Valentina Martelli

Julio Larrea Jiménez

Mucio Continentino

Elisa Baggio-Saitovitch

Kamran Behnia

Thermal transport and phonon hydrodynamics in strontium titanate

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

Thermal transport and phonon hydrodynamics in strontium titanate

Valentina Martelli 1 , Julio Larrea Jiménez 2 , Mucio Continentino 1 , Elisa Baggio-Saitovitch 1 and Kamran Behnia 3,4 (1)Centro Brasileiro de Pesquisas Físicas, 22290-180, Rio de Janeiro, RJ, Brazil We present a study of thermal conductivity, κ, in undoped and doped strontium titanate in a wide temperature range (2-400 K) and detecting different regimes of heat flow. In undoped SrTiO3, κ evolves faster than cubic with temperature below its peak and in a narrow temperature window. Such a behavior, previously observed in a handful of solids, has been attributed to a Poiseuille flow of phonons, expected to arise when momentum-conserving scattering events outweigh momentumdegrading ones. The effect disappears in presence of dopants. In SrTi1-xNbxO3, a significant reduction in lattice thermal conductivity starts below the temperature at which the average interdopant distance and the thermal wavelength of acoustic phonons become comparable. In the high-temperature regime, thermal diffusivity becomes proportional to the inverse of temperature, with a prefactor set by sound velocity and Planckian time (τp = k B T ).

Heat travels in insulators thanks to phonons. This has been described by the Peierls-Boltzmann equation, which quantifies the spatial variation in phonon population caused by the temperature gradient. In recent years, thanks to improved computing performance and new theoretical techniques, a quantitative account of intrinsic thermal conductivity of semiconductors is accessible to first-principle theory [1]. When most scattering events conserve momentum and do not decay heat flux, collective phonon excitations, dubbed relaxons, become fundamental heat carriers [2]. This hydrodynamic regime of phonon flow, identified decades ago [START_REF] Sussmann | Proc. Phys. Soc[END_REF][START_REF] Guyer | [END_REF][5][6], is gaining renewed attention in the context of graphene-like twodimensional systems [11,12].

The perovskyte SrTiO 3 is a quantum paraelectric [13], which owes its very existence to zero-point quantum fluctuations. First-principle calculations find imaginary phonon modes [14], which hinder a quantitative understanding of the lattice thermal transport [15]. This insulator turns to a metal upon the introduction of a tiny concentration of dopants. The metal has a dilute superconducting ground state [16] and an intriguing roomtemperature charge transport [17]. Its thermal conductivity has remained largely unexplored, in contrast to electric [18] and thermoelectric [19] transport.

In this Letter, we present an extensive study of thermal conductivity, κ, of undoped and doped SrTiO 3 crystals and report on three new findings. First of all, in a narrow temperature range, thermal conductivity evolves faster than cubic. This behavior had only been reported in a handful of solids [6] and attributed to a Poiseuille flow of phonons. We argue that the emergence of phonon hydrodynamics results from the multiplication of momentumconserving scattering events due to the presence of a ferroelectric soft mode, as suggested decades ago [7]. It lends support to previous reports on the observation of the second sound in this system [8,9], which has been controversial [10]. Second, our study finds that a ran-dom distribution of dopants drastically reduces thermal conductivity below a temperature which tunes the heatcarrying phonon wavelength to the average interdopant distance. Finally, we put under scrutiny the thermal diffusivity of the system near room temperature and link its magnitude and temperature dependence to the so-called Planckian scattering time [20], in the context of the ongoing debate on a possible boundary to diffusivity [21,22].

The cubic elementary cell of strontium titanate encloses a TiO 6 octahedra and has strontium atoms at its vertices (Fig. 1a). Neutron and Raman scattering studies have identified two distinct soft modes. The first is associated with the antiferrodistortive (AFD) transition, which leads to the loss of cubic symmetry at 105 K [23] by tilting two adjacent TiO 6 octahedra in opposite orientations. It is centered at the R-point of the Brillouin zone (Fig. 1b). The second soft mode [24], located at the zone center, is associated with the aborted ferroelectricity. Fig. 1c presents the temperature dependence of the two modes established by converging spectroscopic tools [23,25,26]. In common solids, only acoustic branches can host thermally-excited phonons at low temperatures. Here, phonons associated with these soft modes remain relevant down to fairly low temperatures.

We used a standard one-heater-two-thermometers technique to measure the thermal conductivity of commercial single crystal of Sr 1-x Nb x TiO 3 [27]. The results, presented in Fig. 1d, reveal different regimes of heat transport classified by previous authors [START_REF] Guyer | [END_REF]6,12]. Simply put, thermal conductivity is the product of specific heat, mean-free-path, and velocity [START_REF] Berman | Thermal conduction in solids[END_REF]. At one extreme, i.e. at low temperature, the phonon mean-free-path saturates, the system enters the ballistic regime and κ becomes cubic in temperature. In the other extreme, at high temperature, the specific heat saturates and thermal conductivity, reflecting the temperature dependence of the mean-free-path, follows T -1 . In this kinetic regime, the wave-vector of thermally-excited phonons is large enough arXiv:1802.05868v3 [cond-mat.mtrl-sci] 27 Mar 2018 The cubic Brillouin zone and its high-symmetry points. c) The temperature dependence of the two soft modes according to the neutron scattering studies [23], hyper-Raman [25] and Brillouin scattering spectroscopy [26]. d) Thermal conductivity of a SrTiO3 crystal(closed red squares) in a log-log plot (For a linear plot see Fig. 3a). Different regimes of thermal transport are identified. Solid lines represent the expected behaviors in these regimes. An additional window due to enhanced Umklapp scattering opens up in the vicinity of the Antiferrodistortive (AFD) transition.

to allow Umklapp scattering events. Well below the Debye temperature, such events become rare and κ increases exponentially. This is this Ziman regime.

The AFD transition has visible consequences for heat transport. First of all, it attenuates κ near T AF D , impeding a smooth evolution between T -1 and exponential regimes. The R-point soft mode associated with the AFD transition provides additional Umklapp scattering at low energy cost. Interestingly, fitting κ ∝ exp( E D T ) in the Ziman regime, one finds E D 20 K, an energy scale comparable to the AFD soft mode. The second consequence of the AFD transition is to generate multiple tetragonal domains in an unstrained crystal [START_REF] Tao | [END_REF]. Given that the typical size of tetragonal domains is a few microns [30], the upper boundary to the ballistic mean-free-path of phonons can be much lower than the sample dimensions.

We found a κ varying faster than T 3 in a narrow (6 K < T <13 K) temperature window just below the peak. Usually, the ballistic regime ends with a downward deviation of κ from its cubic temperature dependence. This

T(K) STO g) h)
FIG. 2: Thermal conductivity, κ, as a function of T 3 in silicon (a) (after ref. [31]) and in KTaO3(b). In both, κ deviates downward from the T 3 line. (c) In bismuth (after ref. [33]) it deviates upward. In three different crystals of SrTiO3 (d,e,f) the deviation is upward. g) Thermal conductivity and specific heat of SrTiO3 evolve faster than cubic in this temperature range. But in a narrow window, thermal conductivity increases more rapidly. h) The apparent mean-free-path in both Bi and SrTiO3 present a local peak, the hallmark of Poiseuille flow.

happens in silicon [31] (Fig. 2a) or in KTaO 3 (Fig. 2b). This is not the case of bismuth where it shows an upward deviation between the ballistic regime and the peak (Fig. 2c). This has been identified as a signature of Poiseuille flow of phonons [33].

The Poiseuille regime emerges when energy exchange between phonons is frequent enough to keep the local temperature well-defined and Umklapp collisions are so rare that the flow is mainly impeded by boundary scattering. Without viscosity, no external temperature gradient would be then required to sustain the phonon drift [5]. This picture, developed decades ago [START_REF] Sussmann | Proc. Phys. Soc[END_REF][START_REF] Guyer | [END_REF][5], requires a hierarchy of time scales. The time separating two normal scattering events, τ N , should become much shorter than the time between boundary scattering events, τ B , and the latter much shorter than the time between resistive scattering events, τ R , which are due to either Umklapp or impurity scattering. The same hierarchy (τ N τ B τ R ) is required for second sound, a wave-like propagation of temperature and entropy, which has been observed in bismuth as well as in other solids displaying Poiseuille flow [6].

We confirmed a faster than cubic κ in three different SrTiO 3 crystals (Fig. 2 d-f). Here, the identification of this behavior with Poiseuille flow is less straightforward since the specific heat of SrTiO 3 also evolves faster than cubic between 4K and 20K [32]. This is because the Debye approximation is inadequate in the presence of soft modes and one needs to consider Einstein terms of the soft optical modes. In order to address this concern, we measured the specific heat of our cleanest crystal and found that the thermal conductivity increases faster than specific heat (Fig. 2g). The effective mean-free-path, P h = 3κCp vs , extracted from the specific heat, C p , and the sound velocity, v s , was found to show a peak comparable to what was found in bismuth [33] (Fig. 2h). In both cases, P h presents a local maximum 1.3 times the Knudsen minimum. The magnitude of the latter is slighly smaller than the crystal dimensions in bismuth, and to the typical size of tetragonal domains in strontium titanate, which have been found to be of the order of a micrometer [30]. As far as we know, the only available explanation for a local peak in P h is Poiseuille flow.

Neither in bismuth nor in strontium titanate, the chemical purity is exceptionally high. The same is true of black phosphorus, where a faster-than-cubic κ was recently observed [34]. Therefore, in these cases, in contrast to He crystals, the Poiseuille flow is presumably caused by a large three-phonon phase space [36] for momentumconserving (compared to momentum-degrading) scattering events. We note that the low-temperature validity of the τ N τ R inequality in strontium titanate was previously confirmed by low-frequency light-scattering experiments [9]. Anomalies detected by Brillouin scattering experiments [26] are believed to be caused by strong anharmonic coupling between acoustic and optical modes at low temperatures. A strong hybridization between acoustic and transverse optical phonons was theoretically confirmed [35] and is expected to flatten the phonon dispersion. This would pave the way for frequent normal momentum exchange. It would also pull down the phonon velocity, providing an alternative explanation for an unusually short apparent mean-free-path.

Let us turn our attention to the effect of atomic substitution. Fig. 3a shows thermal conductivity of SrTi 1-x Nb x O 3 . The magnitude of κ smoothly decreases with increasing dopant concentration. Only at lower temperatures, additional contribution by electrons outweighs the reduction in lattice thermal conductivity. In this range, we resolve a finite T -linear component in thermal conductivity of metallic samples due to the electronic component of thermal conductivity, κ e . This is in agreement with a previous study focused on temperatures below 0.5 K [37], which verified the validity of the Wiedemann-Franz(WF) law in the zero-temperature limit, namely: κ e ρ/T = L 0 , where ρ is the electric resistivity and L 0 = 2.45 × 10 -8 V 2 /K 2 is the Lorenz number. Assuming the validity of the WF law at finite tem- peratures, one can separate the electronic, κ e , and the phononic, κ ph , components of the total thermal conductivity. At finite temperature, because of inelastic scattering, one expects κ e ρ/T L 0 ≤ 1 and electric resistivity provides only a rough measure of κ e , which, as seen in Fig. 3b, becomes rapidly much smaller than κ ph with rising temperature.

The first consequence of the disorder, introduced by this tiny substitution for κ ph , is the loss of the faster than cubic regime associated with Poiseuille flow. As seen in Fig. 3b, reminiscent of what was observed in doped silicon and germanium [START_REF] Carruthers | Proc. R. Soc. Lond. A[END_REF], doping drastically damps κ ph at low temperature. The temperature-dependence of attenuation of phonon thermal conductivity caused by substitution: ∆κ ph = 1 -κ ph (x = 0)/κ ph (x = 0), presented in Fig. 3c displays a regular pattern. For small substitution (x= 0.0004), the lattice thermal conductivity is reduced by 8 percent at room temperature, by as much as 70 percent at 20 K and by 20 percent at 3 K. In other words, the maximum attenuation occurs in an intermediate tem-perature window. With increasing Nb concentration, the pattern is similar, but it shifts to higher temperatures. As seen in the lower panel of Fig. 3c, our measurements on an oxygen-reduced and a calcium-substituted sample produce similar patterns. Since Ca substitution [START_REF] Rischau | [END_REF] keeps the system an insulator, one can conclude that the drastic reduction in lattice conductivity is mainly due to the random distribution of substituting atoms and not to the scattering by mobile electrons.

A rigorous account of the temperature dependence of ∆κ ph is missing. We note, however, that ∆κ ph drastically enhances at a temperature, which shifts upward as the the concentration increases (see upward arrows in Fig. 3c). Consider that with decreasing temperature, the typical wave-vector of thermally-excited phonons shrinks, following: q ph = k B T vs . Therefore, at high-temperature, the phonon wave-length is shorter the average distance between dopants and the effect of disorder is limited. The random distribution of dopants begins to matter when the phonon wavelength becomes comparable to the average interdopant distance. In contrast to electrons, Anderson localization of phonons [40] is not expected to impede diffusive transport [41]. Theoretically, tiny level of disorder is sufficient to transform some phonon modes from propagating waves (propagons) to diffusons, which travel diffusively, or to fully localized locons [42]. One expects phonons with a wavelength much shorter or much longer than randomness length to be less affected. As a consequence, attenuation is to be more pronounced in the temperature window where the most-concerned phonons happen to be dominant thermally-excited carriers of heat. For each concentration, n, a temperature, T qn = hv s / dd k B , can be defined, which corresponds to equality between the typical acoustic phonon wavelength, λ ph = 2π/q ph and interdopant distance, dd = n -1/3 . As one can see in Fig. 3c, T qn is close to where ∆κ ph becomes large. Such a crude picture based on the Debye approximation, should not be taken too literally in presence of soft modes.

In principle, Ab Initio calculations [1] can give an account of heat transport near room temperature. Recently, two groups [43,44] succeeded in determining the phonon spectrum of strontium titanate free of the commonly-found imaginary frequencies [14] and computing the intrinsic lattice conductivity of the cubic phase. Fig. 4a compares our high-temperature data with these calculations [43,44] as well as previous experimental reports [15,45,46]. As one can see in the figure, there is a broad agreement between experimental results. Theoretical calculation using the Generalized Gradient Approximation (GGA) [43] are very close to the experimental data above 250 K. On the other hand, the experimental slope matches more the theory based on microscopic anharmonic force constants [44].

Let us conclude by a short discussion of thermal diffusivity, D = κ Cp in this regime. We can extract D by combining our thermal conductivity data and the specific heat. Fig. 4b presents the temperature de- pendence of thermal diffusivity. One can see that, at room temperature and above, thermal diffusivity tends to be proportional to T -1 . Our data is in good agreement with reported values of thermal diffusivity at high temperature [47]. In the vicinity of room temperature and above, thermal diffusivity becomes proportional to the inverse of temperature. The thermal diffusivity of a good conductor of heat, silicon and a very bad one, PbTe, are also shown. Remarkably, in the two bad conductors, the magnitude and the temperature dependence of D in the high-temperature regime can be expressed in a very simple way:

D = sv 2 s τ p (1)
Here, τ p = k B T is the Planckian scattering time [20], and s a dimensionless parameter (See table I). In PbTe and SrTiO 3 , s is close to unity and the temperature dependence is set by τ p . It emerges as a useful parameter for comparing the thermal conductivity of different cubic insulators. In many perovskytes, recently studied by Hofmeister [47], D has a comparable magnitude and temperature dependence. On the other hand, in a highly conducting cubic insulator such as silicon, D is much larger and drops faster, presumably because the phase space for three-phonon scattering Umklapp events [36] is smaller.

Eq. 1 is strikingly similar to the suggested universal boundary on diffusivity suggested by Hartnoll [21], with sound velocity replacing the Fermi velocity. The experimental motivation for Hartnoll's proposal [21] was the fact that τ p is the average scattering rate of electrons in numerous metals with linear resistivity [20]. Is there a boundary to thermal transport by phonons in insulators? In other, words, is there a fundamental reason for s to remain larger than unity? These are the questions raised by our observation.

JLJ acknowledges the Science Without Borders program of CNPq/MCTI-Brazil and VM and KB acknowledge FAPERJ fellowships (Nota 10 and Visitante). KB is also supported by Fonds ESPCI and by a Quan-tEmX grant (GBMF5305) from ICAM and the Gordon and Betty Moore Foundation. We thank B. Fauqué, Y. Fuseya, S. A. Hartnoll system Vm(cm 3 /mol v sl (100) (km/s) vst(100) (km/s) κ300K (W/m.K) C300K (J/cm ) and transverse sound velocity (vst) in three cubic solids together with their room-temperature thermal conductivity (κ300K ), specific heat (c300K ) and thermal diffusivity (D300K ). FIG. 9: Specific heat and thermal conductivity for strontium titanate, silicon and lead telluride. For silicon, the specific heat was reported in Ref. S5 and thermal conductivity in Ref. S6. For PbTe, the specific heat was reported in S7 and S8 and thermal conductivity in S9.

( 2 )

 2 Institute of Physics, University of São Paulo, CEP 05508-090, São Paulo, SP, Brazil (3) Laboratoire Physique et Etude de Matériaux (CNRS-UPMC), ESPCI Paris, PSL Research University, 75005 Paris, France (4)II. Physikalisches Institut, Universität zu Köln, 50937 Köln, Germany (Dated: January 26, 2018)

FIG. 1 :

 1 FIG.1: a) Crystal structure of strontium titanate; b) The cubic Brillouin zone and its high-symmetry points. c) The temperature dependence of the two soft modes according to the neutron scattering studies[23], hyper-Raman[25] and Brillouin scattering spectroscopy[26]. d) Thermal conductivity of a SrTiO3 crystal(closed red squares) in a log-log plot (For a linear plot see Fig.3a). Different regimes of thermal transport are identified. Solid lines represent the expected behaviors in these regimes. An additional window due to enhanced Umklapp scattering opens up in the vicinity of the Antiferrodistortive (AFD) transition.

FIG. 3 :

 3 FIG.3: a) κ as a function of temperature in SrTi1-xNbxO3. b) Electronic, κe and phononic, κ ph , components of the thermal conductivity in three doped samples compared to undoped strontium titanate. Note the persistence of a T 3 behavior over a wide temperature window with a drastically reduced magnitude. c) Relative attenuation in phonon thermal conductivity, ∆κ ph = 1 -κ ph (x =0) κ ph (x=0) in SrTi1-xNbxO3 (top)and in Sr1-xCax TiO3 (x=0.0045) and in SrTiO 3-δ (n=7× 10 17 cm -3 ). Small arrows represent Tqn (See text).

FIG. 4 :

 4 FIG.4: a) Thermal conductivity at high-temperature compared to previous experimental reports[15,45,46] and theoretical calculations[43,44]; b) Thermal diffusivity, D, extracted from thermal conductivity and specific heat data as a function of temperature in SrTiO3 (solid blue circles) together with data from ref.[47] (open circles), compared to silicon and PbTe. Solid lines represent D = sv 2 s τP (See text).

3 FIG. 5 :FIG. 6 :FIG. 7 :

 3567 FIG. 5: Thickness dependence thermal conductivity of SrTiO3: sample thickness t =0.5mm (full black circles) and t =0.15mm (open red circles). The two curves differ only in the Poiseuille regime.

FIG. 8 :

 8 FIG.8: Panel(a): Electrical resistivity between 1.7 and 300K as a function of temperature for four different samples. Panel (b): low temperature electrical resistivity as a function of T 2 , the solid line is a guide for the eye representing the T 2 behaviour at low temperature. Above a finite temperature, resistivity deviates upward from this quadratic behavior. quadratic behaviour widens with increasing doping[S4].

TABLE II :

 II and A. Kapitulnik for stimulating discussions. The samples SrTi1-xNbxO3 investigated in the present work. The table reports on the nominal content of Nb per formula unit (x), the charge carrier concentration (n) determined by Hall resistivity, resistivity at 300K (ρ300K ), resistivity at 2K (ρ2K ), RRR = ρ 300K ρ 2K , Hall coefficient (RH ), ρ0 and A coefficients obtained fitting the low temperature data with the function ρ = ρ0 + A • T 2 , thermal conductivity at 300K (κ300K ), thermal conductivity at 2K (κ2K ).

	x	n	ρ300K	ρ2K	RRR	RH	ρ0	A	κ300K		κ2K
		(cm -3 )	(mΩcm) (mΩcm)		(cm -3 /C) (mΩcm) (µΩcm/K 2 ) (W/cmK) (W/cmK)
	0.0004 0.53(4)•10 19	277	0.088	2579	1.18	0.078(4)	0.85(1)	0.1		0.0013
	0.001 1.4(1)•10 19	59.1	0.08	system D300K (mm 2 /s) v sl (100)(km/s) s 739 4.33 • 10 -1 0.077(4) 0.26(1) 0.099 0.0011
	0.01	9.4(7)•10 19	9.46	0.064	SrTiO3 148 6.59 • 10 -2 0.060(2)	4.0 0.062(1)	0.093	7.87 7.5•10 -4 2.6
	0.014	n.a.	4.22	0.087	49	n.a.	PbTe 0.075(5)	1.9 0.030(8)	0.098	3.59	0.0014	5.9
	0.02	2.5(9)•10 20	3.53	0.077	46	Si 2.45 • 10 -2 0.072(4)	91 0.029(8)	0.088	8.43	0.0011	51
						TABLE I: Room-temperature thermal diffusivity (D300K ),
						longitudinal sound velocity (v sl ) and the parameter s quan-
						tifying the slope of high-temperature thermal diffusivity, in
						three cubic solids.

TABLE III :

 III Molar volume (Vm), longitudinal (v sl

							3 K) D300K (cm 2 /s)
	SrTiO3	35.7	7.87	4.9	11.0	2.75	0.04
	Si	12.1	8.43	5.84	150	1.65	0.91
	PbTe	40.9	3.59	1.26	2.37	1.25	0.019

Appendix A: Methods Thermal conductivity and specific heat measurements were performed into the commercial system Dynacool PPMS (Physical Property Measurement System) that allows to perform experiments between 1.7 and 400K.

Thermal conductivity was measured with a standard two-thermometers-one heater configuration. The power supplied through the heater established a temperature difference (∆T = T 1 -T 2 ) along the sample that was kept below 1% of the average temperature (T av = (T 1 +T 2 )/2). This condition guarantees to have a negligible thermal flow along the electrical wires of the sensors and the heater, and to obtain an accurate determination of the thermal conductivity. The major source of experimental error comes from the determination of the geometrical factor that can be up to 6%.

Specific heat was measured with the standard commercial platform compatible with Dynacool. Temperature rises were limited to 1-2% of base temperature. Due to the low heat capacity of SrTiO 3 at the lowest temperature, a minimum sample mass of 45mg was necessary to obtain a sizeable contribution of the sample heat capacity respect to the addenda (N-apiezon).

Appendix B: Samples

The samples investigated (SrTiO 3 , SrTi 1-x Nb x O 3 , Sr 1-x Ca x TiO 3 and KTaO 3 ) in this work are all commercial single crystal specimens. For SrTiO 3 , two of the three samples of SrTiO 3 came from two different batches of the same supplier (sample-1 and sample-3), whereas the other one came from a second supplier (sample-2).

The carrier density was determined by measuring the Hall resistivity and was found to be in good agreement with the nominal Nb content. Table I summarizes relevant values of both electrical and thermal transport measurements. ρ 0 and A are obtained fitting the resistivity curve at the lowest measured temperature with the function ρ = ρ 0 +AT 2 (see solid lines in Fig. 8b).

Appendix C: Thickness dependence of thermal conductivity

Sample-3, initially 500µm-thick, was thinned down to 150µm in order to perform the thickness-dependence measurement of thermal conductivity (Fig. 5). We observed that the thermal conductivity decreases in the ballistic regimes and in the temperature range that we identified as Poiseuille, but not at the peak temperature and above.

The lower thermal conductivity at low temperatures implies a lower mean free path, when phonon thermal conductivity is expressed as k = 1 3 c ph v ls . As discussed in the main text, the mean free path of SrTiO 3 in the ballistic regime is much lower than the sample size and of the order of magnitude of the domain size [S1]. Therefore, the modest reduction of the mean free path by reducing thickness suggests a either a correlation between domain and sample dimensions or the existence of a small subset of phonons which can travel across domain boundaries.

Appendix D: Thermal conductivity and its sensitivity to disorder Fig. 6 presents the thermal conductivity of the three samples compared with an early study [S2]. One can see an overall agreement between the data sets. The peak thermal conductivity, κ peak (showed by downward arrows), appears to be sample-dependent. Since controlled substitution drastically affects κ peak (see below), it is reasonable to assume that κ peak is very sensitive to disorder and is largest in the cleanest samples. Fig. 7 compares the thermal conductivity of one of our cleanest SrTiO 3 samples with a reduced (SrTiO 3-δ ), two niobium-doped and one calcium substituted sample. One can see that the effect of oxygen reduction and Ca substitution is similar to the effect of Nb substitution, which was studied in extenso (See Fig. 3 of the main text). An extremely low level of atomic substitution drastically reduces the κ peak . Ca substitution, oxygen reduction and Nb substitution have a very similar conseqiences. Now, the latter two turn the system to a metal, but the former does not introduce mobile electrons). Therefore, as argued in the main text, the main reason for the observed reduction in lattice thermal conductivity is not electron scattering but disorder.

Appendix E: Electric resistivity

Electrical resistivity and Hall-resistivity were also carried out in the PPMS system. The data are similar to what was reported previously [S3, S4]. Figure 8 shows the resistivity of SrTi 1-x Nb x O 3 as a function of temperature (panel (a)) and as a function of T 2 (panel (b)). As found previously [S4], resistivity follows a T 2 behavior at low temperature and then a faster than T 2 (close to cubic) at higher temperatures.

Appendix F: Thermal diffusivity in three cubic solids Figure 9 shows the thermal conductivity and specific heat of SrTiO 3 , Si and PbTe. This data was used to extract the thermal diffusivity of these three cubic semiconductors shown in Fig. 4 of the main text. To quantify the specific heat per volume, the molar volume was used as specified in Table II. The magnitude of s is the main text was extracted using equation 1 and taking v s to be the longitudinal sound velocity along 100 as specified in the table.


Johannes Engelmayer

Xiao Lin

Fulya Koç

Christoph P Grams

Joachim Hemberger

Kamran Behnia

Thomas Lorenz

Ferroelectric order versus metallicity in Sr 1-x Ca x TiO 3-δ (x = 0.009)

We report on a thermal-expansion study of the ferroelectric phase transition in insulating Sr1-xCaxTiO3 (x = 0.009) and its evolution upon increasing charge-carrier concentration up to n 60 × 10 19 cm -3 . Although electric polarization is screened by mobile charge carriers, we find clear signatures of the ferroelectric phase transition in the thermal-expansion coefficient α of the weakly doped metallic samples. Upon increasing n, the transition temperature TC(n) and the magnitude of the anomalies in α rapidly decrease up to a threshold carrier density n above which broadened anomalies remain present. There is no indication for a sign change of α as is expected for a pressure-dependent quantum phase transition with n as the control parameter. Thus, the ferroelectriclike transition is either continuously fading away or it transforms to another low-temperature phase above n , but this change hardly affects the temperature-dependent α(T ) data.

I. INTRODUCTION

Perovskite titanates ATiO 3 with divalent A-site ions contain tetravalent titanium with an empty 3d shell, such that these materials typically form band insulators. Some members of this A 2+ Ti 4+ O 3 family are ferroelectric such as BaTiO 3 , PbTiO 3 , CdTiO 3 1-4 , whereas others like SrTiO 3 , CaTiO 3 , and EuTiO 3 show quantum paraelectric behavior [5][6][START_REF] Lemanov | Perovskite CaTiO3 as an incipient ferroelectric[END_REF][START_REF] Müller | SrTiO3: An intrinsic quantum paraelectric below 4 K[END_REF][START_REF] Katsufuji | Coupling between magnetism and dielectric properties in quantum paraelectric EuTiO3[END_REF][START_REF] Kamba | Magnetodielectric effect and optic soft mode behaviour in quantum paraelectric EuTiO3 ceramics[END_REF][START_REF] Engelmayer | Charge transport in oxygen-deficient EuTiO3: The emerging picture of dilute metallicity in quantum-paraelectric perovskite oxides[END_REF] , i.e., a ferroelectric longrange order is suppressed by quantum fluctuations. Although a ferroelectric transition is absent in both SrTiO 3 and CaTiO 3 6, [START_REF] Hulm | The Dielectric Properties of some Alkaline Earth Titanates at Low Temperatures[END_REF] , mixing Sr and Ca on the A site (Sr 1-x Ca x TiO 3 ) induces ferroelectricity already for tiny calcium substitutions 0.0018 ≤ x < 0.02 with increasing Curie temperature T C (x). For larger x, relaxor ferroelectric behavior is observed and finally the material becomes antiferroelectric above x 0.12 [START_REF] Bednorz | Sr1-xCaxTiO3: An XY Quantum Ferroelectric with Transition to Randomness[END_REF][START_REF] Ranjan | Novel Features of Sr1-xCaxTiO3 Phase Diagram: Evidence for Competing Antiferroelectric and Ferroelectric Interactions[END_REF][START_REF] Ranjan | Antiferroelectric phase transition in (Sr1-xCax)TiO3 (0.12 < x ≤ 0.40): I. Dielectric studies[END_REF][START_REF] Ranjan | Antiferroelectric phase transition in (Sr1-xCax)TiO3: II. X-ray diffraction studies[END_REF][START_REF] Mishra | Powder neutron diffraction study of the antiferroelectric phase transition in Sr0.75Ca0.25TiO3[END_REF] . The high-temperature structure of Sr 1-x Ca x TiO 3 is cubic (space group P m 3m, No. 221), but upon cooling it changes to the tetragonal, centrosymmetric space group I4/mcm (No. 140) at an x-dependent transition temperature T s (x), which is significantly larger than the ferroelectric ordering temperature T C . Because the finite polarization in the ferroelectric state requires the absence of an inversion center, the transition at T C necessarily involves a further symmetry reduction. The crystal structure in the ferroelectric phase was found to belong to the orthorhombic point group mm2 [START_REF] Bianchi | Raman scattering of ferroelectric Sr1-xCaxTiO3, x = 0.007[END_REF][START_REF] Kleemann | Optical detection of symmetry breaking on a nanoscale in SrTiO3:Ca[END_REF][START_REF] Carpenter | Structural relationships and a phase diagram for (Ca,Sr)TiO3 perovskites[END_REF] and for x = 0.02, 0.04 the space group Ic2m (No. 46) was specified by x-ray diffraction [START_REF] Mishra | Low temperature x-ray diffraction study of the phase transitions in Sr1-xCaxTiO3 (x = 0.02, 0.04): Evidence for ferrielectric ordering[END_REF] .

Charge-carrier doping in pure SrTiO 3 by, e.g., a partial removal of oxygen or substitution of Sr (Ti) by La (Nb), induces metallic conductivity [START_REF] Frederikse | Electronic Transport in Strontium Titanate[END_REF][START_REF] Tufte | Electron Mobility in Semiconducting Strontium Titanate[END_REF][START_REF] Frederikse | Hall mobility in SrTiO3[END_REF][START_REF] Lee | Optical absorption and transport in semiconducting SrTiO3[END_REF][START_REF] Spinelli | Electronic transport in doped SrTiO3: Conduction mechanisms and potential applications[END_REF] and, for certain carrier concentrations, even superconductivity [START_REF] Schooley | Superconductivity in semiconducting SrTiO3[END_REF][START_REF] Lin | Critical doping for the onset of a two-band superconducting ground state in SrTiO 3-δ[END_REF] . In systems with both, calcium substitution and electron doping (Sr 1-x Ca x TiO 3-δ ), the T C -related anomalies of the ferroelectric insulating parent compound persist within the metallic and superconducting phase [START_REF] De Lima | Interplay between antiferrodistortive, ferroelectric, and superconducting instabilities in Sr1-xCaxTiO 3-δ[END_REF][START_REF] Rischau | A ferroelectric quantum phase transition inside the superconducting dome of Sr1-xCaxTiO 3-δ[END_REF] .

Rischau et al. investigated the evolution of this ferroelectriclike transition with charge-carrier concentration n for Sr 1-x Ca x TiO 3-δ with x = 0.0022 and x = 0.009 [START_REF] Rischau | A ferroelectric quantum phase transition inside the superconducting dome of Sr1-xCaxTiO 3-δ[END_REF] . Based on minima in the resistivity data ρ(T ), a decreasing T C upon increasing n was derived and a disappearance of the ferroelectriclike phase above a critical carrier density n c that depends on the calcium content x. The mechanism behind this behavior remained unclear, but it was suggested that it could result from Friedel oscillations causing destructive interference of Ca-induced dipoles [START_REF] Rischau | A ferroelectric quantum phase transition inside the superconducting dome of Sr1-xCaxTiO 3-δ[END_REF] . In fact, such a mechanism was discussed theoretically already much earlier by Glinchuk et al. [START_REF] Glinchuk | Ruderman-Kittellike interaction of electric dipoles in systems with carriers[END_REF][START_REF] Glinchuk | The possibility of Kondo-like effect in systems with nontunneling off-center ions[END_REF] .

Rowley et al. discussed the appearance of ferroelectric order in insulating quantum paraelectrics in the context of quantum criticality [START_REF] Rowley | Ferroelectric quantum criticality[END_REF] where for SrTiO 3 the quantum control parameter can be tuned either by stress [START_REF] Uwe | Stress-induced ferroelectricity and soft phonon modes in SrTiO3[END_REF] , by chemical substitution like in Sr 1-x Ca x TiO 3 , or by oxygen-isotope exchange [START_REF] Itoh | Ferroelectricity Induced by Oxygen Isotope Exchange in Strontium Titanate Perovskite[END_REF] . In such a scenario, the charge-carrier concentration n represents an additional control parameter towards a metallic ground state. The presence of a quantum phase transition is intrinsically tied to a diverging Grüneisen parameter [START_REF] Zhu | Universally Diverging Grüneisen Parameter and the Magnetocaloric Effect Close to Quantum Critical Points[END_REF] . For pressuredependent quantum phase transitions, this holds for the Grneisen ratio Γ = α/c p with the thermal-expansion coefficient α and the specific heat c p . In the vicinity of a quantum critical point Γ exhibits a sign change [START_REF] Garst | Sign change of the Grüneisen parameter and magnetocaloric effect near quantum critical points[END_REF] , which results from a sign change of α, because c p is always positive [START_REF]The divergence and sign change of α/cp require a finite pressure dependence of the underlying quantum phase transition[END_REF] . Experimentally, such sign changes of α are observed in diverse materials where the control parameter is either a magnetic field [START_REF] Küchler | Divergence of the Grüneisen Ratio at Quantum Critical Points in Heavy Fermion Metals[END_REF][START_REF] Gegenwart | Metamagnetic Quantum Criticality in Sr3Ru2O7 Studied by Thermal Expansion[END_REF][START_REF] Baier | Magnetoelastic Coupling Across the Metamagnetic Transition in Ca2-xSrxRuO4 (0.2 ≤ x ≤ 0.5)[END_REF][START_REF] Lorenz | Thermodynamics of the coupled spin-dimer system TlCuCl3 close to a quantum phase transition[END_REF][START_REF] Lorenz | Diverging Thermal Expansion of the Spin-Ladder System (C5H12N)2CuBr4[END_REF][START_REF] Breunig | Spin-1 2 XXZ Chain System Cs2CoCl4 in a Transverse Magnetic Field[END_REF][START_REF] Breunig | Quantum criticality in the spin-1/2 Heisenberg chain system copper pyrazine dinitrate[END_REF] , a chemical or hydrostatic pressure [START_REF] Grube | Magnetic and Structural Quantum Phase Transitions in CeCu6-xAux are Independent[END_REF] , or the charge-carrier concentration [START_REF] Meingast | Thermal Expansion and Grüneisen Parameters of Ba(Fe1-xCox)2As2: A Thermodynamic Quest for Quantum Criticality[END_REF] .

Here, we present a detailed study of the evolution of the ground state of Sr 1-x Ca x TiO 3 with x = 0.009 as a function of the charge-carrier density n varying from the insulating parent compound to n 6 × 10 20 cm -3 . Based on thermal-expansion measurements, we investigate the evolution of both the structural transition temperature T s and the T C -related transition as a function of carrier concentration n. With increasing n, T s essen-tially linearly decreases and the structural transition remains well defined over almost the entire doping range. In contrast, T C (n) and the corresponding anomalies in α rapidly decrease in the low-n range up to a threshold carrier concentration n , above which broadened anomalies remain present up to the highest n. There is no evidence for a sign change in α as a function of n. This either suggests the absence of a sharp quantum phase transition, because the ferroelectriclike transition in Sr 1-x Ca x TiO 3 is continuously vanishing, e.g. by varying from long range to short range, or the symmetry of the low-temperature order changes at some critical charge-carrier content n without being directly reflected in the macroscopic uniaxial expansion α(T ).

The discussion of our results is split into two parts. Section III A discusses the symmetry changes at the structural and the ferroelectric transition of the pristine insulating material in order to clarify how the macroscopic uniaxial expansion α(T ) is related to a (partial) twinning occurring at both transitions, which sets the basis for the discussion of the evolution of both transitions as a function of the charge-carrier concentration in III B.

II. EXPERIMENTAL

A commercial Sr 1-x Ca x TiO 3 single crystal with x = 0.009 was used for this study. The nominal Ca content was confirmed by secondary ion mass spectrometry (SIMS) as described in [START_REF] De Lima | Interplay between antiferrodistortive, ferroelectric, and superconducting instabilities in Sr1-xCaxTiO 3-δ[END_REF] . The crystal was cut into cuboid pieces with all faces being cubic {100} planes and dimensions optimized for Hall effect measurements (typically 0.5 × 2.5 × 5 mm). In order to induce electron doping the samples were annealed under vacuum ( < ∼ 10 -5 mbar) for 1 to 2 hours at temperatures between 700 • C and 1000 • C depending on the intended charge-carrier concentration. Resistivity and Hall effect measurements were carried out via a standard six-probe method using the commercial physical property measurement system (PPMS by Quantum Design) and most of these data have been published in Ref. [START_REF] Rischau | A ferroelectric quantum phase transition inside the superconducting dome of Sr1-xCaxTiO 3-δ[END_REF] . On the same samples we studied the uniaxial thermal expansion by measuring the length change ∆L(T ) using a homebuilt capacitance dilatometer. The samples were continuously heated from liquid-helium temperature to 180 K at a rate of about 0.1 K min -1 and the thermal-expansion coefficient α = (1/L 0 )(∂∆L/∂T ) was determined numerically. The heat-capacity measurement of pristine Sr 1-x Ca x TiO 3 was performed using the microcalorimeter option of the PPMS.

III. RESULTS AND DISCUSSION

A. Pristine Sr0.991Ca0.009TiO3 the temperature ranges around both the structural and ferroelectric transition temperature T s and T C , respectively, which were determined from thermal-expansion data [see Fig. 1 (b) and(c)]. In agreement with previous publications [START_REF] De Lima | Interplay between antiferrodistortive, ferroelectric, and superconducting instabilities in Sr1-xCaxTiO 3-δ[END_REF][START_REF] Mccalla | A Unified View of the Substitution-Dependent Antiferrodistortive Phase Transition in SrTiO3[END_REF] the structural transition at T s is clearly visible by a small but distinct c p anomaly, but no anomaly can be resolved in c p (T ) around T C .

α i / T ( 1 0 -8 K -2 ) α= β/ 3 ( b ) T s T C a c c t ≈ 2 a c a t ≈ a c √ 2 ( c ) A x e s ( Å ) T ( K ) 〈1 0 0 〉 c L l o n g L m e d i u m L s h o r t c t / 2 a t / √ 2 ( a )
Figure 1 (b) shows the thermal-expansion coefficients α i /T of pristine Sr 0.991 Ca 0.009 TiO 3 measured along all three of the cubic 100 directions (solid lines) that are parallel to the sample's dimensions L i .

Figure 1 (c) displays the corresponding uniaxial length changes ∆L i /L 0 adjusted to the cubic lattice parameter at 150 K [START_REF] Mishra | A combined X-ray diffraction and Raman scattering study of the phase transitions in Sr1-xCaxTiO3 (x = 0.04, 0.06[END_REF] . The α i are identical at high temperatures and show pronounced anomalies around 139 K and 27 K. The upper temperature can be identified with the cubic-to-tetragonal transition temperature T s . Pure SrTiO 3 becomes tetragonal around T s 105 K [START_REF] Rimai | Electron Paramagnetic Resonance of Trivalent Gadolinium Ions in Strontium and Barium Titanates[END_REF][START_REF] Lytle | X-Ray Diffractometry of Low-Temperature Phase Transformations in Strontium Titanate[END_REF][START_REF] Ohama | The temperature dependence of the lattice constant of SrTiO3 around the 105 K transition[END_REF][START_REF] Sato | The lattice constant vs. temperature relation around the 105 K transition of a flux-grown SrTiO3 crystal[END_REF] and in Sr 1-x Ca x TiO 3 , T s increases with increasing x [START_REF] Ranjan | Antiferroelectric phase transition in (Sr1-xCax)TiO3: II. X-ray diffraction studies[END_REF][START_REF] Carpenter | Structural relationships and a phase diagram for (Ca,Sr)TiO3 perovskites[END_REF][START_REF] De Lima | Interplay between antiferrodistortive, ferroelectric, and superconducting instabilities in Sr1-xCaxTiO 3-δ[END_REF][START_REF] Mishra | A combined X-ray diffraction and Raman scattering study of the phase transitions in Sr1-xCaxTiO3 (x = 0.04, 0.06[END_REF][START_REF] Mitsui | Dielectric and X-Ray Studies of CaxBa1-xTiO3 and CaxSr1-xTiO3[END_REF] . The transition temperature T s 139 K of our pristine sample with x = 0.009 is in agreement with findings in [START_REF] De Lima | Interplay between antiferrodistortive, ferroelectric, and superconducting instabilities in Sr1-xCaxTiO 3-δ[END_REF][START_REF] Mccalla | A Unified View of the Substitution-Dependent Antiferrodistortive Phase Transition in SrTiO3[END_REF][START_REF] Lemanov | Phase transitions in SrTiO3-based solid solutions[END_REF] . The high-temperature cubic phase has the space group P m 3m (No. 221) while in the tetragonal phase it is I4/mcm (140) [START_REF] Unoki | Electron Spin Resonance of Fe 3+ in SrTiO3 with Special Reference to the 110 • K Phase Transition[END_REF] . This transition is antiferrodistortive due to a tilting of the TiO 6 octahedra around the c axis [START_REF] Unoki | Electron Spin Resonance of Fe 3+ in SrTiO3 with Special Reference to the 110 • K Phase Transition[END_REF][START_REF] Fleury | Soft Phonon Modes and the 110 • K Phase Transition in SrTiO3[END_REF][START_REF] Shirane | Lattice-Dynamical Study of the 110 • K Phase Transition in SrTiO3[END_REF] corresponding to (a 0 a 0 c -) in the classification of Glazer [START_REF] Glazer | The classification of tilted octahedra in perovskites[END_REF][START_REF] Glazer | Simple ways of determining perovskite structures[END_REF] , where a 0 denotes the absence of a tilting around the a axis and c -represents an anti-phase tilting around the c axis. The inset in Fig. 1 (b) illustrates the octahedra tilt in a top view of the tetragonal ab plane. Titanium ions (blue) define the corners of the cubic unit cell (dotted square). At T s , the oxygen ions (red) move as indicated by the arrows and consequently, the tetragonal unit cell (dashed square) is doubled in the ab plane and rotated by 45 • . Due to the anti-phase tilting in c direction, the c axis is doubled as well. Thus, the cubic axes a c and the tetragonal axes a t , c t are related by a t ≈ √ 2a c and c t ≈ 2a c . It is evident, that the sample's dimensions L i which are parallel to the cubic 100 directions, point along 110 with respect to the tetragonal axes a t . The lower transition at 27 K signals the ferroelectric phase transition that was characterized by P (E) hysteresis loops [START_REF] Rischau | A ferroelectric quantum phase transition inside the superconducting dome of Sr1-xCaxTiO 3-δ[END_REF] . While the structural transition at T s is seen mainly in the sample's short direction L short , the ferroelectric transition at T C predominantly appears along the sample's medium and long direction L medium and L long , respectively.

Structural phase transitions generally involve transformation twinning [START_REF] Cahn | Twinned crystals[END_REF][START_REF] Authier | International Tables for Crystallography[END_REF] . For a cubic-to-tetragonal transition one expects the emergence of three twin domains enabling different α i to partially compensate each other. Therefore, a completely twinned sample should exhibit an isotropic uniaxial thermal expansion ᾱ, which is related to the volume expansion β = 3 ᾱ. In general, β = i α i , where α i denote the uniaxial expansion coefficients along a set of three pairwise orthogonal directions and for a tetragonal lattice, β = 2α at +α ct with the (usually anisotropic) main-axis expansion coefficients α at and α ct along the tetragonal axes a t and c t , respectively. As shown by ᾱ = β/3 (black dashed line in Fig. 1 (b)), the transition at T s is almost volume-conserving, i.e., the expansion in the long direction roughly compensates the contraction along the short and medium direction. A nearly volume-conserving transition is also reported for SrTiO 3 where the temperature-dependent lattice parameters around T s were determined by high-resolution x-ray diffraction [START_REF] Ohama | The temperature dependence of the lattice constant of SrTiO3 around the 105 K transition[END_REF] . The anisotropic α i and the nearly volume conservation indicate highly unequal twinning fractions in our crystal. The usage of a capacitance dilatometer naturally requires the application of a certain uniaxial stress that can be sufficient to achieve a (partial) detwinning of the crystal [START_REF] Niesen | Magnetic phase diagrams, domain switching, and quantum phase transition of the quasi-one-dimensional Ising-like antiferromagnet BaCo2V2O8[END_REF][START_REF] Kunkemöller | Magnetic shape-memory effect in SrRuO3[END_REF] . This is apparently not the case in our sample. Here, the presence of a dominating twin is not triggered by external conditions but rather predetermined by intrinsic crystal defects [START_REF] When | the force applied via the dilatometer acts on the smallest cross-sectional area of our sample and consequently produces the largest pressure[END_REF] .

We can estimate the fraction of the tetragonal axes a t and c t along the sample's dimensions L i by comparing the anomalies of the thermal-expansion coefficients of our measurements with the slope changes of the temperature-dependent lattice parameters around T s from x-ray diffraction measurements. For this comparison, we use the high-resolution x-ray data of SrTiO 3 [START_REF] Ohama | The temperature dependence of the lattice constant of SrTiO3 around the 105 K transition[END_REF] , which at T s show an a t -axis contraction that corresponds to a change ∆α a 8 × 10 -6 K -1 and a c t -axis expansion corresponding to ∆α c -16 × 10 -6 K -1 . By comparing these values to our thermal-expansion anomalies ∆α i at T s we estimate that the sample's short axis L short contains approximately 0.9a t and 0.1c t , whereas the long axis contains 0.5a t + 0.5c t , and the medium axis contains 0.6a t + 0.4c t . From this estimate we reconstruct the tetragonal pure-axes compositions a t = 1.25L short -0.25L long and c t = 2.25L long -1.25L short . Figures 1 (b) and (c) show the temperature-dependent behavior of the pure tetragonal axes a t / √ 2 and c t /2 as dotted lines. Of course, the derived anomalies ∆α a and ∆α c of Sr 1-x Ca x TiO 3 at T s are by construction identical to those observed by the temperature-dependent xray data of the SrTiO 3 lattice parameters [START_REF] Ohama | The temperature dependence of the lattice constant of SrTiO3 around the 105 K transition[END_REF] . The relation ∆α a ≈ -∆α c /2 is, however, independent from this reconstruction and directly follows from the (almost) absent anomaly in the volume expansion β and the tetragonal symmetry. In contrast to the transition at T s , the ferroelectric transition at T C is not volume conserving as is clearly demonstrated by the pronounced anomaly in the averaged uniaxial expansion ᾱ (see black dashed line in Fig. 1 (b)). Furthermore, our reconstructed pure-axis data suggest that the volume-expansion anomaly essentially arises from a c-axis expansion upon cooling below T C , while the transition is roughly area-conserving with respect to the ab plane. The latter is naturally expected for a tetragonal-to-orthorhombic transition with opposite expansion anomalies of similar magnitudes along the orthorhombic a and b axes. Note that the ferroelectric polarization is expected to be aligned along one of these orthorhombic axes, as it has been discussed in [START_REF] Bednorz | Sr1-xCaxTiO3: An XY Quantum Ferroelectric with Transition to Randomness[END_REF][START_REF] Mishra | Low temperature x-ray diffraction study of the phase transitions in Sr1-xCaxTiO3 (x = 0.02, 0.04): Evidence for ferrielectric ordering[END_REF][START_REF] Kleemann | Domain state properties of the random-field xy-model system Sr1-xCaxTiO3[END_REF] . In this context, it is also worth to mention that a tetragonalto-orthorhombic transition increases the number of possible twin domains by a factor of two. However, the uniaxial length change measured along the tetragonal [110] direction cannot distinguish between multi-domain and single-domain orthorhombic samples, because α [110] = (α a + α b )/2 in both cases.

B. Electron-doped samples Sr0.991Ca0.009TiO 3-δ

Figure 2 shows the thermal-expansion coefficients α/T versus T of Sr 0.991 Ca 0.009 TiO 3-δ with different chargecarrier concentrations up to n ≤ 57.9 × 10 19 cm -3 together with α/T of the pristine sample already shown in Fig. 1.

For clarity, the curves are shifted by 7.5 × 10 -8 K -2 with respect to each other. On each sample, we measured α i along L long . Since all samples were obtained by parallel cuts from the original single crystal it appears plausible that the distribution of twin domains does not vary too much over the individual samples. This assumption is essentially confirmed by the fact that all samples, apart from the one with highest n, show clear anomalies of the same sign and similar shape signaling the structural phase transition at T s . With increasing n the transition temperature linearly decreases from T s 139 K in pristine Sr 0.991 Ca 0.009 TiO 3 to 116 K for n = 22.6 × 10 19 cm -3 [see Fig. 3 (b)]. A linear decreasing T s (n) is known from reduced SrTiO 3-δ [START_REF] Bäuerle | Structural phase transitions in semiconducting SrTiO3[END_REF][START_REF] Wagner | Soft modes in semiconducting SrTiO3 -I. The Zone Boundary Mode[END_REF][START_REF] Tao | Nonmonotonic anisotropy in charge conduction induced by antiferrodistortive transition in metallic SrTiO3[END_REF] and a decreased T s was also seen in reduced EuTiO 3-δ [START_REF] Engelmayer | Charge transport in oxygen-deficient EuTiO3: The emerging picture of dilute metallicity in quantum-paraelectric perovskite oxides[END_REF] , suggesting that this is a generic trend in these almost cubic perovskite titanates [START_REF]The decrease of Ts is, however, not a direct consequence of the charge-carrier doping, because n-type doping by chemical substitution like in SrTi1-xNbxO3 increases[END_REF] . An extrapolation of the linear T s (n) dependence to the highest doping n = 57.9 × 10 19 cm -3 matches the kink at 80 K in the α/T curve of the corresponding sample. This suggests that the structural transition remains present in the entire series of Sr 0.991 Ca 0.009 TiO 3-δ samples, but the highest-doped sample apparently has an essentially homogeneous distribution of twin domains and, consequently, the averaged uniaxial expansion ᾱ hardly shows any anomaly at this volume-conserving transition as discussed above.

The transition at T C is clearly identifiable for the lowerdoped samples (left panel of Fig. 2) and shifts from T C 27 K for pristine Sr 0.991 Ca 0.009 TiO 3 down to 18 K for n = 1.3 × 10 19 cm -3 . For the higher-doped samples (right panel of Fig. 2) the α/T anomalies become much less pronounced and rather broad; except for the sample with n = 12.4 × 10 19 cm -3 . Despite this broadening a signature of the transition remains present in α/T of all doped samples. For example, in the more homogeneously twinned sample with n = 57.9 × 10 19 cm -3 the clear minimum of ᾱ/T around 17 K signals the spontaneous volume expansion resulting from the low-temperature transition, whereas ᾱ/T only shows a kink at T s of the hightemperature structural transition, which is volume conserving.

Figure 3 compares the evolution of both transition temperatures T s and T C as a function of charge-carrier density n. The linear shift of T s and the weak broadening of the structural transition with increasing n indicate that both the distribution of oxygen vacancies and the resulting charge-carrier density are rather homogeneous in the studied samples. Nevertheless, the lowtemperature transition shows a complex behavior. The ferroelectric transition of the insulating pristine sample causes a very sharp α/T anomaly, which remains sharp for the low-doped metallic samples but broadens above n = 1.3 × 10 19 cm -3 . As a criterion to define T C , we take the maximum slope of α/T and use the temperature difference to the minimum of α/T as a measure of the transition width, which is shown as error bars in Fig. 3 (a,b). The corresponding T C (n) curve is concave for n ≤ 1.3 × 10 19 cm -3 , as indicated by the dotted line in Fig. 3 (a) and essentially saturates for larger n. As a further measure of this transition, we use a smooth background α bg and calculate ε = (α -α bg ) dT , which yields the spontaneous elongation resulting from the lowtemperature transition. For α bg (T ) we measured the thermal expansion α STO on a single crystal of SrTiO 3 71 , which remains in the tetragonal phase and scaled this α STO (T ) curve such that it matches α(T ≈ 25 K) of the respective Sr 0.991 Ca 0.009 TiO 3-δ sample. This is shown for 2 exemplary charge-carrier contents in Fig. 3 (e,f) and the resulting ε(T ) are displayed in Fig. 3 (g) with every second measurement skipped for clarity. The evolution of ε(T = 4.2 K, n) as a function n is summarized in Fig. 3 (c,d); the corresponding error bars express the sensitivity of ε(T = 4.2 K, n) on variations of the individual scaling factors used for α STO . We see that ε(T = 4.2 K, n) rapidly decreases at n ≤ 1.3 × 10 19 cm -3 and then levels off at a value of about 15 % of the spontaneous elongation measured in the ferroelectric phase of pristine Sr 0.991 Ca 0.009 TiO 3 . This behavior is qualitatively similar to that of T C (n), which saturates, however, at a comparatively larger value of about 60 % of T C of the insulating pristine material. These are the main findings of the present study, which extends our previous work suggesting the existence of a ferroelectric quantum phase transition inside the superconducting dome of Sr 1-x Ca x TiO 3-δ [START_REF] Rischau | A ferroelectric quantum phase transition inside the superconducting dome of Sr1-xCaxTiO 3-δ[END_REF] . As already discussed there, a true ferroelectric order cannot exist in metals, because the mobile electrons screen any static electric polarization. However, characteristic features of the ferroelectric order in Sr 1-x Ca x TiO 3 are still observed upon weak charge-carrier doping suggesting a ferroelectriclike transition that vanishes via a quantum phase transition as indicated by minima in the resitivity data ρ(T, n) [START_REF] Rischau | A ferroelectric quantum phase transition inside the superconducting dome of Sr1-xCaxTiO 3-δ[END_REF] . The aim of the present study was to further clarify this issue via thermal-expansion measurements, which are a sensitive thermodynamic probe to detect and characterize pressure-dependent quantum phase transitions [START_REF] Zhu | Universally Diverging Grüneisen Parameter and the Magnetocaloric Effect Close to Quantum Critical Points[END_REF][START_REF] Garst | Sign change of the Grüneisen parameter and magnetocaloric effect near quantum critical points[END_REF][START_REF] Küchler | Divergence of the Grüneisen Ratio at Quantum Critical Points in Heavy Fermion Metals[END_REF][START_REF] Gegenwart | Metamagnetic Quantum Criticality in Sr3Ru2O7 Studied by Thermal Expansion[END_REF][START_REF] Baier | Magnetoelastic Coupling Across the Metamagnetic Transition in Ca2-xSrxRuO4 (0.2 ≤ x ≤ 0.5)[END_REF][START_REF] Lorenz | Thermodynamics of the coupled spin-dimer system TlCuCl3 close to a quantum phase transition[END_REF][START_REF] Lorenz | Diverging Thermal Expansion of the Spin-Ladder System (C5H12N)2CuBr4[END_REF][START_REF] Breunig | Spin-1 2 XXZ Chain System Cs2CoCl4 in a Transverse Magnetic Field[END_REF][START_REF] Breunig | Quantum criticality in the spin-1/2 Heisenberg chain system copper pyrazine dinitrate[END_REF][START_REF] Grube | Magnetic and Structural Quantum Phase Transitions in CeCu6-xAux are Independent[END_REF][START_REF] Meingast | Thermal Expansion and Grüneisen Parameters of Ba(Fe1-xCox)2As2: A Thermodynamic Quest for Quantum Criticality[END_REF] . The existence of a welldefined ferroelectriclike transition in the metallic samples is clearly confirmed by the pronounced α/T anomalies (see Fig. 2) up to at least n = 0.72 × 10 19 cm -3 , i.e., the transition remains present in metallic samples which finally become superconducting at lower temperature [START_REF] Rischau | A ferroelectric quantum phase transition inside the superconducting dome of Sr1-xCaxTiO 3-δ[END_REF] . On further increasing n, however, the α/T anomalies do not vanish and, in particular, our data do not give any indication for a sign change of the α/T anomalies as a function of n. Thus, the general behavior of α and the corresponding Grüneisen ratio Γ = α/c p is different from other materials showing quantum phase transitions as a function of magnetic field, pressure, or n as external control parameter [START_REF] Küchler | Divergence of the Grüneisen Ratio at Quantum Critical Points in Heavy Fermion Metals[END_REF][START_REF] Gegenwart | Metamagnetic Quantum Criticality in Sr3Ru2O7 Studied by Thermal Expansion[END_REF][START_REF] Baier | Magnetoelastic Coupling Across the Metamagnetic Transition in Ca2-xSrxRuO4 (0.2 ≤ x ≤ 0.5)[END_REF][START_REF] Lorenz | Thermodynamics of the coupled spin-dimer system TlCuCl3 close to a quantum phase transition[END_REF][START_REF] Lorenz | Diverging Thermal Expansion of the Spin-Ladder System (C5H12N)2CuBr4[END_REF][START_REF] Breunig | Spin-1 2 XXZ Chain System Cs2CoCl4 in a Transverse Magnetic Field[END_REF][START_REF] Breunig | Quantum criticality in the spin-1/2 Heisenberg chain system copper pyrazine dinitrate[END_REF][START_REF] Grube | Magnetic and Structural Quantum Phase Transitions in CeCu6-xAux are Independent[END_REF][START_REF] Meingast | Thermal Expansion and Grüneisen Parameters of Ba(Fe1-xCox)2As2: A Thermodynamic Quest for Quantum Criticality[END_REF] . On the one hand, this could mean the absence of a quantum phase transition, if the ferroelectriclike order changes just continuously disappears towards larger n. On the other hand, however, it is also clear from Figs. 2 and 3 that the shape and width of the α/T anomalies strongly change around n = 1.3 × 10 19 cm -3 . This could mean that the symmetry change of the structural transition is different above and below a critical doping n in the range of 1.3 × 10 19 cm -3 . A similar situation is, in fact, observed in the undoped Sr 1-x Ca x TiO 3 where, as a function of x, the symmetry of the low-temperature ordered phase changes from polar ferroelectric (space group Ic2m; No. 46) via relaxor ferroelectric to antiferroelectric with inversion symmetry (P bcm, No. 57) above x 0.12 [START_REF] Bednorz | Sr1-xCaxTiO3: An XY Quantum Ferroelectric with Transition to Randomness[END_REF][START_REF] Ranjan | Novel Features of Sr1-xCaxTiO3 Phase Diagram: Evidence for Competing Antiferroelectric and Ferroelectric Interactions[END_REF][START_REF] Ranjan | Antiferroelectric phase transition in (Sr1-xCax)TiO3 (0.12 < x ≤ 0.40): I. Dielectric studies[END_REF][START_REF] Ranjan | Antiferroelectric phase transition in (Sr1-xCax)TiO3: II. X-ray diffraction studies[END_REF][START_REF] Mishra | Powder neutron diffraction study of the antiferroelectric phase transition in Sr0.75Ca0.25TiO3[END_REF] . Such a microscopic change is not necessarily reflected in the thermalexpansion coefficient. In the simplest case, this could be an experimental problem, because a more or less vertical phase boundary in an n-T phase diagram is difficult to measure as a function of T . More generally, anomalies in the uniaxial expansion α require a finite dependence of the respective transition temperature on uniaxial stress along this direction 72 . In this context, it is also important that the polarization of Sr 1-x Ca x TiO 3 is in the orthorhombic ab plane [START_REF] Bednorz | Sr1-xCaxTiO3: An XY Quantum Ferroelectric with Transition to Randomness[END_REF][START_REF] Mishra | Low temperature x-ray diffraction study of the phase transitions in Sr1-xCaxTiO3 (x = 0.02, 0.04): Evidence for ferrielectric ordering[END_REF][START_REF] Kleemann | Domain state properties of the random-field xy-model system Sr1-xCaxTiO3[END_REF] , such that a change of the in-plane symmetry will have minor influence on the observed α/T anomaly, because it arises from a spontaneous elongation of the c-axis component in our partially twinned crystals, as discussed above in the context of Fig. 1. Thus, detailed structural analysis of the lowtemperature phases for different charge-carrier concentrations would be necessary to resolve this puzzle.

IV. SUMMARY

In conclusion, we present a detailed thermal-expansion study on Sr 1-x Ca x TiO 3-δ (x = 0.009) with chargecarrier density tuned from the pristine, insulating parent compound to n 60 × 10 19 cm -3 . Both the cubicto-tetragonal transition T s and the ferroelectric transition T C display pronounced anomalies in the thermalexpansion coefficient α(T ) of the pristine crystal. As a function of charge-carrier density n, T s decreases linearly from 139 K to 80 K and the related anomalies in α/T remain distinct and of similar magnitude across almost the entire doping range. Despite the presence of mobile charge carriers, the T C -related anomaly survives for n > 0. However, T C decreases rapidly upon increasing n and the associated anomalies in α/T become very broad above n 1.3 × 10 19 cm -3 . Whether the evolution of T C across this carrier concentration is continuous or passes a phase transition at a critical n is not directly seen in α(T ) but needs to be clarified by a structure analysis.

Figure 1

 1 Figure 1 (a) shows the specific heat c p /T of pristine Sr 0.991 Ca 0.009 TiO 3 . The insets depict enlarged views of

  FIG. 1. (a) Specific heat cp/T of pristine Sr0.991Ca0.009TiO3. Insets show enlarged views around the structural and ferroelectric transition temperature Ts and TC, respectively, both indicated by vertical lines. (b) Thermal-expansion coefficients αi/T (thick lines) of pristine Sr0.991Ca0.009TiO3 measured along the sample's dimensions Li which are parallel to the cubic axes 100 , together with the reconstructed αi/T of the tetragonal axes at/ √ 2 and ct/2 (dotted lines) and average linear expansion ᾱ = β/3 (black dashed line). (c) Corresponding uniaxial length change ∆Li/L0 adjusted to the cubic lattice parameter ac measured at 150 K by x-ray diffraction 49 . The sizes and orientations of the cubic and tetragonal unit cells of SrTiO3 are sketched by the insets in (b) with blue (red) balls representing Ti (O) ions.

  FIG. 2.Thermal-expansion coefficient α/T versus T of Sr0.991Ca0.009TiO 3-δ with different carrier densities n. Curves are shifted with respect to each other.

FIG. 3 .

 3 FIG. 3. Phase diagram of Sr0.991Ca0.009TiO 3-δ . Open symbols refer to weak anomalies. (a) Detail view of the lown regime with guide to the eye for TC(n) (dotted curve). (b) Complete range of n with linear fit of Ts(n) (black solid line). Note the scale breaks in panels (a) and (b). (c, d) Spontaneous strain ε as a function of n. (e, f) We used α(T ) of pure SrTiO3 as background (gray curves) and integrated the difference to α(T ) of the respective Sr0.991Ca0.009TiO 3-δ sample, exemplary shown for n = 0 (e) and n = 19.5 × 10 19 cm -3 (f). (g) Temperature dependence of ε of selected samples.

B. J. Kennedy, Q. Zhou, and M. Avdeev, The ferroelectric phase of CdTiO3: A powder neutron diffraction study, J. Solid State Chem. 184, 2987 (2011).

T. Nakamura, P.-H. Sun, Y. J. Shan, Y. Inaguma, M. Itoh, I.-S. Kim, J.-H. Sohn, M. Ikeda, T. Kitamura, and H. Konagaya, On the perovskite-related materials of high dielectric permittivity with small temperature dependence and low dielectric loss,Ferroelectrics 196, 205 (1997

). 6 I.-S. Kim, M. Itoh, and T. Nakamura, Electrical conductivity and metal-nonmetal transition in the perovskite-related

ACKNOWLEDGMENTS

We acknowledge support by the DFG (German Research Foundation) via project number 277146847 -CRC 1238 (Subprojects A02, B01 and B02). This work is part of a DFG-ANR project funded by Agence Nationale de la Recherche (ANR-18-CE92-0020-01) and by the DFG through projects LO 818/6-1 and HE 3219/6-1. X. L. acknowledges support by the Alexander von Humboldt Foundation and Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ19A040005. * Present address: School of Science, Westlake University, 18 Shilongshan Road, 310024, Hangzhou, China † tl@ph2.uni-koeln.de 1 A. von Hippel, Ferroelectricity, Domain Structure, and Phase Transitions of Barium Titanate, Rev. Mod. Phys. 22, 221 (1950). 2 V. G. Bhide, K. G. Deshmukh, and M. S. Hegde, Ferroelectric properties of PbTiO3, Physica 28, 871 (1962). 3 P.-H. Sun, T. Nakamura, Y. J. Shan, Y. Inaguma, and M. Itoh, The study on the dielectric property and structure of perovskite titanate CdTiO3, Ferroelectrics 217, 137 (1998).


F A Schoulten

A A Mariano

R Vauché

S Bourdel

J Gaubert

N Dehaese

H Barthelemy

IR-UWB Pulse Synthesizer Based on a Sample & Hold Envelope Shaping Circuit

Keywords: FCC regulations, IEEE standards, Impulse Radio, Low power, Pulse Generator, Spectral masks, UWB

This paper presents an ultra-wideband impulse radio pulse synthesizer circuit based on a gated voltagecontrolled oscillator and an envelope shaping circuit, which deploys a sample & hold envelope synthesis. The envelope shaping circuit is mainly composed by parallel shift registers loaded in a parallel manner. The use of a sample & hold envelope enables the reduction of circuit complexity, size area and leakage power. The pulse synthesizer is designed in FD-SOI 28 nm CMOS technology with a supply voltage of 1 V. It can be programmed to address all the IEEE 802.15.4 standardized channels and respects the FCC indoor spectrum requirements. Moreover, it can be turned-on and turned-off which allows to generate nanosecond pulses, where the generated pulse amplitude, central frequency, width, and repetition frequency can be tuned.

I. INTRODUCTION

Since the Federal Communications Commission (FCC) has authorized Ultra-Wideband (UWB) communications in the 3.1-10.6 GHz frequency band, several technologies have been developed in order to exploit it, mostly the Impulse Radio (IR) communications [1][START_REF] Hamalainen | UWB Theory and Applications[END_REF]. It is essential to develop low-power and high data rate transmission circuits in order to attend the demand of the high-quality data transfer within short-medium range coverage, given that wireless transmission chains usually consume the majority part of the system power budget [START_REF] Liu | A 9mW 6-9GHz 2.5Gb/s proximity transmitter with combined OOK/BPSK modulation for low power mobile connectivity[END_REF]. The ultra-wideband impulse radio (IR-UWB) is a technique which consists in low duty cycle short duration pulses transmission and has demonstrated low-power consumption and high data rates capabilities thanks to its large spreading factor and lower power spectral density. It is also well-suited for applications as Wireless Personal Area Network (WPAN), and Wireless Body Area Network (WBAN) [START_REF] Goavec | An efficient method of power spectral density estimation for on-chip IR-UWB transmitter self-calibration[END_REF]. Since the transmitter only needs to be operational when the pulse is emitted, the IR-UWB signal presents a gated-cycled nature, resulting in high energy efficiency [START_REF] Haapala | A fully integrated digitally programmable pulse shaping 6.0-8.5 GHz UWB IR transmitter frontend for energy harvesting applications[END_REF].

Several IR-UWB pulses generation techniques were reported in literature. A first one is based on an edge combiner, which uses simple logic gates to combine two or more delayed logic signals [START_REF] Sparrow | High rate UWB CMOS transceiver chipset for WBAN and biomedical applications[END_REF][START_REF] Vauche | A 100 MHz PRF IR-UWB CMOS Transceiver With Pulse Shaping Capabilities and Peak Voltage Detector[END_REF]. It is simple to integrate in CMOS technology and also presents a low-power consumption. A second one consists in the multiplication of a carrier generated by an oscillator with a baseband pulse, where this architecture has presented a better power consumption and pulse magnitude than the first technique, although the oscillator has leakage issues [START_REF] Artemio-Schoulten | Pulse Design Method for Complexity Reduction of IEEE IR-UWB Pulse Synthesizers[END_REF][START_REF] Park | An all-digital 12 pJ/pulse IR-UWB transmitter synthesized from a standard cell library[END_REF]. Ring oscillators can be used as the local oscillator to be modulated for the pulse generation. The IR-UWB circuits are susceptible to process-voltage-temperature (PVT) variations which reduce their performances,regarding the emitted pulse power spectrum density (PSD) shape and amplitude. Furthermore, the output pulse waveform shaping control is important to ensure that the output pulse PSD fits in the FCC and IEEE standards.

Gaussian pulse shape allows to optimize the bandwidth and sidelobe rejections leakage to respect the FCC spectral masks. However, this pulse shape claims a more complex circuit design and presents higher power consumption. On the other hand, the triangular-shaped pulse has a more straightforward design, lower power, and area requirements. However, to reduce more the complexity of the pulse shaping method, sample & hold envelopes can be applied [START_REF] Artemio-Schoulten | Pulse Design Method for Complexity Reduction of IEEE IR-UWB Pulse Synthesizers[END_REF].

In this paper, an IR-UWB pulse synthesizer, that covers the 3.1-10.6 GHz FCC band and fulfils the IEEE 802.15.4 standard requirements is proposed. The IEEE 802.15.4 standard defines 15 channels divided in 3 groups (sub-GHz, low, and high band), where each channel has a central frequency and a 3 dB attenuation bandwidth. For a particular group, an UWB device must cover at least one specific channel, where the mandatories ones are 0, 3, and 9. The proposed circuit is a synthesizer based on a local oscillator with a pulse shaper circuit, where this last one is the focus of this work. The envelope control circuit consists of several shift registers charged in parallel manner, targeting to shape the generated pulse whose envelope approximates the targeted envelope using a on sample & hold method. Indeed, the envelope control circuit modulates the amplitude of the pulses generated by the oscillator. The envelope shape and the pulse duration are critical to have a pulse spectrum that fits in spectrum masks defined by the IEEE 802.15.4 standard, given that those delimits the UWB pulse frequency bandwidth and its power spectrum density (PSD). Finally, since the pulse synthesized is based on a gated oscillator, it can be powered down between two consecutives emitted pulses, hence consuming only during a pulse transmission. This paper is organized as follows. Section II discusses several IR-UWB pulses envelopes, mainly the sampled & hold, that is applied on the pulse synthesizer. Section III focuses on the envelope control circuit architecture designed with a FD-SOI 28 nm design kit. Finally, simulation results in section IV demonstrate that pulses generated by the synthesizer address IEEE 802.15.4 standardized channels and respect the FCC regulations.

II. IR-UWB PULSE ENVELOPPES

In the IR-UWB pulse generator design, the shape of the transmitted pulse envelope is critical to achieve a pulse frequency response which complies with the standard spectrum masks. The pulse envelope has a major effect over pulse spectrum side lobes levels. For an oscillator generating sinewave signal, the pulse waveform p(t) can be expressed as follows:

( ) ( ) ( ) cos 2 M p t x t f t   = - ()
where x(t) represents the pulse envelope, fM the frequency for which the pulse spectrum density is maximum, and φ the sinewave phase. The mono-lateral Fourier transform of the emitted pulse can be written as:

( ) ( ) ( )

1 exp 2 M P f X f f j + = - - ()
where X(f) is the envelope Fourier transform. Three types of envelopes shapes are addressed in this work: rectangular, triangular, and sample & hold triangular. The most trivial to generate, the rectangular envelope based on a rectangular function ΠT(t), thereby expressed as follows:

( )

R T p t x t A    =         ()
where A and τp are the generated pulse amplitude and width, respectively. This envelope performs a rectangular windowing over the signal generated by the local oscillator.

The rectangular envelope Fourier transform is:

( ) ( ) sinc R p p X f A f    = ()
The duration τp of p(t) can be expressed as a function of the oscillation frequency fm as follows:

2 p M N f  = ()
where N is the integer number of sinusoidal half-periods of the generated pulse. However, the minimal pulse width, in relation of the bandwidth BW-XdB, can be written:

min 2 p XdB BW   - - = ()
where α is a coefficient that has a value of 1.8, 3 and 10.8 when the targeted attenuation XdB is 3 dB, 10 dB and 18 dB, respectively. Fig. 1 presents a pulse waveform and spectrum, addressing the channel 3 (BW-3dB = 499 MHz) from IEEE 802.15.4 standard, having a rectangular envelope. From Fig. 1, it can be noticed that the signal spectrum fits into the IEEE 802.15.4 mask, nevertheless the signal bandwidth for an attenuation of 10 dB is not wide enough to be considered as an UWB signal, where the minimum required is 500 MHz or 20% of the carrier frequency according to FCC regulations [START_REF] Hamalainen | UWB Theory and Applications[END_REF].

The triangular envelope is defined by the following expression:

( )

2 1 0 2 0 p T p t A if t x t if otherwise      -        =      ()
The triangular envelope Fourier transform can be computed as:

( ) ( ) 2 sinc T p p X f A f    = ()
For the triangular envelope, the pulse minimum width τp can be calculated in [START_REF] Sparrow | High rate UWB CMOS transceiver chipset for WBAN and biomedical applications[END_REF], however with α with a value of 2.56, 4.49, or 5.68 when the attenuation is 3 dB, 10 dB or 18 dB respectively. Fig. 2 displays a pulse waveform and spectrum, addressing the channel 3 (BW-3dB = 499 MHz) from IEEE 802.15.4 standard, having a triangular envelope.

From Fig 2, it can be noticed that the spectrum from Fig. 2 fits into the IEEE 802.15.4 masks and presents side-lobes with lower power spectrum density than the rectangular A digital realization composed of sampled envelopes arises as an exciting and simpler option because the digital cells are not well-suited to produce analog pulse shapes, even though an analog realization of pulse shaper presents better tuning performance. The sample & hold envelope is a combination of rectangular time shifted functions, such that:

( ) 1 0 1 1 2 2 K S C e e T e e k K K x t x k t k     - = - -     = -   - +          ()
where τe is the duration of an elementary pulse, K the number of elementary pulses, xC(t) a waveform defined in a closed interval from -t0 to t0, that represents a non-discrete envelope [START_REF] Artemio-Schoulten | Pulse Design Method for Complexity Reduction of IEEE IR-UWB Pulse Synthesizers[END_REF]. In this work, the non-discrete envelope is the triangular one. Shifting into frequency domain, the hold & sample envelope Fourier transform can be expressed as:

( ) ( ) 1 0 1 exp sinc 2 K S e e C k e K k X t j f f X f      - =     -   = - -              ()
Fig. 3 presents an emitted pulse waveform and spectrum, addressing the channel 3 from IEEE 802.15.4 standard, having a sampled & hold envelope. From Fig. 3, it can be noticed that the spectrum from Fig. 3 fits into the IEEE 802.15.4 mask, although it presented stronger side-lobes than the triangular envelope. Therefore, the sample & hold envelope is feasible to be implemented in the IR-UWB pulse synthesizer, reducing its complexity design.

III. ASSOCIATED PULSE SYNTHESIZER CIRCUIT DESIGN

A. Pulse synthesizer architecure

The IR-UWB pulse synthesizer design is performed to comply with the FCC regulations and the IEEE 802.15.4 standard. Moreover, it targets programmable capabilities for the pulse discrete amplitude, width, and central frequency, thus being capable to address several standardized channels. This circuit has an architecture based on carrier generation and envelope shaping (Fig. 4). There are three main blocks: The oscillator controls the pulse central frequency by generating the carrier signal, whereas the shaping circuit controls the pulse amplitude and width. The buffer stage duty is to maintain a good integrity of the VCO signal the entire frequency band of interest. The DAC performs a PAM modulation on the carrier signal using an 8-bit bus from the envelope shaping circuit. The modulated signal is the IR-UWB signal, that has a sample & hold envelope and a discrete amplitude. Besides, the proposed architecture allows to and turned-off the carrier generation circuits and the DAC to reduce the power consumption.

B. Envelope control circuit design

Done in full custom CMOS logic, a digital circuit has been designed whose output is an 8-bit data bus, that addresses the DAC, controlling the IR-UWB pulse amplitude level. This control is done through an 8-byte memory composed by parallel shift registers with parallel loading. Fig. 4 shows the main elements of the envelope shaping circuit: an AND logic gate, a fast memory, and a buffer stage, composed by 8 buffers in parallel.

Among the input signals, the pulse data is an 8-byte data bus, where each byte corresponds to one elementary pulse amplitude level. Thus, this signal stocks data for envelope shaping of a pulse composed by 8 sampled pulses. The Fig. 5 shows the chronograms of the envelope control circuit. "D0" to "D7" correspond to sampled pulses amplitude level codified in 8 bits. When no pulse is transmitted, the 8-bit data bus has null value. The AND logic gate control if the external clock signal arrives at the fast memory. If the "ENABLE CLK" has low logic level, the fast memory clock signal has a low logic level, otherwise, it has the same logic level of the external clock signal. The "ENABLE CLK" has a high logic level for 8 clock cycles. The "LOAD BIT PULSES" is the signal that defines when the fast memory loads the 8-byte data bus. The both signals which are the two first signals displayed in Fig. 5 should not have high logic level at same time. A clock period corresponds to one elementary pulse duration. The fast memory design is based on 8 parallel shift registers. The shift registers are loaded in a parallel manner. The bit 0 of each byte is generated using one shift register, the bit 1 using another shift register, successively until the bit 7 that requires the eighth shift register. Fig. 6 shows the 8-bit shift register design. The register is composed by D flip-flops, a buffer, NAND gates and inverters. The first sampled pulse data load the last D flip-flop, whereas the last sampled pulse data the first D flip-flop. For example, considering that "Bit load" has a high logic level, if "dataP7" signal has a high logic level, the first D flip-flop will be set, hence its output has high logic level. After the shift register load, the pulse amplitude data will propagate through the D flip-flops. After 8 clock cycles, all D flipflops will have an output signal with low logic level. Here, all the generated pulses have an envelope sampled from a triangular continuous envelope, with K = 6. The Figs. 7, 8 and 9 shows IR-UWB pulses and its related power spectrum density (PSD), addressing the IEEE 802.15.4 standardized channels 1, 14 and 15, respectively. Table I indicates the IR-UWB pulses parameters for every addressed channel.

In Figs. 7,8, and 9, it can be noticed that the three presented generated pulses respect the IEEE 802.15.4 standard masks. Even though some IR-UWB pulses did not respect the FCC indoor masks, in particular for 0.5-2 GHz bandwidth, this is not a major concern because the UWB antenna filters the signal within this frequency bandwidth. In terms of power consumption, the simulated pulses from The power consumption of the envelope shaping circuit when it generates pulses shown on Figs. 7, 8, and 9 is 16.83 pJ/pulse, 14.17 pJ/pulse and 9.30 pJ/pulse respectively and the overall pulse synthesizer power consumption is 182.35 pJ/pulse, 298.09 pJ/pulse and 120.43 pJ/pulse, respectively. Table II

V. CONCLUSION

In this paper, an IR-UWB pulse synthesizer based on a local oscillator and an envelope shaping has been presented. It is mainly composed by a gated voltage-controlled oscillator, an 8-bit DAC, and an envelope control circuit formed by parallel shift registers. The envelope applied in the pulse synthesizer is the sample & hold, which requires a less complex envelope shaping circuit architecture. The pulse synthesizer has been designed in FD-SOI 28 nm CMOS technology from STMicroelectronics. The simulation results show that this circuit is capable to address the IEEE standardized channels and respect the FCC indoor regulations.

Fig. 1 .

 1 Fig. 1. IR-UWB pulse with rectangular envelope compliant with the channel 3 of IEEE 802.15.4 standard (BW-3dB = 499 MHz and τp = 6.75 ns).

Fig. 2 .

 2 Fig. 2. IR-UWB pulse with triangular envelope compliant with the channel 3 of IEEE 802.15.4 standard (BW-3dB = 1355 MHz and τp = 3.55 ns).

Fig. 3 .

 3 Fig. 3. Example of IR-UWB pulse with sampled & hold envelope compliant with the channel 3 of IEEE 802.15.4 standard (K = 6, BW-3dB = 499 MHz and τp = 4 ns).

  (a) carrier generation (Voltage Controlled Oscillator and buffer stage); (b) envelope shaping; and (c) the digital to analog converter (DAC). The first and last blocks have been done in Current Mode Logic (CML), that is a differential digital logic family, commonly used in high-date rate integrated systems. As a matter of simplicity, the VCO, the buffer stage and the DAC are not detailed in this work. The main focus of this paper is the envelope control circuit.

Fig. 4 .

 4 Fig. 4. IR-UWB synthesizer proposed design architecture.

Fig. 5 .

 5 Fig. 5. Chronograms of the envelope control circuit.

Fig. 6 .

 6 Fig. 6. Schematic of the proposed shift register with parallel loading

Fig. 7 .

 7 Fig. 7. IR-UWB pulse for the channel 1 of the IEEE 802.15.4 standard.

Fig. 9 .

 9 Fig. 9. IR-UWB pulse for the channel 15 of the IEEE 802.15.4 standard.

  presents a state of art comparison of the recent pulse synthesizer designs that have been reported in the literature. The IR-UWB pulse synthesizer presented in this work presents the widest Fc range.

TABLE I

 I

	.	GENERATED IR-UWB PULSE PARAMETERS
	Channel		1		14	15
	VCO voltage control (mV)	500.58 192.88	216.26
	Central Frequency (MHz)	3494	9984	9484.8
	BW-3dB (MHz)		499	499	1355
	Sampled pulse duration (ps)	716	651	256
	Pulse total duration (ps)	4296	3906	1536
	TABLE II.	STATE OF THE ART COMPARISON
		This Work a	[5]			[7]	[8]	[9]
	Techno.	FD-SOI 28 nm	65 nm		130 nm	65 nm	90 nm
	Power Supply (V)	1	1.2			1.2	1.2	0.9
	Fc (GHz)	3-12	6.5-8		3-5	3-5	3.1-5
	BW3dB (GHz)	0.5-1.3	0.5			0.5-1.3	0.5-1.3 0.5-1.4
	Amplitude (V)	0.53	0.45			0.45	0.4	0.09-0.12
	Eac (nJ/pulse)	0.16	0.048		0.146	1.9	NA
	Static power (μW)	27	0.48			100	NA	170
	Architecture	LO based	Edge Combiner	Edge Combiner	LO based	LO based
			a Simulation results with parasitic capacitances