Uli Buchenau
email: buchenau-juelich@t-online.de

Reiner Zorn

Michael Ohl

Andreas Wischnewski

Dielectric and thermal relaxation in the energy landscape

Keywords: dielectric materials, glass transition, liquids, thermal properties

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Broadband dielectric spectroscopy [1] is the most versatile method to study the flow process in molecular liquids. However, the quantitative nature of the relation between dielectric signal (i.e. molecular reorientation) and shear flow is not yet clear. The classical Debye picture and its extension to viscoelasticity [2,3] considers the molecule as a small sphere with a hydrodynamic radius r H immersed in the viscoelastic liquid. It predicts a slow dielectric decay, about a factor of fifty slower than the mechanical shear stress decay. The measured dielectric decay is on the average a factor of ten faster than this Debye prediction [START_REF] Böttcher | Theory of Electric Polarization[END_REF]. A thorough quantitative analysis of dielectric and shear data in seven glass formers [START_REF] Niss | [END_REF]6] showed a general qualitative agreement with the Gemant-DiMarzio-Bishop extension [2,3] of the Debye scheme, but a rather poor quantitative fit. Very recently, an alternative to the Debye model and its extensions was proposed [7], which gave a much better fit for glycerol [1,8] and propylene carbonate [9] data. The proposal was based on the growing evidence for the identity of thermal and dielectric relaxation functions [7,[START_REF] Birge | [END_REF][12][13] (dielectric hole burning experiments [14,15] even suggest that dynamically distinct domains in the liquid are associated with a time constant characterizing both the dielectric and the thermal behavior).

The present paper begins with a reminder of the textbook introduction to dielectrics [START_REF] Böttcher | Theory of Electric Polarization[END_REF] in terms of the decay of the dielectric polarization after a switch-off of the electric field. In the Debye scheme, the decay of the dielectric polarization is due to the rotational diffusion of the molecules with a Debye-Stokes-Einstein diffusion constant. But one can easily generalize the formalism to other decay mechanisms. In particular, we consider the recent proposal [7] of an initial and a retarded part of the flow process. We show that one must expect such a division in the energy landscape picture. The thermal relaxation sees both the initial and the retarded final part of the process. The equality of thermal and dielectric relaxation times is understandable in terms of an energy landscape argument. The equality itself is checked for salol, using literature data. In SI units, the electrostatic equations read

D 0 = 0 (0)E 0 = 0 E 0 + P 0 , (1)
where 0 is the vacuum permittivity, E 0 is the electric field, D 0 is the displacement field and P 0 is the polarization per unit volume. E 0 , D 0 and P 0 are vectors, so the static dielectric constant (0) is a second rank tensor. Here, however, we limit ourselves to isotropic liquids or glasses, so (0) reduces to a scalar. Similarly, we neglect the conductivity contribution, assuming that it can be simply subtracted.

The electrostatic equations are generalized to frequency-dependent equations [START_REF] Böttcher | Theory of Electric Polarization[END_REF]

D(ω) = 0 (ω)E(ω) = 0 E(ω) + P (ω) (2)
with frequency-dependent fields and dielectric constant, respectively. Thus one has

(ω) = 1 + 1 0 P (ω) E(ω) . (3)
The next step introduces a useful simplification for ∞ , the dielectric constant at infinite frequency. Since one is mainly interested in the relaxation in the frequency range below 100 GHz, one considers all processes above this frequency as immediate processes. This includes the electronic polarizability (time scale 10 -15 seconds) as well as the vibrational contributions (time scale 10 -12 seconds), so

∞ = n 2 + ∆ vib , ( 4
)
where n is the refractive index and ∆ vib is the contribution of the vibrations (molecular librations) to the dielectric constant. Since this contribution is nonzero, one expects ∞ > n 2 in a reasonable fit (this is in fact a main problem of the Gemant-DiMarzio-Bishop extension [2,3] of the Debye scheme [START_REF] Niss | [END_REF]). Consider an electric field E 0 staying constant from t = -∞ until t = 0. At time zero, the field is switched off (Fig. 1). The Fourier transform of this field reads For such a field, the polarization has a constant value

E(ω) = - i ω E 0 . (5)
P 0 = ( (0) -1) 0 E 0 (6)
for times t < 0. At t = 0, the polarization drops instantaneously to P 0 -P ∞ , where

P ∞ = ( ∞ -1) 0 E 0 (7)
At t > 0, P (t) decays with

P (t) = (P 0 -P ∞ )Φ(t), (8)
where Φ(t) is a function which begins with 1 at time zero and drops to zero at infinite time.

If one compares this dielectric polarization decay with the mechanical shear stress decay, one finds a fundamental difference. After switching off the electric field, the polarization decays without any externally applied force. In the mechanical case, one applies a small external shear strain at time zero, keeps the strain constant and one observes the decay of the shear stress. The mechanical shear stress decay is described in linear response by a time-dependent shear modulus G(t). The dielectric case is different. In order to describe it in terms of a time-dependent dielectric constant, one needs to go to the opposite case of Fig. 1, switching the electric field on at time zero and watching the rise of (t) from ∞ to 0 with the function 1 -Φ(t). As a consequence, if one wants to compare decay with decay in order to see which quantity decays faster, one needs to compare the mechanical modulus with the dielectric susceptibility and not with any dielectric modulus, at variance with a recent proposal [START_REF] Niss | [END_REF]. We come back to this point after the treatment of the Gemant-DiMarzio-Bishop relation [2,3].

The Fourier transform P (ω) of P (t) is given by

P (ω) = - i ω P 0 + (P 0 -P ∞ )F[Φ(t > 0)], (9)
where F[Φ(t > 0)] denotes the Fourier transform of the decay function Φ(t). Inserting eqs. ( 5) and (9) into eq. ( 3), one gets

(ω) = 1 + 1 0 P 0 E 0 + P 0 -P ∞ E 0 iωF[Φ(t > 0)] . (10)
The relation between Φ(ω) and Φ(t) is the same as the one between G(ω) and G(t)

Φ(ω) ≡ -iωF [Φ(t > 0)] = ω ∞ 0 Φ(t)(sin ωt + i cos ωt)dt. (11)
Φ(ω) is a complex function which is 1 for infinite frequency and decreases to zero as the frequency goes to zero. With this definition, and inserting eqs. ( 6) and (7) into eq. [START_REF] Böttcher | Theory of Electric Polarization[END_REF], we arrive at the final result

(ω) -∞ (0) -∞ = 1 -Φ(ω). ( 12
)
The Debye decay mechanism is the rotational diffusion of the molecules, with the diffusion constant given by the Debye-Stokes-Einstein equation

D trans = k B T 6πηr H = 4 3 r 2 H D rot , (13)
where D trans is the translational diffusion constant of the molecule and D rot is its rotational diffusion constant. r H is the hydrodynamic radius of the molecule (note that the hydrodynamic Debye-Stokes-Einstein equation is in principle derived for larger objects than a single molecule). For continuous rotational diffusion, the relaxation time for the Legendre polynomials is

τ L,rot = 1 L(L + 1)D rot , ( 14
)
where L is the order of the Legendre polynomial. For the dielectric signal, L = 1, one obtains the Debye relaxation time

τ D = 4πηr 3 H k B T . ( 15
)
The decay function for the dielectric polarization is a single exponential (Debye process)

Φ(t) = exp(-t/τ D ) (16)
which leads to the well-known Debye equation

(ω) -∞ (0) -∞ = 1 1 + iωτ D . (17)
To extend the Debye equation to viscoelasticity [2,3], one replaces the static viscosity η by a frequencydependent function η(ω), which is in turn related to the frequency-dependent shear modulus G(ω)

iωη(ω) = G(ω) ≡ G ∞ g(ω). ( 18
)
Here G ∞ is the infinite frequency shear modulus and g(ω) is a normalized function like Φ(ω), going from zero to 1 with increasing frequency.

The ratio between viscosity and infinite frequency shear modulus defines the Maxwell time

τ M = η G ∞ , (19)
the characteristic shear relaxation time.

If we replace the viscosity η in the τ D of the right hand side of the Debye equation by the expression of eq. ( 18), we get the Gemant-DiMarzio-Bishop expression

1 1 + (4πG ∞ r 3 H /k B T )g(ω) ≡ 1 1 + c r g(ω) , (20)
which contains the dimensionless ratio c r between the Debye relaxation time τ D and the Maxwell time τ M . The Gemant-DiMarzio-Bishop expression, eq. ( 20), does not approach zero for infinite frequency as the Debye expression in eq. ( 17), because the molecule is still able to turn in a balance between the torques exerted by the electric field and the finite restoring force of the sheared elastic medium, respectively. In

∆ vib = (0) -n 2 1 + c r . (21)
Thus ∆ vib is already taken into account, and the Gemant-DiMarzio-Bishop equation [2,3] takes the form

(ω) -n 2 (0) -n 2 = 1 1 + c r g(ω) . ( 22
)
With a hydrodynamic radius of 0.2 nm, a G ∞ of 2 GPa and a temperature of 200 K one calculates c r = 50, so one expects a small vibrational component and nearly two decades difference between τ D and the Maxwell time. The first of these expectations is in fact found, but the second fails by an average factor of ten [START_REF] Böttcher | Theory of Electric Polarization[END_REF][START_REF] Niss | [END_REF]7,16].

Note that the failure is only in the factor and not in the temperature dependence. Unlike the translational diffusion, which decouples from the viscosity below some critical temperature [16][17][18], the rotational relaxation time follows essentially the temperature dependence of the viscosity over the whole temperature range.

On the basis of the Gemant-DiMarzio-Bishop equation (22), it has been recently argued [START_REF] Niss | [END_REF] that one should compare G(ω) with 1/( (ω) -n 2 ), the so-called "rotational modulus". This is true if the equation is valid; it is not true if one has a different decay mechanism of the dielectric polarization. Take, for instance, the Debye model of eq. ( 17). The true relaxation peak lies at 1/τ D in (ω) as it should, but shifts with varying ∆ vib in the rotational modulus 1/( (ω) -n 2 ).

An alternative to the Debye-Gemant-DiMarzio-Bishop scheme is a recent proposal [7] for the structural relaxation in the energy landscape [20] (for a review of the energy landscape concept see [21]) The proposal is based on the experimental finding [13] of two time scales in the undercooled liquid. The faster one is the shear stress decay, described by g(ω). The slower one is the decay of the structural potential energy, seen in dynamic heat capacity measurements or in transient grating experiments [19].

To understand this behavior, let us assume that the shear stress decay occurs via thermally activated processes in the energy landscape [20,21], without specifying the exact nature of the thermally activated processes. They might be "flow by shoving" [22] or some other mechanism [23]; for our purpose, it suffices to assume that the shear decay happens by repeated passages of the system from one minimum of the energy landscape to another. The energy landscape picture provides a natural explanation for the dynamical heterogeneity seen in numerous experiments [24]. One can show that an initial and a retarded part of the flow process follow from the energy landscape concept under reasonable assumptions.

To see this, consider two structural rearrangements occurring in different parts of the sample. Each of them can be characterized by an asymmetric double-well potential in the corresponding configurational coordinate, leading from one potential energy minimum to the other. Since the two structural rearrangements are far apart on a microscopic scale, the two minima of the first structural rearrangement change only slightly by a jump in the other double-well.

In a macroscopic sample, there is a very large number of possible structural rearrangements. Each of them is a local process, and each of them influences every other one.

The consideration shows that one has two relaxation times in each double-well potential, a lifetime of the population of the two minima and a lifetime of the double-well potential itself. For the latter one, the Maxwell time is crucial, because it sets a lower boundary for the time scale on which the core region of the relaxation can change its shape. If the core region can change its shape, the core relaxation can disappear or make way for a different relaxation. This relaxation life time is not determined by a single jump, but by the joint effect of all the other relaxations in the sample, a diffusional motion through phase space. The dielectric data seem to see this final relaxation time. In fact, the dielectric relaxation time in glycerol and OTP is close to the spin-echo relaxation time at the first sharp diffraction peak [7], the decay time of the short range order.

Naturally, the energy landscape itself does not change in time; nevertheless, we believe that the concept of a decay of a local double-well potential, for simplicity assumed to be the same for each double-well in the system, catches an essential feature of the motion of the system in the fixed energy landscape. Qualitatively, such a picture is also compatible with NMR findings [25] in toluene and glycerol, showing a small number of larger-angle rotational jumps (attributable to the hopping part) and a larger number of small-angle rotational jumps of the molecules (attributable to the diffusional part).

The final part of the equilibration is not (or maybe only partially) seen in the stress decay, because it can only happen after the stress decay. As long as there is still some appreciable shear rigidity, higher and higher barriers are jumped over in order to equilibrate the remaining average stress to zero. This mechanism determines the Maxwell time, and a corresponding Maxwell barrier. Roughly speaking, the part of the motion through phase space due to the initial change of conditions changes from hopping to diffusion after the Maxwell time. If an energy barrier is higher than the Maxwell barrier, it is not jumped over, but it flows away.

Here, we follow ref. [7] in assuming that the retarded part of the process can be described as the decay of an energy stored in a harmonic spring rG ∞ in series with a time-dependent spring G(t) (see Fig. 2). The lower r, the more retarded is this final stage of the equilibration. r is the ratio of the energy stored in G(t) to the one stored in the harmonic spring, which does only decay by flowing over into the spring G(t).

Note that not the structural potential energy itself decays, but its difference to an average temperaturedependent value determined by the entropy of the inherent structure of the energy landscape. Our assumptions imply that the difference between the potential structural energy of a given minimum of the energy landscape and the average value has always the ratio r between its harmonic part and its long-range stress component.

In the decay of the structural potential energy of the undercooled liquid, one expects to see both the initial hopping and the final diffusion. The previous paper [7] took only this second retarded action into account. But naturally one has to take the initial process into account as well. Each jump in an asymmetric doublewell changes the structural potential energy. This forces one to introduce an additional dimensionless parameter f , the fraction of the structural potential energy which equilibrates via the initial hopping. In a globally connected energy landscape [21], this parameter f would be one, but in a real energy landscape it is expected to be considerably smaller.

We assume that this initial part has the same time dependence as the shear modulus G(t). Then the Fourier transform of the normalized decay function of the structural potential energy is

Φ(ω) = f g(ω) + (1 -f ) 1 + r 1 + r/g(ω) , ( 23
)
where

g(ω) = G(ω)/G ∞ .
If the dielectric polarization has indeed the same time dependence as the structural potential energy difference, we have to insert this Φ(ω) into eq. ( 12) and obtain

(ω) -∞ (0) -∞ = f (1 -g(ω)) + (1 -f ) 1 -g(ω) 1 + g(ω)/r . ( 24
)
Experimentally, one finds the loss peak of the dielectric constant close to the one of the heat capacity [7,[START_REF] Birge | [END_REF][12][13]32]. In fact, one can argue that the electric field introduces an imbalance between the different structural realizations of the undercooled liquid at the given temperature. If one mirrors the sample at a plane perpendicular to the electric field, one gets a state with the same structural potential energy, but with opposite electric dipole moment. These two states have the same energy in the absence of the electric field, but a different energy in the electric field. Thus the switch-off of the field leaves an imbalance in the structural entropy. It should equilibrate by the same mechanism as the imbalance introduced by a temperature jump. On the basis of this argument, one understands the coincidence of the dielectric relaxation time with the dynamic heat capacity and the transient grating one.

If the decay mechanism of the dielectric polarization is not the diffusion of a single molecule, but the passage from one of the minima of the energy landscape to another, one needs no longer distinguish between the internal field at the molecule and the externally applied field, because the externally applied field acts directly on the electric dipole moment difference of the two minima; all interaction effects are already included in this dipole moment difference. The complicated many-body problem of a system of interacting electric dipoles [START_REF] Böttcher | Theory of Electric Polarization[END_REF] enters only via the temperature dependence of (0).

It is interesting to consider the limiting cases of very small and very large r in the retarded term of eq. ( 24). For very small r, the denominator dominates the behavior and we approximate the Gemant-DiMarzio-Bishop case of eq. ( 22). For very large r, the harmonic spring is like a rigid connection and the relaxation peaks of G(ω) and (ω) become identical. Note that this limit can never be attained by the Gemant-DiMarzio-Bishop extension of the Debye scheme. There, it corresponds to the limit of very small c r , in which the relaxation peak disappears, leaving only the vibrational component.

Dielectric, thermal and rotational relaxation times in salol

For the comparison of dielectric and thermal relaxation times, we add a new example, salol (glass temperature 213 to 217 K, mode coupling critical temperature 255 to 265 K), to the three examples glycerol, propylene carbonate and OTP discussed in the previous work [7]. All measured relaxation times are compared to the Maxwell time, calculated from fits of the infinite frequency shear modulus and of the viscosity. The infinite frequency shear modulus is taken from light scattering Brillouin data. As it turns out, it is better to use longitudinal sound wave data, relating the transverse sound velocity v t to the longitudinal v l by v l /v t ≈ 1.8. The mistake of this approximation is small on the scale of factors of ten considered here. The advantages are (i) the longitudinal infinite frequency sound velocity is much more easily determined (ii) one has data up to the highest temperatures. To take an example, G ∞ from transverse Brillouin OTP data [26] extrapolates to zero at 348 K (not at 308 K as stated erroneously in the previous paper), while one still has data to compare up to 380 K.

In salol, the density follows the relation [27] ρ = 1451.6 -0.857T (25) with ρ in kg/m 3 and T in Kelvin. The longitudinal infinite frequency sound velocity [28] v l = 2400

T g T 0.88 m/s (26)
with the glass transition temperature T g = 218K. With our recipe for the ratio v l /v t , this yields

G ∞ = 2.2 T g T 1.96 GP a. (27)
The viscosity [29][30][31] is parametrized in terms of two overlapping Vogel-Fulcher laws Figure 3 compares dielectric [32,33], heat capacity [32] and transient grating [35] relaxation times to rotational relaxation times extracted from longitudinal Brillouin light scattering data [34]. The times were either recalculated [36] or refitted in terms of a Kohlrausch function exp(-(t/τ KW W ) β ).

log η = log η 0i + B i T -T 0i (28
Fig. 3 contains no mechanical shear relaxation times, but one knows that these have to lie below the Maxwell time, because for a shear modulus following the Kohlrausch function exp(-(t/τ shear ) β )

τ shear τ M axwell = β Γ(1/β) . (29)
Usually, β lies between 0.4 and 0.6, so the ratio should be between one third and two thirds; the shear Kohlrausch relaxation time should be a factor 1.5 to 3 shorter than the Maxwell time. Fig. 3 shows once again that the rotational times keep close to the entropy relaxation times in the heat capacity and in the transient grating measurements, thus supporting the proposed energy landscape mechanism for the relaxation of the molecular orientation. In principle, the transient grating measurements should also be evaluated in the Ansatz of Pick and Dreyfus [37,38], because the signal comes from a grating of both temperature and local molecular orientation. However, an experimental separation [39] of the contributions demonstrated again the equality of the two relaxation functions within experimental error.

Note that the method developed by Dreyfus and Pick [37,38] to determine rotational relaxation times pluses dielectric [32]; crosses dielectric [33]; full triangles heat capacity [32]; full diamonds rotational relaxation times extracted from Brillouin light scattering [34]; full circles transient grating data [35]. from light scattering Brillouin data seems to work remarkably well. According to their Ansatz, one should see the second Legendre polynomial. For continuous rotational diffusion, their relaxation times should be a factor of three smaller than the dielectric ones. One does not see this factor of three; the two sets of data are remarkably close to each other over ten decades in relaxation time, even making the same wiggles compared to the Maxwell time. In the energy landscape mechanism, one does not expect the factor three; there, it depends on the jump angle distribution which can be different for different glass formers [21,25]. The marked temperature dependence of the ratio between rotational relaxation time and Maxwell time indicates a temperature dependence of the retardation parameter r in eq. (24). The rise at low temperature is also seen in glycerol and in propylene carbonate [7]. The salol kinks at 260 and 240 K have been also seen in the "Stickel plot" [33] of the dielectric data (a plot of the inverse square root of the temperature derivative of the peak frequency versus temperature, where one sees the deviations from a perfect Vogel-Fulcher behavior).

From the point of view of the mode coupling theory [40], the kink at 260 K should mark the crossover from an energy landscape behavior below T c to simple liquid behavior above. But Fig. 3 shows that the separation of time scales persists into the liquid domain in salol. This time scale splitting is not the twostage scenario of the mode-coupling theory, because both time scales move together with the Maxwell time. In fact, in ref. [41] this time scale splitting was also observed for propylene carbonate and discussed in terms of the mode coupling theory. The α-process of the theory was not attributed to the slower, but to the faster process.

Summary

The textbook relation between dielectric polarization decay and dielectric constant is applied to the simple Debye case, to its extension to viscoelasticity and to a newly proposed energy landscape mechanism. Data in salol show once again the equality of dielectric and structural relaxation times, supporting the energy landscape mechanism.

We thank Catherine Dreyfus and Robert Pick for helpful discussions.

Figure 1 .

 1 Figure 1. Polarization decay after switch-off of the electric field at time zero.

  has in this case

Figure 2 .

 2 Figure 2. Spring model for the decay of the structural potential energy of an undercooled liquid.

  i = 1 and i = 2, respectively. The first of these two is valid below a temperature T 1 , the second above a temperature T 2 ≤ T 1 . Between T 2 and T 1 , one takes a linear interpolation between the two to ensure continuity. For salol, log η 01 = -36.9 and log η 02 = -3.68 (η in Pas), B 1 = 7500 K and B 2 = 151 K, T 01 = 59 K and T 02 = 225 K, T 1 = 256 K and T 2 = 248 K.

Figure 3 .

 3 Figure3. Kohlrausch-Williams-Watts relaxation times in salol, normalized to the Maxwell time as described in the text. Symbols: pluses dielectric[32]; crosses dielectric[33]; full triangles heat capacity[32]; full diamonds rotational relaxation times extracted from Brillouin light scattering[34]; full circles transient grating data[35].

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Ronan Lefort
email: ronan.lefort@univ-rennes1.fr

Denis Morineau

Régis Guégan

Alain Moréac

Claude Ecolivet

M Guendouz

Structure and relaxation processes of an anisotropic molecular fluid confined into 1D nanochannels

Keywords: ellipsometry, liquid crystals, nanomaterials, neutron scattering, porous media, silicon-based materials Brillouin scattering

teaching and research institutions in France or abroad, or from public or private research centers.

Confinement effects on the molecular dynamics of simple liquids have been mostly scrutinized in porous materials presenting cavities of nanometric dimension [START_REF]Proceedings of the 2nd International Workshop on Dynamics in Confinement[END_REF]. A large variety of materials has been used, often silicates within different geometries, either isotropic (fractal aerogels, interconnected vycors, aerosils or gelsils) or template-controlled (MCM-41, SBA-15). Several experimental difficulties are encountered in such studies due to the unavoidable powder averaging of the measured observables and to the huge surface to volume ratio that is reached. Indeed, the surface contribution to the free energy of the confined liquid cannot be ignored anymore, and the awaited ultimate finite-size effects can be masked by dominant interfacial solid-liquid interactions [START_REF] Radhakrishnan | [END_REF]. Experimental evidence of such surface contribution was reported on simple liquids like toluene [3], as a stretching of the dynamical self-correlation functions toward longer times, and a non-vanishing static contribution within experimental resolution. These data were interpreted by a very large distribution of correlation times, and supported by numerical simulations [3]. This onset of glass-like dynamics in simple liquids appears as an essential consequence of confinement, but is not the only one, for also the structure and the thermodynamical stable and metastable states are affected [4].

These interrelated effects remain partly misunderstood even for simple fluids. A large effort toward a better understanding of more complex confined fluids is timely, for they are introduced in an increasing number of innovative applications of nanosciences (lab-on-chip, nanofluidics...). Liquid crystals (LC) are test-systems in order to investigate fluids with an increasing degree on complexity. Their structure is characterized by anisotropic order parameters giving rise to typical sequences of nematic or smectic phases. However, their molecular nature allows one to confidently assign dynamical contributions in terms of rotations, translational diffusion or librations, accessible to a large number of techniques like dielectric spectroscopy [5], neutron quasielastic scattering or light scattering (photon correlation spectroscopy, optical Kerr effect [6,7]). Both these structural and dynamical aspects have been widely studied on bulk samples, and make LC's model systems particularly suited to experimental investigations of confinement effects on complex fluids. Like more simple liquids, the thermodynamics of confined LC's is also profoundly modified. One of the most striking effect is the instability of the translational smectic ordering with respect to the quenched disorder field introduced by an irregular solid/LC interface [8,9]. This leads to a very important depression of the melting temperature of the confined crystalline phase, and the suppression of the nematic to smectic continuous transition, replaced by a frustrated short-range (nanometric) smectic ordering [10].

We report in this paper new experimental data depicting structural and dynamical features of a confined LC in a nanoporous material of very low dimensionality. Fully hydrogenated liquid crystal (LC) 4-octyl-4-cyanobiphenyl (8CB) was purchased from Sigma-Aldrich and used without further purification. The sequence of phase transitions for bulk 8CB is from low to high temperatures : Crystal -smectic A (SmA) (294 K), SmA -nematic (N) (306 K), and N -Isotropic (I) (313 K). In the SmA phase, the molecular packing is organized in parallel layers separated by c.a. 31 Å, for an average molecular length of about 19 Å. Columnar porous silicon (pSi) matrices were obtained from a heavily doped (100) oriented silicon substrate by electrochemical etching [11,12]. A porosity of 60 %, made of a parallel arrangement of not-connected channels (diameter ≈ 300 Å) is obtained, aligned in a layer of thickness ≈ 30 µm. This very large aspect ratio induces a quasi one-dimensional character of the porous structure, preferentially aligned on the macroscopic lengthscale. Thermal oxidation of the structure can be further performed, leading to a transparent layer well suited to investigations by optical techniques. 8CB was confined into pSi by capillary wetting from the liquid phase under vapour pressure in a vaccuum chamber and at a temperature of 60˚C, well above the N-I transition temperature. The excess of LC was then removed by squeezing the wafers between Whatman filtration papers. The complete filling of the pores was characterized by confocal microRaman spectroscopy [13].

Neutron Scattering

Neutron diffraction experiments were performed on the G6.1 double-axis diffractometer (cold neutron guide) at the Laboratoire Léon Brillouin (LLB, CNRS/CEA, Saclay), with an incident wavelength of 4.71 Å. Quasielastic incoherent neutron scattering experiments were carried out on the time-of-flight (TOF) spectrometer G6.2 at the LLB, with a resolution of 107 µeV FWHM. The covered momentum transfer (Q) range was about 0.4 to 1.9 Å-1 . The temperature was controlled to better than 0.1 K over a range from 100 K to 340 K using a cryoloop. Numerical deconvolution of TOF spectra were carried out following standard procedures using the "QENSH" software provided by the LLB. For both diffraction and TOF experiments, eight filled porous silicon wafers were stacked parallel to each other in a cylindrical aluminium cell, representing a total amount of confined LC of about 20 mg.

Spectroscopic Ellipsometry

Ellipsometry experiments were carried out with a Horiba Jobin-Yvon UVISEL spectrometer, in the infrared range (λ from 800 to 2000 nm). The measured ellipsometric angles Ψ and ∆ are related respectively to the amplitude ratio and to the phase difference between the complex reflexion coefficients (r p and r s ) of the sample (resp. parallel or perpendicular to the incident plane). From these two angles, new quantities are defined according to :

I s = sin(2Ψ)sin(∆); I c = sin(2Ψ)cos(∆) (1)
These functions I s and I c were fitted versus the wavelength using the Horiba "DELTAPSI2" software provided with the apparatus. The sample was modelled as a porous silica layer top of a silicon substrate. The dielectric dispersions of silicon and silica were described using standard library data. The dielectric dispersion of the organic liquid crystal was modeled by a uniaxial anisotropic single oscillator. The porosity of the layer was described using a classical anisotropic Bruggeman model [15] mixing silica and void dispersions for the empty matrix, or silica and 8CB dispersions for the filled matrix.

Brillouin Scattering

Brillouin scattering was performed with a triple pass tandem of Fabry-Perot (Sandercock model) with a Kr + ion laser (Coherent) at a wavelength of 647 nm. Laser power was kept at a few tens of mW in order to avoid a too big heating due to absorption of the bulk silicon supporting the film. At too high counting rates a shutter obturates the photomultiplier inducing discontinuities in the spectra. Typical spectra were accumulated during several hours. Spectra were fitted by an apparatus function convoluted to damped oscillator profiles (Brillouin lines) and a Debye relaxor.

Results and Discussion

All results presented hereafter were obtained close to room temperature, in the smectic phase of 8CB (bulk or confined) [16]. In this temperature range, the liquid crystal 8CB confined in pSi experiences smectic ordering that remains short-range (SRO, ξ ≈ 150 Å) due to quenshed disorder effects imposed by solidliquid interfacial interactions [10,12]. These effects destroy the bulk second order nematic to smectic phase transition. As a consequence, the Bragg peak associated to this smectic SRO and appearing at 2θ B ≈ 8.6( at a working wavelength of 4.7 Å) is significantly broader than the experimental resolution.

One-dimensional ordering

Figure 1 displays ellipsometric spectra measured on empty porous silica and porous silica filled with 8CB.

The interference fringes are due to multiple reflections at interfaces of the porous layer. It is therefore possible to deduce an estimation of the layer thickness, and typical values between 2.5 and 5 µm (depending on the sample) were found, in good agreement with independent SEM results. A simple comparison of both curves shows strong differences in the amplitude modulation of those fringes, as a consequence of the dispersion of the optical index of the confined liquid crystalline phase. This observation can be used as a direct characterization of a proper filling of the porous layer with 8CB. A semi-quantitative analysis was performed using the model described in section 2.3. Best fits agree with an optical uniaxial birefringence of the filled porous layer of about ∆n ≈ 0.15, with an optical axis aligned with the direction of the nanopores. The important differences in the ellipsometric data allow to separate unambiguously this additional birefrengence due to the liquid crystal from the natural optical anisotropy of the empty porous layer. These two contributions can be further more distinguished through the temperature dependence of the liquid crystal contribution. From the orientation of this measured optical anisotropy, it can be concluded that the orientational order parameter of the confined 8CB strongly couples to the one-dimensional character of the porous matrix.

Figure 2 shows the intensity of the Bragg peak associated to the smectic SRO measured at constant q B ≈ 0.19 Å-1 , while rotating the sample plane by an angle θ S with respect to the neutron incident beam. The best fit using an arbitrary lorentzian form indicates that the maximum of the Bragg reflexion is peaked at θ 0 ≈ 4.6˚, which corresponds to the sample orientation θ S ≈ θ B where the Q vector is parallel to the nanochannels of the porous silicon layer. The linewidth is about 17˚HWHM. Within this rather small mosaicity, the director of the smectic layers is obviously well aligned with the silicon channels axis.

Rotational dynamics

Both ellipsometry and neutron diffraction results show that 8CB confined in porous silicon displays in the meantime orientational and translational order parameters that are macroscopically one-dimensionally oriented through a strong coupling with the peculiar low dimensionality of the matrix. Accordingly, the molecular dynamics of the confined phase might be also affected by this anisotropic ordering. Whereas translational diffusion should in some extent depend on the local properties of the smectic SRO domains, the rotational molecular relaxations should be more sensitive to orientational parameters. In this section, we report Brillouin and quasielastic neutron scattering results, that complementary probe the molecular relaxations in the picosecond time range.

Figure 3 shows the comparison of Brillouin spectra obtained on porous silica filled with 8CB or on the separated constituents. The 8CB spectrum is a depolarized one which does not show, as expected, any Brillouin doublet, but allows the best observation of the quasielastic region. The black line is the best fit according to the procedure detailed in section 2.4. The Brillouin doublets appearing on the spectra have different origins : the one at the lowest frequency is a longitudinal mode propagating tangentially to the surface, whereas at the highest frequency appears a longitudinal mode propagating mainly along the pores perpendicular to the surface. The response of the bulk liquid crystal consists in a single quasielastic lorentzian line of about 5 GHz HWHM (i.e. τ c ≈ 30 ps). The symmetry of the experimental setup suggests to assign this line to uniaxial rotation of the molecules around their long axis. The response of an empty porous silica matrix reveals strong Brillouin modes, sitting on a very broad quasielastic contribution (more than 35 GHz). Comparatively, the Brillouin spectrum of a filled porous layer looks like a combination of these individual responses, characterized by an enhancement of the silica response (due to the large scattering efficiency of 8CB) and by a broadening of the acoustic modes. This damping is due to the coupling of the solid matrix to the liquid crystal through its viscosity. The silicon matrix brings its rigidity and has a strong influence on the elastic constants of the material, while the continuity of displalcement at the interface silica/LC must be fulfilled. This condition results in a dissipative coupling between the solid and the liquid phases that is responsible for anomalous damping of modes polarized in the direction of the pores and can be described by poroelasticity theory [14]. A quasielastic contribution of about the same width as for the bulk 8CB can be seen on the data, suggesting that the uniaxial molecular rotations are only weakly affected by the confinement. Quasielastic neutron scattering experiments were performed, in order to probe additional relaxations. Figure 4 shows the TOF spectra measured on bulk 8CB and 8CB/pSi. Spectra were fitted using one lorentzian plus one elastic (resolution limited) components. For bulk 8CB, the quasielastic contribution has a linewidth of about 0.25 meV (τ c ≈ 3 ps). Its occurence at large values of momentum transfer together with the presence of an elastic contribution would suggest that it also reflects localized reorientations (rotations, end-chain relaxations...). For 8CB confined in porous silicon, figure 4(b) reports a significant (relative) increase of the elastic contribution, together with a decrease of the quasielastic linewidth down to 0.16 meV. This moderate slowing down of the fast reorientational motions agrees with the Brillouin results, and confirms that rotational molecular dynamics seem to be only weakly disturbed by the anisotropic confinement in porous silicon. On the other hand, the increase of the elastic contribution might have different origins: first, some of the molecular modes that contributed to the bulk quasielastic spectrum can be slowed down enough under confinement so that they can no more be detected within the experimental resolution. Second, also the geometry of the rotational dynamics probed in the picosecond range can be different for bulk or confined LC. In that case, different trajectories result in different elastic incoherent structure factors. Probably, these effects can happen together, as the result of a heterogeneous propagation of a surface induced slowing-down through the inner pore volume.

Conclusion

We report in this note new experimental results on a smectic liquid crystal confined into a low dimensional, macroscopically oriented porous silicon layer, which has been introduced recently [10,13]. The combination of a large panel of complementary techniques allows one to probe in parallel the structural and dynamical behavior of the confined molecules. It is shown that both orientational and translational order parameters of the confined liquid crystal couple to the anisotropy of the silicon matrix, and give rise to unusual structural properties reminiscent for some aspects of electric or magnetic field effects on bulk liquid crystals. The picosecond molecular dynamics of this oriented short-range ordered confined smectic phase were probed by light and neutron scattering techniques. On this time scale, only a slight confinement induced slowing down is detected, indicating that the local dynamics involving vibrations, rotations along the long molecular axis or chain-end librations remain mostly bulk-like. However, the increase of elastic intensity for the confined sample suggests that other modes could experience a heterogeneous spatial distribution of amplitudes or correlation times. This can be mostly expected for restricted reorientations occurring around the nanosecond, that might be much more sensitive to neighboring orientational parameters. Also, slow translational diffusion might be strongly affected by the interaction at the solid interface, and the coupling with the one dimensional character of the matrix. Further experiments are needed in order to characterize this point.

Figure 1 .

 1 Figure 1. Infrared light ellipsometric spectra of (a) an empty porous silica layer, and (b) a porous silica layer filled with 8CB. Experimental data (I s : filled circles, I c : open circles) are compared to the fitted model (solid line).

Figure 2 .

 2 Figure 2. Neutron cross section diffracted at the smectic Bragg position (Q = 0.19 Å-1 ) by 8CB confined into porous silicon. The inset sketches the experimental setup. The sample macroscopic orientation is defined by the angle θ S between the silicon wafer plane and the incident beam.

Figure 3 .

 3 Figure 3. Brillouin spectra measured on (a) bulk 8CB (b) empty porous silica, and (c) porous silica filled with 8CB.

Figure 4 .

 4 Figure 4. Quasielastic spectra measured by time-of-flight neutron scattering at 295 K and Q = 1.83 Å-1 , on (a) bulk 8CB, and (b) 8CB confined in porous silicon. The solid line is a best fit using a model including an elastic peak (dotted line) and a lorentzian line (dashed line).

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Acknowledgements

The authors are thankful to J.-M. Zanotti and B. Frick, for expert advice on neutron scattering experiments and fruitful discussion, and also M. Stchakovsky for precious help on spectroscopic ellipsometry measurements.


L Chocinski-Arnault

A Riviere

Isothermal mechanical relaxations in single and mixed alkali silicate glasses

Keywords: internal friction, silicate glasses Keywords (user supplied): Isothermal mechanical spectroscopy, mixed alkali effect Isothermal mechanical spectroscopy, internal friction, mixed alkali effect, inorganic glass

Very low frequency damping measurements carried out in various mixed alkali silicate glasses (1-x)Na 2 O.xLi 2 O.3SiO 2 with x = 0, 0.25, 0.5 and 0.75 point out new experimental features. A new relaxation peak occurring at very low temperature is evidenced in the single alkali glass (x = 0) in addition to the two classical relaxation peaks previously described and assigned respectively to the motion of alkali ions (ionic peak) and to a local rearrangement of the glassy structure involving non-bridging oxygen (NBO peak). For the three mixed alkali glasses, the mixed alkali relaxation peak is shown to be superimposed to two satellite peaks corresponding to the ionic and NBO peaks. Moreover, Arrhenius plot of the mixed alkali peak shows that the apparent activation energy is different below and above 340 K. These experimental results are discussed using recent models proposed for the alkali effect and a model developed previously for mechanical relaxation in ionic glasses below T g .

Introduction

Various peaks are previously observed in single alkali silicate glasses: a first one, occurring near room temperature when the frequency of vibration is of the order of 1 Hz, is generally attributed to the action of the applied stress on alkali ions and called "ionic peak" [1][2][3][4]. The origin of a second peak observed at higher temperature is still debated: it appears only with the presence of non-bridging oxygen and of water. This peak is often called "nonbridging oxygen peak" (NBO peak) [2,[4][5][6][7]. Mixed alkali silicate glasses exhibit a very large internal friction peak [4,5,[8][9][10][11][12]. Some authors maintain that this peak takes the place of the ionic peak and others that it is superimposed to the peaks observed in the single silicate glasses. Another point of discussion concerns the results obtained for mixed alkali glass with a very low ratio of a second alkali ion [5,8,11]: a small relaxation peak is described by the authors as a mixed alkali peak but in fact this peak can correspond to the NBO peak. In order to clarify this situation, new internal friction experiments were performed using the isothermal mechanical spectroscopy technique allowing very accurate measurements, for four silicate glasses of (1-x)Na 2 O⋅xLi 2 O⋅3SiO 2 type with x = 0, 0.25, 0.5 and 0.75.

-Experimental procedure

Isothermal mechanical spectroscopy (IMS) consists in measuring internal friction in a very wide frequency range (10 -4 Hz -40 Hz) after stabilization of the temperature and of the sample microstructure. For materials with high stiffness, a good accuracy in damping (± 10 -4 ) can be obtained only using an inverted forced pendulum working in the subresonant frequency range [13,14]. Such a pendulum must have a very high resonance frequency (200 -300 Hz) to allow the forced measurements until 40 Hz. In comparison with measurements at a fixed frequency, IMS presents decisive and basic advantages and allows a better The apparatus used for this work was already described in detail [START_REF] Rivière | Low frequency techniques[END_REF]. Measurements were carried out in a vacuum. In the case of the forced vibration technique, internal friction Q -1 is equal to tan ϕ where ϕ is the phase lag between the applied stress and the resulting strain. Isothermal experiments started three hours after each temperature change. The measurement frequencies (f) were ranged between 40 Hz and 10 -4 Hz and ten discrete frequencies per decade were used. Internal friction was also measured at the free frequency of the pendulum (≈ 200 Hz) by the free decay method. The maximal strain amplitude was ε M = 5×10 -6 . The dimensions of the specimens were 1.5mm × 5mm × 50 mm.

Experimental results and interpretation

Single alkali silicate glass

Isothermal spectroscopy experiments performed in the single alkali silicate glass Na 2 O⋅3SiO 2 exhibit three mechanical relaxation peaks. In fact, a new relaxation peak is evidenced at very low temperature (Fig. 1). The corresponding activation energy (11.8 kJ/mole, 0.12 eV) corresponds to a local motion of sodium ions inside their sites. For an anelastic behavior, the applied stress does not act directly on the ions but the elastic deformation of the vitreous structure can induce change in ion position. The classical ionic peak corresponding to the jump of sodium ions to a vacant neighboring site and the non-bridging oxygen peak were also evidenced in the sample. Fig 2 displays the Arrhenius plots (Napierian logarithm of the peak maxima versus the corresponding inverse measurement temperature) for the three relaxation peaks. The relaxation parameters (the apparent activation energy E a and the limit relaxation time τ 0 ) are: E a ≈ 58 kJ/mol (0.6 eV) and τ 0 = 8 ×10 -14 s for the ionic peak and E a ≈ 94 kJ/mol (0.95 eV) and τ 0 = 1.5×10 -11 s for the non-bridging oxygen peak.

[insert Figs. 1 and 2 here]

Mixed alkali silicate glasses

For the three mixed alkali silicate glasses (1-x)Na 2 O⋅xLi 2 O⋅3SiO 2 , isothermal internal friction measurements exhibited an apparent peak of large amplitude. Fig. 3 shows the internal friction spectrum versus frequency for each glass (x=0.25, 0.5 and 0.75) at the same temperature 350K. It is noticeable that the height and the location of this apparent maximum depend on the glass composition. From the experiments performed at different temperatures, this apparent peak shifts to higher frequency with increasing temperature and covers about 3 decades. However, for each composition, the height depends on the temperature of the measurement as shown in Fig. 4 corresponding to the 0.5Na 2 O⋅0.5Li 2 O⋅3SiO 2 mixed alkali silicate. In such case, the relaxation peaks plotted against the inverse temperature are not symmetrical. More often such relaxation spectra are complex and a relaxation peak is generally superimposed on a background or other peaks. For relaxation spectra plotted against the temperature, the use of symmetrical functions to deduce the relaxation peak introduces systematically a mistake on the determination of the temperature of the peak maximum. On the contrary, isothermal experiments exhibit always symmetrical peaks which can be easily and accurately subtracted from the experimental spectra. [insert Figs. 3 and4 here] From the isothermal experiments performed in frequency sweep, internal friction can also be plotted against the temperature T at a given frequency, as illustrated in Fig. 5 3 the 0.5Na 2 O⋅0.5Li 2 O⋅3SiO 2 glass: the internal friction spectrum can be separated into a high intensity peak (mixed alkali peak) at about 330K and, at higher temperature (∼410K), a lower intensity peak (NBO peak) which appears as a shoulder at the bottom of the high peak. Fig. 5 shows also the decrease of the modulus accompanying each of the relaxation phenomena. So, the NBO peak is still present in mixed alkali glasses. The ionic peak is also evidenced in mixed alkali silicate glasses as shown in Fig. 6. This peak corresponds to the jump of both sodium and lithium ions. The relaxation strength is much larger for the single alkali silicate glass than for the 0.5Na 2 O⋅0.5Li 2 O⋅3SiO 2 glass and, in this case, only few ions jump directly to a neighboring site. [insert Figs. 5 and6 here] In Fig. 7 the Arrhenius plots corresponding respectively to the ionic and NBO peaks obtained for the single alkali glass and the three mixed alkali glasses are reported. The plots corresponding to the NBO peaks are quite similar for all the glasses. On the contrary, Fig. 7 shows that the ionic peak shifts to lower frequency (at fixed temperature) for the mixed alkali glasses. This fact was previously mentioned [4,5,8,[10][11][12].

The Arrhenius plots corresponding to the mixed alkali peak are displayed in Fig. 8. For each sample, the experimental points cannot be aligned in a single straight line and the Arrhenius spectra correspond to two different energy activation values; the transition temperature being about 340 K.

[insert Figs. 7 and8 here] The experimental relaxation parameters are displayed in Table 1 for the three mixed alkali glasses.

[insert Table 1 here]

Discussion and conclusion

The use of isothermal mechanical spectroscopy provides new and more accurate experimental results: -a new relaxation peak was evidenced for the first time at very low temperature in single alkali silicate Na 2 O⋅3SiO 2 glass; the associated activation energy (11.8 kJ/mole, 0.12 eV) allows to assign this peak to very small motion of alkali ions inside their sites.

-contrary to previous works [10,12,[START_REF] Mohyuddin | [END_REF], our experimental results show that NBO peak is still present in mixed alkali silicate glasses; this peak is quite similar in single and mixed alkali glasses and corresponds to a relaxation inside the vitreous structure.

-as predicted in a theoretical approach suggested by Meyer and Bunde [START_REF] Meyer | Proceedings of the Workshop on Non-equilibrium Phenomena in Supercooled Fluids, Glasses and Amorphous Materials[END_REF], the ionic peak is also still present in mixed alkali silicate glasses and corresponds to jumps of both sodium and lithium ions. The very small relaxation strength indicates that only few ions can jump directly to a vacant neighboring site.

-the shift of the ionic peak to higher temperature for the mixed alkali glasses are in agreement with previous papers [1][2][3].

-the Arrhenius plots corresponding to the mixed alkali peaks indicates a transition temperature (about 340 K). The very large relaxation strength of the mixed alkali peak can be explained by recent theories about the mixed alkali effect [START_REF] Maass | [END_REF]19,20]. According to these theories, the both kinds of mobile ions have different jump paths. The mutual interception of these jump paths involves a freeze of the ions and consequently a decreasing of the electrical conductivity. So, a great number of sodium ions are close to a lithium one (and vice versa). If the elastic deformation allows one ion to jump, the second one will also jump quite automatically. Such correlative

  at 0.0032Hz for

Fig. 1 -Fig. 2 -

 12 Figure captionsFig.1-Very low temperature internal friction peak in single alkali silicate glass Na 2 O⋅3SiO 2 measured at 145 K ( ) and 154 K ( ).Fig.2-Arrhenius plot of the three peaks observed in single alkali silicate glass Na 2 O⋅3SiO 2 : very low temperature peak ( ), ionic peak () and NBO peak ( ).

Fig. 3 -

 3 Fig. 3 -Mixed alkali peak measured at 350 K in the three mixed alkali silicate glasses (1-x)Na 2 O⋅xLi 2 O⋅3SiO 2 for x = 0.25 ( ), 0.5 () and 0.75 ( ).

Fig. 4 -

 4 Fig.4-Mixed alkali peak in 0.5Na 2 O⋅0.5Li 2 O⋅3SiO 2 mixed alkali silicate glass measured at 350 K ( ), 375 K () and 396 K ( ).

Fig. 5 -Fig. 6 -Fig. 7 -

 567 Fig. 5 -Mixed alkali silicate glass 0.5Na 2 O⋅0.5Li 2 O⋅3SiO 2 -Internal friction ( ) and modulus effect ( ) measured at 0.0032 Hz.

Fig. 8 -

 8 Fig. 8 -Arrhenius plot of the mixed alkali peak in the three mixed alkali silicate glasses (1-x)Na 2 O⋅xLi 2 O⋅3SiO 2 : x = 0.25 ( ), 0.5 () and 0.75 ( ). The arrow indicates the transition temperature.

  Fig.1

  Fig.2

  Fig.3

  Fig.4

  Fig.8

  139 kJ/mole (1.4 eV) τ 0 = 1×10 -20 E a = 104 kJ/mole (1.08 eV) τ 0 = 2×10 -15

Table 1 -

 1 Relaxation parameters for the three mixed silicate glasses

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Yidir Androussi
email: ydir.androussi@univ-lille1.fr

Tarik Benabbas

Slawomir Kret

Vincent Ferreiro

Alain Lefebvre

A Transmission Electron Microscopy study of composition in Si1-xGex / Si (001) quantum dots

Keywords: quantum dots, transmission electron microscopy user supplied): chemical composition, displacement field Transmission electron microscopy, quantum dots, chemical composition, displacement field

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Self-assembled quantum dots (QDs) in heteroepitaxial semiconductor systems have recently been an area of intense study because many electronic devices based on QDs have been theoretically proved to possess better electronic and optical properties than quantum-well devices. Knowing the structural parameters of QDs including the shape, size and chemical composition at different stages of QD island growth is thus important for understanding the structure-property relationship of the QDs as well as revealing information on QD growth mechanisms [1]. However, the accurate characterization of coherent islands is not trivial due to their small sizes and the coupling effects between composition and strain field. An additional complexity is that the island shape and composition are a function of substrate temperature (during growth), of coverage and of the type of crystal growth technique.

Ge/Si (001), in particular, has served as a model system because it is a simple two-component system and due to the hope of combining these QDs easily with existing Si technology. Morphology evolution of uncapped QDs has been extensively investigated and it is reasonably well understood [2][3][4], but despite its impact on optical and electronic properties, island composition evolution is still in debate. The determination of local composition can be carried out with techniques that average over many islands. These include high resolution x-ray diffraction [5], x-ray anomalous scattering [6,7] and x-ray absorption fine structure [8]. Selective etching coupled with atomic force microscopy [9] and electron-microscope-based methods make it possible to measure composition variations throughout individual islands. These methods are scanning tunneling microscopy [10] and transmission electron microscopy (TEM) techniques including high resolution imaging combined with finite element analysis [11], electron energy loss spectrometry (EELS) [12][13][14] and xray energy dispersive spectrometry (EDS) [15][16]. For each of the above TEM studies, very thin cross-section specimens are required. However, it is difficult to prepare cross-section TEM specimens, especially for samples with low QD densities, and even then the section might include any part of the QD. As a result, there is an advantage in using plan-view samples. Besides, plan-view samples have a larger sampling region that provides a superior statistical basis and contrary to cross-section samples, the surface relaxation effects are negligible in them. That is why plan-view specimens have been successfully used to extract alloying information in Si 1-x Ge x /Si QDs from diffraction contrast images [17].

It has been recently shown that information on the chemical composition of coherently strained islands can also be obtained by TEM when fringes -which we have called "moiré-like fringes"-are observed in plan-view images of In x Ga 1- x As/GaAs QDs [18][19][20]. The same type of fringes has also been observed in SiGe/Si QDs [21][22]. This paper is aimed at showing that these moiré-like fringes can be used to study the chemical composition of such QDs. Finite- element (FE) calculations are performed in order to take account of various complex compositional variations. It is then shown how the resulting strain fields (and consequently the compositional inhomogeneities) can be studied via the dynamical contrast of TEM plan-view images of QDs.

EXPERIMENTAL PROCEDURE

The growth of the self-assembled islands is carried out in an ultra-vacuum chemical vapor deposition growth chamber with silane and germane diluted in hydrogen used as precursors [23]. Four monolayers of Ge are deposited at 600 or 700°C on Si (001). Atomic force microscopy (AFM) experiments are carried out in air using a Nanoscope III Multimode microscope from Digital Instruments operating in the tapping mode. Integrated silicon tips with a radius of curvature of about 10 nm and cantilevers (model TSEP) with a nominal spring constant 30 Nm -1 are used. The (512 x 512 pixels) images are obtained with a 100 x 100 µm piezoelectric scanner and with a 0.4 Hz scanning frequency.

[Insert figure 1 about here] The finite-element (FE) calculations are performed using a "home-made" program so as to calculate displacement fields R taking account of possible compositional variations in Si 1-x Ge x /Si QDs. They are carried out in the case of lens-shaped domes (Fig. 1) with values of b and h as indicated in table 1. The compositional variations are introduced by local variations of the elastic constants and for each finite element, a virtual thermal expansion coefficient is introduced so as to simulate a local lattice mismatch that will occur during the FE calculation by raising the temperature by 1K. The thermal expansion coefficient α has thus to fulfill the relation a(x) = a(0) (1 + α ∆T) where a(x) is the bulk-material lattice parameter corresponding to the Si concentration x of the finite element.

[Insert table 1 about here] Plan-view TEM thin foils are mechanically thinned from the substrate side by wedge polishing. TEM images are taken with a Philips CM30 microscope (operated at 300 kV) and directly captured with a cooled slow-scan chargecoupled device (CCD) Gatan camera.

The two-beam dynamical TEM contrast of QDs is simulated by solving the Howie and Whelan equations based on the so-called column approximation [START_REF] Hirsch | Electron Microscopy of Thin Crystal[END_REF]. The displacement field R is computed at equispaced points in the [001] direction by carrying out the Lagrange interpolation procedure on the data points provided by finite-element calculations. It should be noted that these calculations make it possible to readily vary the shape of the islands and their aspect ratio. A bimodal dot distribution with square-based pyramids and domes is observed by atomic force microscopy at 700°C (see for instance both types of islands in Fig. 2) whereas only domes are observed at 600°C. The average sizes of both types of islands are indicated in table 1 as a function of temperature. The aspect ratio ρ = 0. 20 ± 0.02 was found to be constant for all the observed D 2 islands.

RESULTS

F

[Insert figure 2 about here] The characteristics of plan-view dynamical TEM dark-field images of strained islands have been fully described in the case of coherently strained In x Ga 1-x As/GaAs [19,[START_REF] Benabbas | [END_REF] or Si 1-x Ge x /Si islands [21,22]. They all exhibit a black/white contrast, as shown in Fig. 3 and4 in the case of pyramids (P) grown at 700°C and in the case of domes grown at 600°C (D 1 ) or 700°C (D 2 ). When the islands display a sufficiently high aspect ratio (h/b > ~ 0.2), moiré-like fringes are found to be superimposed to the black/white contrast (see D 2 domes in Fig. 4). This is fully consistent with what has been demonstrated in the case of In x Ga 1-x As/GaAs islands [19]. These images were obtained for QDs on the electron entrance surface, with a diffraction vector g = 220 parallel to the interface and with w g = s g ξ g = -1.10 ± 0.05 (s g is the deviation parameter and ξ g the extinction distance). This negative value of w g was used because it was found to give the highest contrasts and because it was consistent with the general rules established by Katerbau [26] for lattice defects near the specimen surfaces. Following these rules, the contrast of such defects depends on the imaging mode (bright or dark field), on the defect position (near the electron entrance or the electron exit surface of the specimen) and on the sign of w g . The various cases are summarized in table 2 : for bright-field images, the contrast is reduced (enhanced) for w g <0 (w g >0) whatever the defect position. On the contrary, for dark-field images, the reduction (or enhancement) simultaneously depends on the sign of w g and on the defect position. There is thus an advantage to take darkfield images (rather than bright-field images) either with QDs near the electron entrance (with w g <0) or near the electron exit surface (with w g >0): in both cases, the contrast is enhanced for QDs whereas it is reduced for defects resulting from the specimen thinning on the opposite surface.

[Insert figure 3 about here] [Insert figure 4 about here] [Insert table 2 about here] Uncapped Si 1-x Ge x /Si islands grown either by molecular beam epitaxy or by chemical vapor deposition generally display aspect ratios inferior to ~ 0.3 and two types of compositional heterogeneities. In the first type, a diffuse interface with Si/Ge mixing is observed, the island composition is homogeneous away from the intermixed interface and a laterally constant composition is maintained in the r direction [7,13]. In the second type, the islands contain a Si-rich core covered with a Ge-rich shell and the composition is not laterally constant [6,17]. It should be noted that a uniform composition has been found in low mismatch Si 0.8 Ge 0.2 /Si QDs [27]. Islands grown by liquid phase epitaxy are not taken into account in our study because they display higher aspect ratios (~ 0.5) [28]. We have thus calculated dark-field (220) TEM images of D 2 domes for homogeneous compositions and for two types of composition variations. In the first type, a laterally constant composition is maintained in the r direction (Fig. 5a) whereas compositional gradients are introduced in the z direction either with a linear ("linear/linear" model) or S-shaped profile ("linear/S" model). In the second type, various models have been tested with parabolic contour lines in the (r,z) plane and linear dependence on the z axis ("parabolic/linear" model) (Fig. 5b). The linear (S) dependence of indium composition x as a function of z is indicated in Fig. 5c (d).

[Insert figure 5 about here] Fig. 6 shows calculated dark-field (220) TEM images of Si 1-x Ge x /Si domes with the same geometrical characteristics as D 2 domes and either with a homogeneous composition (with various values of germanium content x) (Fig. 6a) or with the models of compositional inhomogeneities defined in Fig. 5 (Fig. 6b-d). All the images have the same characteristics (i.e.black/white contrast and superimposed moiré-like fringes). L is defined as the distance between the centers of white and black lobes, and σ is defined as the mean periodicity of moiré-like fringes (the periodicity of these fringes is not constant within any experimental or calculated images and that is why σ has been defined as a mean periodicity). A (σ, L/b) diagram can be established for various homogeneous or heterogeneous compositions, and with the following parameters in the case of Fig. 7: aspect ratio ρ = 0.20, thin foil thickness t = 349 nm, A 220 = 0.0315 (the anomalous absorption coefficient), ξ e 220 = 62.7 nm (the effective extinction distance of Si 0.5 Ge 0.5 , calculated using the Vegard's law), w 220 = -1,10. It should be noted that moiré-like fringes are not found for low values of x (x < ~ 0.4 for homogenous composition and x l or x S < ~ 0.6 for heterogeneous compositions). The main result in this figure is that islands with a homogeneous composition can be theoretically unambiguously distinguished from islands with compositional inhomogeneity. However the experimental uncertainties on the measured values of σ and L/b (spread out in the rectangle in Fig. 7) make it difficult to distinguish between islands with an homogeneous composition x = 0.4 and islands with a "parabolic/linear" heterogeneous composition.

[Insert figure 6 about here]

We have then considered the influence of the uncertainties with which the above parameters (t, A 220 , ξ e 220 , w 220 and ρ) can be determined. It is thereafter [Insert figure 7 about here]

The anomalous absorption coefficients were respectively taken to be A 220 = 0.0144 for Si and A 220 = 0.0485 for Ge: these values were obtained from those calculated at 100 kV [29] or 120 kV [30] by extrapolating to 300 kV, using the analysis by Metherell and Whelan [31]. Vegard's law was used in the case of Si x Ge 1-x alloys.

[Insert figure 8 about here]

As an example, Fig. 8 shows the variations of (σ, L/b) hom. points as a function of A 220 varying in the range 0.0144 (Si) -0.0485 (Ge): inspecting this figure shows that for any Si 1-x Ge x /Si alloy, the positions of these points are rather insensitive to the uncertainties with which A 220 can be calculated. The influence of w 220 is depicted in Fig. 9. This figure shows that the experimental error in this parameter (± 0.05) has little influence on the (σ, L/b) hom. points, mainly through variations in L.

[Insert figure 9 about here] The effective extinction distances were respectively taken to be ξ e 220 = 74.8 ± 0.02 nm for Si and ξ e 220 = 50.7 ± 0.11 nm for Ge in keeping with the calculations of Doyle and Turner [32] or Lu et al [33]. The Vegard's law was used in the case of Si 1-x Ge x /Si alloys. Calculations show that the positions of (σ, L/b) hom. points are insensitive to the uncertainties with which ξ e 220 can be calculated.

[Insert figure 10 about here] [Insert figure 11 about here] Calculated dark-field (220) TEM contrasts were found to be very dependent on foil thickness t. Defining the moiré-like fringe contrast C as I 2 -I 1 / I 2 + I 1 where I 2(1) is the intensity of the first dark (bright) fringe in the dark lobe, Fig. 10 shows, for instance in the Si case, that the variation of C as a function of t is periodic and with an approximate ξ e 220 period. Only experimental images with the highest contrasts (> ~ 0.50) leading to precise measurements of σ and L, were considered, i.e. images corresponding to the range 330-360 nm for the 300-380 nm period. The specimen thickness was measured in the vicinity of every studied QD and the location of the corresponding electron microprobe was subsequently checked with the resulting contamination spot. As an example, for t = 349 nm situated in this range, Fig. 11 shows the variations of (σ, L/b) hom. points corresponding to the uncertainty ± 4 nm with which t can be measured using conventional CBED analysis [34,35]. The influence of this uncertainty on (σ, L/b) hom. points is found to be negligible.

[Insert figure 12 about here] Fig. 12 shows the variations of (σ, L/b) hom. points as a function of aspect ratio (0.15 < ρ < 0.30). Examining this figure shows that for a homogeneous x = 0.4 composition, moiré-like fringes are not found with ρ = 0.15, which is consistent with the fact that moiré-like fringes are never observed in the case of D 1 domes. It should be noted that for any given composition x, σ is very sensitive to the variations of ρ and that it is thus important to determine the aspect ratio as accurately as possible. In the case of D 2 domes, the effect of the uncertainty ± 0.02 on the determination of ρ is illustrated in Fig. 12.

DISCUSSION AND CONCLUSION

The above analysis shows that all the parameters (t, A 220 , ξ e 220 , w 220 and ρ) influencing the contrast of moiré-like fringes can be measured or calculated with a sufficient precision and have thus little influence on the calculated values of σ and L/b. However, the experimental uncertainties on the measured values of σ and L/b (spread out in the rectangle in Fig. 7-9 and 11-12) make it difficult to distinguish between islands with a homogeneous x = 0.4 composition and islands with a "parabolic/linear" heterogeneous composition. That is why an additional x-ray energy dispersive spectrometry investigation has been performed [36] and it has shown that D 2 domes have an approximately homogeneous x = 0.4 composition. This result is consistent with those of Schülli et al [7] who have studied the influence of growth temperature on interdiffusion in uncapped SiGe/Si islands grown by molecular beam epitaxy: the aspect ratio is found to be constant (0.22) between 620 et 750°C and the maximum Ge content rapidly decreases from about 70 to 22% for growth temperatures between 620 and 800°C and is approximately 45% at 700°C. Our geometrical approach shows how it is possible to distinguish quantum dots with a homogeneous composition from those with compositional inhomogeneity. However this does not make it possible to choose between various compositional variations and precise determination of these variations should take account of the relative intensities of black/white lobes and of moiré-like fringes.

Formatted:

  Font color: Red Formatted: Font color: Red Formatted: Font color: Red Formatted: Font: Symbol Formatted: Font: Symbol Deleted: indium Deleted: Defining L as the distance between the centres of the white and black lobes, and σ the periodicity of moiré-like fringes for every calculated imagecase of the (σ, L/b) hom. points corresponding to homogeneous compositions.

Table 2 :

 2 Enhancement (+) or reduction (-) of dynamical TEM contrast for lattice defects near the specimen surfaces.

14 Figure 1 :

 141 Figure captions

Figure 2 :

 2 Figure 2: Tapping mode (1.2 µm x 1.2 µm) image of Si 1-x Ge x /Si QDs grown at

Figure 3 :

 3 Figure 3: Experimental dark-field (220) TEM image of D 1 dome-shaped Si 1- x Ge x /Si QDs grown at 600°C; w 220 = -1.10.

Figure 4 :

 4 Figure 4: Experimental dark-field (220) TEM image of Si 1-x Ge x /Si QDs grown at 700°C; w 220 = -1.10, t = 349 nm. (a): a pyramid (P) and two domes (D 2 and D' 2 ). Only the domes display moiré-like fringes. (b): enlargement of the D 2 dome.

Figure 5 :

 5 Figure 5: Various models of compositional inhomogeneity. (a): linear variation

Figure 6 :

 6 Figure 6: Calculated dark-field (220) TEM images of dome-shaped Si 1-x Ge x /Si QDs with a homogeneous composition (a), or with various compositional

Figure 7 :

 7 Figure 7: Calculated (σ, L/b) diagram for dome-shaped Si 1-x Ge x /Si QDs, either

Figure 8 :

 8 Figure 8: Variations of (σ, L/b) hom. points as a function of A 220 . Aspect ratio ρ = 0.20, t = 349 nm, ξ e 220 = 62.7 nm, w 220 = -1,10. The rectangle located in the

Figure 9 :Figure 10 :

 910 Figure 9: Variations of (σ, L/b) hom. points for various values of w 220

Figure 11 :

 11 Figure 11: Variations of (σ, L/b) hom.. points for various values of t ranging from 345 to 353 nm. Aspect ratio ρ = 0.20, A 220 = 0.0315, ξ e 220 = 62.7 nm, w 220 = -

Figure 12 :

 12 Figure 12: Variations of (σ, L/b) hom. points for various values of ρ ranging from

Table 1 :

 1 Average sizes of domes (D 1 and D 2 ) and pyramidal islands (P) for various growth temperatures ; b=base, h= height.

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ACKNOWLEDGEMENTS

This work was partially supported by the French " Région Nord-Pas de Calais", by the European FEDER ("Fonds Européen de Développement Régional") and the "Action Concertée Nanosciences/Nanotechnologies". Many thanks are due to Drs D. Bouchier and L.H. Nguyen (Institut d'Electronique Fondamentale, Orsay, France) for providing the SiGe/Si samples.