N Pham † ‡

P Roca

Cabarrocas

A Hadjadj
email: aomar.hadjadj@univ-reims.fr

A Beorchia

F Kail

Hydrogen evolution during deposition of microcrystalline silicon by chemical transport

Keywords: amorphous silicon, diffusion, hydrogenation, microcrystalline silicon, plasma processing amorphous silicon, diffusion, hydrogenation hydrogen diffusion, amorphous silicon, microcrystalline silicon, hydrogen solubility

We have exposed a freshly deposited boron-doped hydrogenated amorphous silicon (a-Si:H) layer to a hydrogen plasma under conditions of chemical transport. In situ spectroscopic ellipsometry measurements revealed that atomic hydrogen impinging on the film surface behaves differently before and after crystallization. First, the plasma exposure increases the hydrogen solubility in the a-Si:H network leading to the formation of a hydrogen-rich subsurface layer. Then, once the crystallization process engages, the exceeding hydrogen starts to leave the sample. We have attributed this unusual evolution of the exceeding hydrogen to the role of the grown hydrogenated microcrystalline (µc-Si:H) layer that gradually prevents the atomic hydrogen coming from the plasma to reach the µc-Si:H/a-Si:H interface. Consequently, the hydrogen solubility, initially increased by the hydrogen plasma, recovers its initial value of an untreated a-Si:H material. To support the idea that the out-diffusion is a consequence and not the cause of the growth of the µc-Si:H layer, we have solved the combined diffusion and trapping equations that govern the hydrogen diffused into the sample,

It is well-known that hydrogen diffusion is a crucial step in film growth by chemical vapour deposition of hydrogenated amorphous silicon (a-Si:H) thin films. The interaction of atomic hydrogen with a-S:H films has been widely investigated in order to understand the kinetics of H incorporation in the film [1]. The hydrogen injection is performed either from a solid source (a deuterated a-Si:H layer) or from a gas-phase source (a H-or D-plasma or heated filament). The long range motion of hydrogen is also often evoked in the light-induced metastability in a-Si:H [2]. A better understanding of H interaction with a-Si:H will facilitate the growth of a-Si:H related materials and devices with improved electronic properties and stability.

The solid phase transformation of a-Si:H to hydrogenated microcrystalline silicon (µc-Si:H) in low temperature plasma conditions has been the subject of a large number of studies [3][4][5][6][7][8][9][10]. All the growth models of µc-Si:H agree on the fact that hydrogen is an essential ingredient in the plasma enhanced chemical vapour deposition (PECVD) of such a material [3,[11][12][13][14][15]. However, its exact role is still subject of broad controversies. The reasons preventing the extraction of an unambiguous relationship between the crystallites formation and the hydrogen evolution are related to the complexity of the overall PECVD process: growth via different Si precursors, hydrogen etching of weak Si-Si bonds, exothermic surface reactions of hydrogen, …

In previous studies [16,17], we have shown that when an a-Si:H thin film is exposed to a hydrogen plasma, two processes can occur. When the walls of the reactor are clean, the hydrogen plasma leads to the etching of the a-Si:H film deposited on the heated substrate and the formation of a H-rich subsurface layer. On contrary, if the walls were previously coated with a-Si:H, the H plasma leads to the deposition of µc-Si:H by chemical transport [START_REF] Kail | Proceedings of 19 th European photovoltaic Solar Energy Conference[END_REF][START_REF] Kail | [END_REF]. In this last case, silicon is etched from the reactor walls by hydrogen and deposited on the heated substrate, leading to the formation of a µc-Si:H film. With the beginning of the microcrystalline layer deposition, both the thickness and H excess in the subsurface layer decrease. The hydrogen out-diffusion, inconsistent with a normal diffusion process, continues until the H-rich subsurface layer completely disappears. Although this phenomenon is, without any doubt, intimately linked to the amorphous microcrystalline phase-transformation of the a-Si:H network, the mechanism of such a relationship remains unsolved. The purpose of this article is to prove that the out-diffusion of hydrogen can be explained thanks to the role of the grown µc-Si:H layer that gradually reduces the hydrogen supply from the plasma and then decreases the solubility of H in Si until it recovers the value of an untreated a-Si:H.

Experimental

Boron-doped a-Si:H is deposited in a conventional 13,56 MHz radio frequency (rf)

PECVD system on 1737 a Corning glass substrate. The deposition conditions were: 71 mTorr of pure silane, 3W of rf power and a substrate temperature of 230 °C. The pdoping was obtained by adding to silane 5 mTorr of trimethylboron diluted at 2 % in hydrogen. After deposition, the film is immediately exposed to the hydrogen plasma to avoid any oxidation. Indeed a thin native oxide layer can act as an efficient barrier against the hydrogen diffusion [20]. The H plasma treatment conditions were: 1 Torr of H 2 and 30 W of rf power.

At regular intervals of time, the H plasma is stopped and an ellipsometry spectrum is recorded to check the induced changes in the film structure. In situ UV-visible spectroscopic ellipsometry (SE) measurements were performed within the 1. sample has an infinite thickness and a smooth surface [START_REF] Azzam | Ellipsometry and Polarized Light[END_REF]. More quantitative information can be derived from the SE measurements using optical models and the Bruggeman effective medium approximation (BEMA) theory [START_REF] Aspnes | [END_REF]. The BEMA theory is well-adapted for the interpretation of the SE measurements performed on a-Si:H and µc-Si:H materials [START_REF] Aspnes | [END_REF]23]. The dielectric function of the deposited a-Si:H is described by the Tauc-Lorentz dispersion law [24] given by:

+ = + = g E g du E u u E E E C E E E E C E A E 2 2 2 1 1 2 2 2 0 2 2 0 2 ) ( 2 ) ( ) ( 1 ) ( ) ( ) ( (1)
where E 0 and 1 ( ) are respectively the resonance energy of the Lorentz oscillator and the real part of the dielectric function at high photon energies. E g is an optical band gap useful to compare the optical properties of different samples. It is lower than the usual Tauc gap [25]. stands for the Cauchy principal part of the integral.

The µc-Si:H material is described by a 3-phase system: microcrystallites (reference data measured by Jellison et al. [26]), amorphous silicon (dispersion law of Eq. ( 1)) and void. To determine the amount of Si-Si bonds converted to an excess Si-H bonds (i.e. an excess of the hydrogen content) by the H plasma treatment, we have followed the approach of Mui and Smith who assumed that the basic polarizable units are Si-Si 4 n H n tetrahedron [27]. The evolution of the volume fraction f H of the Si-Si 3 H tetrahedron follows that of the hydrogen excess in the film [28]. The effective dielectric function of the subsurface layer affected by the H plasma is calculated using a mixture of the same dielectric function as the bulk (Eq. ( 1)) plus an excess of hydrogen.

Results and discussion

Spectroscopic ellipsometry results

Ellipsometry is a powerful tool to study in situ the kinetics of the H induced modifications. Fig. 1 shows the evolution of the imaginary part of the pseudo-dielectric function < 2 > of a p-type a-Si:H film, at various stages of its exposure to a H plasma, in the case where the reactor walls are coated with a-Si:H. The as-deposited film presents a < 2 > spectrum characteristic of hydrogenated amorphous silicon with a broad peak around 3.6 eV. For hydrogen plasma exposure times such as t < 7 min (Fig. 1a), a slight shift of interference fringes towards high energies indicates a decrease in the film thickness due to the etching process. Moreover, the H in-diffusion leads to a gradual decrease of the < 2 > peak amplitude along with a slight shift towards high energies. At 10 minutes the amplitude of < 2 > reaches its lowest value, corresponding to the formation of a porous layer on the film surface where nucleation of crystallites can take place. Beyond a hydrogen plasma exposure duration of 15 min (Fig. 1b), the < 2 > spectrum becomes that of a microcrystalline silicon, characterized by the appearance of a broad shoulder at 4.2 eV. The growth of the µc-Si:H layer gives rise to a shift of the interference fringes to low energy synonymous with an increase in the film thickness. At the same time, the increase in the < 2 > peak amplitude reveals a continuous change and rearrangement in the deposited µc-Si:H layer structure making it more compact. In order to confirm the microcrystalline character of the grown layer, we have deposited a 190 nm-thick a-Si:H film on a GaAs substrate and exposed it to a H plasma during 45 minutes. The cross-sectional transmission electron microscopy (TEM) image of Fig. 2 clearly shows the formation of a 65 nm-thick µc-S:H layer. The electron diffraction pattern corresponding to the grown layer (inset of Fig. 2) is that of a µc-Si:H structure. Fig. 3 shows the multilayer optical model used to fit the SE spectra of Fig. 1. The model was supported by other ex situ characterization techniques [17][START_REF] Kail | Proceedings of 19 th European photovoltaic Solar Energy Conference[END_REF][START_REF] Kail | [END_REF]28]. The surface and subsurface modifications, due to atomic H impinging on the film and diffusing through it, lead to the formation of a H-rich subsurface layer accompanied with the etching of the film surface. These effects start at the ignition of the hydrogen plasma. The etching rate as well as the time needed for the complete formation of the H-rich subsurface layer (from a few seconds to a few minutes) depends on the hydrogen plasma conditions, the substrate temperature and the film doping [17][START_REF] Kail | Proceedings of 19 th European photovoltaic Solar Energy Conference[END_REF][START_REF] Kail | [END_REF]. With the beginning of the microcrystalline layer deposition, both the thickness and the H excess in the subsurface layer decrease indicating the out-diffusion of hydrogen from the sample. At the steady state, the excess of the Si-Si 3 H tetrahedron units in this subsurface layer (f H ) can reach 20 -30 % [17,[START_REF] Kail | Proceedings of 19 th European photovoltaic Solar Energy Conference[END_REF]. In the case of boron-doped a-Si:H films, it amounts to say that the hydrogen content in the subsurface layer increases from 16 at. %, in the asdeposited state, to 19 -21 at. % after the H-plasma treatment.

We have reported in Fig. 4 the changes, versus the H plasma treatment time (t), of the thickness (d Total ) of the whole treated film together with that (d Bulk ) of the bulk a-Si:H layer (Fig. 4a), the thickness (d H ) and the H-excess volume fraction (f H ) of the H-rich subsurface layer (Fig. 4b) as well as the thickness (d Cryst ) and the volume fraction of crystalline phase (f Cryst ) of the deposited µc-Si:H layer (Fig. 4c). Fig. 4a shows that the hydrogen diffusion into the film leads, on the one hand, to the formation of the H-rich subsurface at the expense of the a-Si:H bulk layer and, on the other hand, to the etching of the film. The etching process, with a rate of 1 nm/min, lasts the first 5 -6 minutes of plasma treatment. The two parameters that characterize the H-rich subsurface layer (Fig. 4b) show a 3-step evolution: i) a sharp increase during the first 1 -2 minutes, ii) a plateau that goes on until t = 9 -10 minutes and iii) a complete disappearance after 1 -2 minutes. The two first steps include the etching phase [8] as well as the beginning of the µc-Si:H layer deposition by chemical transport [9,10]. The last step occurs once a 25 nm-thick µc-Si:H layer is deposited with a crystalline fraction of 20 -30 %. The hydrogen out-diffusion is also obvious through the 20 nm increase in the bulk a-Si:H layer (Fig. 4a). The results displayed in Fig. 4 allow us to draw the following conclusions: i): The µc-Si:H layer is made from silicon growth precursors coming from the plasma phase. It does not result, from a solid phase transformation of the initial a-Si:H layer but only from the top part of it. ii) The nucleation of crystallites occurs within a 10 nm-thick region below the surface. iii): The out-diffusion of hydrogen from the H-rich subsurface layer accompanies the deposition of the µc-Si:H layer.

The need for an incubation (or a transition) layer for the µc-Si:H growth from an a-Si:H layer has been demonstrated by several experimental studies [4,5,8,9,29]. The transition layer was first interpreted as a porous a-Si:H layer [4,5,29]. Later, it was identified as a hydrogen saturated layer [16,30]. In conditions of the solid phase transformation of a-Si:H to µc-Si:H (chemical annealing or high hydrogen chemical potential), the formation of the µc-Si:H film is often attributed to the out-diffusion of hydrogen by forming molecular hydrogen [8]. The energy released from the formation of hydrogen molecules may enhance the relaxation of Si atoms in order to form crystallites. Consequently, the crystallization results from the out-diffusion of hydrogen.

Our experimental results (Fig. 4) lead to the conclusion that the out-diffusion of hydrogen is a consequence and not the cause of the growth of the µc-Si:H layer. The trap-limited hydrogen diffusion model that we shall develop in the next section aims to prove that the grown µc-Si:H layer, gradually stops the atomic hydrogen supplied by the plasma and then induces the observed out-diffusion of hydrogen.

Hydrogen capture and release model

Our model is based on the same principle that governs the dissolution of a gas in a liquid. It is well-known that gases increase in solubility with an increase in pressure.

Indeed, Henry's law [START_REF] Atkins | General Chemistry[END_REF] states that the solubility of a gas in a liquid is directly proportional to the pressure of the gas above the surface of the solution. Carbonated beverages provide the best example of this phenomenon. The carbonated beverage is bottled under pressure to increase the carbon dioxide dissolved in solution. When the bottle is opened, the pressure above the solution decreases, and as a result, the solution effervesces and some of the carbon dioxide bubbles off. An other example is given by deep sea diving. When the diver rises from the depths too fast, the nitrogen gas dissolved in the blood will come out of solution as the pressure is released and form bubbles in the blood and other body fluids.

If we consider that H atoms will behave towards the a-Si:H film in the same way of a gas with a liquid, we shall take advantage of such a phenomenon to interpret the timedependence of the H-rich subsurface layer of Fig. 4b. In other words, in presence of the H plasma, the chemical potential of hydrogen rises leading to the H in-diffusion and trapping within the a-Si:H film. The growth of the µc-Si:H will consume most of H impinging atoms and then will reduce the amount of hydrogen reaching the µc-Si:H/a-Si:H interface. The gradual decrease in the mobile H amount will, consequently, make We shall treat the formation of the H-rich subsurface layer within a one-dimensional diffusion of hydrogen into a semi-infinite material (x 0). The mobile hydrogen (C m (x, t)) can be captured with a rate and released with a rate r at hydrogen traps. We shall call N the concentration of those available traps. The a-Si:H film surface is fixed at the coordinate x = 0 and the substrate extends to x = . The coordinate x = 0 will refer to the µc-Si:H/a-Si:H interface once the deposition process of the µc-Si:H layer by chemical transport starts. If we neglect the shift of the film surface during the etching process, the concentration of hydrogen diffusing into the sample and that of hydrogen trapped (C (x, t)) are governed by the following set of equations:

( ) ( ) ( ) t t x C x t x C D t t x C m H m = , , , 2 2 (2)
( ) ( ) ( ) ( ) t x C t x C t x C N C t t x C r m m , , , (0) 1
, = (3)
The capture rate can be linked to the density of traps N and their capture radius R by [START_REF] Waite | [END_REF]:

H N D R 4 = (4)
The following points concerning the diffusion model must be emphasized: (iii) Although high values (> 10 10 cm 2 s 1 ) of the hydrogen diffusivity D H at the very initial stage of hydrogen plasma treatment have been reported by several authors [20,[33][34][35], we assume that, during all the plasma treatment, the transport of atomic H is characterized by a constant D H . Moreover, a quite important concentration-dependence of D H has also been reported [36].

(iv) We have considered only one trapping level and a uniform and constant density of traps N . We shall not make any allusion to the microscopic nature of the trapping sites. In this way, contrary to some H diffusion models in p-type a-Si:H [37], we shall not differentiate the dopant atoms, that can trap hydrogen, from other types of hydrogen traps.

(v) Eqs. ( 2) and (3) do not refer to the charge state of the diffusing H even if some recent reported results indicate that H is very likely to be positively charged in the p-type a-Si:H [38][START_REF] Beyer | Semiconductors and Semimetals[END_REF][START_REF] Fedders | [END_REF][41]. An electric field effect (due, for example, to the local electric field that could arise at the µc-Si:H/a-Si:H heterojunction) cannot be excluded [42].

(vi) As we did in a previous paper [17], we shall associate the hydrogen excess in the subsurface layer determined by SE measurements with the concentration of trapped hydrogen C (x, t). Indeed, a non-bonded H does not give a significant contribution to the dielectric function of the film [20]. we neglect the reemission process, Eq. ( 3) can be easily integrated leading to the following expression of C (x, t):

= t m t' t' x C N exp N t x C 0 d ) , ( 1 ) , ( (5)
To find an analytical solution C m (x, t) of Eq. ( 2), one has to drop the coupled term. This simplification amounts to underestimate the trapping rate . At the beginning of the H plasma exposure, the initial conditions for the set of equations ( 2) and ( 3) are: C m (x, 0) = C m (0) for x 0, C m (x, 0) = 0 for x > 0 and C (x, 0) = 0. As the sample thickness always remains larger than the hydrogen diffusion length, one boundary condition is to consider that C m ( , t) = 0 and C ( , t) = 0 for t 0. The rapid formation of the H-rich subsurface layer (almost one minute) suggests a high value for the time factor in the exponential term of Eq. ( 5). If we call C m (0) the concentration of mobile H at the film surface, large values are expected for the ratio C m (0)/N . In fact, in the case of "normal"

or unattenuated H plasma, the flux of atomic hydrogen at the film surface is around 10 14 cm 2 s 1 [43]. With an etching rate of 1 nm/min a value of C m (0) > 10 21 cm 3 is expected.

Moreover, values as high as 10 20 -10 21 cm 3 were experimentally measured for the H surface concentration [36,[44][45][46][47].

Since the formation of microcrystalline nucleus, H atoms coming from the plasma will rather be involved in the growth of the crystalline grains and, consequently, will not reach the µcSi:H/a-Si:H interface. Consequently, the deposition of the [49].

These authors showed that the µc-Si:H formation reduces the incorporation of Si-H and Si-H 2 bonds in the a-Si:H bulk layer. Katiyar and Abelson have exposed a freshly deposited a-Si:H film to H plasma [9]. They observed a decrease in the 2000 cm 1 infrared mode which accompanies the formation of µc-Si:H on the a-Si:H film.

Thus, once the µc-Si:H layer starts to grow at t = T 1 = 5 min (Fig. 4c), less and less hydrogen atoms reach the µcSi:H/a-Si:H interface. As a result, the H chemical potential, initially raised by the H plasma, will decrease during the growth of the µc-Si:H layer.

To express the fact that, after a time !, hydrogen atoms cannot any more reach the µcSi:H/a-Si:H interface, we can use the following boundary condition:

1 1 1 exp ) 0 ( ) 0 ( ) , 0 ( T t T t C T t C t C m m m > ! = ( 6
)
With the mentioned initial and boundary conditions, an exact analytical solution of Eq.

(2) can be found:

( )

1 1 1 1 ) , f( ) , f( ) , f( (0) ) , f( (0) ) , ( T t , T t x , T t x t, x C T t t, x C t x C m m m > + = (7)
where the function f(x, t, !) is defined by:

t t D 2 x D x t t D 2 x D x t , t , x H H H H (8)
The exponential behaviour of C (x) profile tail (see Fig. 5), allows us to define an effective diffusion length L D by:

e t C t L C D 1 ) , 0 ( ) , ( = (9)
If we identify the mean diffusion length L D with the thickness d H of the H-rich subsurface layer, we shall be able to determine both the diffusion coefficient D H and the capture rate (and consequently the density of traps N ). Indeed, one can determine these two parameters (D H and ) by minimizing the following quantity:

$ j H e t , d C 2 1 ) ( (10)
The summation j is over the (t, d H ) experimental couples for times such as t 9 min (before the H-rich subsurface layer starts to vanish, see Fig. 4b). Assuming C m (0)/N = 10 and using T 1 = 5 min and ! = 1 min, the fit of the experimental results leads to: D H = 2.09x10 14 cm 2 s 1 and = 1.35x10 [20]. They estimated a value of 10 3 s 1 for .

We do not know the capture radius for a hydrogen trapping centre R . If we assume it constant and roughly equal to the size of a dangling bond R % 0.2 nm [17,44], then we derive a density of traps N = 2.57 x 10 19 cm 3 . Such a value of N is two orders of magnitude higher than that determined in the case of undoped a-Si:H exposed to a H plasma at the same temperature [17]. Furthermore, N is about one order of magnitude higher than the defect density (mainly isolated dangling bonds) induced by the boron doping in a-Si:H [50].

In Fig. 5 are plotted the concentration profiles C m (x) and C t (x) at different times obtained using the values of D H = 2.09x10 14 cm 2 s 1 , = 1.35x10 1 s 1 , C m (0)/N = 10, T 1 = 5 min and ! = 1 min. Before the deposition of the microcrystalline layer (t < T 1 ), both C m and C rapidly extend and reach a saturation limit. We have already reported such a behaviour in H plasma conditions where only the etching of a-Si:H takes place [17]. This rapid diffusion agrees with the abrupt formation of the H-rich subsurface layer (Fig. 4b). Beyond the etching phase and during the crystallization process (t > T 1 ), the sample starts to run out of mobile hydrogen. Consequently, the H trapped profile remains unchanged.

The integrated amount of mobile (or trapped) hydrogen Q m, within the whole film thickness can be defined by:

F o r P e e r R e v i e w O n l y 16 = 0 d ) ( ) ( x t , x C t Q , m , m (11)
Fig. 6 shows the time-dependence of the Q m /Q ratio, deduced from the hydrogen profiles of Fig. 5. Although Q m /Q gradually decreases for t < T 1 , the amount of mobile hydrogen remains higher than that trapped in the H-rich subsurface layer. Within this range of time, the capture process of hydrogen is the prevailing one. Beyond t = T 1 , the ratio Q m /Q undergoes a drastic fall. Such a sharp depletion in mobile hydrogen marks the end of the predominance of the hydrogen capture process. According to Figs. 4 and6, the H-rich subsurface layer starts to vanish for plasma exposure times higher than 10 min. This when Q m /Q < 0,1.

Kinetics of hydrogen release

The growth of the µc-Si:H layer prevents the H atoms coming form the plasma to reach the µc-Si:H/ a-Si:H interface. Consequently, by analogy with the gas dissolution in a liquid, the H pressure above the a-Si:H surface vanishes leading to a decrease in the H solubility in the a-Si:H network.. According to Fig. 4, this process starts at t = T 2 = 8 -9 min and lasts a time & =1 -2 min. During this period of time, the H reemission becomes predominant at the expense of the trapping process. In this case, the integration of Eq. (3) leads to:

[ ] 2 2 2 ) ( exp ) ( ) ( T t T t T Q t Q r = (12)
The characteristic time & can be identified as the inverse of the release rate r . Thus, a value of & =1 -2 min leads to r = 0.8 -1.7x10 2 s 1 . In the case of the hydrogenation of intrinsic a-Si:H at 250 °C with a heated filament, An et al. have found a quite lower reemission rate ( r < 2x10 7 s 1 ) upon extinguishing the heated filament [20]. In Fig. 7 we compare the calculated value Q (t) with the experimental d H (t) x f H (t) product at different H plasma exposure times. Both experimental and calculated values are normalized with respect to t = T 2 . The calculations were performed using: D H = 2.09x10 14 cm 2 s 1 , = 1.35x10 1 s 1 , C m (0)/N = 10, T 1 = 5 min and ! = 1 min, for the H trapping process, and r = 1.7x10 2 s 1 (& = 1 min) and T 2 = 9 min for the H reemission process. Fig. 7 displays a good agreement between calculated and experimental results within the whole time-range.

We do not know exactly how the released H atoms do leave the subsurface layer.

Probably hydrogen desorbs by simultaneous rupture of two Si-H bonds and forms H 2 molecules. A direct abstraction of a bonded H by a mobile atomic H, like a Eley-Rideal mechanism, cannot be excluded since such a mechanism involves a low activation barrier energy [52]. The hydrogen molecules may leave the film by rapid diffusion through voids, that could be formed by the H plasma treatment [51] in a-Si:H or grain boundaries of the µc-Si:H layer [1,53,54]. The low sticking coefficient of H 2 on Si at moderate temperature [55], favours the rapid out-diffusion of hydrogen. Moreover, some theoretical works point out a number of arguments that relate the possibility of molecular hydrogen playing an important role in the observed transport of hydrogen in a-Si:H [56][57][58]. Some works might even find H 2 more mobile than H [START_REF] Abtew | Los Alamos National Laboratory[END_REF]. The formation of H 2 molecules in a silicon network, after a post-hydrogenation, has been demonstrated by many experimental works. In the case of crystalline silicon, molecular hydrogen occupies the cavities induced by the hydrogenation [START_REF] Ma | [END_REF]. In the case of microand nanocrystallized silicon films the presence of molecular hydrogen is detected in surface cavities as well as at grain boundaries [61]. Moreover, in hydrogen effusion experiments, the low temperature ( 400 °C) effusion peak, in a-Si:H and µc-Si:H deposited at low temperature, is often attributed to the rapid diffusion of molecular hydrogen (H 2 , D 2 and HD) [1]. It is also correlated to the presence of an interconnected void network in material deposited at low temperature. The low temperature effusion process is found not to be limited by diffusion as the effusion peak temperature is independent of film thickness. Further investigations are needed to probe other tracks in order to clarify how does happen the observed out-diffusion of hydrogen.

Summary

We have investigated the hydrogen plasma treatment of p-type a-Si:H films in conditions of µ-Si:H deposition by chemical transport. We dealt with the kinetics of hydrogen excess evolution during the plasma exposure and particularly its out-diffusion when the crystallization of the film starts. To interpret the experimental results, we have solved the combined diffusion and trapping equations that govern the H-diffusion into the sample. We have used a specific boundary condition that accounts for the reduction of the atomic H flux with the growth of the µc-Si:H layer. The proposed model explains the experimentally observed evolution of hydrogen during the plasma treatment.

Moreover, it supports the idea that the out-diffusion is a consequence and not the cause of the growth of the µc-Si:H layer.

F

5 eV - 5 eV

 5 photon energy range. The experimental ellipsometry angles and are converted -dielectric function < > using the equation for a single interface. The pseudo-dielectric function reduces to the dielectric function when the reflecting

  Deleted: SiDeleted: at least not predominantly, Deleted: H-rich subsurface

  potential in the a-Si:H film . Then, the exceeding hydrogen will vanish.

  The model only aims to interpret the evolution of the subsurface layer during the H plasma treatment. It does not deal with the deposition process of the microcrystalline layer.(ii) All H atoms involved in Eqs. (2) and (3) are additional H atoms incorporated in the a-Si:H film during the H plasma treatment. The initial tightly bonded H atoms are not concerned.

12 3. 2 . 1

 1221 Kinetics of hydrogen trapping. Fig.4bsuggests that the formation of the H-rich subsurface layer corresponds to the phase where the trapping process is predominant. If

Formatted:

  Font: SymbolDeleted: This effect can be expressed by Deleted: at the film surface (x = 0)

Deleted: 0, 8 Deleted: 1

 81

Figure 1 :

 1 Figure 1: Imaginary part of the pseudo-dielectric function of the a-Si:H film thickness during the H plasma exposure, (a): before and (b): after the µc-Si:H layer deposition. The arrow indicates the position of the broad shoulder which is a signature of the µc-Si:H material.

Figure 2 :

 2 Figure 2: Cross-section TEM of an 190 nm-thick a-Si:H film deposited on a GaAs substrate and exposed to a H plasma during 45 minutes. The treatment leads to the formation of a 65 nm-thick µc-Si:H. The inset displays the electron diffraction patern of the crystallized layer.

Figure 3 :

 3 Figure 3: Schematic of the multilayer optical model used to fit the SE measurements of Fig. 1: (a) : In its as-deposited state, the a-Si:H film is constituted by a bulk and a surface roughness. (b) : From the ignition of the H plasma, a H-rich subsurface layer starts to form accompanied by an etching of the film. (c) and (d) : When µc-Si:H deposition by chemical transport starts, an outdiffusion of hydrogen leads to the disappearance of the H-rich subsurface layer.

Figure 4 :

 4 Figure 4: Time-evolution of (a) : the thickness of the whole film d Total and the thickness of the a-Si:H bulk layer d Bulk , (b) : the thickness d H and the Hexcess fraction f H of the H-rich subsurface layer and (c) : the thickness d Cryst and the crystalline volume fraction f Cryst of the µc-Si:H layer deposited by chemical transport during the H plasma exposure. The curves are provided as guides for the eye. The arrows indicate the values of the etching duration T 1 and the time T 2 at which the hydrogen out-diffusion starts.

Figure 5 :FormattedFigure 6 :Fig. 5 .

 565 Figure 5: Concentration profiles C m (x, t) and C (x, t) at different H plasma treatment times. The values used for the calculations are: D H = 2.09x10 14 cm 2 s 1 , = 1.35x10 1 s 1 , C m (0)/N = 10, T 1 = 5 min and ! = 1 min.

Figure 7 : 5 Figure 5 N

 755 Figure 7: Experimental (d H x f H ) and calculated (Q ) integrated amount of trapped hydrogen versus the time of the H plasma exposure. The values used for the calculations are: D H = 2.09x10 14 cm 2 s 1 , = 1.35x10 1 s 1 , C m (0)/N = 10, T 1 = 5 min and ! = 1 min, for the trapping process (solid curve), and r = 1.7x10 2 s 1 , T 2 = 9 min and & = 1 min for the reemission process (dashed curve). The data are normalized with respect to their values at t = T 2 .

  Figure 6 N. PHAM et al.

  [48] may explain the observed out-diffusion of hydrogen. Many researchers have reported the transformation of a-Si:H to µc-Si:H with a concomitant out-diffusion of hydrogen. Using in situ SE and infrared attenuated total reflection spectroscopy measurements, Fujiwara et al. studied the initial layers from which µc-

	from growth precursors Si:H nucleates, in a rf PECVD process, with increasing the [H 2 ]/[SiH 4 ] dilution
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	13

µcSi:H layer

  1 s 1 . The value of D H is consistent with those case of diffusion from the plasma[46]. As a comparison, in the case of intrinsic a-Si:H, we have found that D H decreases from 3.0x10 14 to 2.4x10 13 cm 2 s 1 when the temperature of H plasma treatment is raised from 100 to 250 °C[17]. At the same time, decreases from 5.5x10 2 to 2.7x10 2 s 1 . An et al. have exposed intrinsic a-Si:H to atomic H generated by a filament heated in H 2 with a substrate temperature of 250 °C

	reported in the
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Marc Heggen

Michael Feuerbacher

Quasicrystal plasticity in the framework of a constitutive model: Interaction of the microstructural parameters at high strain-rates

Keywords: constitutive equations, dislocations, plasticity, quasicrystalline alloys, quasicrystals user supplied): constitutive equations, dislocations, plasticity Al-Pd-Mn quasicrystal, plastic deformation, work softening, work hardening, constitutive model, plasticity

Quasicrystal plasticity in the framework of a constitutive model: Interaction of the microstructural parameters at high strain-rates

INTRODUCTION

The plasticity of quasicrystals shows very particular features such as the lack of work hardening at high strains [3,4,5,6,7]. This behaviour was shown to be accompanied by a decrease of the dislocation density [5,8,9]. A quantitative model of quasicrystal plasticity based on a system of constitutive equations involving the stress-strain behaviour and the dislocation density evolution was presented by Feuerbacher and co-workers in 2001 [1]. This model reproduces the characteristic features of work softening and the decrease in dislocation density. In a later work, the model was successfully applied to a wide range of experimental parameters in which various stress-strain behaviour was observed [2]. Here, work softening after the lower yield point is observed at medium strain rates of 10 -5 s -1 (730 and 760 °C) and 10 -4 (760 °C). At strain rates of 10 -3 s -1 (730 and 760 °C) and 10 -4 s -1 (730 °C), respectively, a work-hardening regime is observed after the lower yield point, giving rise to a second stress maximum at about 5 % strain. This second maximum is followed by work softening at higher strains. In contrast, at the lowest strain rates of 10 -6 s -1 and 0.7 × 10 -6 s -1 the stress remains at an almost constant level after the lower yield point and neither work softening nor hardening is found. The stress-strain curves computed according to the constitutive model reproduce all essential features of the experimental curves remarkably well. However, the interplay of the microstructural parameters of the constitutive model yielding such complex deformation behaviour, in particularly combining stages of work hardening and softening, is weakly understood.

In the present paper, we briefly recall the constitutive model of quasicrystal plasticity and discuss the results of a deformation experiment at 10 -3 s -1 showing work hardening as well as work softening. The evolution of the microstructural parameters dislocation density and order parameter, their interaction with other microstructural factors like the dislocation velocity, and

CONSTITUTIVE MODEL

The constitutive model of Feuerbacher et al. [1] employs two microstructural parameters, the dislocation density and the order parameter . These are the basic parameters determining the stress-strain behaviour. It is assumed that the degree of order is decreased by dislocations moving through the quasicrystalline structure. The decrease of order is proportional to the increase of plastic strain plast and the current state of order . On the other hand diffusional reordering processes restore the order of the quasicrystal. We assume that the increase of order with time is proportional to (1-), which is a measure of the order destroyed. Therefore, the evolution of the order parameter is described by the Ansatz

0 0 ) 1 ( ) ( dt d d plast plast + = , ( 1
)
where 0 is a characteristic time constant for the reordering process.

The macroscopic plastic strain rate plast & is given by the Orowan equation [10]

bv m s plast = & , (2)
where m s is the average Schmidt factor, is the dislocation density and b the average magnitude of the dislocation Burgers vector [START_REF] Feuerbacher | Mechanische Eigenschaften von ikosaedrischen Al-Pd-Mn Quasikristallen[END_REF]. The dependence of the dislocation velocity on the applied stress is described by the power law

4 m B v = ˆ, (3)
where B and m are temperature-dependent parameters and $ is an internal variable that can be regarded as a "reference stress" [START_REF] Kocks | [END_REF], which reflects the current microstructural state of the material. The reference stress contributions are given by $ ( ) ( )

µ = + + 0 b . ( 4
)
The first term, 0 , is a constant background friction stress. The second term describes the stress contribution due to dislocation-dislocation interactions (Taylor part) and is proportional to the square root of the strain-dependent dislocation density. The parameter is of the order of unity and µ is the shear modulus [START_REF] Estrin | Unified Constitutive Laws of Plastic Deformation[END_REF]. The third term represents a quasicrystal-specific friction stress.

This stress contribution is proportional to the strain-dependent order parameter . The parameter describes the flow-stress contribution originating from the destruction of order by the movement of dislocations in the quasicrystalline material.

The total strain is the sum of the plastic strain plast and the elastic strain elast = / E, where E denotes Young's modulus. Combining these equations with Eqs (2) and (3) we obtain

( ) = & m s bB m E d d / 1 . (5)
To determine the evolution of the dislocation density, , we consider dislocation multiplication and dynamic recovery, which both depend on plastic strain, and we account for static recovery, 5 [START_REF] Kocks | [END_REF]. The first term corresponds to the athermal storage of dislocations and involves a multiplication parameter M. The second term, entering with a negative sign, describes the annihilation of moving dislocations (dynamic recovery). Y is a characteristic annihilation distance that depends on the order parameter according to

) ( 1 0 0 + = Y Y , (7)
where 0 0 µb Y = is a constant.

RESULTS AND DISCUSSION

In order to elucidate the interplay between the microstructural parameters, we will follow their evolution in detail and investigate the correlation with the stress-strain behaviour using a curve at 10 -3 s -1 and 730°C. Figure 1 shows an experimental (solid line) and a computed stressstrain curve (broken line). The latter curve was computed by numerical integration of the set of equations ( 1), ( 5) and ( 6) using the parameters in Table 1. The free parameters were determined by fitting a complete set of stress-strain curves at four different strain rates and the corresponding experimental dislocation densities (cf. [2]). Before reaching the upper yield point, the dislocation velocity increases strongly with increasing stress (cf. Equation 3) -plastic straining of the sample is setting in. With the increase of the plastic strain rate, dislocation storage according to Equation ( 6) commences. The storage of dislocations is reflected in the strong increase of the dislocation density (Figures 2 d ande) and the increase of the Taylor contribution.

The strong increase of the dislocation density after the upper yield point leads to a strong decrease of the dislocation velocity (Figure 2 c) according to the Orowan equation ( 2). Hence the total stress decreases according to Equation (3). As a consequence, the elastic strain contribution decreases as well leading to a plastic strain rate that is higher than the applied strain rate (Figure 2 b). After the upper yield point at 1.28 %, the maximum plastic strain rate of 1.62 × production and strain rate on the one hand and dislocation density on the other hand are both expressed by which enters Equation ( 6). The strong increase of the dislocation density (cf. Figure 2 d) leads to an increase of dislocation-dislocation interactions and a strong increase of the Taylor contribution according to Equation ( 4) so that the total stress increases at strains between 2.3 and 6.0 %. With the increase of the dislocation density the annihilation by dynamic recovery increases as well. As Equation ( 6) shows, dynamic recovery outruns dislocation multiplication at high dislocation densities.

With the onset of plastic deformation, disordering of the structure is initiated according to Equation (1), which is directly reflected in the sharp drop of the order parameter at 1.4 % in Figure 2 f. The decrease of order leads to a decrease of the quasicrystal part to the reference stress according to Equation (4). Furthermore, the decrease of order influences the dislocation velocity and the stress-strain curve via Equations ( 3) and (2). Additionally, the decrease of order leads to a higher annihilation rate of dislocations, since the annihilation distance increases with decreasing order parameter (Equation 7). As a result, the dislocation density as well as the Taylor contribution of the stress decrease at high strains according to Equations ( 6) and (4).

Moreover, Fig. 2 a shows that the maximum of the Taylor contribution (white arrowhead) is located at higher strains than the second maximum of the stress-strain curve (black arrowhead).

This shift of the stress maximum to lower strains is caused by the decreasing quasicrystal contribution. In conclusion, a complex deformation behaviour with stages of work hardening and softening is observed at high strain rates. This complex behaviour is comprehensively described in the framework of the constitutive model. The interplay of the microstructural parameters dislocation density and order parameter and the stress-strain curve was demonstrated. Taylor hardening, due to the strong increase of the dislocation density is the reason for an increase of the stress after the lower-yield stress. However, disordering of the structure during plastic deformation counteracts hardening in two different ways. On one hand it directly influences the stress-strain curve via the quasicrystal stress part, which is only a small fraction of the total stress. On the other hand disordering leads to a higher annihilation rate of dislocations and leads to a decrease of the dislocation density. As a result, the Taylor contribution of the stress decreases and work softening sets in. Table 1: Parameters used for the computation of stress-strain curves and the evolution of the dislocation density and the order parameter. "Fixed" parameters were determined by independent experiments and "free" parameters were varied to fit the calculated to the experimental curves.

  on the macroscopic plastic deformation behaviour are discussed in terms of the constitutive model. We show that the constitutive model allows for a comprehensive treatment of the characteristic phenomena of the plasticity of icosahedral Al-Pd-Mn.

  which depends on time only. The change of with plastic strain is described by

Figure 2 a

 2 shows the stress contributions according to Equation (4), the constant background friction stress, the Taylor part, and the quasicrystalspecific stress part, to the total stress. The upper curve represents the total stress according to the broken line of Figure 1. The proportions of different stress contributions change according to the evolution of the microstructural parameters and . The individual contributions can be calculated by multiplying the corresponding terms of Equation (4) with the stress dependent factor ) ( ˆ) ( , which relates the fraction of the respective contribution of the reference stress ˆto http://mc.manuscriptcentral.com/pm-stress . The lower curve in Figure 2 a is the Taylor contribution to the total stress, the middle curve represents the Taylor contribution plus the quasicrystal contribution ) ( to the total stress. Hence, the difference between the lower curve and the middle curve corresponds to the quasicrystal contribution. Furthermore, the difference between the middle curve and the upper curve corresponds to the contribution of the background stress 0 to the total stress. To make connection to the microstructural parameters, we show the plastic strain rate plast & (figure 2 b), the dislocation velocity v according to Equation (3) (figure 2 c), the dislocation density and the rate of dislocation production and annihilation (figures 2 d and e), and the order parameter (figure 2 f) as a function of the total strain . The evolution of the Taylor contribution to the total stress in Figure 2 a reflects the evolution of the dislocation-density shown in Figure 2 d. At the beginning of the deformation experiment the stress increases elastically. The dislocation density is low and hence the Taylor contribution to the stress is small.

  s -1 is reached at 1.45 %. The rate of dislocation production and annihilation shows two maxima. The first one at 1.68 % corresponds to the maximum plastic strain rate, and the second one at 2.64 % corresponds to the maximum dislocation density. The relation between dislocation

Figure 1 :

 1 FIGURE CAPTIONS

Figure 2 :

 2 Figure 2: (a) Computed stress contributions, i.e. the Taylor part (light grey), the quasicrystal part (white) and the background stress (dark grey), at 730 °C and & = 10 -3 s -1 . The arrowheads indicate local maxima of the stress contributions. (b) Plastic strain rate, (c) dislocation velocity, (d) dislocation density, (e) rate of dislocation production and annihilation, and (f) order parameter.

Figure 1 :Figure 2 :

 12 Figure 1: Experimental (solid line) and computed (broken line) stress-strain curves at 10-3 s-1 and 730 °C. Both curves show the appearance of a second stress maximum at around 5 % strain. 705x554mm (72 x 72 DPI)

TABLE / TABLE CAPTION 730 °C

 /

	Parameter	Value	free/fixed	Ref.
		130 MPa	free
	0	65 MPa	free
	0	0.1	free
	0	2 × 10 4 s	free
	B	4.5 × 10 -9 m s -1	free
	M	0.38	free
	Y0	7.8 × 10 -9	free
	µ	80 GPa	fixed	[11]
	B	1.83 × 10 -10 m	fixed	[11]
		2 × 10 11 m -2	fixed	[11]
	E	80 GPa	fixed	[14]
	M	4.8	fixed	[11]
	N	5	fixed	[14]
	ms	0.4	fixed	[15]

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M De Boissieu
email: marc.de-boissieu@ltpcm.inpg.fr

Phason modes in quasicrystals Phason modes in quasicrystals

Keywords: diffraction methods, quasicrystals Keywords (user supplied): Phason, diffuse scattering Quasicrystals, phason modes, diffuse scattering AMS Subject Classification:

come

Introduction

Right after the discovery of quasicrystals, the concept of phason has been introduced and became the subject of many investigations. Whereas the concept of phasons is well defined for incommensurately modulated phases, in the quasicrystal community the name phason is used to address quite different physics: there are phason modes, phason 'jump' or phason hopping, phason strain, phason disorder, random phason disorder… just to quote some of the wording which one can find in the literature. This being said, in the entire quasicrystals (QC's hereafter) research field phasons are believed to play a role, starting from the growth mechanism up to their physical properties. Studies on the QC growth have introduced phason as either an inherent part of the growth process and the resulting structure (random tiling), or have invoked phason annealing behind the growth front. Structural quality of quasicrystal is often refereed to with the degree of 'phason strain' in the system. Atomic structure determination has been the subject of intensive investigations leading to a debate on random tiling versus perfect QC, or entropy versus energy stabilizing mechanisms, debate in which phason plays a crucial role. A generalized elasticity theory has been developed for QC's with the introduction of phason elastic constants: diffuse scattering and phason modes have been studied in this perspective. Some QC's structural phase transitions have been shown also to be driven by phason modes. Of course dynamics of QC's is intimately related to phason dynamics, especially long wavelength phason modes. Physical properties studies have also invoked phasons: the spiky electronic density of states might be smoothed out by phason disorder. Plastic deformation of QC's is intimately related to the introduction of generalized dislocations (in high dimension) which induce both a phonon and phason distortion field and the notion of phason walls. The concept of phason has even been used to analyze crystal approximant defects. In summary phasons are everywhere in QC's studies! In this paper we review studies carried out on long wavelength phason modes. It is interesting to put these studies on phason modes in QC's in the broader perspective of phason modes in aperiodic crystals, where aperiodic crystals encompasses incommensurately modulated phases, intergrowth or composites structures and quasicrystals (A detailed introduction can be found in [START_REF] Janssen | Aperiodic Crystals. From modulated phases to quasicrystals[END_REF], and a detailed review in [START_REF] De Boissieu | Physics of Quasicrystals[END_REF]). Indeed, although they might be very different systems, phason modes originate from the same description in the high dimensional picture of aperiodic crystals. We will first introduce in section 2 the hydrodynamic theory of aperiodic crystal, where a unified description of phason modes can be derived. Going from a continuous phenomenological description, as the one carried out in the hydrodynamic theory, to a detailed atomistic description is a difficult problem which is far from being achieved for QC's. We will thus first present what is understood in a few incommensurately modulated phases, where atomistic model are at hand. In particular results on phason dynamics of the ThBr4 incommensurate phases will be presented section 3. In section 4, we give a rapid overview on theoretical models and simulations related to phason modes. Section 5 is dedicated to the review of experimental results, first in the i-AlPdMn phase for which there is the largest set of data, then for other quasicrystals and the relationship with crystal approximant and finally for decagonal phases. We end this review by some conclusion and perspectives in section 6.

Hydrodynamic theory of aperiodic crystals.

The name phasons was coined in 1971 by Overhauser [START_REF] Overhauser | [END_REF] in the context of charge density wave (CDW) systems. Studying the lattice excitation of an incommensurate CDW, he demonstrated that because the free energy of the system is invariant under a phase shift of the modulation there are new lattice excitations which he named phasons, showing up around the satellite reflections and whose dispersion is acoustic like. The nature and damping of those modes has been intensively studied from the microscopical view. We first present general results which can be derived from the hydrodynamic theory of aperiodic crystals.

Hydrodynamics is based on symmetry breaking analysis and deal with long wavelength excitations for which the frequency L goes to zero as a function of some power of the excitation wavevector q. The basis for the analysis of all aperiodic phases is best understood in the high dimensional picture, where the high dimensional space decomposes into the parallel (or physical) and the perpendicular space. For all aperiodic phases the free energy of the system is invariant under a rigid translation of the parallel space in the perpendicular direction, as a direct consequence of the cut space having an irrational slope. From mode counting arguments it can be shown that for all aperiodic crystals, there are phason modes which are not propagative modes but collective diffusive modes for which one can write:

F o r P e e r R e v i e w O n l y -iL = D Phason q 2 (1)
where D Phason is the phason diffusion constant, which means that the phason mode decay exponentially with time with a characteristic time N inversely proportional to q 2 . We stress here that the diffusive nature of phason modes also holds for incommensurately modulated phases and is not specific to quasicrystals. In fact, as illustrated in section 3, anharmonicity is crucial to reconcile the hydrodynamic and microscopic picture of phason modes [START_REF] De Boissieu | Physics of Quasicrystals[END_REF]4]. Indeed in the case of incommensurately modulated phases, phason modes have a finite lifetime independent of q, leading to two regimes: a long wavelength one for which the phason is diffusive and a shorter wavelength one for which the phason is a damped propagating mode.

This hydrodynamic theory relies on the hypothesis of a continuum approximation for which an infinitesimal displacement of the parallel space along the perpendicular direction does not cost energy. This has been studied in details in the Frankel-Kontorova type model and lead to the so-called analyticity breaking or to the existence of a phason gap.

Finally it is worth noting that phason modes can be viewed as modes with a wavevector in the parallel space and a polarization in the perpendicular space [5], as illustrated Figure 1. We stress again that modes are not propagative in this picture.

Phason modes in the incommensurately modulated ThBr4 phase.

The ThBr4 phase undergoes a phase transition at 95 K from a tetragonal phase to a 1D incommensurately modulated phase with a wavevector q=0.32c*. The diffraction pattern shows weak satellite reflection, whose intensity is well reproduced by a simple sinusoidal modulation. The lattice excitation of this phase has been studied by inelastic scattering whose results has been analysed in the framework of the Landau soft mode theory [6,7]. Above Ti, one of the optical mode shows a softening (Figure 2) with a minimum frequency at the incommensurate wavevector. In the incommensurate phase, the phason mode appears, with a linear dispersion around the satellite reflection (Figure 2). It is important to note that the slope of this dispersion is different from the sound velocity found in ThBr4, pointing to a new excitation which is specific to the incommensurately modulated state. The remaining optic like branch is a mode which induces variation of the amplitude of the modulation (named amplitudon). The phason mode can also be viewed in the high dimensional description as a mode having a wave-vector in the physical space and a polarisation in the perpendicular direction. In the frequency range shown Figure 2, this is a damped propagating mode. However, unlike the case of phonon, the phonon lifetime is constant and independent of q. This means that in the very long wavelength limit the phason mode is overdamped and become diffusive, in agreement with the hydrodynamics theory. Of course the found phason diffusion constant is rather rapid as expected from a process which is driven by small atomic displacements around an average position. There are very few systems for which such phasons modes have been evidenced. Another interesting system is the NaNO 2 crystal, in which the transition is of the order-disorder type and the elementary process is relaxational in nature, so that the phason mode only shows diffusive type behavior [8].

The origin of the incommensurability has been also the subject of many theoretical studies. One of he most elaborated model is the so called discrete frustrated Q 4 model or DIFFOUR model. This is a model combining a discrete basis structure and continuous degrees of freedom. A key ingredient in this model is the first and second neighbors interactions with opposite direction leading to 'frustration' [START_REF] Janssen | Aperiodic Crystals. From modulated phases to quasicrystals[END_REF]. The phase diagram of this model can be computed. As a general rule, the low T phase is a locked phase in which the modulation wavevector takes a rational value, the incommensurately modulated phase being at intermediate temperature.

In this picture the origin of the incommensurate phase contains two terms: the frustration term on one hand (which however can be 'resolved' on a rational wavevector) and the entropy term gained by phason fluctuation in the incommensurate phase.

There are many open problems in this field, which are not discussed here: analyticity breaking, phason gap, sliding modes. The interested reader is referred to ref [START_REF] De Boissieu | Physics of Quasicrystals[END_REF].

Phason modes: theory and simulation in quasicrystals

Soon after the discovery of quasicrystals, several groups pointed out the importance of phason modes [9][10][11][12]. This opened the way for a series of theoretical and modeling studies. A complete development of hydrodynamics and elasticity of quasicrystals can be found in the review by Lubensky [START_REF] Lubensky | Aperiodicity and order : Introduction to quasicrystals[END_REF]. The Landau theory of phase transitions driven by phason modes has also been developed [START_REF] Ishii | [END_REF]15]. This gives the possible group sub-group relationship for a QC to crystal transformation. Relations between the phason elastic constant imposed by the QC thermodynamic stability have been derived [16,17]. In a way completely analogous to phonons which give rise to the well known thermal diffuse scattering [18][19][20], phason modes lead to diffuse scattering in the diffraction pattern which is located beneath the Bragg peaks with three main characteristics [21]: (i) the diffuse scattering decays as 1/q 2 , its intensity is Q per dependent, (iii) its shape anisotropy is given by the phason elastic constants (the ratio K2/K1 of the two phason elastic constants for icosahedral phases). This anisotropy might also be temperature dependent close to phase transitions [16,17]. As shown in section 4, the study of diffuse scattering turned out to be a very efficient way of experimentally evidencing phason modes. One has thus a rather complete phenomenological description of phason modes.

One important question is however the relation between this continuous theoretical description and the microscopic description of quasicrystals. This is a difficult problem since for an atomistic modeling of phason modes one need (i) an atomic model, (ii) the corresponding Hamiltonian describing atomic interactions, and (iii) a plausible description of the elementary 'phason move'. Moreover, any realistic simulation will be faced with the very long time scales needed for phason modes and difficult to achieve in simulations.

Most of the simulations have been carried out within the random tiling model proposed by Elser and Henley [START_REF] Henley | Quasicrystals: the state of the art[END_REF]. In this picture atoms or clusters of atoms are arranged in a tiling, whose tiles can be reshuffled. One difficulty is finding a tiling allowing this reshuffling in a simple way and allowing for the exploration of all the phase space. Most of the simulations have been done using either a Penrose type tiling for decagonal phases, and a Amman (or 3D Penrose) tiling for icosahedral phases. The first simulation has been carried out in a very simple model in which atoms are located only on the vertices of the 3D Amman icosahedral tiling which constitute a rigid framework (there are no atom interaction). Although this is a rather crude model, the important outcome of this study, is that the simulated diffuse scattering matches what is predicted by the hydrodynamics theory [START_REF] Tang | [END_REF] showing that phason modes are related to the configurational entropy gained by the tile reshuffling (see also [24]). More sophisticated models have been since then designed, in particular for decagonal phases, where realistic models combined with ab-initio based calculation have been proposed (see M. Widom et al, this conference). Entropy stabilised structure have even been predicted [25] in the BMgRu system. For icosahedral phases, up to now simulations have been restricted to model systems. Using an Hamiltonian favoring 3-fold connections, Mihalkovic and Henley designed a model whose phason elastic constants are changing with temperature, eventually leading to a crystal approximant [26]. In a simple model for which the ground state is a perfect QC, Dotera et al. have shown that free energy is quadratic as a function of the phason strain only above a critical temperature. Below this temperature, the hydrodynamic and elasticity theory no longer apply [27]. Finally, it is worth to note that phason modes are intimately related to chemical disorder, a point which has been studied in [28,29]. In the debate on entropy versus energy stabilization of QC, the entropy contribution to the system free energy is of course crucial. This phason entropy has been determined in a few model systems [25,[30][31][32]. Along these lines, the recent work of Engel [33] and Trebin (see this proceeding) on molecular dynamics simulation of a 2D model system using double well pair potential interaction, is particularly interesting since it provides a phase diagram, illustrating the respective role of energy and entropy in the system stabilization.

In summarizing this section on phason theory in QC's, it can be said that whereas the phenomenological hydrodynamics and elasticity of QC's has been fully developed, a few realistic models are available for decagonal phases, but an atomistic understanding and modeling of phason modes is not yet achieved especially for icosahedral phases.

Experiments and phason modes

A detailed presentation on experimental findings can be found in [START_REF] De Boissieu | Physics of Quasicrystals[END_REF]. We give in the following an overview of the main experimental results, and its comparison with the hydrodynamic theory. As explained in section 4, the hydrodynamic theory of quasicrystal has been checked experimentally mainly using diffuse scattering measurements. Indeed diffuse scattering is related to the Fourier transform of the two body correlation function and carry the information on phason modes. A few experimental difficulties have to be pointed out. First there are several possible origins for the diffuse scattering in a metallic alloy, and one has to discriminate them. This is generally done by simulating their respective contribution in reciprocal space. In particular the very peculiar Q per dependence and shape anisotropy of phason diffuse scattering is a key signature. Together with the phason diffuse scattering, there is also the thermal diffuse scattering (TDS) due to phonons. The TDS contribution can be suppressed using neutron scattering and an energy analysis with sufficient resolution. In the case of X-ray, this contribution can be evaluated, provided measurements are made on an absolute scale, i.e. the measured signal is converted in a differential cross section per atom. Moreover, if the diffuse scattering is on an absolute scale, it allows the determination of the phason elastic constant on an absolute scale. Finally, having diffuse scattering measurements on an absolute scale is the only way to compare the 'amount' of diffuse scattering for different samples.

We first summarize results obtained in the i-AlPdMn phase for which a complete study has been carried out. We then continue by results in other icosahedral phases, and end up with a few results obtained in decagonal phases.

The i-AlPMn phase.

The i-AlPdMn phase is particularly interesting since it can be grown as centimeter size single grains [34][35][36]. Although the phase diagram is intricate, there is a narrow phase field for which the icosahedral phase is stable down to room temperature. Czochralski or Bridgman grown single grain quasicrystal, once annealed, have an exceptional structural quality, in term of mosaic spread and residual phason strain, leading to dynamical diffraction [37][38][39]. The large single grain size, allowed studies with neutron scattering and energy analysis, so that the TDS contribution was suppressed. A clear diffuse scattering signal shows up around the Bragg peaks, with all the characteristics of phason modes diffuse scattering: 1/q 2 decay, Q perp dependence of the intensity and shape anisotropy. In fact the observed signal is perfectly reproduced using the hydrodynamic theory and two phason elastic constants whose ratio K2/K1=-0.52 [40]. These findings have also been confirmed by absolute scale measurements using synchrotron radiation X-ray scattering. Using the absolute scale measurement it has been shown that the TDS contribution is negligible, and that the amount of diffuse scattering is independent of the heat treatment, ruling out an interpretation in term of defects: the observed diffuse scattering is related to equilibrium long wavelength phason modes (Figure 3). The K1/k B T and K2/k B T elastic constants, extracted from this measurement were found equal to 0.1 and -0.052 atom -1 respectively [41], values which are useful for any quantitative comparison with models. At room temperature, one does not expect the phason modes to be dynamics. This was shown by a T study of the diffuse scattering, which showed a rather unexpected result, i.e. that the diffuse scattering diminishes as the temperature is increased, while the shape anisotropy also decreases. The T dependence of the diffuse scattering is very well reproduced by a softening of the phason elastic constant towards a 3-fold type instability, as the temperature is decreased from 780°C down to 500°C [42]. Below 500°C, the diffuse scattering intensity does not change anymore, showing that phason fluctuations are most likely quenched in at RT. This phason softening is most likely a signature of pre-transitional fluctuations, as expected in the random tiling or entropy stabilized scenario, the transition being avoided because of kinetics reasons. Indeed, for a very nearby chemical composition, the icosahedral phase is only stable at high temperature, and transform around 700°C in a F2 M phase, which remains quasiperiodic but with a tetrahedral point group symmetry [43][44][45].

From these results, one expects thus phason dynamics to show up above 500°C. The dynamics has been studied using X-ray photon correlation spectroscopy (XPCS) at the ESRF [46], a third generation synchrotron. Indeed, the brilliance of the X-ray light is such that a coherent X-ray beam can be produced. The coherent Xray beam acts as a laser light, and when measuring the phason diffuse scattering, instead of a smoothly decaying intensity one observes large intensity fluctuations named speckle pattern. The speckle pattern is related to the phason fluctuations, so that any time dependence of the phason fluctuations will lead to changes in the speckle pattern. This is best observed by computing the time intensity correlation function. At a temperature of 400°C, the correlation function is time independent, whereas at 650°C, the correlation function decays exponentially with time (Figure 4, left panel). From this decay one can extract a characteristic time N, which is found to depend on the phason wavevector q. As expected for a diffusive mode, the characteristic time increases as 1/q 2 , from which a phason diffusion constant equal to 1.510 -16 m 2 s -1 has been determined [46,47] (Figure 4, right panel). A few measurements carried out at 600°C indicate that the process is most likely thermally activated, with an activation energy of the order 3 eV. This diffusion constant is slower than the Mn self diffusion (1.10 -14 m 2 s -1 , 2 eV) [START_REF] Mehrer | Quasicrystals -Structure and Physical Properties[END_REF], indicating that phason modes involve exchange of chemical species on neighboring sites, with thus a longer time scale. The measured time scale and activation energy are very close to what has been obtained by M. Feuerbacher et al. in the study of a phason wall relaxation left behind a moving dislocation [START_REF] Feurbacher | [END_REF] (see also these proceedings).

At much longer wavevector (i.e. short wavelength) Coddens et al. have evidenced a very fast process by measuring quasielestic neutron scattering, interpreted as an atomic jump. From data analysis they concluded that this is an assisted jump, an atom going from one site to an unoccupied one. A detailed study on a single grain, lead to the conclusion that there are 2 atoms, probably Pd, jumping simultaneously over a distance of about 0.3 nm [50]. This last process has not been related to the details of the atomic structure. Moreover, the time scale found in this study, is much too fast to be compatible with what has been observed at longer wavelength, so that most likely other process have to be invoked to explain the long wavelength phason dynamics.

Although the atomic structure of the i-AlPdMn phase is rather well understood, there is no model for phason modes in this phase. Some hint of what is the effect of a long wavelength phason mode might be nevertheless obtained looking at the 6D picture of the i-AlPdMn phase. The most detailed model, has been obtained by Yamamoto et al. [51], with the same rough features as those obtained in other models [52,53]. The structure is built up with 3 atomic surfaces (AS), 2 large one's on the nodes and a smaller one (Pd) on one of the body center. The two large AS are decomposed (in a complex way) in smaller parts or sub-domains, each of which consist of Pd, Al, Mn or a disordered sites (Al/Mn …), and have a small component in the parallel space. When looking at the effect of a translation of the cut space along the perpendicular direction, the effect on the 3D structure might be decomposed in two situations. In the first one, which corresponds to the case where atomic surfaces are connected and satisfy the closeness condition (see Katz and Gratias [START_REF] Katz | Lectures on Quasicrystals[END_REF]), an atom 'jumps' from one position to another 'unoccupied' one having almost the same local environment. This situation corresponds to the LS to SL substitution in the Fibonacci sequence. For the i-AlPdMn QC, this concerns the external part of the AS and leads to short 'jump' distances of about 0.15 nm along 5-fold axis for Al atoms for instance. This is this situation which has been most of the time considered as a 'phason jump'. There is however a second case, corresponding to the inner part of the large atomic surfaces, which generally are not connected: as a result when the cut is moved along the perpendicular direction, there will be some chemical specie exchange (say Pd and Mn), between two sites which might be far apart. In other word a phason mode is coupled to chemical disorder, or more precisely, is chemical disorder which occurs on specific sites. This is most likely the limiting process in phason mode diffusivity, which makes them very slow.

Other icosahedral phases.

Diffuse scattering studies have been carried out on several other quasicrystalline phases. The main result is that for all known quasicrystals, even the so called 'perfect' one's, phason diffuse scattering has been observed. The shape anisotropy (related to the ratio of the K1 and K2 phason elastic constants), and the amount of diffuse scattering (related to the magnitude of the phason elastic constants) is however strongly sample dependent. This is for instance the case of the isostructural quasicrystals, i-AlPdMn, i-AlCuFe [START_REF] Boudard | [END_REF] and i-AlPdRe [56]. Although they share a similar atomic structure the diffuse scattering shape anisotropy is quite different, with a 5-fold type diffuse scattering for instance in the i-AlCuFe phase, in agreement with a phason driven phase transition toward the rhombohedral phase with an intermediate 5-fold modulated phase [57][58][59].

The recently discovered binary CdYb phase [60] and related family in the Zn-Mg-Sc system [61] has also been the subject of investigation. Phason diffuse scattering has been observed in the CdYb icosahedral phase [56]. Of particular interest is the comparison carried out between the i-ZnMgSc and the 1/1 cubic ZnSc approximant. Whereas the diffuse scattering observed in the ZnSc cubic approximant can be fully interpreted by a TDS contribution, in the i-ZnMgSc quasicrystal there is a supplementary contribution to the diffuse scatteruing arising from phason modes [62]. We stress here again the importance of having carried out absolute scale measurements, which allowed an exact computation of the TDS contribution. This result nicely demonstrates that the phason diffuse scattering is related to the long range quasiperiodic order, since both the QC and its approximant share the same local environment in term of atomic clusters. Moreover, in both cases, the inner tetrahedron corresponds to a site where the 'phason' jump occurs [63], experimentally illustrating the known result that one needs a quasiperiodic long range order for the long wavelength phason to take place. Finally it was found that the amount of diffuse scattering is much smaller in the i-ZnMgSc phase than in the i-AlPdMn one, as a consequence of larger phason elastic constants. This lead to a very large number of Bragg peaks, having a large Q per component, which are observed in the diffraction pattern (Figure 5).

In this review, we did not touch upon the phonon-phason coupling term. Although this term certainly exists, the observed diffuse scattering is in general dominated by the pure phason contribution. Note however, an interesting method to determine this constant proposed by Edagawa in comparing crystal approximant and quasicrystal lattice parameters [64][65][66]. Using this method, the phonon-phason constant could be evaluated in a few systems.

Decagonal phases

Decagonal phases are described as a periodic packing of a few quasiperiodic layers. As such, TEM imaging technique can be used, especially when seen down the periodic axis. However, the 3D character of these phases should not be forgotten, especially when discussing phason modes.

The first evidence for phason modes was obtained by Edagawa et al. in the d-AlCuCo [67,68]. In an insitu HRTEM experiment carried out at 1123 K, they observed time dependent contrast changes in the images, which could be mapped out onto a 2D Penrose tiling, with an edge length of about 2 nm. The contrast change corresponds to a 'phason tile flip' in the Penrose tiling. Images where obtained on rather thick part of the sample, so that what is observed corresponds to an atomic column 'flip', and thus correlated motion, and since the edge length of the tiling is 2 nm, the observed contrast is most likely a cluster column rather than an atom column. The flip occurs with an intermediate state, and long time scales of several tens of seconds between two flips. Whether the observed flip is related to the hydrodynamic theory of quasicrystal could not be proved in a further detailed analysis [68].

Single atomic flips on a distance of the order 0.1 nm have been evidenced by Abe et al. using annular dark-field scanning transmission electron microscopy (ADF-STEM) at 1100K on a d-AlNiCo sample. They observed changes in the image contrast, which is interpreted as an anomalous Debye-Waller effect, anharmonic in nature. By comparing the experimental images with a model and its high dimensional picture, they proved that the anomaly is caused by atoms 'jumping' from one site to another one with almost the same local environment, at a distance of about 0.1 nm and corresponding to the border of the atomic surfaces where the closeness condition apply. This is, up to now, the only experiment which has clearly revealed this small distance atomic jump. Of course the technique does not give any clue on the time scale of those jumps. This observation is consistent also with molecular dynamic simulation, where an anomaly on Al diffusion has been predicted [69].

A large variety of diffuse scattering phenomena have been observed in decagonal phases. The diffuse scattering is strongly composition dependent and changes with temperature. Diffuse scattering has been observed both in between the quasiperiodic Bragg layers (as a result of a short range order along the periodic axis, doubling of the unit cell) and within the quasiperiodic Bragg layers. Various interpretation have been given in term of columnar cluster disorder, or Huang scattering… [70][71][72][73][74] . It is only recently that the phason diffuse scattering has been studied [75,76]. Looking at the overall diffuse scattering distribution, the authors have extracted the phason and phonon/phason elastic constant, on a relative basis (measurements were not carried out on an absolute scale). Results have been also interpreted by mean of a random distribution of a cluster, which reproduces well the short range order extracted from a Patterson analysis of the diffuse scattering data. Although this model by its construction can not reproduce the long wavelength phason modes, it is a first step in understanding phason diffuse scattering in decagonal phases. Finally it is worth to note that in situ studies of the diffuse scattering showed that the diffuse scattering diminishes when the temperature is increased in a way similar to what was observed in i-AlPdMn (although the origin of the diffuse scattering is not attributed to phason modes in decagonal phases).

Other quasicrystals.

Recently, dodecagonal quasicrystals have been discovered in soft condensed matter [77] or polymer phases [78]. These are particularly interesting systems for several reasons. First, the length scale and interaction between the building blocks is fundamentally different from those in play in already known QC. Second, models used for the dodecagonal QC are known to have fractal surfaces. This is a challenging problem for simulation, since in this case it has been shown that the fractal shape of the atomic surfaces induces a very complex process called the zipper [79]. Whether this process is really playing a role in experimentally found systems is of great interest. Finally, the entropy gained by tile re-arrangment is known theoretically, so that some estimate of the configurational entropy might be carried out.

To conclude this short section, it is worth noting the recent photonic quasicrystal, which allows to introduce dislocation like defects and phason in a completely new system, for which the Hamiltonian is again very different from other systems [80]. This offers the opportunity of studying phason modes in a controlled set up.

Conclusion

In conclusion we have seen that phason modes have been evidenced in a few aperiodic phases. Experimental results are consistent with the hydrodynamics theory, which is best derived in the high dimensional description of aperiodic crystals. Phason modes can be viewed as modes with a wavevector in physical space and a polarization in the perpendicular space, but as a result of the hydrodynamic theory, phason modes are always diffusive mode, provided the wavelength is large enough, i.e. phason modes are not propagative modes, but are decaying exponentially with time.

The degree of understanding of phason modes at the atomic scale is however quite different in the systems investigated. For a few incommensurate displacive modulated phases, it has been found that two regimes exists: in a restricted wavelength range, phason modes are damped propagative mode, measurable by inelastic neutron scattering and resulting from a soft phonon mode transition toward an incommensurate phase. Models strongly suggest that the incommensurability arises because of frustration. At longer wavelength, phason modes become diffusive. In the study of an incommensurate density modulation (order disorder type) phason modes have been shown to have a relaxational character.

For quasicrystal, phason modes have been evidenced experimentally by X-ray or neutron diffuse scattering measurements. In all icosahedral phases studied so far, the phason diffuse scattering has been evidenced. However the magnitude and the sign of the phason elastic constant derived from such measurement is strongly sample dependent, even for isostructural phases. This suggests that phason modes in that case are strongly related with chemical disorder. The dynamic nature of phason modes has been studied only in the i-AlPdMn phase using XPCS, where results are in agreement with hydrodynamics and the phason diffusion constant has been extracted. The very slow phason diffusion constant is compatible with results obtained by dislocation relaxation studies. Finally, the temperature study of the diffuse scattering in the i-AlPdMn lead to result which are consistent with a softening of the phason elastic constant as a signature of 3fold type pretransitional fluctuations.

The above results can be qualitatively explained by the shape and chemical distribution of the 6D models which have been derived in a few systems: indeed whereas the external shape of the atomic surfaces fulfils the so called closeness condition, this is generally not the case for the inner part of the atomic surfaces. This unavoidably induces chemical disorder on specific sites of the quasicrystalline structure: this is certainly the slow limiting process in phason mode relaxation. Of course, realistic atomic modeling is necessary for a better understanding of phason modes. Although some realistic simulations have been carried out in decagonal phases, the most promising system is the CdYb one, where both detailed atomic structure [63] and realistic pair potential are now available [81]. Moreover a detailed T study of the diffuse scattering in this system would be of great interest.

Phason modes remain one of the most fascinating subjects in the aperiodic crystal field. Understanding of phason modes would certainly benefit from an interdisciplinary approach encompasses all aperiodic phases, with an exchanges of the knowledge gained in the various field. Indeed there are only a few incommensurately modulated systems for which phason modes have been evidenced, the question of the sliding modes in composites (or intergrowth) being an open one. Diffuse scattering studies have been mainly used for QC but could certainly be used for modulated or composite structures. The hypothesis used in the hydrodynamic theory, the possible analyticity breaking transition, the relation between approximant (or lockin phase) and the aperiodic phase, all those problems would certainly benefit from a unified and broad approach. Finally, the still open and fascinating questions of why aperiodic order sets in, what is the driving force stabilizing the long range aperiodic order, can now be tackled using realistic models. The dependence is linear, from which a phason diffusion constant is extracted (see text). From [46] and [47].

Figure 5: Comparison of the diffuse scattering in the i-ZnMgSc and i-AlPdMn as measured along a 2-fold axis. In both cases the diffuse scattering intensity is expressed in absolute scale unit. The amount of diffuse scattering is larger in the i-AlPdMn phase than in the i-ZnMgSc one. The number of large Q per , weak reflections is much larger in the i-ZnMgSc phase. The vertical bars stand for the position of theoretically calculated reflection with a maximum Q per value of 7 and 2.5 in the i-ZnMgSc and i-AlPdMn phases respectively (from [62])

Figure 1 :

 1 FIGURE CAPTIONS

Figure 2 :

 2 Figure 2: Experimental dispersion relation as obtained in the ThBr4 by inelastic neutron scattering above the transition (left) and in the incommensurate regime (right) (from [6]). The soft mode condenses towards the position of the satellite reflection, left panel. The right panel displays only the phason and amplitudon modes as measured close to the satellite reflection. Note that the slope of the phason mode is significantly different from the slope of the acoustic branch.

Figure 3 :

 3 Figure 3: Left panel: diffuse scattering measured in the i-AlPdMn phase in a 5-fold plane. Right panel, simulated map, using phason diffuse scattering and the 2 phason elastic constants (see text).

Figure 4 :

 4 Figure 4: Measurement of the phason dynamics in the i-AlPdMn phase using XPCS, at 650°C. Left panel: exponential time decay of the phason mode. The wavevector and characteristic time are indicated in the inset. Right panel: Evolution of the inverse characteristic time as a function of the square of the phason wavevector.The dependence is linear, from which a phason diffusion constant is extracted (see text). From[46] and[47].

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Acknowledgements: I would like to thanks S. Francoual, R. Currat and T. Janssen for fruitful discussions.