the capacitance curves within the Cohen and Lang model [7]. The typical thermal evolution of the capacitance C(T )— imaginary part of the admittance—of the films is presented in Fig. 1 (small gaps samples) and Fig. 2 (wide gaps samples). Both curves exhibit a low temperature plateau stretching here over the temperature range 20–150 K followed at higher temperature by an increase of the capacitive response up to one order of magnitude (thermal emission processes). The ageing effect is clearly present in both series consisting of a capacitance increasing with time. Nevertheless, these likely relaxational processes are characterized by two very different time scales: the evolution is
R. Bouzerar et al. / Thin Solid Films 482 (2005) 90–93
91
defect production–annihilation kinetics. Oppositely, this kinetics might influence the relaxational dynamics. We propose in this section a simplified phenomenological model accounting for the dependence of the stress relaxation upon the defect content of the films. 4.1. The simplified electron Hamiltonian
Fig. 1. Thermal evolution of the capacitive response (in F) of the small gaps samples at an excitation frequency f=130 Hz. The lower curve was picked up after the deposition. The other ones were recorded 6 and 8 weeks later.
clearly faster in the wide gaps series (typically a few hours) and far much longer (up to several weeks) in the small gaps samples.
3. Ageing effect and structural relaxation In this section, we focus on the ageing effects. These effects suggest a coupling between the electronic properties of the films and the mechanical ones. More precisely, the ageing effect might reflect the relaxation of the built-in stress field. We will show in this section how the overall density of states of the films depend on their stress state and how the electronic properties influence the relaxational dynamics. In amorphous systems, the stress relaxation reflects a local ordering of the microstructure by releasing the bonds’ length and orientation distorsions (this is one of the main contributions to the stress field). If it was the only process (ordering) to occur, we should have observed a substantial decrease of the density of states at the Fermi level. We thus hypothesize that the reported increase of the capacitance with time reflects defect creation accompanying the stress field relaxation. Nevertheless, the nature of these defects (paramagnetic, five- or sevenfold rings, etc.) cannot be deduced from the present experiments. Indeed, a study of the effect of annealing on the films’ properties coupled to ESR measurements could be a good test for this hypothesis since annealing is a good way to achieve the stress relaxation and determine the interplay between ordering and disordering processes.
In a dynamical regime, the stress tensor Aij (including the residual stress) of any solid medium acquires a timedependent part, the so-called P Y viscousYstress component [8] r ijV so that it reads ij ðr ; t Þ ¼ rij ðr Þ þ rijVðt Þ where the first term is the static residual stress field distribution. In the weak amplitude stress limit, the viscous stress depends Bvj i linearly on the strain rate tensor e˙ ij u 12 Bv þ where the Bxj Bxi Y vj ðr ; t Þ are the components of the velocity field at any point and at time t. This reads rijV ¼ gijkl e˙ kl where the Einstein’s convention of summation over repeated indices was used. In an amorphous and homogeneous medium (that is the case of our films), the four-rank viscosity tensor g ijkl possessing only two independent components g and n simplifies to: 1 ð1Þ rijV ¼ 2g e˙ ij e˙ ðt Þdij þ n˙e ðt Þdij 3 P where e˙ ðt Þ ¼ l e˙ ll is the trace of the strain rate tensor corresponding to a volume variation rate. We now assume of the system occupies a wavefunction / that any electron Y Y rY þ Rl localized around an atom at position Rl . Because of the structural relaxation process, the atomic positions vary slightly with time. The total interaction HamiltonianP of the electrons with the atoms reads then H ¼ int l;k P Y Y Y where the fRk g1VkVN are the atomic rY þ R R l
k
positions where indices l and k run over neighbouring atomic positions due to the strongly localized nature of the electronic states. The atomic relative positions can be Y Y Y decomposed into two terms, namely Rkl ¼ Rl Y0 Rk ¼ Y0 Rkl þ uY RY0 ; t uY RY0 ; t where the terms Rkl refer l k to the realization of the ideal local short-range order
4. A model for the coupling between electronic and mechanical properties The ordering process is thus limited by these defects arising from the stress field relaxation. According to this point of view, the stress relaxation process controls the
Fig. 2. Thermal evolution of the capacitive response (in F) of the wide gaps samples at an excitation frequency f=130 Hz taken after the deposition and a few hours later.
92
R. Bouzerar et al. / Thin Solid Films 482 (2005) 90–93
(local relaxation of the distorted bonds). The relative Y Y displacement field uY ðRl0 ; tÞ uY ðRk0 ; tÞ ¼ d uYl ðt Þ d uYk Y Y0 Y Y0 ðt Þ þ U ðRl ; tÞ U ðRk ; tÞ; contains a fast varying component (phonon contribution) generating small amplitude random fluctuations of the short-range order. We will give up this noise term by averaging the displacement field over a long enough time. The second term is a slowly varying displacement responsible for the relaxation of the structure which might be rewritten in first approximation, X Y0 Y Y Y0 ðU ðt Þ U ðt ÞÞa c ð Rkl Þb eab ð Rk ; tÞ ð2Þ l
k
b
Making use of the elastic constants, this expression can be equivalently expressed with the stress tensor (Aab ) components leading to the P final form of theP approximate interaction Y Y Hamiltonian Hint c l;k PðrY ; Rkl0 ; ab ðRk0 ; tÞÞuK. We will address now the problem of the calculation of the density of states, essential for the interpretation of the capacitive data. The small amplitude of the viscous stresses (the relaxation is very low) allows a power expansion of this Hamiltonian up to first order, X Y Y0 X Y Y0 Hint c Pð r ; Rkl ; rab ðR 0k ÞÞ þ bklij rijVðRk ; tÞ ð3Þ l;k l;k;i;j ! Y P BP Y r ;Rkl0 ; P ab where the symbols bklij ¼ are in fact B
ij
rV¼0
the components of a tensor operator. The extra term in the Hamiltonian acquires a time dependence induced by the viscous stress tensor. This term corresponds in fact to the above-mentioned slow evolution of the environment of the electrons. The b operators depend on the static part of the stress tensor, and consequently on the structural realization of disorder within our system. 4.2. A simplified treatment of the coupling The treatment of this slowly varying perturbation being difficult, we will adopt a phenomenological point of view which allows us to bypass this necessity of modelling of the structural disorder. With this aim in view, we assume that the viscous stresses vary slowly in space. Within this approximation, the local electronic density of states reads, Y Y G E; r cG 0 E U r ; t ð4Þ where G 0 is the density of states of the system in the absence of viscous stresses and the function X Y Y U r ;t ¼ h/E jbˆ klij r j/E irijVðt Þ ð5Þ l;k;i;j
is expressed as a linear combination of the components of the viscous stress whose coefficients are the expected values of the b operators in the electron eigenstates / E of energy E. In fact, this result is obtained through a perturbative expansion of the viscous term truncated to first order.
R The mean electronic density is n¯ ðt Þ ¼ VNðtÞ u VI film n Y R r ; t Þd3 rY with a local density n ðrY ; t Þ ¼ Spect f ð E ÞG 0 ðE UðrY ; t ÞÞdE . Using the usualR properties of the Fermi– l P Dirac function f, we obtain n¯ c l G0 ð E UðrY ; t ÞÞ dE, the bar indicating an averaging over the film volume. Because of the low relaxation rate, the function U is small (that is |U|Vl) allowing an expansion in powers of U truncated to first order in such a way that N N c G0 ðlÞU¯ ðt Þ ð6Þ V V0 This might be rewritten in the form N V 1ðtÞ V10 c DV ðt Þ U¯ ðt ÞG0 ðlÞ or, more explicitly, noticing that N V2 ¼ P 0 D V ðt Þ=V0 ¼ l ell ðt Þueðt Þis the global relative deformation, XP U¯ ðt Þ ¼ bklij rijVðt Þ ¼ n¯ 0 eðt Þ=G0 ðlÞ ð7Þ n¯ ¼
l;k;i;j
where the tensor b is evaluated around the Fermi level Ecl. This relationship is interesting since it establishes a kind of linear variation of the viscous stress with the global deformation analogous to the Hooke’s law but with a mechanical modulus depending on the density of states at the Fermi level l. In fact, Eq. (7) is a relaxation equation as can be seen by inserting Eq. (1) into it: XP XP 2g bklij e˙ ij þ ðn 2g=3Þ˙e ðt Þ bklii ¼ n¯ 0 eðt Þ=G0 ðlÞ l;k;i;j
l;k;i
ð8Þ The first term in the left-hand side can be decomposed into two terms by distinguishing the i=j and ipj contributions: X P XP 2g bklij e˙ ij þ n˙e ðt Þ bklii ¼ n¯ 0 eðt Þ=G0 ðlÞ ð9Þ l;k;i;j;ipj
l;k;i
where we noticed that for a quasi-homogeneous film 1 e˙P ii ¼ 3 e˙ ðt Þ for each index i. This requires the condition P l;k;i bklii b0 (since nN0 as expected for a viscosity) to hold for any relaxation process to be present (this is in fact a kind of mechanical stability P condition), characP terized by a time scale sfnj l;k;i blkii jG0 ðlÞ=¯n 0 proportional to the viscosity and the density of states at the Fermi level. The connection with defect configurations is made natural by this expression, and more especially by the dependence of the time scale s upon the density of states G 0(l). This connection with defects is reinforced by the dependence upon viscosity which is a signature of the defect creation/annihilation kinetics.
5. Connection with the defect kinetics Following Robertson [3], we will define a defect configuration of an amorphous system as an atomic configuration creating states deep in the bandgap, which is around the Fermi energy l. It is usually believed that the two essential contributions to the overall density of localized
R. Bouzerar et al. / Thin Solid Films 482 (2005) 90–93
states in the bandgap arise from structural distorsions (that is frozen-in spatial fluctuations of the bond lengths and orientations) and from these defects. Structural distorsions lead to a localisation of the electronic states close to the band-edges which tail into the pseudogap. These states are characterized by a DOS G1 which is predominant near the band-edges and which overlaps only weakly near the Fermi level. Conversely, the deep states arising from defect configurations overlap strongly near the Fermi level (deep traps) where it gives birth to a defect band. The DOS G2 associated with these defects is the dominating contribution around l. It is reasonable to assume that the two types of bands do not overlap in such a way that the overall DOS is the sum of the above-mentioned contributions. In spite of this simplification, it is clear that the characteristic time s is dominated by the defect content. A high enough defect density, or equivalently G 0(l), induces a slow relaxational dynamics (time s long) since the atomic configurations responsible for the defect structure require a high energy to be relaxed. We intend now to derive from Eq. (9) a phenomenological equation controlling the defect kinetics. Eq. (9) shows that only states around the Fermi level l are involved in the relaxational dynamics. As was previously noticed, these states dominate the defect band. The defect density at any point ofR the film should evolve slowly with time as ND ðrY ; t Þ ¼ Defband G0 ð E UðrY ; t ÞÞdE since for deep states G 0(E)cG 2(E). As these defect states are lying around E=l, we assume this band to be dnarrowT enough of small width U for theP average defect density to be simplified into N¯ D ðt ÞcU G0 ðl UðrY ; t ÞÞ . U is in fact a characteristic energy whose nature depends on the type of defect. The evolution of the density is thus dN¯ D =dt ¼ U
P 0 Y
BU BG ðl Uðr ; t ÞÞ : Bt Bl
ð10Þ
After a straightforward expansion, the last expression reads,
93
property in Eq. (10), we obtain the kinetic equation governing the defect creation/annihilation rate: XP bklij r˙ ij V þ U G0W ðlÞ dN¯ D =dt ¼ U G0V ðlÞ klij
XP b2klij rijVr˙ ijV
ð12Þ
klij
6. Conclusion There are mainly three interesting aspects in our model. The first consists of the prediction of the relaxation time PoneP sfnj l;k;i blkii jG0 ðlÞ=¯n 0 of the viscous stress, which appears to increase with the DOS at the Fermi level. This point agrees with the observation of a slower relaxation of the denser films, characterized by a highest DOS. The second point regards the dependence of the defect creation/ annihilation rate upon the shape of the density of states around the Fermi energy l, that is, the first and second derivatives of the DOS. The behaviour of the solutions is actually in progress. A final consequence of our model is the direct connection between the capacitance of the films as revealed by admittance spectroscopy and the defect creation kinetics. Indeed, the squared capacitance per unit area reads at low enough temperature (low temperature plateau) P e e q2 P Y 0 r C 2 ¼ e 0 e r q2 G l / r ; t c ND ðt Þ U
ð13Þ
It is thus clear that an increase of the capacitance with time reflects a defect creation and oppositely annihilation kinetics will manifest as a decrease.
References
P
dU¯ BU þ U G0W ðlÞ U ð11Þ dN¯ D =dt ¼ U G0V dt Bt P P P P Y Y ˙Vi Vj V in which U BU ikjll iVjVkVlV bklij ðr ÞbkVlViVjV ðr Þ rijVðt Þr Bt ¼ and G8V and G 0W are the first and second derivatives of the DOS. The kinetic Eq. (11) is rather complex because of the second term of the r.h.s. If the b klij are independent components (we just have to remember that these terms acquire a random part due to their dependence upon the static stress influenced by disorder), we can write P P bklij ðrY ÞbkVlViVjV ðrY Þ ¼ b2klij dkkV dllV diiV djjV . Making use of this
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