Kalathiparambil Kishor Kumar

S Barbosa

L Couëdel

C Arnas

Kishor Kumar

C Pardanaud

F R A Onofri

In-situ characterisation of the dynamics of a growing dust particle cloud in a direct-current argon glow discharge

Keywords: numbers: 52.27.Lw, 52.25.Os, 52.40.Hf

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In-situ characterisation of the dynamics of a growing dust particle cloud in a direct-current argon glow discharge

Introduction

Dusty or complex plasmas are partially ionised gases which contain solid dust particles that acquire an electric charge due to their interactions with the surrounding ions and electrons.

They can be either injected or grown directly inside the plasma. For example, nanoparticles (NPs) are produced as candidate analogue of dust in diffuse interstellar medium, in radio-frequency (RF) discharges using hydrocarbon gases [1]. In the context of dense plasmas, NPs were surprisingly found in dust samples collected in carbon-based wall tokamaks [2,3]. The possibility of formation in the coldest regions, near the tokamak wall was therefore investigated [4,5,6]. Generally, NP formation [7,8,9,10,11], charging and transport [12,13,14,[START_REF] Hayashi | L806 Figure A1. a) Schematic of the LES acquisition system: (1) tungsten cathode, (2) stainless-steal grounded anode, (3) glass half cylinders, (4) UV fused silica windows, (5) highly-stabilized Halogen-Deuterium lamp, (6) optical fibers, (7) on-line intensity attenuator, (8) achromatic coupling and collimating optics[END_REF] were extensively studied experimentally in silane-based RF discharges for semi-conductor applications such as the improvement of photovoltaic devices [16].

It was shown that their growth started by the formation of molecular precursors which were coming from trapped negative ions resulting from the dissociation of reactive gases in the plasma [17,18,19]. After nucleation, NP growth was attributed to successive agglomerations of the smallest NPs. This mechanism was found to inhibit further nucleations resulting in NPs with narrow size dispersions [20,21]. Finally, the agglomeration stopped due to electrostatic repulsion and the growth continued at constant NP number concentration by surface deposition of molecular species. It was recently demonstrated that in direct-current (DC) sputtering discharges, in which the nucleation was initiated by sputtered metallic atoms, the growth mechanisms were more complex [22]: i) small particles were always observed, indicating a constant nucleation and ii) the growth by agglomeration did not seem to saturate. Moreover, a strong dispersion in the appearance time of the second NP generation was reported [23]. These experiments were performed using a tungsten cathode as source of metal atoms. The incident DC discharge power being rather low, the cathode surface oxide was sputtered away after few tens of seconds. During this time interval, the evolution of the discharge and plasma parameters was the result of a competition between the influence of the cathode surface state and the influence of the charging of growing NPs. The influence of the latter became dominant on the plasma and discharge parameters (loss of plasma electrons) only when the oxide layer was removed [22]. Scattering of a vertical laser sheet going through the plasma just above the anode revealed that the NP cloud was compressed and pushed towards the anode during the discharge. Scanning electron microscopy (SEM) images of substrates exposed to the plasma for given durations showed that NPs were continuously falling down on the anode during the discharge [24]. These observations were explained by the fact that the electrostatic force at the negative glow-anode sheath boundary could not balance the ion drag, gravity, and thermophoresis forces for NPs of more than a few tens of nanometres in diameter. From electron microscopy images, it was shown that up to ∼ 500 s, only mono-modal or bi-modal populations of NPs were collected [22]. However, in-situ characterization of the growing NP cloud at different heights between the cathode and the anode remained necessary to unveil completely the growth and transport processes of metallic NPs in a DC discharge. For this purpose, we decided to combine imaging of the NP cloud by the scattering of a laserlight sheet with light extinction spectrometry (LES) which permitted the direct measurement of the particle size distribution (PSD) and number concentration, N , of the NP cloud at different positions in the discharge. This diagnostic is based on the extinction of a collimated white light beam whose spectral range extended from ultra-violet to infra-red ranges. It has the capability to detect NPs smaller than 20 nm and is effectively used to measure PSDs and N in colloidal suspensions [25]. This optical diagnostic presents various advantages with respect to the Mie-scattering ellipsometry diagnostic, often used in dusty plasmas [START_REF] Hayashi | L806 Figure A1. a) Schematic of the LES acquisition system: (1) tungsten cathode, (2) stainless-steal grounded anode, (3) glass half cylinders, (4) UV fused silica windows, (5) highly-stabilized Halogen-Deuterium lamp, (6) optical fibers, (7) on-line intensity attenuator, (8) achromatic coupling and collimating optics[END_REF]21,26,27]. In particular, in the latter case, it is experimentally more challenging as it requires accurate measurements of the ellipsometric angles and the iterative data procedure used to find the particle parameters (PSD, local density, refractive index) is not trivial and requires strong assumptions. Otherwise, a reduction of the number of parameters can be considered. For instance, the choice of a model providing the time evolution of the particle radius can help in this way [START_REF] Hayashi | L806 Figure A1. a) Schematic of the LES acquisition system: (1) tungsten cathode, (2) stainless-steal grounded anode, (3) glass half cylinders, (4) UV fused silica windows, (5) highly-stabilized Halogen-Deuterium lamp, (6) optical fibers, (7) on-line intensity attenuator, (8) achromatic coupling and collimating optics[END_REF]21].

In the present paper, we report a complete study on the growth and transport of tungsten NPs in DC sputtering discharges, in similar conditions to our previous experiments [22,23,24]. Sections 2 and 3 are devoted to the description of the experimental set-up and different diagnostics (electron microscopy, Raman spectroscopy, laser-light scattering and LES). A special attention is given to the LES diagnostics whose resolution was numerically evaluated. Section 4 presents our experimental findings and finally, section 5 is an overall discussion with conclusions.

Plasma-discharge: experimental set-up

Tungsten NPs were grown in a DC argon glow discharge initiated between a ∼10 cm diameter tungsten cathode and a stainless-steal grounded anode (see Fig. 1). The inter-electrode distance was also 10 cm. Two glass half-cylinders were used to confine the plasma. A 1 cm gap was kept between them for optical diagnostics. A static argon pressure of 0.6 mbar (no gas flow) was set during the experiments. The electrode assembly was contained in a cylindrical vacuum chamber of 30 cm diameter and 40 cm length. An oil diffusion pump achieved a base pressure of ≤ 10 -6 mbar. A manually operated gate valve was used to isolate the chamber from the oil diffusion pump during experiments. The vertical position of the center of the laser-light sheet can be adjust between 1 cm and 9 cm above the anode. The field of view of the camera is adjusted to the laser-light sheet position. Note that the LES diagnostics and the laser-light scattering imaging were not implemented simultaneously.

A regulated power supply (Sefelec HT15A, 1 kV, 300 mA) was used to bias the cathode. The discharge current density was kept at a constant value (0.53 mA•cm -2 corresponding to a current of 40 mA). The current variations were less than 0.05 %. The discharge parameters were such that the plasma mainly consisted of a negative glow. Under the chosen operating conditions, the cathode was sputtered and tungsten NPs were grown. More details about the experimental set-up and the evolution of discharge parameters during NP growth are given in Ref. [22].

Diagnostics

Electron microscopy

The anode disc had a 1.6 cm diameter hole at the centre through which the tungsten NPs produced during the discharge were collected on stainless steel substrates. The substrate holder could be manipulated from one of the axial ports of the chamber, such that each substrate could be positioned under the hole in the anode. Two methods of collection were used:

• (i) The substrate holder was moved under the anode so that successive substrates were exposed to the plasma during a given time interval.

• (ii) The substrate saw the extinction of the discharge. Note that in this case, the substrate could, or not be exposed to the plasma during the whole discharge duration.

NPs were then analysed ex-situ using a scanning electron microscope. NPs have also been transferred from the stainless steal substrates to transmission electron microscopy (TEM) grid in order to perform high-angle annular dark-field imaging (HAADF) with a scanning transmission electron microscope (STEM). This technique, highly sensitive to atomic number variations in analysed samples, was used to check the tungsten content of the NPs.

Raman spectroscopy

Raman spectroscopy analyses were performed on collected NPs in order to determine if they were oxidised. The Raman spectrometer was a commercial Horiba-Jobin-Yvon HR LabRAM apparatus ( ×50 lens, λ R =514.5 nm, laser power at the surface ≈1 mW•µm -2 ). This technique consists in measuring the wavelengths of photons produced by inelastic scattering of the incident light with the analysed sample. The energy difference between incident and scattered photons corresponds to a vibrational energy which can be used as a fingerprint of a given material. Tungsten oxides are known to display optical Raman active bending and stretching modes in the ranges 100-300 cm -1 and 700-800 cm -1 , respectively [28] whereas pure tungsten does not display any optical Raman active mode.

Laser-light-sheet scattering method

The NP cloud dynamics was studied using the laserlight-sheet scattering method [START_REF] Onofri | Chapter II: Optical particle characterization[END_REF][START_REF] Onofri | [END_REF]. A vertical laser sheet of 2 cm height (40 mW, wavelength λ L = 532 nm, linear polarisation at 45 • with respect to the horizontal plane (i.e. scattering plane)) was passing through the electrode assembly in the gap between the half-cylinders (see Fig. 1). The light scattered by the dust particles was recorded at 90 • by a 12 bits megapixel resolution digital CCD camera (BASLER ACE acA1300-60gm). A low frame rate of 1 frame per second and a long exposure time of 0.5 s were necessary to compensate for the low intensity of the scattered light. An interference filter centred on the laser wavelength was placed in front of the camera lens to reject most of the plasma emission. The position of the center of the laser sheet was adjusted between 1 cm and 9 cm above the anode during the experiments. N and the normalized PSD, n, were recovered in-line using the LES technique, which consists in analysing the spectral transmittance T (λ) of a broadband and collimated light beam passing through the plasma containing the studied NP cloud. If I 0 and I t are the spectral intensities of the incident and the exiting beam respectively, T (λ) can be defined as follows:

T (λ) = I t (λ) -I plasma (λ) -I b (λ) I 0 (λ) -I b (λ) (1)
where λ stands for the considered wavelength, I plasma for the intensity of the plasma emission in the direction of propagation of the incident beam and I b for the noise of the detection system (i.e dark noise and residual background optical noise). If the detection of multiple scattered photons is negligible, the measured transmission is simply an exponentially decreasing function whose argument is the product of the probing length L (i.e the cloud width), and the cloud turbidity τ . The latter is the product of N and the mean extinction cross section C ext of the particles illuminated by the probing beam. C ext is an integral quantity that depends on the extinction cross section of each particle (shape, size, refractive index, etc.) and its statistical weight in the particulate medium. It is more convenient to write the LES transmission equation in its linearised and discretised form [25]:

ln(T (λ i )) = -N L MD j=1 C ij n j (2)
where C ij is the statistical averaged extinction crosssection for the particle size class j (with j = 1, 2, . . . M D , where M D is the maximum number of considered size classes) and for the wavelength λ i (with i = 1, 2, . . . K i , where K i is the maximum number of considered wavelengths); n j is the probability density associated with the particle-size class centred on the diameter D j .

The particle shape model used in the present study to describe tungsten NPs is a sphere (see section 4.1 for validation). Consequently, all the elements of C ij can be calculated using the Lorenz-Mie theory [START_REF] Onofri | [END_REF]. Particle inhomogeneities, such as porosity or tungsten oxide spherical inclusions, can also be taken into account using the Maxwell-Garnett effective media approximation. More details and comparisons with other light scattering theories can be found in Refs. [START_REF] Onofri | [END_REF]25]. To illustrate the effects of particle inhomogeneities, the evolution of C ext is plotted in Fig. 2 for pure (tungsten mass fraction, f W = 100%), porous (f W = 74%, the remaining being empty) and oxidized (f W = 74% and f W O3 = 26%) tungsten NPs. The differences among the cases are mostly observed in the UV-range, where the size parameter (πD/λ) of these Rayleigh-like particles is maximum, as well as the imaginary part of their refractive index [31].

Inversion procedure of LES measurements

Solving Eq.2 requires to solve a discrete ill-posed problem [31,25] and thus, to implement in the inversion algorithm a specific regularization scheme. The classical regularization methods, such as the Philips-Twomey and Tikhonov methods or the truncated singular value decomposition, did not work properly when applied to the present experimental data. It mainly resulted from the weakness of the signal-to-noise ratio of experimental transmission spectra because of the relative low value of N in the discharge. In the present study, to get more robust inversions, additional physical inputs were required. In Refs. [22,24], it was shown that up to 500 s, PSDs were mono-modal or bi-modal depending on the collection method. Based on these SEM analyses, PSDs were thus assumed to be of a particular type: unimodal or bimodal one. A mode n(D) was simply described by a log-normal distribution depending of two parameters: a mean diameter D and a standard-deviation σ. The mode boundaries were deduced by truncating the PSD at 1/1000 of its global maximum [25]. It followed that, for a bimodal distribution (i.e. two distinct particle populations n 1 (D) and n 2 (D)), five parameters needed to be estimated: two mean diameters D 1 and D 2 , two standard deviations σ 1 and σ 2 , and a constant α quantifying the relative weight in number of each distribution:

n(D) = αn 1 (D) + (1 -α)n 2 (D), with 0 ≤ α ≤ 1.
All of these parameters could be deduced from a classical [25] iterative and weighted least squares algorithm that compares experimental and the theoretical extinction spectra (i.e. Eq. 2).

Sensitivity of the LES inversion procedure

In this section, numerical results providing physical insights and estimations of the LES technique resolution are reported.

Five test cases, either unimodal or bimodal, are considered. Their statistical parameters and the associated logarithm of the synthetic transmittance spectra , ln(T), are reported in Table 1 and Fig. 3a respectively. These spectra have been calculated for pure tungsten particles using the Lorenz-Mie theory. To focus on the effects of the particle size on the transmittance spectra, they were all calculated for a constant particle number concentration.

For synthetic unimodal populations (see case (a) and case (b), Table 1), an increase of the particle size leads to a decrease in transmistance, in particular at short wavelengths. In the case of a synthetic bimodal particle clouds, the shape of ln(T) spectra still depends on the size of the particles but also on the proportion α of a population with respect to the other one. A low proportion of small particles (α ≤ 0.1) changes only slightly transmission rate at short wavelengths, as show the spectra associated with the cases (b) and (c) (Table 1). Larger proportion of small particles in the synthetic cloud (α 0.9) leads to an increase of the transmission rate (see spectra associated with the cases (d) and (e)).

To further test the capability of LES inversion procedure to distinguish between one or two particle populations, a white noise with an amplitude corresponding

) [-] f W =100% f W =74% f W =74%, f WO3 =26% Pure tungsten case (b) (b) (a) case (b) case (a)
Figure 3. a) Evolution of the logarithm of synthetic transmittance spectra calculated for tugnsten spherical particles. The associated PSD parameters are reported in Table 1. b) Evolution of synthetic ln(T) spectra calculated in the case of an unimodal population (case (b) in Table 1) for pure, porous or oxidized tungsten NPs.

to 4% (i.e. SN R ≈ 14dB) of the minimum transmission was added to the transmittance spectra shown in Fig. 3a. Note that this noise amplitude is representative of the experimental one. The error between the recovered (after inversion) and the nominal statistical parameters of these synthetic PSDs are reported on Table 2. With an inversion of the first type (i.e researching unimodal PSDs), the statistical parameters of cases (a) and (b) are perfectly recovered, while those of bimodal populations (cases (c) ... (e)) are naturally poorly recovered. Note that for low values of α, using a unimodal assumption leads to rather satisfactory estimation of D 2 and σ 2 , i.e the particle population with the largest size and highest concentration. This is consistent with the previous observation regarding spectra of bimodal particle clouds containing only a low concentration of small particles. On the contrary, using a second type inversion (i.e researching bimodal PSDs) allows a good reconstruction of all of these synthetic PSDs (see Table 2).

LES measurements and inversions also depend on the particle composition (through the particle complex refractive index). As an illustration, Table 3 reports the errors between the parameters of the recovered and the nominal PSDs in the case of an unimodal distribu-Table 1. Parameters of synthetic PSDs associated with spectra of Fig. 3. Dm and σm are the statistical parameters of the unimodal populations equivalent to the bimodal populations (they are obviously unchanged for unimodal populations).

Dependent model: nominal parameters

Cases D 1 σ 1 D 2 σ 2 α D m σ m [nm] [nm] [nm] [nm] [-] [nm] [nm]
Unimodal cases case (a) Table 2. Error ("Err") between the recovered and nominal mean diameters and standard deviations of synthetic spectra of Fig. 3a.

A negative error indicates an underestimation of the statistical parameter. In cases (c -e), for unimodal inversion method, the error between recovered and nominal parameters are calculated using the mean values: Dm, and σm.

Unimodal Bimodal Cases

Err D Err (σ)

Err D 1 Err (σ 1 ) Err D 2 Err (σ 2 ) Err (α) [nm] [nm] [nm] [nm] [nm] [nm] [-] case (a) 2.0 0.0 2.0 0.0 - - 0.0 case (b) -1.0 -1.0 -1.0 -1.0 - - 0.0 case (c) 0.6 -4.8 2.0 0.0 -1.0 -1.0 0.0 case (d) 1.9 -5.6 2.0 0.0 -1.0 -1.0 0.0 case (e) -0.7 1.3 2.0 0.0 -1.0 -1.0 0.0 Table 3.
Error between the recovered and nominal mean diameters and standard deviations of synthetic spectra of Fig. 3b. 1): the tungsten particles were alternatively pure (tungsten mass fraction, f W = 100%), porous (f W = 74%) or partly oxidized (f W = 74% and f W O3 = 26%). As mentioned above, for these calculations, the Maxwell-Garnett effective medium approximation was used to account for the effects of pores or WO 3 inclusions. As before, a white noise with a relative amplitude of 4% is added on the associated ln(T) spectra. It turns out that, globally, an overestimation of the particle density leads to an underestimation of the mean diameter and to a slight underestimation of the standard deviation. WO 3 inclusions do not have a large effect on the synthetic ln(T) spectra.

Assumed compositions

Pure Porous Oxidized Err D Err (σ) Err D Err (σ) Err D Err (σ) [nm] [nm] [nm] [nm] [nm] [nm] Pure, f W = 100% -1.0 -1.0 -2.0 20
In Fig. 4, the recovered evolution of the mean diameter as a function of the relative noise amplitude of the synthetic transmission rate corresponding to the PSD of case (b) is plotted. When the relative amplitude is lower than 4%, the statistical parameters of case (b) are almost perfectly reconstructed, which is not the case for higher relative noise amplitude. In particular, an increase of the noise amplitude leads to an overestimation of the value of σ 1 . However, the error on D 1 does not exceed 15% even for noise amplitude 12%. The absolute error on N also increases with the noise amplitude but the recovered values are within 30% of the real one for noise amplitude 12%.

Given all the numerical tests, it can be concluded that, on the conditions that one has a minimum knowledge on the NPs (shape, composition) and makes reasonable assumptions on the PSD shape, LES is an accurate diagnostic even in noisy environments.

LES Experimental setup

The LES set-up (See Appendix A) was composed of a highly-stabilized Halogen-Deuterium lamp (Ocean Optics, DH-2000-DUV), solarisation resistant optical fibres with a 100 µm core, an on-line intensity attenuator, achromatic coupling and collimating optics (50 nm focal length parabolic mirrors), a diaphragm Evolution of the absolute error on N as a function of the relative noise amplitude. Synthetic spectra are calculated using the PSD of case (b) (see Table 1). and a CCD spectrometer (Oceans Optics, Maya Pro). This low noise and high dynamic range spectrometer has an enhanced response in the UV range and a global half-height resolution of 2-5 nm in the spectral range 220-1040 nm where LES measurements were performed. Full optical calculations have demonstrated that for the considered particle size and concentrations, and because of the small aperture angle of the LES detection system (less than 1 o ), multiple scattering is negligible. Optical choppers were used to chop the probing beam of 1 mm in diameter, in order to record periodically: the optical and electronic signals, the reference signal and the plasma light emissions. This allowed us to record one corrected transmission spectrum (see Eq.2) every 2 s.

Results

Electron microscopy

SEM results have been described in details in previous articles [22,24]. Here is a quick outline of the main results obtained:

• NPs were pushed towards the anode located at the bottom of the discharge set-up and could get detrapped and fall on the anode.

• The later the substrate was exposed to the plasma, the bigger were the collected NPs. The biggest NPs could have diameter up to 150 nm for discharge duration of 500 s.

• Small particles (mean diameter ∼30 nm) were always observed on the substrate when the discharge was switched off (method (ii), see section 3.1). These results indicated constant nucleation and growth of NPs in the upper part of the discharge (cathode side).

Electron microscopy studies of particles collected after a 300 s plasma indicate that NPs are roughly spherical, and look mainly like a compact aggregate of tungsten crystallites (d ≃ 2 -4 nm) (see Fig. 5), justifying partially our hypothesizes on the particle shape model for LES data inversion (see Fig. 7 in Ref. [25]).

Raman spectroscopy

Measurements were performed directly on the cathode and on stainless steel samples covered with NPs. Weak tungsten oxide signatures were observed by Raman spectroscopy. In order to check if NPs were weakly oxidised or not, they have been heated under an oxygen containing atmosphere. The growth of tungsten oxide bending and stretching modes was observed only for temperature higher than 700 • C. Final Raman scattering intensities were multiplied by a factor of ∼25 showing that the weak bands, observed at room temperature were due to surface oxide, probably formed during the venting of the chamber, after the experiments. It is worth reminding that we have shown in Sec.3.4.3 that a small amount of oxide in the NPs has a negligible impact on LES transmittance spectra. Therefore, for the rest of this study, it was assumed that the NPs in the plasma were composed of tungsten crystallites and, consequently, complex refractive index of pure tungsten [START_REF] Palik | Handbook of Optical Constants of Solid[END_REF] was used to inverse LES measurements. In an earlier study [24], laser-light-sheet scattering measurements just above the anode revealed that the dust particle cloud was compressed and pushed towards the anode during the discharge. For a better understanding of the dynamics of the growing process, further experiments with different positions of the laser-light sheet above the anode have been performed. In each experiment, no NPs could be detected by our laser-light scattering system during the first 30 s after plasma ignition.

In Fig. 6a, a snapshot of the scattered laser-light 45 s after plasma ignition is shown. The centre of the laser-light-sheet was positioned at 2 cm of the cathode surface (i.e. 8 cm from the anode). For a better visualisation of the boundary of the dust cloud, an edge detection was applied to the raw image (Fig. 6b). The top edge of the cloud had a very convex shape and an apparently dust-free hole centred on the discharge axis was visible (no NP detected by laser-light scattering). In Fig. 6a, one can see that there was a strong plasma emission in this region. It corresponded to the expanding luminous "blue glow" appearing at the centre of the tungsten cathode a few seconds after plasma ignition [22]. During the hemispherical expansion of this glow, the NP cloud was pushed toward the bottom and the side of the discharge gap explaining its convex shape.

In Fig. 7a, three snapshots are shown for a laserlight sheet positioned 3.4 cm above the anode. The scattered light intensity increased with time until the dust cloud was pushed down and toward the side of the discharge and, as observed close to the cathode, an apparently "dust-free" hole was expanding in the discharge. This behaviour was observed at all investigated heights between the electrodes and, as already stated, could be linked to the expanding luminous "blue glow" observed on the cathode centre a few second after plasma ignition (see Fig. 6) [22]. This effect was clear at t = 210 s after plasma ignition: in Fig. 7a the top edge of the NP cloud is convex. Note that the intensity gradients observed in Fig. 7 are the result of a convolution between the vertical intensity profile of the laser-light sheet, with the NP size/number concentration gradients in the region illuminated by the laser-light sheet. Since the scattering regime for the considered NPs is in between the Rayleigh regime and the lower bound of the Mie scattering regime, the NP scattering efficiency is proportional to D 6 for the smallest and tends to (but does not reach) the limit of D 2 for the largest NPs. That is any detailed interpretation of the continuous changes in the intensity gradients (or "inhomogeneities") observed in Fig. 7a, has to be done with caution.

In Fig. 7b, the distance of the upper edge of the NP cloud is plotted as a function of time. Note that even though every experiment was carried out with the same discharge parameters, there was a strong dispersion in the measured time (±50 s for the cloud top edge positioned at 3 cm above the anode). This is due to the high sensitivity of NP growth to the initial conditions (base pressure, cathode surface state, etc.). This effect was also observed in radio-frequency discharge in which NPs are grown by sputtering of a polymer layer on the electrode [START_REF] Mikikian | [END_REF]. The general behaviour of the NP cloud was however always the same. The maximum height of the dust cloud edge was ∼7.5 cm above the anode at t≃30 s after plasma ignition. Soon after the NP cloud could be detected, it started to be pushed toward the anode and the edge of the discharge. For ∼100 s, the movement of the NP cloud was quite slow and the height variation of the upper edge was ∼1 cm. Then, in ∼200 s the upper edge position felt from ∼6.5 cm to less than 1 cm, meaning that a major part of the plasma seemed free of dust. However, our previous studies have shown that small NPs (hardly detectable by laser-light scattering ) were growing in this region [24].

Light extinction spectrometry

In Fig. 8a, the temporal evolution of the transmittance at different wavelengths is shown when the probing beam was positioned 2.8 cm above the anode. As expected for NPs, the transmittance at short wavelengths was more affected by the presence of the growing tungsten NPs than at long wavelengths (i.e. T ≃ 1). In Fig. 8b, evolutions of the transmittance at λ = 250 nm for five experiments with the same discharge parameters (two measurements at h = 5.3 cm and three measurements at h = 2.8 cm) are shown. While not exactly the same, due to the high sensitivity of the experiments to the initial conditions, one can see that the measurements at the same height were similar. The overall shape was comparable for both heights but the kinetics was faster in the measurements made at the highest positions. For instance, the minimum of transmittance occurred at t≃170 s at h = 5.3 cm while it occurred at t≃220 s at h = 2.8 cm. These minima in transmittance could be linked to the opening of the apparently "dust-free" hole at the investigated height. The times of the minima matched indeed within errorbars the position of the upper edge of the dust particle cloud on the discharge axis (see Fig. 7b). Before reaching a transmission minimum, the NP number concentration and/or NP size in the LES beam have increased which explained the transmittance decrease. From this time, the path length of the LES beam through the cloud visible by laser-light-sheet scattering started to decrease and the transmittance started to increase. Note that it was previously reported [24] that small NPs were present in this hole. Extinction being very sensitive to the NP size, the larger having the greatest effect, replacing large NPs by smaller one is almost similar in effect to a dust-free zone.

It may appear surprising that LES transmittances at higher probing positions in the plasma were smaller than at lower LES positions. However, as the cloud was pushed toward the side of the discharge, some particles were constantly lost through the vertical slits between the half glass tubes used for the optical diagnostics, explaining the higher transmittance at lower beam positions.

In Figs. 9a andb, the recovered evolution of the PSD and N are presented. They correspond to the transmittance spectrum shown in Fig. 8a. Six different phases could be extracted in the measured NP growth and dynamics §. • Before ∼70 s, no information could be accurately extracted from the transmittance spectra. It indicates that NPs in the LES beam were very small and possibly in very low number concentration.

• From ∼70 s to ∼170 s, the PSD was found to be clearly unimodal, as the inversion procedure forcing a bimodal distribution gave two identical modes with the same weight. During this phase, the mean particle diameter increased from ∼20 nm to ∼30 nm. N was nearly constant (∼ 2•10 14 m -3 ) even if a smooth decrease (over ∼ 12 s) in N , coinciding with higher values of σ 1 , was observed at t ∼ 80 s. This increase in σ 1 could be attributed to local higher values of the relative noise in LES measurements (see results of the synthetic study presented in section 3.4.3, more particularly in Fig. 4) that could be due to very intense and fluctuating plasma emissions occuring after ignition [22]. As a transmittance spectrum depends on a convolution between N and the PSD, large recovered values of σ 1 (i.e. a large faster distribution) imply small recovered values of N . Moreover, the PSD remained mono-modal after 80s which favours the hypothesis of an inversion artefact.

• At t ∼170 s, an inflection point is observed on the transmittance rate at all wavelengths, while the PSD became abruptly bi-modal: each mode which were initially identical are suddenly well separated. Simultaneously, the total concentration in number decreased abruptly (in less than ∼ 6 s). The number concentration of the small (i.e original) mode particles decreased also abruptly, while the concentration of the large mode particles, which just appeared, was very small.

These observations suggest an agglomeration of the small particles at constant volume.

Nevertheless as the agglomeration process was not instantaneous, small particles were not totally consumed and some remained in the LES beam. After this time, the mean diameter of the particles in the large mode stayed almost constant while the mean diameter of the particles of the small mode steadily increased.

Two mechanisms could explain the apparent growth of the NPs of the small mode: surface growth by molecular sticking or agglomeration. If the NPs of the small mode were growing by surface deposition or by agglomeration with very small undetectable particles, their number concentration should be constant. It was not the case here, Fig. 9 clearly shows a decrease of their number concentration. In addition, it can be shown using the theory of Ref. [34], that tungsten vapour concentration is maximum around 1 cm below the cathode and is rapidly decreasing when going towards the anode. Consequently, surface growth on the anode side, where LES measurements were performed, was most probably negligible. A more likely explanation is that they were consummed by agglomeration to create the NPs of the large mode. Even if the agglomeration of the 35 nm particles should normally be suppressed because they are negatively charged, it may still exist due to a mechanism called the polarizationinduced ion flux asymmetry (PIFA) [35,36]. In the PIFA mechanism, it is proposed that a NP becomes polarised due to the influence of the electric field created by the neighbouring grain which leads to an asymmetry of the ion flux onto the surface grain resulting in a force in the direction of the electric field. PIFA mechanism thus allows negatively charged large particle to agglomerate with smaller negatively charges one. It was verified experimentally in a RF discharge [36]. Furthermore, as NPs were getting deeper in the negative glow, i.e. closer to the anode, the electron temperature decreased and so did the NP charge (in absolute value) reducing the repulsive force. Finally, the agglomeration hypothesis is favoured by the sudden appearance of the large mode in concordance with a sudden decrease of the number concentration of the NPs of the small mode. One should also note that the two modes of the PSD are log-normal and only the mean sizes are given in Fig. 9. While the PIFA mechanism allows the agglomeration of negatively charged particles, it still favours the agglomeration of the smallest one with the largest one. Consequently, smallest NPs were consumed in priority and, in return, the mean diameter of the small mode increased. Experiments at different heights showed that this phase occurs quicker when closer to the cathode. A possible explanation for the triggering of the agglomeration phase is that when large particle pass through a cloud of smaller particle, agglomeration is enhanced [37,35,36].

• At ∼220 s, the transmittance reached a minima at all wavelengths and from this time N started to decrease rapidly. When looking at the two modes of the PSD, their mean size remained almost unchanged but the weight of the smaller particles decreased rapidly. As discussed previously, this corresponded to the opening of the "dust-free" region in the LES beam. From this time, the path length of the LES beam through the cloud (visualised by laser-light-sheet scattering) started to decrease and the transmittance started to increase due to a decreasing number density. Note that in the region freed by the first generation of NPs, new NPs were starting to grow [24]. They however remained undetectable with the LES diagnostics as, extinction being very sensitive to NP size, large NPs had the greatest effect and the overall effects was almost similar to a "dustfree" zone. It also explains why the weight of smaller mode of the PSD was more affected. As stated earlier, the evolution of the LES signal was similar for the different investigated positions of the probing beam. Only the dynamics was faster when closer to the cathode. After inversion of the LES spectra, another difference appeared. In Fig. 10, the size of the particle at the minimum of transmittance for different LES beam height are presented. The PSD was bimodal at all investigated heights. No trend is observable for the mean size of the NPs belonging to the small mode which remain around 40 nm while the mean size of the particle belonging to the large mode is increasing when getting closer to the anode. This is a clear indication for a continuous growth of NPs while getting transported towards the anode. Earlier measurements of the particle collected on the anode showed that the NPs can reach size up to 150 nm [24]. Such large size were not measured by LES, strongly suggesting that NP growth continued below the minimum investigated LES beam height. This is also compatible with the earlier hypothesis of an agglomeration phase enhanced by the penetration of large NPs into the cloud of smaller particles, the former growing further while transported towards the anode. • From ∼250 s to ∼350 s (see Fig. 9), the two modes of the PSD were almost superimposed (∼50 nm and ∼60 nm). The number concentrations of both mode were equivalent (≤ 0.1 • 10 14 m -3 ) and the total number concentration was slowly decreasing. These observations were the consequence of two effects: the agglomeration process which consumed the smallest particle and the opening of the "dust-free" zone in the LES beam. However, it has to be noted that the two modes of the PSD remain distinct from each other as the largest particles of the original small mode are probably too big to agglomerate, even considering the PIFA mechanism. • From ∼350 s to ∼400 s, N increased rapidly while the mean NP diameter felt down to ∼35 nm (monomodal distribution). On the transmittance spectrum, it corresponded to a new decrease of the transmittance. This period of time corresponded to the expulsion of the last large NPs of the first generation and consequently, an easier detection of the second generation of NPs. It is in accordance with our earlier experiments [24]: large NPs were expelled from the plasma and new small ones were synthesized in the region freed from the first generation (mainly above as the NP cloud is pushed down) where nucleation could take place again from sputtered atoms. This behaviour is similar to the one observed in radio-frequency discharges where a "void" usually appears in the plasma center, in which new dust particles are grown [38,[START_REF] Mikikian | [END_REF]39]. • From ∼400 s, the PSD became again bi-modal with an associated decrease of N n in a very similar manner to what happened at t ∼ 170 s. These were signs of an agglomeration process occurring in the second generation NP cloud.

Conclusion and perspectives

In this article, a detailed characterisation of a growing NP cloud in a DC sputtering low pressure argon glow discharge with a tungsten cathode was performed. The shape and composition of NPs with ex-situ diagnostics (SEM,TEM.HAADF-TEM and Raman spectroscopy) was determined. The particles were found to be mainly compact spheroids of pure tungsten crystallites which allowed us to use the complex refractive index of pure tungsten for the LES diagnostics. Laserlight scattering of a vertical laser sheet positioned at different heights in the inter-electrode gap showed that the NP cloud was pushed towards the edge of the discharge and the anode leaving a zone in which nucleation of new NPs could take place. In addition, the evolution of the PSD and N were studied in-situ by LES. This diagnostic has revealed a phase of NP growth by agglomeration, followed by the appearance of a second NP generation. The agglomeration phase was identified by a sudden change of the PSD from monomodal to bi-modal followed by a gradual disappearance of the small NPs. Results have also confirmed that the second generation of NPs appeared in the space freed by the first NP generation. At the minimum of transmittance, while the PSD was bi-modal at all investigated positions between the electrode, the mean size of the NPs belonging to the large mode of the PSD increased when getting closer to the anode. On the contrary, the mean size of the NPs belonging to the small mode remained approximately constant. This results suggests that agglomeration was triggered when the biggest NPs passed through a cloud of smaller NPs. They thus grew until they reach the anode side of the discharge. It shows that the largest particles are agglomerates of smaller ∼35 nm particles and is consistent with our results reported in earlier studies using electron microscopy measurements to reconstruct PSDs of particle collected at the anode centre [22,23,24]. Furthermore, as the NP growth continued below the LES scanning positions, it explains the apparent dissimilarities observed between the in-situ optical and the ex-situ electron microscopy PSD measurements. Future investigations will combine local measurements of plasma properties with the in-situ PSD diagnostics, to confirm some interpretations drawn in the present paper and to correlate NP growths and dynamics to the local evolution of the plasma parameters.

Figure 1 .

 1 Figure 1. (Colour online) Schematic of the experimental set-up.The vertical position of the center of the laser-light sheet can be adjust between 1 cm and 9 cm above the anode. The field of view of the camera is adjusted to the laser-light sheet position. Note that the LES diagnostics and the laser-light scattering imaging were not implemented simultaneously.

3. 4 .

 4 Light extinction spectrometry (LES) 3.4.1. LES principle

Figure 2 .

 2 Figure 2. Spectral evolution of the extinction coefficient Cext of spherical particles of (a) D = 29nm, (b) D = 65nm and of various compositions: pure tungsten, tungsten with spherical vacuum inclusions (f W = 74%), tungsten with tungsten oxide spherical inclusions (f W = 74%, f W O3 = 26%).

  λ [nm] ln(T) (× 10 -3

  on N, with (N 1 ) nominal = 10 14 [m-3 ] Recovered mean diameter

Figure 4 .

 4 Figure 4. (a) Evolution of the recovered first two moments of the PSDs as a function of the relative noise amplitude. (b)Evolution of the absolute error on N as a function of the relative noise amplitude. Synthetic spectra are calculated using the PSD of case (b) (see Table1).

Figure 5 .

 5 Figure 5. Images of a NP collected after a 300 s plasma. a) SEM. b) HR-TEM. c)HAADF-STEM. The fringe patterns in b) are the crystallites. The white dots in c) are the tungsten crystallites.

4. 3 .Figure 6 .

 36 Figure 6. Typical snapshot of the scattered laser-light 45 s after plasma ignition : a) raw image, b) same as a) with an edge detection filter applied to the image. The horizontal black arrow shows the position of the lower edge of the laser sheet which was positioned 2 cm below the cathode. The oblique white arrow shows the top edge of the NP cloud. The dashed vertical line indicates the position of the discharge axis.

Figure 7 .

 7 Figure 7. (colour online) a) Snapshots of the video of the scattered laser-light at different times after plasma ignition. The dashed green lines show the upper and lower edges of the vertical laser sheet. The centre of the laser sheet was 3.4 cm above the anode. b) Distance of the upper edge of the NP cloud (1 st generation) to the anode on the discharge axis as a function of time extracted from laser-light scattering data. The latter were accumulated over 21 experiments. The red line represents a fitted mean position of the upper edge of the NP cloud. The dashed green lines are the errorbars at 3 standard deviations.

Figure 8 .

 8 Figure 8. a) Evolution of the transmitance at different wavelengths as a function of time, 2.8 cm above the anode for a 500 s discharge. b) Transmittance at 250 nm at different heights above the anode: gray lines, h = 2.8 cm; black lines, h = 5.3 cm. Each plain, dashed, or dotted line corresponds to a separate experiment.

  Figure 9.a) Temporal evolution of mean diameters (represented by squares or by circles) and the associated standard-deviations (represented by bars) measured by LES at the height h = 2.8 cm from the anode. A bimodal PSD is detected, its large mode is represented by the circle line, while its small mode is represented by the square line. b) Temporal evolution of N , as well as αN and (1 -α)N associated with the small and large modes of the PSD, respectively.

Figure 10 .

 10 Figure 10. Evolution of the particle size reached at the minima in transmittance.

§ Note that these phases occurred independently of the LES scanning position, the kinetics at the highest position being only

Acknowledgments

This work was supported by the INSIS of CNRS via PEPS (Projets Exploratoires Premier Soutien) grant. The authors would also like to thank S. Khrapak and the A*MIDEX foundation for their financial support in the framework of ANR-11-IDEX-001-02 and ANR-11-LABX-0092 (Labex MEC). We are grateful to M. Cabié for her technical support in electron microscopy.

Appendix A. LES set-up

The LES set-up (See Fig. A1) was composed of a highly-stabilized Halogen-Deuterium lamp (Ocean Optics, DH-2000-DUV) (5), solarization resistant optical fibres (6) with a 100 µm core, an on-line intensity attenuator (7), achromatic coupling and collimating optics (50 nm focal length parabolic mirrors) (8-9), a diaphragm (10) and a CCD spectrometer (Oceans Optics, Maya Pro) (12). Optical choppers (11a -11b) were used to chop the probing beam of 1 mm in diameter and a typical probing length of 10cm, in order to record periodically: the optical and electronic signals, the reference signal and the plasma light emissions. This allowed us to record one corrected transmission spectrum (see Eq. 2) every 2 s.


Manel Moussavou

Michel Lannoo
email: michel.lannoo@im2np.fr

Nicolas Cavassilas

Demetrio Logoteta

Marc Bescond
email: bescond@iis.u-tokyo.ac.jp

Physically based Diagonal Treatment of Polar Optical Phonons Self-Energy: Performance Assessment of III-V Double-Gate Transistors

published or not. The documents may come

INTRODUCTION

During more than 40 years the impressive growth of semiconductor industry was based on the scaling of the transistor, which represents the elementary brick of integrated circuits. However, since the 90 nm node, the device scaling in itself is no more capable to maintain this continuous technological development. Additional technological boosters (e.g, high-κ dielectric and mechanical strain) were first incorporated to improve the electrical characteristics of transistors [START_REF] Kuhn | Considerations for ultimate CMOS[END_REF]. Multiple gate Metal Oxide Semiconductor Field-Effect-Transistors (MOSFETs) have also been developped to counteract the short channel effects when reaching the nanometer scale [START_REF] Colinge | Multiple-gate SOI MOSFETs[END_REF][START_REF] Autran | A simulation analysis of FIBL in decananometer double-gate MOSFETs with high-κ gate dielectrics[END_REF].

Another important option is to consider alternative channel materials with better electrical properties than Si [START_REF] Kim | Effects of surface orientation on the performance of idealized III -V thin-body ballistic n -MOSFETs[END_REF][START_REF] Xu | GaSb inversion-mode PMOSFETs with atomic-layer-deposited Al2O3 as gate dielectric[END_REF][START_REF] Fang | Ultrathinbody high-mobility InAsSb-on-insulator field-effect transistors[END_REF]. Due to their very high electron mobility, III-V materials have been recognized to be good channel material candidates [START_REF] Alamo | Nanometre-scale electronics with III -V compound semiconductor[END_REF][START_REF] Tomioka | A III -V nanowire channel on silicon for high-performance vertical transistors[END_REF]. While several technological [START_REF] Kim | 30 nm InAs pseudomorphic HEMTs on an InP substrate with a current-gain cutoff frequency of 628 GHz[END_REF] and physical [START_REF] Fischetti | Simulation of electron transport in high-mobility MOSFETs: Density of states bottleneck and source starvation[END_REF] challenges must be overcome to incorporate III-V devices in future CMOSFETs, they are considered as a promising solution to meet the power constraints arising in several applications. The recent International Roadmap for Devices and Systems (IRDS) pointed out the achievement of high performances p-type strained III-V MOSFETs as key near-term challenges to address for the device scaling [START_REF]IRDS home page[END_REF].

A drawback of using III-V materials for a monolithic integration is the large imbalance between electron and hole mobilities (e.g., in bulk InAs, the electron mobility is µ e = 40 × 10 3 cm 2 • V -1 • s -1 while the hole mobility is µ h = 500 cm 2 • V -1 • s -1 ) [START_REF] Heyns | Ultimate scaling of CMOS logic devices with Ge and III-V materials[END_REF]. Since CMOS circuits require both n-and p-channels with reasonably matched performances, the future III-V CMOS technology is required to narrow the gap between n-and p-type MOS-FETs.

The use of mechanical strain represents a relevant technique to reach this aim [START_REF] Suthram | Strain additivity in III-V channels for CMOSFETs beyond 22nm technology node[END_REF][START_REF] Kim | Electronic band structure calculations for biaxially strained Si, Ge, and III-V semiconductors[END_REF][START_REF] Park | Performance comparisons of III-V and strained-Si in planar FETs and nonplanar FinFETs at ultrashort gate length (12 nm)[END_REF][START_REF] Bedell | Strain scaling for CMOS[END_REF]. Therefore, comprehensive investigations on III-V pMOSFETs are urgently needed in order to better predict and optimize their performances with respect to the choice of the material, mechanical strain and crystallographic orientation.

Quantum transport modeling can play a central role to determine the best device configuration. Most theoretical studies have so far explored the n-type III-V MOS-FETs [START_REF] Yokoyama | III-V-semiconductor-on-insulator nchannel metal-insulator-semiconductor field-effect transistors with buried Al2O3 layers and sulfur passivation: Reduction in carrier scattering at the bottom interface[END_REF][START_REF] Pethe | 2006 16th Biennial University/Government/Industry Microelectronics Symposium[END_REF][START_REF] Doornbos | Benchmarking of III-V n-MOSFET maturity and feasibility for future CMOS[END_REF][START_REF] Kim | CMOS performance benchmarking of Si, InAs, GaAs, and Ge nanowire n-and pMOSFETs with Lg= 13 nm based on atomistic quantum transport simulation including strain effects[END_REF] while few investigations focused on their p-type counterparts. Indeed, modeling hole transport in semiconductor nanostructures is typically a more complex task, as it requires to take into account the coupling between several bands [START_REF] Shin | Full-quantum simulation of hole transport and band-to-band tunneling in nanowires using the k.p method[END_REF][START_REF] Cavassilas | Multiband quantum transport simulations of ultimate p-type double-gate transistors: Influence of the channel orientation[END_REF]. The situation is even more complicated when strain and phonon scattering are included [START_REF] Cavassilas | Multiband quantum transport simulations of ultimate p-type double-gate transistors: Effects of hole-phonon scattering[END_REF][START_REF] Nguyen | Quantum modeling of the carrier mobility in FDSOI devices[END_REF][START_REF] Lee | Quantum treatment of phonon scattering for modeling of three-dimensional atomistic transport[END_REF].

Non-Equilibrium Green's Function (NEGF) formalism is one of the most established approach to treat quantum transport in nano-devices [START_REF] Datta | Electronic Transport in Mesoscopic Systems[END_REF][START_REF] Ferry | Transport in Nanostructures[END_REF][START_REF] Haug | Quantum Kinetics in Transport and Optics of Semiconductors[END_REF]. It is able to describe inelastic scattering, like electron-phonon interactions, through the concept of self-energy [START_REF] Mahan | Many-Particle Physics[END_REF]. For numerical convenience the electron-phonon scattering is usually treated within a local approximation, since most of the NEGF codes are based on a recursive algorithm [START_REF] Luisier | Atomistic full-band simulations of silicon nanowire transistors: Effects of electronphonon scattering[END_REF][START_REF] Lopez Sancho | Highly convergent schemes for the calculation of bulk and surface Green functions[END_REF], which requires a diagonal (i.e. local) description of the self-energy. This assumption was demonstrated to be physically sound for interactions with acoustic and nonpolar optical phonons [START_REF] Svizhenko | Role of scattering in nanotransistors[END_REF][START_REF] Jin | A three-dimensional simulation of quantum transport in silicon nanowire tran-sistor in the presence of electron-phonon interactions[END_REF][START_REF] Bescond | Size and temperature dependence of the electronphonon scattering by donors in nanowire transistors[END_REF][START_REF] Cavassilas | On the local approximation of the electronphoton interaction selfenergy[END_REF].

However, III-V compound semiconductors are subject to strong polar optical (PO) interactions of the Fröhlich type [START_REF] Fröhlich | Theory of electrical breakdown in fonic crystals[END_REF][START_REF] Appel | Polarons[END_REF][START_REF] Ehrenreich | Electron scattering in InSb[END_REF], which intrinsically have a non-local character. Indeed, this type of electron-phonon interaction decays very slowly in space like a Coulomb potential [START_REF] Fröhlich | Theory of electrical breakdown in fonic crystals[END_REF]. This non-locality increases the numerical complexity of the quantum transport computations in a prohibitive manner. Previous NEGF studies have proposed to model PO phonon scattering through diagonal self-energies multiplied by a scaling factor [START_REF] Klimeck | Numerical approximations for polar optical phonon scattering in resonant tunneling diodes[END_REF], but without providing general prescriptions.

In this paper, by directly starting from the Fröhlich theory of bulk polar optical phonons, we derive an analytical expression of the scaling factor in confined systems, which can be used to mimic the effects of polar optical phonons within a local approximation. We then use our model to calibrate the electron-PO phonon coupling strength in self-consistent NEGF simulations, in order to assess the performances of p-type double-gate (DG) MOSFETs as a function of the channel material, of the crystallographic orientation and of the strain. For all the considered devices, TSC =2.4 nm, TOX =1 nm, and the source and drain doping is 8×10 19 cm -3 . Two gate lengths, LG=15 nm and 7 nm, are considered. VDS=-0.6 V.

We use a two-dimensional (2D) NEGF code selfconsistently coupled with the Poisson equation [START_REF] Dib | Theoretical comparison of Si, Ge, and GaAs ultrathin p-type double-gate metal oxide semiconductor transistors[END_REF][START_REF] Moussavou | Influence of uniaxial strain in Si and Ge p-type doublegate metal-oxide-semiconductor field effect transistors[END_REF]. The valence bandstructure of III-V direct gap materials is described by a 8-band k.p Hamiltonian [START_REF] Nainani | Enhancing hole mobility in III-V semiconductors[END_REF].

The paper is organized as follows: Section presents the NEGF framework. In Section , an expression of the PO phonon self-energy is derived, first in the bulk case and then for a thin film DG MOSFET. In Section , the diagonal approximation is introduced, and an analytic expression of the scaling factor is derived. In Section , we present and discuss the results of our numerical simulations of III-V DG MOSFETs. Section IV summarizes the key findings.

METHOD

Quantum kinetic equation within the eight-band k • p framework

We consider a 8-band k • p Hamiltonian including the spin-orbit coupling to model the valence bandstructure of the 2D DG pMOSFET shown in Fig. 1. The DG pMOS-FET geometry is sketched in Fig. 1. We model the valence band structure through a 8-band k • p Hamiltonian, which is able to take into account the coupling between the light, heavy and split-off hole bands, and the coupling between them and the conduction band [START_REF] Nainani | Enhancing hole mobility in III-V semiconductors[END_REF][START_REF] Bahder | Eight-band k.p model of strained zincblende crystals[END_REF]. The device is assumed translationally invariant in the transverse y direction. Accordingly, the Hamiltonian can be parametrized in terms of the wave vector component k y . In order to obtain accurate results, in numerical computations, a large enough number of suitably spaced values of k y has to be considered. We chose k y of the form 2πn/L y , with n an integer and L y =10 nm.

Mechanical strain is modeled through the Bir-Pikus Hamiltonian, and by assuming bulk values of deformation potentials [START_REF] Bir | Symmetry and Strain Induced Effects in Semiconductors[END_REF]. The bandstructure material parameters are extracted from reference 45 and reported in Table I. To solve the transport problem in the presence of inelastic interactions, the Keldysh retarded and lesser/greater-than Green's functions are calculated for each transverse mode k y by solving the following equations (in matrix notation):

[G R (k y , )] = [I] -[H 2D (k y )] -[U ] -[Σ R L (k y , )] -[Σ R scat (k y , )] -1 (1)
[G ≶ (k y , )] =[G R (k y , )] [Σ ≶ L (k y , )] +[Σ ≶ scat (k y , )] [G R † (k y , )], (2)
where is the hole energy, I is the identity matrix, [H 2D (k y )] the 2D Hamiltonian for a given transverse mode and [U ] the 2D potential (in terms of hole energy). To ensure accurate calculations, 10 discrete values of k y are considered in the simulations. In Eqs.( 1) and ( 2), [Σ R,≶ L ] are the retarded and lesser/greater-than selfenergies of the leads, respectively. Moreover, [Σ

R,≶ scatt ] = [Σ R,≶ ac ] + [Σ R,≶ op ] + [Σ R,≶ pop ], where [Σ R,≶ ac ], [Σ R,≶ op ], [Σ R,≶
pop ] are the acoustic, optical and PO phonon self-energies, respectively. The self-energies of the leads are calculated exactly [START_REF] Svizhenko | Effect of scattering and contacts on current and electrostatics in carbon nanotubes[END_REF], while the scattering ones are treated within a perturbative approach, through the self-consistent Born approximation (SCBA) [START_REF] Lee | Efficient quantum modeling of inelastic interactions in nanodevices[END_REF]. Within the SCBA and by considering the elastic approximation, the hole-acoustic phonon self-energies are expressed as

[Σ R,≶ ac (k y , )] =iIm{ ky ξ∈{x,y,z} [M ac,ξ ] [G R,≶ (k y , )][M † ac,ξ ]}, (3)
where [M ac,ξ ] describes the hole-acoustic-phonon coupling. The optical phonon lesser/greater-than and retarded self-energies read [START_REF] Aeberhard | Microscopic nonequilibrium theory of quantum well solar cells[END_REF] [Σ ≶ op (k y , )] = iIm{ ξ∈{x,y,z}

[M op,ξ ] × N op [G ≶ (k y , -ω op )] +(N op + 1)[G ≶ (k y , + ω op )] × [M † op,ξ ]} (4) [Σ R op (k y , )] = iIm{ ξ∈{x,y,z} [M op,ξ ](N op [G R (k y , -ω op )] + (N op + 1)[G R (k y , + ω op )] + 1 2 ([G > (k y , -ω op )] -[G > (k y , + ω op )]))[M † op,ξ ]}, (5)
where ω op is the optical-phonon energy, N op = 1/(exp( ω op /k B T ) -1) is the corresponding phonon distribution, [M op,ξ ] represents the hole-optical-phonon coupling and a dispersionless approximation has been considered. We solve Eqs. ( 1) and ( 2) through the recursive algorithm described in Refs. 49 and 50. This approach allows one to efficiently compute the diagonal and first off-diagonal blocks of the Green's functions, from which the charge and current densities are obtained. This considerably lightens the computational burden with respect to a direct numerical implementation of Eq. ( 1), but requires to resort a local approximation for the scattering self-energies. If this approximation is applied to PO phonons, the electron-phonon coupling needs to be renormalized, in order to avoid an underestimation of the scattering rates. In the next Section, we will define a diagonal approximation that locally conserves the electron-PO phonon scattering rates, and derive easy-to-handle analytical expressions for the corresponding scaling factor.

Polar optical phonon self-energy

We start from the second order expression of the lesser/greater self-energy of the electron-phonon interaction [START_REF] Haug | Quantum Kinetics in Transport and Optics of Semiconductors[END_REF]:

[Σ ≶ ( )] = q 2µω ∂H ep ∂u q × (N pop + 1)G ≶ ( ± ω) + N pop G ≶ ( ∓ ω) × ∂H † ep ∂u q (6)
In Eq. ( 6), H ep is the electron-phonon interaction Hamiltonian, q denotes the phonon wave vector, and u q and µ are the amplitude of the normal mode relative displacement with respect to the equilibrium position and the reduced mass of the atoms in the unit cell, respectively.

We evaluate Eq. ( 6) by calculating H ep = eφ, where e represents the elementary charge and φ the energy due to polar optical phonons. We first consider the bulk situation (Subsection ), and then we derive the case of a quantum well (Section ).

Bulk formulation

In this Subsection, we outline the calculation of the bulk electron-PO phonon interaction, by following the Fröhlich approach [START_REF] Fröhlich | Theory of electrical breakdown in fonic crystals[END_REF]. We consider ionic crystals with two atoms per unit cell. In the long wavelength limit, u q (R) expresses the relative displacement of the two atoms in the unit cell at R. The associated ionic polarization P i (R), defined as the average dipole moment per unit volume, can be written to the first order in the displacement as

P i (R) = e * Ω u q (R), (7)
where e * is the effective charge and Ω the volume of the unit cell. We assume that only longitudinal PO phonons couple with electrons. In this hypothesis,

u q (R) = u q q q e iq•R √ V (8)
where, in the long wavelength limit q → 0, the displacement can be considered as a continuous function of R normalized over the volume V of the system. As div(εE) = div(E + 4πP i ) = iq(E q + 4πP iq ) = 0, u q induces an electrostatic potential φ q corresponding to an electric field E q = -4πP iq = -iqφ q (in the electrostatic esu-cgs unit system):

φ q = - i q 4πe * Ω u q . ( 9
)
One still has to determine e * . For this, we first notice that a finite static u q can be induced by an external field E b . For longitudinal displacements this gives

µω 2 L u q = e * E b ( 10
)
where µ is the reduced mass and ω L is the longitudinal phonon frequency. According to Eq.( 7) this is equivalent to

P i = (e * ) 2 µω 2 L Ω E b . (11)
On the other hand,

E + 4πP = E b and E = E b /ε. Thus P = 1 4π 1 - 1 ε E b . ( 12
)
where P = P e + P i , and P e is the electronic polarizability. Eq. ( 12) is in general valid for fields with a harmonic time dependence e iωt . In static conditions (ω=0, ε = ε(0)), both P e and P i contribute to P, while, when ω → ∞ (ε = ε(∞)), the contribution of P i is negligible. Therefore, we can calculate P i as

P i = - 1 4π 1 ε(0) - 1 ε(∞) E b . (13)
The comparison between Eqs.( 11) and [START_REF] Suthram | Strain additivity in III-V channels for CMOSFETs beyond 22nm technology node[END_REF] gives

(e * ) 2 = µω 2 L 1 ε(∞) - 1 ε(0) Ω 4π . (14)
Eqs. ( 9) and ( 14) provide an expression of φ q in terms of u q , which, however, does not take into account the screening by the carriers. The latter can be easily included via linearized theories such as the Debye or Thomas-Fermi approximations.

In both cases, we have to replace 1 q in equation ( 9) by q q 2 +q 2 s where q 2 s is given either by Debye formula

q 2 D = 4πe 2 ε(∞)k B T n 0 , (15)
valid at low carrier concentration n 0 , or by the Thomas-Fermi expression

q 2 T F = q 2 D 3 2 k B T F (16)
valid at high carrier concentrations, i.e when the Fermi level F is located inside the band and its distance from the band edge is > k B T , where k B is the Boltzmann constant and T is the absolute temperature. The electronphonon coupling is thus given by -eφ q e iq.R √ V which, with equation ( 9) modified for screening gives

H ep = i eq q 2 + q 2 s 4πe * Ω u q e iq•R √ V . (17)
In fact, there are two sources of electron screening, one by the valence electrons, the other one by the excess carriers. For the first one the plasmon frequency is around 15 eV, much higher than the typical optical phonon energies of order 50 meV. One can thus safely use static dielectric screening in this case which amounts to divide the phonon induced potential by the dielectric constant. The situation is not the same for the excess carriers since the squared plasma frequency scales as n/εm * where n is the excess electron concentration, ε the dielectric constant and m * the effective mass. For the weak concentrations considered here this can bring the corresponding plasma frequency in the same range as the phonon frequency. Qualitatively this would reduce q s (which represents the strength of the screening) with respect to its static value.

In practice, we consider cases for which the static q s is negligible so that this effect can be discarded. By combining Eqs. ( 6) and ( 14), we obtain

[Σ ≶ P OP,bulk ( , r 1 , r 2 )] = M 2 V {(N pop + 1)G ≶ ( ± ω L , r 1 , r 2 ) + N pop G ≶ ( ∓ ω L , r 1 , r 2 )} × q e iq(r1-r2) q 2 (q 2 + q 2 s ) 2 , ( 18
)
where Σ ≶ is expressed in real space with coordinates r 1 and r 2 and

M 2 = 2π ω L e 2 1 ε(∞) -1 ε(0) .

Extension to the case of a quantum well

We consider first the case of a system in which phonons are confined only in the z direction, between z = 0 and z = L z . In this work we have followed the general approach of Mori and Ando on electron-optical phonon interaction in heterostructures [START_REF] Mori | Electron-optical-phonon interaction in single and double heterostructures[END_REF]. For a double heterostructure they show that one can separate the eigenmodes into confided LO modes with vanishing polarization field at the interface and interface modes decaying exponentially away from the interface. We concentrate here on the first category. A second point is related to the confinement effect on the phonon modes. This was shown to be important for acoustic modes due to their important dispersion curve [START_REF] Ramayya | Electron transport in silicon nanowires: The role of acoustic phonon confinement and surface roughness scattering[END_REF]. However optical phonons exhibit almost no dispersion so that it is equivalent to treat them as Einstein oscillators, the confinement effect being negligible. As discussed previously we consider longitudinal PO displacements confined to the region extending between z = 0 and z = L z which are solutions of ε(ω) = 0. We express the potential φ in terms of a complete set of bulk basis functions satisfying the correct boundary conditions at z = 0 and L z , i.e no polarization at the two interfaces:

φ(R) = q φ q 2 L z A sin(q z z)exp(iq • r ) ( 19
)
where R = (r , z), q = (q , q z ), and A is the transverse cross section and q z = nzπ Lz . As before, longitudinal modes correspond to E = -4πP i = -∇φ. As P i is still given by ( 7) each phonon mode u q (R) is related to φ q by 4πe * Ω u q (R) = φ q 2 L z A q n cos(q z z) + iq sin(q z z) e iq •r .

(

) 20
From Eq.( 20) it is straightforward to see that the eigenmodes u q (R) are orthogonal (i.e. u q (R) • u q (R) = δ qq ) and

u 2 q = 1 V |u q (R)| 2 d 3 R = q Ω 4πe * φ q 2 (21)
with q = q 2 z + q 2 . This means that

∂φ(R) ∂u q = 4πe * Ωq 2 V sin(q z z)exp(iq • r ) (22)
which leads exactly to the same expression for ∂Hep ∂uq as Eq.( 17) for the bulk, except for the replacement of

1 V e iqzz by 2
V sin(q z z). This result can be extended to the case of a system where phonons are confined in two directions x and z, by further replacing e iqxx with √ 2sin(q x x). In the following, we will mainly focus on this case, as it models the phonon confinement between the source and drain contacts and the top and bottom gate oxide in the DG MOSFET geometry presented in Section . Below, we report the corresponding expression of the lesser/greater than Green's function

[Σ ≶ P OP ( , r 1 , r 2 )] = M 2 G ≶ ( , r 1 , r 2 )I(r 1 , r 2 )
with :

G ≶ ( , r 1 , r 2 ) = (N pop + 1)G ≶ ( ± ω L , r 1 , r 2 ) + N pop G ≶ ( ∓ ω L , r 1 , r 2 ),
and :

I(r 1 r 2 ) = 4
V q sin(q x x 1 )sin(q x x 2 )sin(q z z 1 )sin(q z z 2 )

× q 2 (q 2 + q 2 s ) 2 e iqy(y1-y2) , q x = n x π L x , q z = n z π L z , q y = 2n y π L y . (23)
Scaling factor for 2D double-gate geometry

In this Section, we will introduce the diagonal approximation for the PO phonon self-energies mapped on the discretized simulation domain of DG MOSFETs (Subsection 1 ). Then, we will derive an analytical expression for the scaling factor and we will provide simplified formulas allowing for a rapid numerical computation (Subsection and , respectively). We also report an analytical expression of the scaling factor for the case of a one-dimensional confinement and benchmark it value against previous numerical studies (Subsection ). For the sake of simplicity, the derivation is carried out for electrons, but the final results fully apply also to holes.

The discretized diagonal approximation

Let us denote by ρ=(x, z) the position vector in the x-z plane. We define a discretized version of Σ ≶ P OP (y 1 , ρ 1 ; y 2 , ρ 2 ) on a square lattice of parameter a as

Σ ≶ P OP,i,j (y 1 , y 2 ) = χ i (ρ 1 )Σ ≶ P OP (y 1 , ρ 1 ; y 2 , ρ 2 )χ j (ρ 2 ) =a 2 Σ ≶ P OP (y 1 , a i ; y 2 , a j ). ( 24
)
where χ i (ρ)=1/a is a normalized averaging function in the unit cell around the lattice site a i . Since our system is uniform along the y direction we first Fourier transform Eq.( 23) and apply Eq.( 24) with a i = a j = (x i , z i ). The diagonal elements of Σ ≶ P OP then read as:

[Σ ≶ pop,ii ( , k y )] = 4 V M 2 a 2 q G ≶ i,i ( , k y -q y ) sin 2 (q x x i )sin 2 (q z z i ) q 2 (q 2 + q 2 s ) 2 (25)
We average the squared sine functions in Eq. ( 25) to 1/2 and transform the discrete summation over q x and q z into an integral. Care must however be taken that, due to the confinement, q x and q z are discrete quantities (see Eq. ( 23)). We have found that an approximation procedure is

1 L x L z qx= nπ Lx >0 qz= mπ Ly >0 q 2 (q 2 + q 2 s ) 2 1 π 2 q M 0 dq x q M 0 dq z q 2 + q 2 0 (q 2 + q 2 0 + q 2 s ) 2 q 2 0 = π 2L x 2 + π 2L z 2 ( 26
)
q M is the size of the Brillouin zone. After integration in polar coordinates, we finally obtain

[Σ ≶ pop,ii ( , k y )] = M 2 a 2 4πL y qy G ≶ i,i ( , k y -q y ) × ln q 2 M + q 2 y + q 2 0 + q 2 s (q 2 y + q 2 0 + q 2 s ) -q 2 s ( 1 q 2 y + q 2 0 + q 2 s - 1 q 2 M + q 2 y + q 2 0 + q 2 s ) . (27)
Based on this formula, in the next Subsection we will proceed to the calculation of the scaling factor.

The scaling factor

We define the scaling factor S by requiring that the diagonal approximation conserves the total scattering rates. Its Fourier transform is thus exact in the small k limit. It has thus all the physical ingredients of the full second order expansion of the electron Greens function which considers that the phonons remain in thermal equilibrium. Here the numerical accuracy is checked by comparison with a previous 1D full calculation. A more detailed study of the validity of the local approximation is out of the scope of the present paper and will be the subject of a forthcoming work. We quantify the total scattering rate on a lattice site a i and at energy as j Γij ( ), where Γ = lim ky→0 [Σ < pop ( , k y ) -Σ > pop ( , k y )] is the broadening function integrated over the y direction. This expression takes into account all the phononassisted intersite hoppings. We therefore set

S i ( ) = j Γ i,j ( ) Γ i,i ( ) (28)
and we notice that S is, in principle, site-and energydependent.

To proceed further we need to consider this broadening function in more detail.

According to Eq. ( 23),

Σ < P OP ( ) -Σ > P OP ( ) = M 2 [G < ( ) -G > ( )]I (29)
In the spirit of the Boltzmann transport equation we will evaluate S in equilibrium conditions. Therefore, we assume

G < ( ) =iA( )f ( ) G > ( ) = -iA( )[1 -f ( )], ( 30
)
where f is the Fermi function and A( ) is the spectral function. By injecting Eq. ( 30) into Eq. ( 29) and by using Eq. ( 23), we obtain

G < ( ) -G > ( ) = i [N pop + f ( + ω L )] A( + ω L ) + [N pop + 1 -f ( -ω L )] A( -ω L ) = iA( ) (31)
The continuous expression of Γ thus reads

Γ( , ρ 1 , ρ 2 ) = M 2 A( , y 1 , ρ 1 , y 2 , ρ 2 )I(y 1 , ρ 1 , y 2 , ρ 2 ) = M 2 qy A( , q y , ρ 1 , ρ 2 )I(q y , ρ 1 , ρ 2 ), (32)
where the second equality follows by a Fourier transformation with respect to (y 1 -y 2 ) and by taking the limit k y → 0, and

A( , q y , ρ 1 , ρ 2 ) = 2π 2 L x L z k δ( -qy -k ) × sin(k z z 1 )sin(k z z 2 )e ikx(x1-x2) . (33)
In order to simplify the notation, in the following we will refer to Γ, Γ( , ρ 1 , ρ 2 ) = M 2 qy A( , q y , ρ 1 , ρ 2 )I(q y , ρ 1 , ρ 2 ), [START_REF] Bescond | Size and temperature dependence of the electronphonon scattering by donors in nanowire transistors[END_REF] instead than to Γ. In terms of Γ ij , S reads

S i = j [N pop + f ( + ω L )]Γ ij ( + ω L ) + [N pop + 1 -f ( -ω L )]Γ ij ( -ω L ) [N pop + f ( + ω L )]Γ ii ( + ω L ) + [N pop + 1 -f ( -ω L )]Γ ii ( -ω L ) . ( 35
)
Derivation of Γ ii .

From Eqs. ( 33) and ( 34) with ρ 1 =ρ 2 , after averaging over z, we obtain

Γ ii ( ) = M 2 a 2 4πL y 2π L x L z qy,kx,kz δ( -qy -kx -kz ) × ln q 2 M + q 2 y + q 2 0 + q 2 s (q 2 y + q 2 0 + q 2 s ) -q 2 s 1 q 2 y + q 2 0 + q 2 s - 1 q 2 M + q 2 y + q 2 0 + q 2 s . (36)
We notice that performing the average over z actually removes the dependence of the diagonal elements of Γ on the site index i. In Eq. ( 36) the energies kz are quantized with k z = nπ Lz > 0 while qy and kx are continuous quantities. We can thus replace the sum 2π LxLz qy,kx by an integral 1 2π dk x dq y . We use polar coordinates to transform it to 1 2π ρdρdθ with ρ 2 = k 2

x + q 2 y and q 2 y = ρ 2 cos 2 θ. With the further change of variable ρ = 2 2m * ρ 2 , ρdρ = m * 2 d ρ , we see that the delta function gives 1 ifkz > 0. Finally, by averaging the functions cos 2 θ to 1/2, we obtain:

Γ ii ( ) = M 2 a 2 m * 4πL z 2 kz<k ln q 2 M + q 2 0z ( ) q 2 0z ( ) -q 2 s 1 q 2 0z ( ) - 1 q 2 M + q 2 0z ( ) (37)
with

q 2 0z ( ) = q 2 0 + q 2 s + k 2 -k 2 z 2
, q 2 0 defined by Eq.( 26), and k 2 -k 2 z > 0 . The strong confinement along z in a DG MOSFET geometry implies widely separated subbands. Therefore, only the lowest subband, corresponding to k z = π/L z , contributes to the sum in Eq. ( 37), and we can set

(k 2 - k 2 z )/2 = m * / 2 ( -1 )
, where 1 is the energy of the bottom of the lowest subband.

It is interesting to cast this result under a more usual form, in terms of parameter α which mesures the strength of the Fröhlich interaction

α = e 2 m * 2 ω L 1/2 1 ε ∞ - 1 ε 0 ( 38
)
From Eq.( 18) we can write:

M 2 π m * 2 k L = 2α ω L with k L = 2m * 2 ω L 1/2 .
(39) By using Eq.( 39) and by setting

k z = k 1 = π/L z , we obtain Γ ii ( ) = α ω L 2π (k 1 k L a 2 ) ln q 2 M + q 2 0z q 2 0z -q 2 s 1 q 2 0z - 1 q 2 M + q 2 0z (40) with q 2 0z ( ) = q 2 0 + q 2 s + k 2 -k 2 1 2 . Derivation of j Γ ij ( ) .
Here it is useful to map the sum over j into an integral:

j Γ ij = a 2 j Γ(a i , a j ) = Γ(a i , r)dr. ( 41
)
From Eqs. ( 32), ( 33) and ( 41) we obtain

j Γ ij = M 2 2 L z +∞ -∞ dk x kz dxdz kz,qy δ( -qy -k ) × 4 V sin(k z z i )sin(k z z)e ikx(xi-x)
× qxqz sin(q z z i )sin(q z z)sin(q x x i )sin(q x x) q 2 (q 2 + q 2 s ) 2 [START_REF] Nainani | Enhancing hole mobility in III-V semiconductors[END_REF] In this expression, the integrals over z and x lead to δ kz,qz and δ kx,-qx -δ kx,qx , respectively. We obtain:

j Γ ij = M 2 4π V q δ( -q )sin 2 (q z z i )sin 2 (q x x i ) q 2 (q 2 + q 2 s ) 2 , ( 43
)
where q x = nxπ Lx > 0, q z = nzπ Lz > 0, q y = 2πny Ly ≶ 0. After averaging the squared sine functions in Eq. ( 43) to 1/2 and transforming the sum over q x and q y into an integral, by following steps analogous to the derivation of Eq. ( 37), we arrive at

j Γ ij = M 2 π m * 2 π L z kz k 2 k 2 + q 2 s . (44)
As for Γ ii , by averaging over x and z, we have obtained an expression independent of the lattice site index i. In terms of α, k 1 and k L defined before, and by restricting the sum over k z to the first subband, we finally obtain:

j Γ ij = 2α ω L k 1 k L k 2 (k 2 + q 2 s ) 2 . ( 45
)
Numerical value of the scaling factor in the case of a quantum well

The numerical evaluation of the scaling factor in Eq. ( 35) can be simplified by taking into account that the transport occurs only through the lowest subband. We first consider two different limits:

1. High charge-densities: the transport occurs at energies close to the Fermi level F , which lies within the lowest confined subband: F > 1 . As 1 ω L 30 meV, we can write

S = j Γ ij ( F ) Γ ii ( F ) (46)
By using Eqs. ( 40) and [START_REF] Vurgaftman | Band parameters for III-V compound semiconductors and their alloys[END_REF], and taking into account that q M q 0z , we find

S = 2π (k F a) 2 1 1 + q 2 s k 2 F 2 ln q M q 0z -1 2 q 2 s q 2 0z , (47)
where k F = (2m * F )/ is the Fermi wave vector. 2. Low charge densities: F is below the bottom of the lowest confined subband, and the transport energies are within few k B T from 1 . Therefore, also in this case we can use Eq. ( 47), provided k F is replaced with k 1 = (2m * 1 )/ = π/L z .

For both high and low charge densities, it can be shown that q 0 z k 1 and q s k 1 , which allows Eq. ( 47) to be further simplified. In the case of low charge density we find

S = 2π (k 1 a) 2 1 ln 2q M k 1 . ( 48
)
In the considered systems, where the confinement energy contribution dominates over those associated to the unconfined directions, k F k 1 , and Eq. ( 48) is also valid in the high charge density case. Moreover, it is easy to argue that this result also applies to charge density values intermediate between the considered limits. Eq. ( 48) predicts that S is essentially independent of the effective mass, and dominated by k 1 , or, equivalently, by L z . To exemplify, Table II reports the numerical values of S corresponding to several different values of L z .

Based on interaction self-energies (Eqs.(3-5) ( 27) and ( 48)), physical quantities like carrier density and electronic current can be calculated [START_REF] Cavassilas | Multiband quantum transport simulations of ultimate p-type double-gate transistors: Effects of hole-phonon scattering[END_REF]. NEGF equations are then self-consistently with Poisson equation to determine the actual electrostatic potential in the active region.

The scaling factor for the 1D case It is obviously of strong interest to test our method against the results of numerical computations able to take into account the full electron-PO phonon scattering selfenergy.

In Ref. [START_REF] Klimeck | Numerical approximations for polar optical phonon scattering in resonant tunneling diodes[END_REF], full self-energy computations were performed in the case of GaAs-AlAs resonant tunneling diodes and compared to a 1D diagonal approximation along the direction of the current. In this study, it was unambiguously shown that the full self-energy results could be recovered within a diagonal approach by using a scaling factor S = 12. By following an analogous approach to that previously described in the case of 2D confinement, we find that the equivalent of Eq.(48) for a system confined along a single direction is with k L defined by equation [START_REF] Klimeck | Numerical approximations for polar optical phonon scattering in resonant tunneling diodes[END_REF]. In the case of GaAs at low temperature, for small electron concentration (k F k L ), with ω L = 33 meV and a discretization parameter of a = 4 Å, this gives S = 12.7, in excellent agreement with the results of Ref. [START_REF] Klimeck | Numerical approximations for polar optical phonon scattering in resonant tunneling diodes[END_REF].

S = 4 πa k 2 F + k 2 L , (49)

SIMULATIONS OF DG P-TYPE TRANSISTORS

Based on the model developed in Section , we now investigate the performance of III-V DG p-MOSFETs through self-consistent NEGF simulations, which couple the kinetic equations introduced in Section with the twodimensional Poisson equation in the device cross-section.

The thickness of the III-V film is assumed to be T SC =2.4 nm and the source and drain extensions (see Fig. 1) are p-doped with a concentration of 8×10 19 cm -3 . We consider as channel materials InAs, InSb and GaSb, as they provide the highest hole bulk mobilities. Performances are investigated as a function of two different crystallographic orientations, < 100 > and < 110 >, and strain constraints (uniaxial tensile and compressive strain), and benchmarked against analogous Si and Ge devices. All the simulations were carried out with a source to drain voltage V DS = -0.6 V. Acoustic-and optical phonon-electron interactions were described through the self-energies defined in Eqs. [START_REF] Autran | A simulation analysis of FIBL in decananometer double-gate MOSFETs with high-κ gate dielectrics[END_REF][START_REF] Kim | Effects of surface orientation on the performance of idealized III -V thin-body ballistic n -MOSFETs[END_REF][START_REF] Xu | GaSb inversion-mode PMOSFETs with atomic-layer-deposited Al2O3 as gate dielectric[END_REF], by using the deformation potential values in Ref. [START_REF] Adachi | Properties of Group-IV, III-V and II-VI Semiconductors[END_REF].

The electron-PO phonon scattering has been included through the diagonal self-energy [START_REF] Ferry | Transport in Nanostructures[END_REF], rescaled by a factor S=111.1 (see Table II).

Figure 2 shows how a uniaxial compressive and tensile strain along either the < 100 > or < 110 > direction, modifies the valence band structure of a 2.4 nm thick InAs film. The main impact on the transport is expected to derive from the modifications undergone by first subband, which is widely separated from the others due to the strong confinement, and therefore dominates the transport. Both in the < 100 > and < 110 > directions, a compressive strain induces a reduction of the effective mass value at k x =0, while a tensile strain leads to an increase of it. We found the same behavior in the case of InSb and GaSb. The values of the effective mass extracted for the three materials as function of the crystallographic orientation and strain are reported in Table III.

In order to assess the impact of strain on the I ON /I OFF ratio, we computed, for each strain constraint, transport direction and material, the OFF-state current I OFF as a function of the ON-state current I ON . To isolate the effect of the strain from those due to the modulation of the electron-phonon interaction, a ballistic transport regime was assumed.

The case of the InAs MOSFET in the < 110 > orientation with L G =15 nm is shown in Figure 3. The shift of the curves reflects the value of the first subband effective mass: devices with a lower effective mass tend to have better I ON /I OFF ratios, and therefore I OFF -I ON characteristics shifted toward higher I ON values. This trend remains unchanged for all the considered materials and transport directions (not shown).

According to these results, in the following we restrict the focus to compressively strained devices. Fig. 4 illustrates the I OFF -I ON characteristics of the InAs, InSb and GeSb MOSFETs. The results for a < 110 >-oriented, compressively strained MOSFET with the channel in Ge, which has been shown to represent a better option than Si for DG devices at L G =15 nm [START_REF] Moussavou | Influence of uniaxial strain in Si and Ge p-type doublegate metal-oxide-semiconductor field effect transistors[END_REF], are also included. Our results indicate that Ge-based devices can outperform the others with respect to low operating power (LOP) requirements. Among the considered III-V devices, the < 100 >-oriented InAs and InSb MOSFETs appear to provide the best performance for LOP and low standby power (LSTP) applications, respectively.

In Figure 5, we plot the I OFF -I ON characteristics in the presence of electron-phonon scattering for the InAs, InSb and Ge MOSFETs. The corresponding ballistic curves are also plotted for comparison. As expected, for all the materials, the impact of the electron-phonon scattering is stronger for high I ON values. However, in the case of Ge, the effect is significantly weaker. More precisely, for a I OFF current of 10 -4 A/m, the electron-phonon interaction induces a reduction of the ON-current of about 65% and 58% in InAs and InSb, respectively, while this value amounts to 24% in Ge. The relative importance of the electron-PO phonon scattering in determining this difference can be quantified by evaluating the I ON reduction in the III-V devices due only to acoustic and non-polar optical phonons. In these conditions, for the InAs devices we found a I ON reduction of around 25%. Similar values are expected for the InSb device. Thus, our results indicate that electron-PO phonon interactions can strongly disadvantage III-V MOSFETs with respect to analogous non-polar material devices. In Fig. 5, this directly translates in a widening of the performance gap between the III-V and Ge devices with respect to ballistic transport conditions. It turns out that the Ge MOSFET can largely outperform the III-V counterparts for LOP applications. InAs and InSb devices are competitive only for LSTP applications, as the impact of the electron-phonon scattering reduces due to the smaller current.

A substantially different scenario arises when shorter gate lengths are considered. The results obtained by repeating all the previous simulations for L G =7 nm indicate that the < 110 >-oriented GaSb MOSFET exhibits in this case the best I ON /I OFF performance (not shown). Fig. 6 illustrates the I ON -I OFF characteristics of this device with and without strain, in the ballistic regime and in the presence of electron-phonon scattering. These results are compared with those obtained for a < 100 >oriented, tensile-strained Si MOSFET, which, at this gate length scale, exhibits better performance with respect to the corresponding Ge-based device [START_REF] Moussavou | Influence of uniaxial strain in Si and Ge p-type doublegate metal-oxide-semiconductor field effect transistors[END_REF]. In the ballistic regime, we see that the Si device clearly outperforms all the III-V devices, for both LOP and LSTP applications. Indeed, the larger effective mass (m*=0. becomes a significant advantage at this channel length, since it mitigates the OFF-current degradation due to the source-to-drain tunneling. [START_REF] Dib | Theoretical comparison of Si, Ge, and GaAs ultrathin p-type double-gate metal oxide semiconductor transistors[END_REF][START_REF] Moussavou | Influence of uniaxial strain in Si and Ge p-type doublegate metal-oxide-semiconductor field effect transistors[END_REF] For the same reason, the < 110 >-oriented, tensile-strained GaSb MOSFET, in which the first subband effective mass (m*=0.107×m 0 ) is larger than for the other GaSb devices, exhibits the second best performance. As in the longer channel case, the inclusion of electronphonon scattering leads to a larger current degradation in III-V devices. This results in a significant improvement of the Si MOSFET performance with respect to the GaSb device, which exhibits, at the same I ON , an I ON /I OFF ratio smaller of at least one order of magnitude.

CONCLUSION

In this work, we proposed a physically sound and numerically efficient NEGF quantum transport methodology to treat non-local polar optical phonon scattering. The methodology is based on a diagonal (i.e. local) selfenergy approximation, in which long range effects are incorporated through the introduction of a scaling factor. We derived a simple analytical expression of this factor for a 2D phonon quantum well. An analytical expression for a 1D well was also provided and checked against previous numerical studies. Based on a 8-band k • p Hamiltonian we used this approach to assess the performance of p-type III-V DG transistors for different channel materials, crystallographic orientations and applied strains. Our simulation results indicate that III-V transistors are hardly expected to outperform their Ge and Si counterparts and that the strong impact of polar optical phonon scattering in such materials is one the main reasons. Ac-cordingly, our results also highlights the need, even in the case of transistors with very scaled gate lengths, to accurately model the electron-PO phonon interaction in order to obtain reliable predictions of the device operation.

FIG. 1 .

 1 FIG.1. Sketch of the considered Double-Gate MOSFETs. For all the considered devices, TSC =2.4 nm, TOX =1 nm, and the source and drain doping is 8×10 19 cm -3 . Two gate lengths, LG=15 nm and 7 nm, are considered. VDS=-0.6 V.

6 FIG. 2 .

 62 FIG. 2. Impact of uniaxial compressive and tensile strain, corresponding to a deformation | |=3%, on the valence band structure of a 2.4 nm-thick InAs film. Both the strain and the band structure are along the direction < 100 > (top), and < 110 > (bottom).

3 FIG. 3 .

 33 FIG. 3. Impact of strain | |=3% on the ballistic IOFF vs ION characteristics of a < 110 >-oriented InAs DG MOSFET with LG=15 nm.

3 FIG. 5 .

 35 FIG. 5. IOFF vs ION characteristics of < 110 >-oriented InAs, InSb and Ge DG MOSFETs under compressive strain (| |=3%), and in the presence of electron-phonon scattering. In all the cases LG=15 nm.

TABLE I .

 I Principal k • p parameters used in this work [45]. Eg Ep ∆so a0 InAs 20.0 8.5 9.2 0.417 21.5 0.39 6.06 InSb 34.8 15.5 16.5 0.23 23.3 0.81 6.48

	γ L 1	γ L 2	γ L 3
	GaSb 13.4 4.7 6.0 0.81 27 0.76 6.09
	Units		eV eV eV	Å

TABLE II .

 II Numerical values of S as a function of Lz. S 81.7 95.9 111.1 127.3 144.5 162.6 181.7 201.8

	Lz 2.0 2.2	2.4	2.6	2.8	3.0	3.2	3.4

TABLE III

 III

	. Effective mass values (×m0) extracted from the
	first valence subband.
		InAs	InSb GaSb
	< 110 > direction
	Unstrained	0.041 0.067 0.061
	Comp.	0.032 0.0469 0.044
	Tens.	0.0548 0.184 0.107
	< 100 > direction
	Unstrained	0.412 0.059 0.061
	Comp.	0.030 0.042 0.047
	Tens.	0.059 0.140 0.094

  FIG.6. IOFF vs ION characteristics for < 110 >-oriented, compressively strained, tensile strained and unstrained GaSb MOSFETs, compared with the corresponding < 100 >oriented, tensile strained Si device. LG=7 nm.

		ITRS IOFF requirement for LOP		10 -1
				L G = 7 nm
				GaSb <110> unstr. ball.
				GaSb <110> tens. ball.
				GaSb <110> comp. ball.
				Si <100> tens. ball.
				GaSb <110> unstr. scatt.
				GaSb <110> tens. scatt.
				GaSb <110> comp. scatt.
				Si <100> tens. scatt.
		ITRS IOFF requirement for LSTP
	0	10 3	2×10 3	3×10 3
		I ON (A/m)