Angelos Nersesian

Wouter Dobbels

Emmanuel Xilouris

Maarten Baes

Simone Bianchi

Viviana Casasola

Christopher J R Clark

Ilse De Looze

Frédéric Galliano

Suzanne C Madden

Manolis E Xilouris

Aleksandr V Mosenkov

Evangelos-D Paspaliaris

Ana Trčka

Probing the spectral shape of dust emission with the DustPedia galaxy sample

Keywords: infrared, galaxies -galaxies, ISM -galaxies, spiral -galaxies, elliptical and lenticular, cD -galaxies, irregulars

INTRODUCTION

Interstellar dust has been well established as one of the main regulators of several physical and chemical processes in the interstellar medium (ISM) in galaxies. Dust grains are mainly formed through the process of grain growth when a critical metallicity is reached in the ISM at the late stage of galaxy evolution (e.g. Hirashita 2012;Draine 2009;Asano et al. 2013;Zhukovska 2014;Galliano et al. 2018Galliano et al. , 2021;;Granato et al. 2021), except at low-metallicity environ-★ E-mail: angelos.nersesian@ugent.be ments (12+log(O/H) < 8) where the dust condensation in the ejecta either by the stellar winds of Asymptotic Giant Branch (AGB) stars or core-collapse SuperNovae becomes more important (e.g. Liang & Li 2009;Asano et al. 2013;Galliano et al. 2018). Cosmic dust makes its presence known through (𝑖) the reddening of the stellar spectrum at the shortest wavelengths (0.1-1 𝜇m); (𝑖𝑖) by emitting the absorbed starlight as thermal radiation at longer wavelengths (3-1000 𝜇m); and (𝑖𝑖𝑖) by depleting heavy elements from the ISM. Specifically, in a typical spiral galaxy dust can absorb, on average, up to 25% of the emitted stellar radiation (e.g. Viaene et al. 2017;Bianchi et al. 2018), with the absorption fraction rising up to even ∼ 87% in the special cases of (ultra) luminous infrared galaxies ((U)LIRGs; e.g. Paspaliaris et al. 2021).

Different processes excite dust grains emitting in the infrared (IR) regime. The mid-infrared (MIR) region of the spectrum is powered mostly by stochastic heating, when individual ultraviolet (UV) photons heat up small dust grains to temperatures above the local thermal equilibrium. On the other hand, dust grains that reached a thermal equilibrium with their local radiation field dominate the far-infrared (FIR) and sub-millimetre (submm) regimes. Since these dust grains are in a thermal equilibrium, their emission could be represented with a modified black-body function, which allows us to infer the dust properties of a given galaxy (i.e. the dust mass, and temperature and emissivity index 𝛽). Another way to probe those properties is by looking at the FIR colour indices (e.g. Boselli et al. 2010bBoselli et al. , 2012;;Cortese et al. 2014;Smith et al. 2019). The different FIR colours are useful indicators of the various temperature components in the FIR regime, but also of the emissivity index 𝛽 which holds key information about the chemical composition of dust grains in the diffuse ISM. Specifically, 𝛽 determines the opacity 𝜅 𝜆 of the dust, with a typical value found to be ∼ 2 for a carbonaceous and silicate grain composition (Draine & Lee 1984). However, the dust emission is subject to many degeneracies such as the one between the dust temperature and the dust emissivity index (Kelly et al. 2012).

Moreover, the spectral shape of the FIR emission varies significantly with the galactic environment and the different intrinsic stellar properties of galaxies, for example the star-formation rate (SFR) or the stellar mass. Pinpointing which physical properties drive the spectral shape of the dust emission is crucial, to interpret the large dispersion seen in many FIR colour-colour diagrams. Many studies have tried to find such correlations but usually end up with weak trends and/or large scatter between the stellar parameters inferred by spectral energy distribution (SED) fitting and the FIR colours (e.g. Boselli et al. 2012;Cortese et al. 2014;Smith et al. 2019). In some cases the large scatter can be attributed to the inconsistent photometry measurements from galaxy to galaxy and from waveband to waveband, incomplete samples etc. Using the peak and width directly from the FIR SED might be slightly easier to interpret their dependence with the physical properties. For example, Rémy-Ruyer et al. (2015) measured the peak and the width of the dust emission in order to derive the relationship of the SED shape with the sSFR and metallicity. However, the main problem with studying the FIR fluxes themselves is that all the bands are strongly correlated. An interesting technique which could be able to offer new clear trends among galaxy intrinsic properties and the shape of the FIR SEDs, and possibly introduce new variables that correlate better with dust properties, is principal component analysis (PCA). PCA is a statistical technique for data visualisation and dimensionality reduction. The principal components by construction they capture the most variance, and so they are more descriptive of the full FIR SED, allowing us to get a better insight regarding the source of dispersion in the FIR SEDs. This method of analysis has been already applied in many astronomical studies: the reduction of dimensionality for SED libraries (Han & Han 2014), the SED decomposition of the N11 massive star-forming complex in the Large Magellanic Cloud (LMC; Galametz et al. 2016), the investigation of the dependencies among the main physical quantities of the KING-FISH galaxy sample (Hunt et al. 2019), the spectral classification of local galaxies (Tous et al. 2020), and to explain the variations in the polycyclic aromatic hydrocarbon (PAH) emission in NGC 2023 (Sidhu et al. 2021), among many others.

A relevant study and motivator for our paper, is the one by Safarzadeh et al. (2016). The authors have used PCA to study how the dust emission of simulated isolated and merging galaxies, across different redshifts, is connected with various global properties. Identifying the best correlations between their PCA results and the physical properties of their galaxy sample, allowed them to predict the shape of the FIR SED. They have found that a double power law in terms of the IR luminosity and dust mass is sufficient to determine the shapes of the galaxy FIR SEDs. Furthermore, they have showed that galaxy sizes do not improve the predictive power of the FIR SEDs, suggesting that the physical size of a galaxy does not prompt the characteristic dust temperature. We closely follow the technique and notation for the PCA, as outlined by Safarzadeh et al. (2016), but instead of using the data of simulated galaxies, we apply their method on the DustPedia galaxy sample (Davies et al. 2017).

The DustPedia galaxies are a unique amalgamation of galaxies all observed by Herschel Space Observatory, and have been treated in a uniform and consistent way (Clark et al. 2018). Our goal is to explain the variance in the FIR SEDs and its link with the intrinsic stellar and dust properties. We should make a note here that it is not our intention to predict the FIR emission. We refer to Dobbels et al. (2020), who used a machine learning framework to predict the FIR emission and the dust properties, based on the available UV to MIR observations of the DustPedia and Herschel-ATLAS (H-ATLAS; Valiante et al. 2016) galaxy samples. This paper is structured as follows. In Sect. 2 we present the dataset used in this study, and in Sect. 3 we show several FIR/submm colour-colour relations. Section 4 describes the concept behind the use of PCA, while in Sect. 5 we discuss the discovered trends between the PCA results and their relation to other global physical properties such as SFR, dust mass, dust temperature, etc. Our main conclusions are summarised in Sect. 6.

THE DATASET

In this paper we delve into the unique and most complete dataset of FIR SEDs in the local Universe, the DustPedia project1 (Davies et al. 2017;Clark et al. 2018). The database includes imagery and photometric data for 875 local galaxies (distances of < 40 Mpc) all observed by Herschel Space Observatory (Pilbratt et al. 2010). Furthermore, infrared data from other missions are available such as Spitzer (Werner et al. 2004), Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010), InfraRed Astronomical Satellite (IRAS; Neugebauer et al. 1984), andPlanck (Planck Collaboration et al. 2011). As for the UV, optical, and near-infrared (NIR) regimes the available data were collected from GALaxy Evolution eXplorer (GALEX; Morrissey et al. 2007), Sloan Digital Sky Survey (SDSS; York et al. 2000;Eisenstein et al. 2011), and 2 Micron All-Sky Survey (2MASS; Skrutskie et al. 2006), respectively. The photometry has been performed for all galaxies by Clark et al. (2018) in a uniform and consistent way. For details about the photometry pipeline we refer the reader to Clark et al. (2018). Then, Nersesian et al. (2019) fitted the panchromatic SEDs (UV-submm) of 8142 DustPedia galaxies using CIGALE3 (Noll et al. 2009;Boquien et al. 2019), and delivered their global physical properties along with their modelled SEDs. The dust properties were estimated assuming The Heterogeneous Evolution Model for Interstellar Solids 4 (THEMIS; Jones et al. 2013Jones et al. , 2017;;Köhler et al. 2014) dust model. The THEMIS framework uses laboratory measurements to build the synthetic optical properties of amorphous hydrocarbon and silicate materials. So far, THEMIS was qualitatively consistent with many dust observables of the diffuse Galactic ISM, including the observed FUV-NIR extinction and the shape of the IR to millimetre dust thermal emission.

Another characteristic information about the DustPedia sample is that it contains galaxies of various morphological types, i.e. ellipticals, lenticulars, spirals and irregulars, as well as of different orientations, with inclinations ranging from edge-on to face-on. The morphological type is parameterised with the Hubble stage 𝑇 that ranges from -5 to 10, taken from HyperLEDA 5 (Makarov et al. 2014). We divide our sample into six main morphological classes, E [-5, -3.5); S0 [-3.5, 0.5); Sa-Sab [0.5, 2.5); Sb-Sc [2.5, 5.5); Scd-Sdm [5.5, 8.5); 10]. In some instances within text we refer to galaxies with 𝑇 < 0.5 as early-type galaxies (ETGs), while galaxies with 𝑇 ≥ 0.5 are defined as late-type galaxies (LTGs).

For the purposes of this study, we have created a sample of 791 galaxies by excluding 23 peculiar galaxies, to secure that our analysis will not be affected by those objects. Among the excluded galaxies there are 19 harbouring an active galactic nucleus (AGN) and 4 jet-dominated radio galaxies. We use 10 wavebands: MIPS 24 𝜇m, IRAS 60 𝜇m, PACS 70, 100, 160 𝜇m, SPIRE 250, 350, 500 𝜇m, and Planck 550, 850 𝜇m. In order for PCA to function properly, the complete FIR SED is required. However, it is quite difficult to obtain a complete set of FIR SEDs for all 791 galaxies. Not all galaxies have fluxes in the same set of filters due to heterogeneous data. Subsequently, we chose to use the Bayesian flux densities for all bands (not only for the missing values), from the SED models produced with CIGALE and which are available at the DustPedia archive. CIGALE estimates the flux posteriors in likelihood-weighted averages over all SED models. The advantage of utilising the Bayesian fluxes, instead of interpolating the observed data, is that they act as some kind of smoothing for bands with large uncertainties; since a realistic library of SEDs is used as a prior (e.g. Dobbels et al. 2020). Figure 1 shows the mean FIR SEDs of each morphological group. Out of those 10 wavebands, on average, 8 show deviations lower than or equal to 15% from their observed counterpart, while the IRAS 60 𝜇m and PACS 100 𝜇m show the largest deviations (29% and 17% respectively) 6 . In Appendix A, we present the Kernel Density Estimation (KDE) of the normalised residuals for each waveband to detect any systematic deviations in the various bands.

THE FIR COLOURS

In this section, we present the flux densities of the DustPedia galaxies in terms of FIR-submm colours. Several of the colour-colour diagrams and their relationships have already been explored for different galaxy samples (e.g. Boselli et al. 2010bBoselli et al. , 2012;;Bendo et al. 2012;Dale et al. 2012;Auld et al. 2013;Smith et al. 2019, and many others). A colour index is defined as the ratio of the flux densities measured in different FIR bands. Out of all the possible colour-colour combinations, in Fig. 2 we show those colour-colour 4 https://www.ias.u-psud.fr/themis/THEMIS_model.html 5 http://leda.univ-lyon1.fr/ 6 A detailed evaluation of how well the flux density in a particular waveband is retrieved by CIGALE is given in Nersesian et al. (2019). diagrams that are sensitive to the different dust emission components. For example, the 𝑓 60 / 𝑓 100 is more sensitive to the warm dust component mainly heated by young stars, and therefore can be used to trace the star-forming activity in galaxies. A colour index such as the 𝑓 60 / 𝑓 250 is indicative of the average dust temperature, since the involved wavelengths sample the peak of the dust emission. Finally, the colour indices 𝑓 100 / 𝑓 500 , 𝑓 160 / 𝑓 500 , 𝑓 160 / 𝑓 850 , 𝑓 250 / 𝑓 500 , and 𝑓 350 / 𝑓 850 sample the Rayleigh-Jeans tail of the dust emission, situating them as good tracers of the coldest dust component (< 15 K) which carries most of the dust mass in galaxies (Siebenmorgen et al. 1999).

A selection of several FIR colour indices and their in-between relationships are shown in Fig. 2. The different colour-coded markers represent galaxies that fall into six broad morphological groups. We rank the strength of the relationships among the colour indices with the Spearman's rank correlation coefficient (𝜌), given in the top-left corner of each panel. Another noticeable trend is an increased dispersion for the colour-colour relations which trace the warm and cold dust component. Yet the scatter for our sample is not as large as described in e.g. Boselli et al. (2012), and the reason is that we use the Bayesian and not the observed flux densities. The IRAS 60 𝜇m band ( 𝑓 60 ) is responsible for the large uncertainties in the first two columns of Fig. 2. The galaxies also appear to be clearly separated in two modes according to their morphological type, especially in the first three columns of Fig. 2. The LTGs fall in the "upper" mode suggesting a colder dust component, while the ETGs fall in the "lower" mode. Interestingly, many of the Sm-Irr galaxies fall in the "lower" mode as well, along with the ETGs. This is a clear indication that the spectral shape of the dust emission varies and, therefore, the dust properties significantly change with the evolution state of galaxies.

In the following sections we will try to investigate which physical parameters are exactly responsible for the variations in the shape of the FIR SED, by analysing the dust emission with PCA.

FITTING THE FIR SEDS WITH PCA

PCA is a widely used method to explore the variance of large datasets and for creating predictive models. A regular application of this method is to reduce the dimensionality of a large dataset and visualise the relation between different parameters. The principal components are defined to be the eigenvectors of the data's covariance matrix, and they are designed to capture as much variance as possible. The first component captures the most variance, followed by the second, and so forth, revealing the internal structure of the data in a way that best explains the variance in the data.

As already mentioned in Sect. 2, we make use of the full FIR SEDs for each galaxy as derived with CIGALE. First, we normalise the SEDs with the respective IR luminosity, defined as the integral of the SED over the wavelength range 8-1000 𝜇m (e.g. Kennicutt 1998). Then we apply the PCA using the python module sklearn, which finds the mean SED of our sample and consequently recovers the principal components that could describe the observed variance. The fraction of the variance in the data that can be explained by the different PCs, determines also the rank of each PC. Given a galaxy j in the sample, it becomes possible to recon- struct its observed FIR SED through the linear combination of the different PCs:

𝜆𝐹 norm 𝜆, 𝑗 = 𝜆𝐹 norm 𝜆 + 𝑁 ∑ 𝑖=1 𝐶 𝑖, 𝑗 × PC 𝑖 , (1)
where 𝜆𝐹 norm 𝜆 is the mean FIR SED, 𝑁 is the number of PCs which is the same as the number of FIR bands that were used, PC 𝑖 is the 𝑖th PC, and 𝐶 𝑖, 𝑗 is the coefficient of the 𝑖th PC for the 𝑗th galaxy. After the identification of the main PCs, we can examine how their coefficients are related to various global physical quantities retrieved from a panchromatic SED fitting.

Figure 3 shows the mean FIR SED of our galaxy sample and the main two principal components which can account for 95% of the variance. Specifically, the first principal component explains 71% of the variance while the second component explains 24%. The decomposition of the FIR SED into PCs is in excellent agreement with the results reported by Safarzadeh et al. (2016). In particular, Safarzadeh et al. (2016) have found that 97% of the variance in their SED sample can be captured by the first two PCs (PC 1 , 75%; and PC 2 , 22%). The general approach to reconstruct the FIR SED of a given galaxy is to multiply each PC with a coefficient unique for each galaxy and then add it to the mean SED of the sample (see Eq. 1). The coefficients of each PC can take either positive or negative values and can determine how the FIR SED varies from the mean SED. We refer to the coefficients of PC 1 as 𝐶 1 and of PC 2 as 𝐶 2 .

For the interpretation of the two PCs we look at Fig. 4, where four SED examples are shown. The top-row shows the effect of PC 1 and the bottom-row of PC 2 . If PC 1 was added with a positive coefficient to the mean FIR SED then the peak of the dust emission shifts towards shorter wavelengths (i.e. warmer dust component; top-left panel), whilst if PC 1 was added with a negative coefficient the peak shifts towards longer wavelengths (i.e. colder dust component; top-right panel). Similarly, if PC 2 was added with a positive coefficient to the mean FIR SED then the peak of the emission be- comes broader (bottom-left panel), while if PC 2 was added with a negative coefficient the FIR SED becomes narrower (bottom-right panel). Therefore, we can infer that PC 1 affects the wavelength at which the dust emission has its peak, whereas PC 2 controls the width of the FIR SED. Another interesting result is presented in Fig. 5, which depicts the relationship between 𝐶 1 and 𝐶 2 or in other words the rela- tionship between the temperature proxy of the SED and its broadness, respectively. Each point represents a DustPedia galaxy and it is colour-coded according to each galaxy's morphological type. The Spearman's rank correlation coefficient (𝜌) is given as well.

We estimated the uncertainties of the coefficients with a bootstrap technique, by re-sampling the SEDs according to their uncertainty distribution. At first glance, the two quantities seem uncorrelated (𝜌 = -0.04 if we consider the full sample). However, if we include the information of the morphology for each galaxy we draw another picture. A mild anti-correlation exists between 𝐶 1 and 𝐶 2 , with 𝜌 = -0.36 if we consider only the LTGs. In particular, the anti-correlation is more tight for galaxies of type Sa-Sab, Sb-Sc, and Scd-Sdm (𝜌 = -0.47). Evidently, these galaxies have in general colder and narrower FIR SEDs. In contrast, the E, S0, and many Sm-Irr galaxies, although they showcase a higher degree of dispersion, they mostly occupy the parameter space which indicates broader FIR SEDs and a warmer dust component. These results could explain the split of the galaxy populations in two modes, visible in the first three columns of Fig. 2. Finally, most of the ETGs reside above the parameter space occupied by LTGs. Thus, a potential application of Fig. 5 is to probe the morphological characteristics of unresolved galaxies in large surveys, like e.g. GAMA (Driver et al. 2009) or COSMOS (Scoville et al. 2007).

DISCUSSION

In this section, we focus on the correlations between the coefficients of PC 1 and PC 2 , which control the peak wavelength and width of the FIR SED for each galaxy, and the various global physical parameters that were retrieved from a panchromatic SED fitting (Nersesian et al. 2019) with CIGALE (see Figs. 6 and7). We specify these parameters as: the dust luminosity (𝐿 dust ), defined as the integral of the total SED over the wavelength range 8-1000 𝜇m; the stellar mass (𝑀 star ); the SFR; the specific SFR (sSFR = SFR/𝑀 star ); the dust mass (𝑀 dust ); the dust-to-stellar mass ratio (𝑀 dust /𝑀 star ); the dust temperature (𝑇 dust ); the bolometric luminosity (𝐿 bolo ); and the fraction of absorbed stellar luminosity by dust ( 𝑓 abs = 𝐿 dust /𝐿 bolo ).

An interesting result that is quite evident from both figures is the clear separation of the two main galaxy populations, i.e. LTGs (blue and green points) and ETGs (red and orange points). Therefore, it is more constructive to discuss our results by distinguishing the galaxies in our sample as ETGs and LTGs.

Starting with Fig. 6 and the top-row, we notice a very weak dependence of 𝐿 dust with 𝐶 1 for the whole sample (𝜌 = -0.11). The strength of the relation does not change if we separate the LTGs from the ETGs. We find a weak positive trend between 𝐿 dust and 𝐶 1 for the LTGs (𝜌 = 0.12); and a negative trend for the ETGs (𝜌 = -0.12). A stronger trend can be achieved for the LTGs (𝜌 = 0.30), by excluding the Sm-Irr types which have a large dispersion. The strongest dependence of 𝐶 1 is with 𝑇 dust (full sample 𝜌 = 0.60). A clear separation exists for LTGs and ETGs, with a tight increasing trend for both galaxy populations. ETGs have, in general, higher dust temperatures than the LTGs as already found by Nersesian et al. (2019) for the DustPedia galaxy sample, and also shown for the KINGFISH (Skibba et al. 2011), HRS (Smith et al. 2012), and HeViCS (Auld et al. 2013) projects. Many of the Sm-Irr galaxies occupy the same parameter space with the ETGs, but the mechanisms that drive the dust temperature are different. In the case of ETGs, both the hardness of the radiation field of an older stellar population, as well as the X-ray emitting hot gas are responsible for the warmer dust (e.g. Natale et al. 2010), whereas in the case of Sm-Irr galaxies the high sSFR (high SFR and low 𝑀 star ) is the main driver of the dust temperature (e.g. Rémy-Ruyer et al. 2015). Finally, we notice a negative relation between 𝐶 1 and 𝑀 dust (𝜌 = -0.31), with ETGs having a moderate negative trend (𝜌 = -0.39), whereas the correlation for the LTGs is negligible (𝜌 = -0.08).

The middle-row of Fig. 6 depicts the relation of 𝐶 1 with the intrinsic stellar properties of our sample. Looking at the individual middle panels, we notice that 𝐶 1 does not exhibit any strong trends with the stellar properties. The 𝜌 values somewhat improve for the LTGs if we exclude the Sm-Irr types (𝐿 bolo , 𝜌 = 0.25; SFR, 𝜌 = 0.27; 𝑀 star , 𝜌 = 0.19), but still the trends remain weak. Although the positive trend of 𝐶 1 with 𝑀 star is pretty faint, it might still indicate the impact of the stellar mass on the peak wavelength of the FIR SED. For example, in ETGs the stellar mass concentration and the X-ray luminosity from hot halo gas increases with increasing stellar mass, resulting in the warmer SEDs. Finally, in the bottom-row of Fig. 6, we show the dependence of 𝐶 1 with three different ratios. Immediately, we notice that all three ratios follow a similar trend with 𝐶 1 (and 𝐶2; see bottom-row of Fig. 7). This is an expected behaviour, since the three ratios correlate with each other. Bianchi et al. (2018) and Trčka et al. (2020) showed a strong correlation of sSFR with 𝑓 abs and the dust-to-stellar mass ratio, respectively. A correlation between 𝑓 abs and the dust-to-stellar mass ratio is also expected, since the dust mass and stellar mass are calculated from their respective luminosities.

Furthermore, a clear separation between the two galaxy populations is visible. Out of the three quantities, the dust-to-stellar mass ratio has the stronger trend with 𝐶 1 (full sample 𝜌 = -0.46). The trend becomes even stronger for the ETGs (𝜌 = -0.49). This anticorrelation suggests that galaxies with a lower dust-to-stellar mass ratio tend to have warmer dust SEDs. In the bottom-left panel, we see that a faint decreasing trend exists between 𝐶 1 and 𝑓 abs (𝜌 = 0.33).

The 𝐶 1 coefficients of the LTGs exhibit a weak correlation with 𝑓 abs (𝜌 = 0.18 or 𝜌 = 0.33 if we exclude the Sm-Irr types), oppositely to the ETGs which show a mild anti-correlation (𝜌 = -0.28) yet with a larger dispersion. Similarly, and despite the distinct clustering of the two galaxy populations in the bottom-middle panel, the relation between the sSFR and 𝐶 1 is quite weak (𝜌 = -0.15). Even by looking at the two galaxy populations separately, we get a faint correlation with sSFR (𝜌 = 0.14) for the LTGs, and a weak anti-correlation (𝜌 = -0.14) for the ETGs. From the bottom-row of Fig. 6, there is evidence that the peak wavelength of the dust emission is affected, to a various degree, by the stellar mass concentration in the case of ETGs (low dust-to-stellar ratio, 𝑓 abs , and sSFR), and by the process of star-formation in the case of LTGs (high dust-to-stellar ratio, 𝑓 abs , and sSFR).

By shifting the attention to Fig. 7 and if we consider the full sample, 𝐶 2 follows an anti-correlation trend with most of the depicted quantities. Among the tightest relations are the one with the SFR (𝜌 = -0.49), and the dust luminosity (𝜌 = -0.51). Yet the strongest anti-correlation that is visible, is with 𝑓 abs (𝜌 = -0.57). Highly star-forming galaxies contain more dust, and therefore higher fraction of the diffuse stellar radiation field is absorbed. Moreover, 𝐶 2 has a moderate negative trend with the dust-to-stellar mass ratio (𝜌 = -0.37), the sSFR (𝜌 = -0.40), and with 𝑀 dust (𝜌 = -0.42). Actually, the relation between the sSFR and the width of the dust SED is in agreement with the results of Rémy-Ruyer et al. (2015), who showed that the sSFR plays a significant role shaping the dust SED. A mild negative trend exists also with 𝐿 bolo and 𝑀 star but only in the case of LTGs. Finally, we note that for ETGs a weak correlation exists with 𝑇 dust (𝜌 = 0.22), and a weak anti-correlation for the LTGs (𝜌 = -0.10 or 𝜌 = -0.20 if we exclude the Sm-Irr types). On the one hand, active spiral galaxies possess more diffuse dust, but they also have dust in the star-forming regions where starlight will be efficiently absorbed, resulting in a wider range of temperatures and subsequently to a broader SED. As the effective dust temperature of LTGs increases the SED becomes narrower. This is purely driven by black body behaviour, since warmer black bodies have a more peaked spectrum. On the other hand, the dust in ETGs is concentrated mostly in their central regions, exposed to the extremely dense radiation field of old stars resulting in a FIR SED with a flatter peak. From these results we infer that galaxies tend to have narrower FIR SEDs with increased SFR, dust luminosity, and dust mass.

In addition, we have information about the global metallicities for a fraction of DustPedia galaxies. These metallicities were derived by De Vis et al. (2019) using six different calibrations. In particular, we use the S calibration of Pilyugin & Grebel (2016) as it is the most reliable at low-metallicity (De Vis et al. 2017). Out of the 791 galaxies in our sample, only 355 galaxies have a metallicity measurement. Figure 8 depicts the coefficients of the first two PCs as a function of metallicity. For 𝐶 1 we find a weak negative trend (𝜌 = -0.15), and a mild anti-correlation for 𝐶 2 (𝜌 = -0.37). We should warn the reader that the correlations presented here may suffer from a bias effect, since we miss the metallicity information for more than half of the galaxies in our sample, and especially for the majority of the ETGs.

Focusing on the top panel of Fig. 8, we notice that there is a weak tendency for low-metallicity galaxies to have higher 𝐶 1 values (i.e. warmer effective dust temperature). Most of the low-metallicity galaxies are Sm-Irr types with positive 𝐶 1 values, indicating that the peak of their dust emission is shifted towards shorter wavelengths. These results are in good agreement with previous studies, which have demonstrated that low-metallicity galaxies have, in general, warmer dust (e.g. Helou et al. 1985;Melisse & Israel 1994;Rémy-Ruyer et al. 2014, 2015). Sm-Irr galaxies are characterised by their clumpy ISM, low stellar mass, and high star formation activity (i.e. high sSFR, see also Fig. 6), which is responsible for the warmer dust temperatures. As galaxies evolve and increase in metallicity, their FIR SEDs become colder and narrower as it is shown in the bottom panel of Fig. 8. Yet the scatter in both panels is large, with several high-metallicity galaxies residing in the parameter space indicative of a hot and broad FIR SED, and vice versa. Our results are consistent with those of Rémy-Ruyer et al. (2015), who showed that the dust emission for most of the metal-rich KINGFISH galaxies is characterised by a cold and narrow SED, whereas low-metallicity dwarf galaxies have warm and broad SEDs.

CONCLUSIONS

We performed PCA on a sample of FIR SEDs of 791 DustPedia galaxies. Our main goal was to identify the driving source of variation in the FIR SEDs of nearby galaxies, as well as to determine which global physical parameters affect the SED shape the most. Our main conclusions are the following.

(i) Through the PCA we find that the first two principal components are sufficient to explain 95% of the variance in the FIR SEDs of our galaxy sample. The first component is associated with the wavelength of the peak of the SED, while the second controls how narrow or broad is the peak. This result is in excellent agreement with the results of Safarzadeh et al. (2016) for a sample of simulated galaxies.

(ii) A strong anti-correlation exists between the coefficients of the first two PCs 𝐶 1 and 𝐶 2 , if we consider galaxies of type Sa-Sab, Sb-Sc, and Scd-Sdm. The parameter space occupied by these galaxies indicates that they have, in general, colder and narrower FIR SEDs. In contrast, the E, S0, and Sm-Irr galaxies, although they showcase a higher degree of dispersion, mostly occupy the parameter space indicative of broader and warmer FIR SEDs.

(iii) 𝐶 1 (i.e. the position of the peak of the dust emission) correlates strongly with 𝑇 dust as expected, since the peak of the dust emission is controlled by the effective dust temperature. 𝐶 1 also showcases a moderate anti-correlation with the dust-to-stellar mass ratio. If we only consider the LTGs, 𝐶 1 has a weak positive trend with 𝐿 dust , 𝐿 bolo , SFR, sSFR, and 𝑓 abs , while there is no trend with 𝑀 star nor with 𝑀 dust . On the other hand, if we only consider the ETGs, we find a mild anti-correlation of 𝐶 1 with 𝑀 dust , and 𝑓 abs , while 𝐶 1 is weakly related to 𝐿 dust , 𝐿 bolo , 𝑀 star , SFR, and sSFR. We conclude that the peak wavelength of the dust emission is affected primarily by the effective dust temperature, and to some extent, by the stellar mass concentration in the case of ETGs (low dust-tostellar ratio, 𝑓 abs , and sSFR), and by the process of star-formation in the case of LTGs (high dust-to-stellar ratio, 𝑓 abs , and sSFR).

(iv) 𝐶 2 (i.e. the width of the dust emission) shows mostly an anti-correlation in relation to the various physical quantities. The stronger anti-correlations are with 𝑓 abs , 𝐿 dust , SFR, 𝑀 dust , and the sSFR. From these results we infer that galaxies tend to have narrower FIR SEDs with increased SFR, dust luminosity, and dust mass.

(v) We find a weak tendency for low-metallicity galaxies (primarily Sm-Irr galaxies) to have higher 𝐶 1 values, i.e. warmer effective dust temperature. As galaxies evolve and increase in metallicity, their FIR SEDs become colder and narrower. These results are consistent with those of Rémy-Ruyer et al. (2015), who showed that the dust emission for most of the metal-rich KINGFISH galaxies is characterised by a cold and narrow SED, whereas low-metallicity dwarf galaxies have warm and broad SEDs.

Figure 1 .

 1 Figure 1. The mean, Bayesian FIR SEDs of each morphological group. The FIR SEDs are scaled for visualisation purposes. The y-axis is in arbitrary units.

Figure 2 .

 2 Figure2. FIR colour-colour diagrams. A FIR colour index is defined as the ratio of the flux densities in two different wavebands. Galaxies are colour-coded according to their morphological type, and divided into six broad morphological groups as indicated in the legend of the figure. The Spearman's rank correlation coefficient (𝜌) is given in the top-left corner of each panel.

Figure 3 .

 3 Figure3. The mean FIR SED of our sample and its first two principal components. The solid black line corresponds to the mean FIR SED, normalised with the IR luminosity. The dashed blue line corresponds to the first PC and the dashed red line to the second PC. The fraction of the variance in the data that is explained by each PC is indicated in the legend. Together, the first two PCs account for 95% of the total variance of the data in our sample.

Figure 4 .Figure 5 .

 45 Figure 4. Examples of reconstructed FIR SEDs with the first two principal components. The solid black line corresponds to the mean FIR SED, normalised with the IR luminosity. The dashed blue line corresponds to the first PC and the dashed red line to the second PC. The solid purple line is the observed SED using the Bayesian fluxes from CIGALE, while the solid green line shows the SED, reconstructed out of the first two PCs (see Eq. 1).At the top-right corner of each panel we show the unique pair of coefficients (𝐶 1 , 𝐶 2 ), for the corresponding galaxy. Any differences between the reconstructed and the observed SEDs are due to the omission of the remaining PCs.

Figure 6 .

 6 Figure6. Dependence of the coefficients of PC 1 with nine global physical parameters of the galaxies in our sample. Top-row: The dust luminosity, the dust temperature, and the dust mass. Middle-row: The bolometric luminosity, the star-formation rate, and the stellar mass. Bottom-row: The fraction of bolometric luminosity absorbed by dust, the specific star-formation rate, and the dust-to-stellar mass ratio. Galaxies of different morphological groups are indicated with different colours and markers as shown in the legend of the figure. The Spearman's rank correlation coefficient (𝜌) is given in the bottom-left corner for the full sample (black font), only the ETGs (red font), and only the LTGs (blue font).

Figure 7 .

 7 Figure 7. Same as Fig. 6 but for the coefficients of PC 2 .

Figure 8 .

 8 Figure 8. Dependence of the coefficients of PC 1 (top panel) and PC 2 (bottom panel) with metallicity. Galaxies of different morphological groups are indicated with different colours and markers as shown in the legend of the figure. The Spearman's rank correlation coefficient (𝜌) is given in the bottom-left corner.

  Figure A1. KDE plots of the ratio between the observed fluxes and the best model fluxes, for our galaxy sample. The KDE plot of each waveband is given in each panel. The vertical line declares the zero value.

  Looking at the 𝜌 values it is clear that all colour indices are closely related. The FIR colours which cover the closest spectral bands are also those with the tightest relations, for example 𝑓 160 / 𝑓 850 vs 𝑓 160 / 𝑓 500 and 𝑓 250 / 𝑓 500 vs 𝑓 160 / 𝑓 850 . We find a weakened relation between the colour indices sensitive to the warm dust component (i.e. 𝑓 60 / 𝑓 100 ) and the colour indices tracing the emission of the cold dust component ( 𝑓 160 / 𝑓 500 , 𝑓 160 / 𝑓 850 , 𝑓 250 / 𝑓 500 and 𝑓 350 / 𝑓 850 ). Similar trends have been found for the Herschel Reference Survey (HRS Boselli et al. 2010a) 7 by Boselli et al. (2012) and Cortese et al. (2014), as well as for the JINGLE survey in Smith et al. (2019).

http://dustpedia.astro.noa.gr

Out of the 875 galaxies, 61 were excluded according to their flagging codes given byClark et al. (2018).

https://cigale.lam.fr/ MNRAS 000, 1-10 (2020)

Most of the HRS galaxies are included in the DustPedia sample.MNRAS 000, 1-10 (2020)

MNRAS 000, 1-10 (2020)

http://www.astropy.org MNRAS 000, 1-10 (2020)

This paper has been typeset from a T E X/L A T E X file prepared by the author.MNRAS 000, 1-10 (2020)

ACKNOWLEDGEMENTS

We would like to thank the anonymous referee for the helpful comments and suggestions. AN, WD, gratefully acknowledge the support of the Research Foundation -Flanders (FWO Vlaanderen). EMX, gratefully acknowledges the support by Greece and the European Union (European Social Fund-ESF) through the Operational Programme "Human Resources Development, Education and Lifelong Learning 2014-2020" in the context of the project "Anatomy of galaxies: their stellar and dust content through cosmic time" (MIS 5052455). DustPedia is a collaborative focused research project supported by the European Union under the Seventh Framework Programme (2007Programme ( -2013) call (proposal no. 606847)) call (proposal no. 606847). The participating institutions are: Cardiff University, UK; National Observatory of Athens, Greece; Ghent University, Belgium; Université Paris Sud, France; National Institute for Astrophysics, Italy and CEA, France. This research made use of Astropy, 8 a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2013, 2018).

DATA AVAILABILITY

All the photometric data and the physical properties that were used in the analysis of this work, are publicly available at the DustPedia archive http://dustpedia.astro.noa.gr.

APPENDIX A: DEVIATION BETWEEN THE OBSERVED AND THE FITTED CIGALE FLUXES

In order to validate our choice of using the fitted CIGALE flux densities instead of the observed fluxes in our analysis, we present the Kernel Density Estimation (KDE) of the normalised residuals for each waveband. A systematic effect in the KDE of a given band could help reveal potential weaknesses in our analysis. Figure A1 reveals that CIGALE is able to reproduce quite accurately the observed fluxes of our galaxy sample. For the majority of the wavebands (7 out of 10), have their KDE of the residuals peaking near ±0.01 dex and with deviations within ±0.35 dex. The bands that show a significant deviation are the IRAS 60𝜇m (±0.38 dex), PACS 70𝜇m (±0.44 dex), and PACS 100𝜇m (±0.4 dex). A shift in the peak of the KDE distribution by 0.09 dex, 0.05 dex, and 0.05 dex is also noted for IRAS 60𝜇m, PACS 70𝜇m, and PACS 160𝜇m, respectively. Nevertheless, these deviations are not severe and would not introduce any systematic effects in our current analysis.


Masiar Sistani

Raphael Böckle

David Falkensteiner

Minh Anh Luong

Martien I Den Hertog

Alois Lugstein

Walter M Weber
email: walter.weber@tuwien.ac.at

Nanometer-Scale Ge Based Adaptable Transistors Providing Programmable Negative Differential Resistance Enabling Multi Valued Logic

The functional diversification and adaptability of the elementary switching units of computational circuits are disruptive approaches for advancing electronics beyond the static capabilities of conventional complementary metal-oxide-semiconductor based architectures. Thereto, in this work the one-dimensional nature of monocrystalline and monolithic Al-Ge based nanowire heterostructures is exploited to deliver charge carrier polarity control and furthermore to enable distinct programmable negative differential resistance at runtime. The fusion of electron-and hole conduction together with negative differential resistance expressivity in a universal

adaptive transistor may enable energy efficient reconfigurable circuits with multi-valued operability that are inherent components of emerging artificial intelligence electronics. KEYWORDS germanium, nanowires, heterostructures, adaptive electronics, reconfigurable transistor, negative differential resistance Alongside with information and communication technologies becoming ubiquitous in everyday life and thus ever more stringent constraints on computing hardware performance, complementary metal-oxide-semiconductor (CMOS) scaling is being pushed towards its physical limits. [START_REF] Tan | Future Internet: The Internet of Things[END_REF] In this context, the functional diversification of the elementary computing units, the transistors, is an alternative approach to advance computational complexity, enabling a hardware platform that inherently encompasses adaptability in emerging self-learning electronic paradigms. [START_REF] Weber | Reconfigurable Nanowire Electronics -A Review[END_REF][START_REF] Radamson | State of the Art and Future Perspectives in Advanced CMOS Technology[END_REF] To leverage circuit performance beyond the limits encountered in miniaturization, the research field of reconfigurable electronics has explored computational enhancement by extending the functionality of transistors. [START_REF] Heinzig | Reconfigurable Silicon Nanowire Transistors[END_REF][START_REF] Glassner | Multimode Silicon Nanowire Transistors[END_REF] This concept is based on the reconfigurable field-effect transistor (RFET) that fuses both basic switching functions necessary for realizing an energy efficient complementary circuit design by dynamically toggling the device either to p-type (hole conduction) or n-type (electron conduction) operation through the implementation of a dedicated program gate electrode. [START_REF] Yu | One-Dimensional Germanium Nanowires for Future Electronics[END_REF] Customized logic and physical synthesis routes have been developed for the design of combinational circuits.

Benchmarks have shown that the use of runtime reconfiguration vs. conventional CMOS can deliver compact circuit topologies e.g. in multi-bit adders and arithmetic logic units having a substantial reduction in transistor count, a smaller area footprint and reduced latency length of critical paths. [START_REF] Rai | A Physical Synthesis Flow for Early Technology Evaluation of Silicon Nanowire Based Reconfigurable FETs[END_REF][START_REF] Havemann | High-Performance Interconnects: An Integration Overview[END_REF] Moreover, it has been shown that exclusive OR (XOR) based gates can be efficiently applied with RFETs 9 allowing for majority based arithmetic operations [START_REF] Amarú | Biconditional BDD: A Novel Canonical BDD for Logic Synthesis Targeting XOR-Rich Circuits[END_REF] that have been hindered so far in CMOS. [START_REF] Gaillardon | Towards Structured ASICs Using Polarity-Tunable Si Nanowire Transistors[END_REF][START_REF] Raitza | Exploiting Transistor-Level Reconfiguration to Optimize Combinational Circuits[END_REF] Another concept that has been broadly discussed to overcome the power density limits and increasing signal propagation delays of modern ultra-scaled chip architectures is multi-valued logic (MVL), replacing conventional binary systems by operation schemes with radices or bases higher than two. [START_REF] Smith | A Multiple Valued Logic: A Tutorial and Appreciation[END_REF] Hence, implementing an operation scheme with higher performance employing fewer devices and interconnects compared to standard CMOS circuits, owing to higher functionality of MVL circuits can be envisaged. [START_REF] Berezowski | Multiple-Valued Logic Circuits Design Using Negative Differential Resistance Devices[END_REF] Recently, a MVL concept based on exploiting the monostable-to-multistable 15 nature of serially connected negative differential resistance (NDR) devices was demonstrated, creating a staircase of holding states. [START_REF] Andreev | Negative Differential Transconductance Device with a Stepped Gate Dielectric for Multi-Valued Logic Circuits[END_REF] An example for a simple yet innovative logic element taking advantage of the NDR characteristic is the monostable-bistable transition logic element (MOBILE), employing two NDR devices connected in series capable to perform both NAND and NOR operations. [START_REF] Berger | Negative Differential Resistance Devices and Circuits[END_REF][START_REF] Liang | Design of AND and NAND Logic Gate Using NDR-Based Circuit Suitable for CMOS Process[END_REF] Furthermore, stacking more than two NDR devices, an efficient, and compact signed-digit NDR based MVL adder combining a more than five-fold improvement in circuit propagation delay and a 15 times smaller area compared to common CMOS based circuits has been proposed. [START_REF] Gonzalez | Standard CMOS Implementation of a Multiple-Valued Logic Signed-Digit Adder Based on Negative Differential-Resistance Devices[END_REF] Distinct physical methods are currently employed to provide NDR, including resonant tunneling diodes that demand complex semiconductor epitaxial structures, which are difficult to integrate with CMOS. [START_REF] Park; Anisha | Si/SiGe Resonant Interband Tunneling Diodes Incorporating Delta-Doping Layers Grown by Chemical Vapor Deposition[END_REF] Further, CMOS backend compatible threshold switching in metal/Nb-oxide insulator/metal structures were shown, but exhibit an inherent device-to-device variability in the applied threshold current to inhibit current controlled NDR. [START_REF] Herzig | Multiple Slopes in the Negative Differential Resistance Region of NbOx-Based Threshold Switches[END_REF] As a third concept, the Gunneffect discovered by J.B. Gunn [START_REF] Gunn | Microwave Oscillations of Current in III-V Semiconductors[END_REF] and C. Hilsum, [START_REF] Hilsum | Transferred Electron Amplifiers and Oscillators[END_REF] which is also known as the transferred electron effect in specific semiconductor systems with a multi-valley band-structure delivers voltage controlled NDR. Besides highly strained Si NWs, [START_REF] Shiri | Hilsum Effect in Mechanically Strained Silicon Nanowires: Tunable Negative Differential Resistance[END_REF] n-type InSb devices [START_REF] Smith | Effect in n-Type InSb[END_REF] and diamond structures, [START_REF] Suntornwipat | Observation of Transferred-Electron Oscillations in Diamond[END_REF] low-dimensional Ge is one of the most promising prospective materials both for the performance enhancement of RFETs and realization of NDR-based MVL through the transferred electron effect, which is enabled by the acceleration of electrons by sufficiently high electric fields. Thereby electrons are scattered from the energetically favorable low effective mass conduction band valley to an energetically close heavy effective mass valley. [START_REF] Butcher | The Gunn Effect[END_REF] Accessible at room-temperature, [START_REF] Böckle | Gate-Tunable Negative Differential Resistance in Next-Generation Ge Nanodevices and Their Performance Metrics[END_REF] the NDR characteristic bears enormous potential for the design of MVL regarding performance metrics such as area, speed, or power. [START_REF] Berger | Negative Differential Resistance Devices and Circuits[END_REF]

RESULTS/DISCUSSION

Here, we present the NDR-mode RFET concept capable of exhibiting both p-and n-type FET reconfigurable operability as well as gate-tunable NDR functionality as deterministically programmed by the applied terminal voltages. Thereby, the functionality of the elementary switching units is strongly enhanced, disruptive computational and analog oscillatory circuits.

The key enabler for this concept is the electronic structure of the quasi-1D monolithic Al-Ge-Al nanowire (NW) heterostructure, enabling low operation voltages, reconfigurable p-and ntype polarity and furthermore accessibility to the transferred electron effect at room temperature as required for providing NDR features. Across the discussion of results, the material choice and suitability of our particular heterostructure will become evident. For device fabrication and as a simple integration vehicle, vapor-liquid solid [START_REF] Wagner | Vapor-Liquid-Solid Mechanism of Single Crystal Growth[END_REF] (VLS) grown, single-crystal, nominally intrinsic Ge NWs with <111> longitudinal axis orientation with diameters of approximately 30 nm are employed. The NWs are enwrapped in a 20 nm thick Al2O3 high-k insulator shell and are further integrated into a field effect transistor (FET) architecture. Although we chose to use a bottom-up approach, our device concept is also applicable to top-down processed structures based on Ge on insulator (GeOI) substrates. [START_REF] Wind | Monolithic Metal-Semiconductor-Metal Heterostructures Enabling Next-Generation Germanium Nanodevices[END_REF] For contact formation, a thermally induced exchange reaction between the Ge NWs and Al contact pads was used to embed Ge channels contacted by quasi-1D monocrystalline Al NW leads [START_REF] Brunbauer | Electrical Transport Properties of Single-Crystal Al Nanowires[END_REF] with abrupt Al-Ge heterojunctions. [START_REF] Kral | Abrupt Schottky Junctions in Al/Ge Nanowire Heterostructures[END_REF][START_REF] El Hajraoui | In Situ Transmission Electron Microscopy Analysis of Aluminum-Germanium Nanowire Solid-State Reaction[END_REF] In contrast to common metal-germanide formation, [START_REF] Marshall | Metal-Germanium Contacts and Germanide Formation[END_REF] the Al-Ge exchange results in a single-elemental metallic segment with a low resistivity of ρ = (131 ± 27) × 10 -9 Ωm and an abrupt metal-semiconductor heterostructure without any measureable or electrically active Al impurities within the Ge segment. [START_REF] El Hajraoui | In Situ Transmission Electron Microscopy Analysis of Aluminum-Germanium Nanowire Solid-State Reaction[END_REF] Further, the Al-Ge contact formation overcomes the difficulty in reproducibility and deterministically defining the phase of the intruded metallic segments as the phase stability of alternative contact materials to nanometer scale Ge, such as NixGe1-x and CoxGe1-x are known to exhibit strong variability and yield issues. [START_REF] Yaish | Kinetics of Nickel Silicide Growth in Silicon Nanowires: From Linear to Square Root Growth[END_REF][START_REF] Tang | Formation and Device Application of Ge Nanowire Heterostructures via Rapid Thermal Annealing[END_REF][START_REF] Breil | Challenges of Nickel Silicidation in CMOS Technologies[END_REF][START_REF] Grzela | Growth and Evolution of Nickel Germanide Nanostructures on Ge(001)[END_REF] Utilizing this device architecture, we demonstrate that Al-Ge-Al NW heterostructures are capable of merging the electrical properties of unipolar p-and n-type FETs into a single type of device with identical technology, geometry and composition. [START_REF] Trommer | Enabling Energy Efficiency and Polarity Control in Germanium Nanowire Transistors by Individually Gated Nanojunctions[END_REF] Consequently, reducing the technological complexity, they enable a dynamic and runtime programming of circuits at the device level. [START_REF] Weber | Reconfigurable Nanowire Electronics -A Review[END_REF][START_REF] Weber | Reconfigurable Silicon Nanowire Devices and Circuits: Opportunities and Challenges[END_REF] In this configuration, the gates atop the abrupt Al-Ge junctions serve as polarity-gates (PGs) setting the charge carrier type in the channel to switch between p-and ntype operation, while the control-gate (CG) can be used to thermionically modulate the draincurrent through the transistor allowing to turn the device on or off.

As seen in Figure 2, setting VPG = -5 V, p-type operation is programmed. Transfer measurements (ID vs. VCG) were performed by sweeping VCG between -5 V to 5 V for fixed bias voltages between VD = -100 mV and 1 V (see respective schematic band diagram in Figure 2). Unipolar p-type operation is set by an upward band bending below the PGs enabling holeinjection into the Ge channel. Regulating the thermionic emission of the injected charge carriers though the Ge channel via VCG, the transistor can be turned on (negative VCG) and off (positive VCG). Switching to n-type operation, all applied voltages are simply reversed, i.e. VPG = 5 V, setting the junction transparent for electrons and effectively blocking the injection of holes into the Ge channel. In analogy to the previous experiments unipolar n-type FET operation is obtained. The on-current of the n-type program shows a high variation over different applied drain voltages VD. This is perfectly correlating with the asymmetric activation energies for pand n-type operation in Al-Ge-Al NW heterostructures, which were obtained from the slope of the activation energy plot of ln(J/T 2 ) vs. 1000/T (Arrhenius plot). [START_REF] Beister | Temperature Dependent Switching Behaviour of Nickel Silicided Undoped Silicon Nanowire Devices[END_REF] A detailed description of the extraction of the activation energies is supplied in the supporting information. Whereas in the p-type operation no evident barrier is visible in the measurement range, a relatively high barrier was extracted for n-type operation (see Figure S1). [START_REF] Böckle | Gate-Tunable Negative Differential Resistance in Next-Generation Ge Nanodevices and Their Performance Metrics[END_REF] Importantly, compared to a single top-gate FET (see Figure S2), our multiple gate FET concept enables a suppression of source to drain leakage for both p-and n-type operation by electrostatically tuning the electrostatic shape along the Ge channel length. Through the blocking barrier formed at the drain junction, static power consumption is strongly suppressed despite the low band-gap material being employed. For comparison, we also investigated single top-gated Schottky FETs, which show a weakly ambipolar p-type characteristic (see supporting information). Regarding the metrics of our three-gate Ge transistor, setting |VPG| = 5 V, a total ION/IOFF ratio of approximately 3 × 10 5 and 4 × 10 2 is achieved for p-and n-type respectively. The transconductance gm = dID/dVCG is obtained, with 90 µS/µm (p-type operation) and 2.1 µS/µm (n-type operation) (see Figure S3). Further, threshold voltages (Vth) of -1.4 V (p-type operation) and 1 V (n-type operation)

were extracted (see supporting information). Applying VD = 1 V, it was possible to achieve Jh = 9.2 × 10 5 A/cm² in p-type operation and Je = 3.3 × 10 3 A/cm² in n-type operation. Compared to the single top-gate device, JOFF could be significantly decreased to Jh OFF = 25 A/cm² and Je OFF = 16 A/cm², about a factor of 10 3 smaller compared to the single top-gate device. Importantly, the on-currents do not show any noticeable change, as the geometry of the Al-Ge-Al heterostructure certainly remained the same. In agreement with device TCAD predictions, [START_REF] Baldauf | Scaling Aspects of Nanowire Schottky Junction Based Reconfigurable Field Effect Transistors[END_REF] we want to note that an enhancement of the key parameters of the proposed Ge based reconfigurable transistor are expected for thinner high-k dielectrics, enhanced interfacial layers and shorter Ge channel lengths. To confirm unipolar device operation of the proposed threegate Ge NW transistor, Figure S4 shows the temperature dependency of p-and n-type programs.

In agreement with thermionic emission theory, [START_REF] Sze | Physics of Semiconductor Devices[END_REF] the transfer curves in the subthreshold region are flattening with increased temperature for both operation types and an increase of the off-current due to a higher amount of thermally generated carriers surpassing the blocking barrier is observed with increasing temperature according to the underlying transport mechanisms in the subthreshold, thermionic emission as well as thermal field emission (see equation S1).

Nevertheless, an ambipolar operation is still well suppressed for both operation modes even up to T = 380 K. Further, the output characteristics for the p-and n-program are shown in Figure S5. Comparing the operation modes reveals a supra-linear output characteristic for low bias voltages of the n-program, with the on-current showing a high variation over different applied drain voltages VD, which is typical for Schottky FETs. [START_REF] Prakash | Understanding Contact Gating in Schottky Barrier Transistors from 2D Channels[END_REF] In contrast, the p-mode characteristics exhibit a linear increase for small biases as in a conventional metal-oxide-semiconductor FET (MOSFET). These results underline the barrier extraction results in Figure S1. The demonstration of a device concept capable of resembling both p-or a n-type FET operation principally enables any logic function in a complementary design. [START_REF] Weber | Reconfigurable Silicon Nanowire Devices and Circuits: Opportunities and Challenges[END_REF] Moreover, the bidirectional nature of each transistor within a circuit enables a runtime reconfiguration of logic functions and is a clear advantage for flexible and adaptive circuit design. 4

Figure 2. Subthreshold transfer characteristic for different bias voltages between VD = 100 mV and 1 V showing unipolar p-and n-type operation programmed by VPG. The schematic band diagrams of p-and n-type device operation are inserted.

Next, we investigate the electron-dominated transport regime of our three-gate Ge transistor. This is motivated by the previous observation of a gate-tunable NDR, where we have investigated the performance metrics and scaling capabilities of back-gated and single-topgated Al-Ge-Al NW FETs. [START_REF] Böckle | Gate-Tunable Negative Differential Resistance in Next-Generation Ge Nanodevices and Their Performance Metrics[END_REF] By applying VPG = 5 V, setting the junction with a higher transparency for electrons, the downward band bending results in an enhancement of electron injection due to the barrier becoming thinner. Nevertheless, distinct to the injection of holes in the p-mode, only the electrons that have higher thermal energies that surpass the relatively high energy barrier are injected by thermionic emission into the semiconductor region. The high Schottky barrier for electrons is the result of our Al-Ge heterostructure, which is not available in other typical metal contacts to Ge, as with conventional metal germanide contacts. Based on the Gunn-effect originally observed by J.B. Gunn [START_REF] Gunn | Microwave Oscillations of Current in III-V Semiconductors[END_REF] and interpreted with the electron transferred effect by H.

Kroemer [START_REF] Kroemer | Theory of the Gunn Effect[END_REF] with the help of the theoretical work of C. Hilsum, 22 and B.K. Ridley -T.B.

Watkins 45 a voltage controlled NDR region is observed in the output characteristics of our devices" Thereto, the injected electrons, that in our case already have a high thermal energy, surpassing the barrier gain even more energy across the channel when accelerated with sufficiently large biases to be scattered from the energetically favorable conduction band valley, characterized by a low effective mass to a heavy effective mass valley nearby. [START_REF] Butcher | The Gunn Effect[END_REF] Although the L-and Γ-minima are energetically close in energy (0.66 eV vs. 0.8 eV), based on the calculation of the coupling constants between the conduction band valleys in bulk Ge (see Table S1), it was shown that the transferred electron effect is more likely to apply involving the Land X-minima rather than the L-and Γ-minima, with respective effective masses of m ! * = 0.082 m0 and m # * = 0.288 m0. [START_REF] Jacoboni | Electron Drift Velocity and Diffusivity in Germanium[END_REF] A schematic illustration of the electron transfer from the L-and X-minima is shown in the lower inset of Figure 3. A more detailed energy band-structure of Ge with all energy band-gaps is shown in Figure S6. Consequently, linearly increasing VD while monitoring ID for different VCG between 0 V and 5 V, Figure 3 reveals voltage controlled NDR features progressive enhancement for higher VCG. To characterize this effect, the most important parameter is the peak-to-valley ratio (PVR), which is defined by I(VPeak) / I(VValley). 17

A large and stable PVR is highly anticipated for logic applications. [START_REF] Berger | Negative Differential Resistance Devices and Circuits[END_REF][START_REF] Liang | Design of AND and NAND Logic Gate Using NDR-Based Circuit Suitable for CMOS Process[END_REF] As the electron concentration in the Ge channel can be modulated by varying VCG, a tunable PVR is observed.

Consequently, starting with VCG = 3 V the PVR increases from 1.2 to 7.7 at VCG = 5 V. In comparison with Ge quantum dot tunneling diodes, [START_REF] Oehme | Ge Quantum Dot Tunneling Diode with Room Temperature Negative Differential Resistance[END_REF] Si and Si/SiGe resonant interband tunneling diodes, [START_REF] Park; Anisha | Si/SiGe Resonant Interband Tunneling Diodes Incorporating Delta-Doping Layers Grown by Chemical Vapor Deposition[END_REF][START_REF] Schmid | Silicon Nanowire Esaki Diodes[END_REF] the room-temperature PVR of our best performing RFET devices is approximately a factor 10 larger. In total the characteristic of more than 10 similar devices has been measured and analyzed. All devices showed a stable and reproducible RFET and NDR characteristic.

Previously, we have also investigated the scalability of NDR in back-and single top-gated Ge NWs. [START_REF] Böckle | Gate-Tunable Negative Differential Resistance in Next-Generation Ge Nanodevices and Their Performance Metrics[END_REF] We found that shrinking the Ge channel length to LGe = 150 nm, a room-temperature PVR of 4.5 can be maintained and VPeak can be reduced down to 0.9 V. Whereas, for this work, due to the three top-gate architecture, we have used longer Ge channels, we forecast that the current levels could be improved using a dense parallel arrangement of NWs, as carried out in Pregl et al. [START_REF] Pregl | Parallel Arrays of Schottky Barrier Nanowire Field Effect Transistors: Nanoscopic Effects for Macroscopic Current Output[END_REF] by contact printing or Wind et al. [START_REF] Wind | Monolithic Metal-Semiconductor-Metal Heterostructures Enabling Next-Generation Germanium Nanodevices[END_REF] by top-down fabrication.

Moreover, as the purpose of the PGs is to lower the injection barrier for electrons (in the case of the NDR-mode), they can be used to tune the NDR. However, this was found to be limited to the peak-current, not VPeak. As we were aiming for maximizing the peak current, we focused on the maximum positive voltage on the PGs of 5 V. We believe that scaling down the gateoxide thickness, the top-gate voltages could be significantly reduced and an even better gating of the barriers at the Al-Ge junctions could be achieved.

Most importantly, as shown in the upper inset of Figure 3, our RFET device is capable of modulating VPeak via VCG between VD = 4.1 V and 2.7 V. Hence, a cascode of devices, each biased with different VCG, would result in an I/V characteristic comprising several overlapping NDR regions that could be used as MOBILE devices, enabling both NAND and NOR operations. [START_REF] Liang | Design of AND and NAND Logic Gate Using NDR-Based Circuit Suitable for CMOS Process[END_REF][START_REF] Chen | Monostable-Bistable Transition Logic Elements (MOBILEs) Based on Monolithic Integration of Resonant Tunneling Diodes and FETs[END_REF] Moreover, this highly desired device behavior could also be effectively used for small footprint and energy efficient computational MVL, where the radix is set by the number of devices connected in series. Notably, as shown by recent circuit simulations, exploiting such characteristics, complex logic functions such as multi-value adders revealing a significantly reduced circuit propagation delay vs. conventional CMOS could be realized. Further, Figure 4 shows the typical temperature dependence of the NDR mode, recorded for applying VCG = 5 V. Due to increased thermal excitation and thus a higher electron concentration in the L-valley, the current peak before the NDR region is shifted to lower electric fields for higher temperatures. Further, associated with a decrease of transfer efficiency due to a smaller scattering mean free path related to thermal excitation of charge carriers, the PVR decreases for increasing temperatures. [START_REF] Berger | Negative Differential Resistance Devices and Circuits[END_REF] Moreover, we note that not only the electron concentration in the L-valley increases at elevated temperatures, but both the high and low effective mass sub bands i.e. the Γ1-and X-valley are also being thermally populated. Thus, the transfer of electrons from the L-to the Γ1-and X-valley does not make any noticeable decrease of current expected from the effective mass difference. Consequently, a superposition of the thermally populated charge carriers and the electrons transferred by the Gunn-effect is evident.

This can be illustrated by plotting the ratio of populations (n) at EΓ1 and EX with respect to EL as a function of temperature (see Figure S7). [START_REF] Conwell | High Field Transport in Semiconductors[END_REF][START_REF]Germanium Band Structure and Carrier Concentration[END_REF] As the ratio nX/nL shows a steep increase vs.

temperature compared to i.e. nΓ1/nL, the Gunn-effect appears weaker at elevated temperatures.

Additionally, the efficiency of the scattering of electrons from the L-to the X-valley degrades.

Despite pioneering work on the temperature dependence of Gunn-effect based on phonon scattering in highly strained Si NWs [START_REF] Shiri | Hilsum Effect in Mechanically Strained Silicon Nanowires: Tunable Negative Differential Resistance[END_REF][START_REF] Shiri | Reversible Modulation of Spontaneous Emission by Strain in Silicon Nanowires[END_REF] and n-type InSb devices, 24 theoretical and experimental investigations of the thermal stability of NDR in Ge is still elusive.

The inset of Figure 4 shows the evaluation of the average PVR for temperatures between T = 295 K and 380 K of five similar devices. Linearly fitting the data, the average PVR decreases from 10.9 at T = 295 K to 1.2 at T = 360 K with a rate of approximately 0.1/K. Further, an average decrease of the current of about 50 % for increasing the temperature from T = 295 K to 360 K. NDR could be observed even for temperatures up to T = 360 K, which is an important prerequisite for realistic logic applications and proves the capabilities of the proposed device concept.

CONCLUSIONS

In conclusion, we have embedded a quasi-1D monocrystalline Al-Ge-Al NW heterostructure into a three-gate transistor architecture capable of delivering charge carrier polarity control and furthermore to enable distinct programmable NDR at runtime. NDR could be observed even for temperatures up to T = 360 K, which is an important prerequisite for realistic logic applications and proves the capabilities of the proposed device concept. Most importantly, our device concept enables a modulation of both the PVR and the position of the NDR peak. Hence, a cascode of devices, each biased with a different voltage on the CG, would result in an I/V characteristic comprising several overlapping NDR regions that could be used as MOBILE devices, enabling both NAND and NOR operations. Consequently, the presented NDR-mode RFET concept offers a platform enabling applications such as static memory cells, fast switching logic circuits or small footprint and energy efficient computational MVL. In this respect, our investigations might provide a significant step towards a beyond CMOS approach enabling functional diversification and alternative computing.

METHODS/EXPERIMENTAL

Device fabrication:

The starting materials were Ge NWs with diameters of approximately 30 nm grown on a Si (111) substrate using the VLS process with germane (GeH4, 2% diluted in He) as precursor and a 2 nm thick sputtered Au layer as growth promoting catalyst. The growth was performed in a low-pressure hot wall CVD chamber. Subsequent to the growth, the Ge NWs were coated with 20 nm high-k Al2O3 using atomic layer deposition. The passivated Ge NWs were drop-casted onto a 100 nm thick thermally grown SiO2 layer atop of a 500 µm thick highly p-doped Si substrate. Al contacts to the Ge NWs were fabricated by a combination of electron beam lithography, 22 s of BHF (7:1) etching and 5 s HI (30%) etching to remove the Al2O3-shell as well as the native Ge oxide shell at the contact area, 100 nm Al sputter deposition and lift-off techniques. A successive thermally induced exchange reaction by rapid thermal annealing at a temperature of T = 624 K in forming-gas atmosphere initiates the substitution of Ge by Al. [START_REF] Kral | Abrupt Schottky Junctions in Al/Ge Nanowire Heterostructures[END_REF][START_REF] El Hajraoui | In Situ Transmission Electron Microscopy Analysis of Aluminum-Germanium Nanowire Solid-State Reaction[END_REF][START_REF] Sistani | Room-Temperature Quantum Ballistic Transport in Monolithic Ultrascaled Al-Ge-Al Nanowire Heterostructures[END_REF] Facilitating this heterostructure formation scheme allows the integration of single-crystalline monolithic Al-Ge-Al NW heterostructures with tunable channel lengths. Further, omegashaped Ti/Au top-gates were fabricated using a combination of electron beam lithography, Ti/Au evaporation (8 nm Ti, 125 nm Au) and lift-off techniques.

High-resolution HAADF STEM:

HAADF STEM was performed on Al-Ge-Al NW heterostructures fabricated on 40 nm thick Si3N4 membranes using a probe-corrected FEI Titan Themis, working at 200 kV. The Al-Ge interface in the shown images is viewed along the [112] direction of observation of the Ge crystal.

Electrical characterization:

The electrical measurements were carried out at room temperature and ambient conditions using a combination of a semiconductor analyzer (HP 4156B) and a probe station. Source-measure units were used for all device terminals. Gate currents were found to be below 1 pA. To

  Figure 1a shows a high angle annular dark field (HAADF) scanning transmission electron microscopy (STEM) image of the Al-Ge-Al NW heterostructure with a Ge channel length of LGe = 1.7 µm. High-resolution (HR) TEM images of the Al-Ge junction of the NW heterostructure enwrapped in the 20 nm thick Al2O3-shell and a zoomed-in image of the abrupt Al-Ge interface, are shown in Figure 1b. To enable p-and n-type FET reconfigurable operability as well as tunable NDR functionality, we add three individual top-gates to our Al-Ge-Al NW heterostructures, two aligning atop of the abrupt Al-Ge heterojunctions and one in the middle of the Ge channel. A colored SEM image of an Al-Ge-Al NW heterostructure embedded in the three-gate FET architecture is shown in Figure 1c.

Figure 1 .

 1 Figure 1. Structure of the adaptable Ge nanowire transistor. (a) HAADF STEM image of an

Figure 3 .

 3 Figure 3. I/V characteristic of a RFET device with LGe = 1180 nm for VPG = 5 V and VCG

Figure 4 .

 4 Figure 4. I/V characteristic recorded for VCG = VPG = 5 V of a RFET device, recorded for

  minimize the influence of ambient light as well as electromagnetic fields, the probe station was placed in a dark box. ASSOCIATED CONTENT Supporting Information Available: Estimated effective energy barrier of an Al-Ge-Al NW heterostructure, analysis of the electrical transport of a top-gated Al-Ge-Al NW heterostructure, evaluation of the transconductance and extraction of the threshold voltage of the three-gate Ge RFET, temperature dependent p-and ntype subthreshold transfer characteristics of the three-gate Ge RFET, more detailed bandstructure of Ge including all energy band-gaps, calculated coupling coefficients between the conduction band valleys in Ge, simplified estimation of the charge carrier population in the Γ 1and X-valley in dependence of the temperature regime of 295 -380 K, examples for NDRmode RFET circuits. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION

ACKNOWLEDGEMENT

The authors further thank the Center for Micro-and Nanostructures for providing the cleanroom facilities.

Corresponding Author

* E-mail: walter.weber@tuwien.ac.at

Author

Notes

The authors declare no competing financial interest.