Introduction

The Semiconductor-On-Insulator (SOI) philosophy offers a great control over the transistor channel: the device body is isolated between two oxides: the gate oxide that all classical MOSFETs have and the so-called buried oxide (BOX) [START_REF] Celler | Frontiers of silicon-on-insulator[END_REF]. The first device parameter that was not at all controlled in classic Bulk MOSFETs was the channel thickness. Reducing the thickness of the silicon body provides a better subthreshold slope [START_REF] Wouters | Subthreshold Slope in Thin-Film SOI MOSFET's[END_REF], a very efficient control of the short-channel effects (SCEs), as well as the drain-induced barrier level (DIBL). Furthermore, the SOI structure allows for application of back-bias to control the threshold voltage value. In addition, the threshold voltage variability can be dramatically reduced, since there is no need for strong doping in the channel. Actually, the film is so lightly doped that it is already partially depleted (PD) or fully depleted (FD) even before applying bias voltages. In this way, ultra-thin body (UTB) SOI MOSFETs can be designed and produced, with a silicon film thickness from around 7 to 14 nm. However, as we will later show, the reduction of the film thickness can lead to serious electrostatic coupling effects between the front and back gate. Finally, by reducing the BOX thickness from the typical 145-300 nm value down to 8-25 nm, the device is called ultra-thin body and buried-oxide (UTBB) FDSOI MOSFET [START_REF] Liu | Ultra-thin-body and BOX (UTBB) Fully Depleted (FD) device integration for 22nm node and beyond[END_REF] and offers an easier back-bias threshold voltage control. FD-SOI technology is considered one of the best candidates for Short Channel Effect (SCE) control in future sub-28 nm CMOS generations, while it remains compatible with standard planar CMOS technology [START_REF] Liu | Ultra-thin-body and BOX (UTBB) Fully Depleted (FD) device integration for 22nm node and beyond[END_REF]- [START_REF] Doris | Extreme scaling with ultra-thin Si channel MOSFETs[END_REF]. The use of a midgap/high-k metal gate stack with undoped SOI films allows for great improvement of variability as compared to bulk technology [START_REF] Fenouillet-Beranger | Fully-depleted SOI technology using high-K and single-metal gate for 32nm node LSTP applications featuring 0.179μm2 6T-SRAM bitcell[END_REF], [START_REF] Weber | High immunity to threshold voltage variability in undoped ultra-thin FDSOI MOSFETs and its physical understanding[END_REF], [START_REF] Sugii | Comprehensive study on vthvariability in silicon on thin BOX (SOTB) CMOS with small random-dopant fluctuation: Finding a way to further reduce variation[END_REF]. The use of ultra-thin body and buried oxide thickness (UTBB) also enables to enhance the technology scalability, providing an ideal subthreshold slope and better drain-induced barrier lowering (DIBL), as well as larger backto-front gate coupling effect useful for threshold voltage V th control.

Besides, low frequency noise and RTS fluctuations which scale with the reciprocal device area, become more important with technology scaling down. They are not only limiting the analog circuit operation, but they could also jeopardize the digital circuit functioning. They could even appear as an ultimate variability source due to dynamic carrier trapping in undoped channel devices. For these reasons, the study of LFN and RTN in UTTB FDSOI is a key issue for technology evaluation and evolution.

Furthermore, it has been repeatedly shown that the LF noise in FDSOI devices can be influenced by the coupling effect between the back and front interfaces [START_REF] Simoen | Critical discussion of the front-back gate coupling effect on the low-frequency noise in fully depleted SOI MOSFETs[END_REF]- [START_REF] Zafari | Modelling and Simulation of Coupling Effect on Low Frequency Noise in Advanced SOI MOSFETs[END_REF]. Due to this coupling, it is difficult to predict precisely the contribution of each interface to the measured noise. Moreover, application of a positive or negative bias voltage to the buried oxide can possibly lead to the appearance of either Lorentzian-type noise [START_REF] Simoen | Critical discussion of the front-back gate coupling effect on the low-frequency noise in fully depleted SOI MOSFETs[END_REF], [START_REF] Lukyanchikova | Influence of an accumulation back-gate voltage on the low-frequency noise spectra of fabricated on ELTRAN and UNIBOND wafers[END_REF], [START_REF] Lukyanchikova | Back and front interface related generation-recombination noise in buried-channel SOI pMOSFET's[END_REF], or a significant increase of the flicker noise level [START_REF] Zafari | Low frequency noise in multi-gate SOI CMOS devices[END_REF], [START_REF] Theodorou | Impact of front-back gate coupling on low frequency noise in 28 nm FDSOI MOSFETs[END_REF]. Thus, the analytical study of the noise sources and their dependence on the bias conditions is crucial for both device characterization and noise modeling.

In this chapter, after a brief theoretical introduction on the fluctuation mechanisms in MOSFETs, we present the most important aspects of our recent research work concerning the noise characterization and modeling of FDSOI MOSFETs, as well as the development of circuit noise simulation methods. The aspects of front/back gate coupling effects on both flicker and generationrecombination noise, as well as the LFN variability phenomenon, are covered in detail.

Theoretical background

Origin of low-frequency noise in MOSFETs a. Carrier number fluctuations and correlated mobility fluctuations (CNF/CMF)

Within the classical carrier number fluctuation approach, the fluctuations in the drain current arise from the fluctuations of the inversion charge located at the gate dielectric interface, and which stem from the variations of the interfacial charge due to trapping-detrapping of free carriers into slow dielectric traps. The interface charge variation Q it can be translated into a flat band voltage variation as V fb =-Q it /(WLC ox ), C ox being the gate dielectric capacitance and WL the device area [START_REF] Ghibaudo | On the theory of carrier number fluctuations in MOS devices[END_REF]. A rigorous detailed analysis should also account for the correlated mobility fluctuations µ eff due to the modulation of the Coulomb scattering rate by the interface charge fluctuations. The overall drain current fluctuations therefore become for the linear operation region [START_REF] Hung | A unified model for the flicker noise in metaloxide-semiconductor field-effect transistors[END_REF], [START_REF] Ghibaudo | Improved Analysis of Low Frequency Noise in Field-Effect MOS Transistors[END_REF].

it eff d fb m d Q µ I V g I        (1)
where g m is the transconductance, µ eff is the effective mobility,  is the Coulomb scattering coefficient (10 4 Vs/C for electrons and 10 5 Vs/C for holes [START_REF] Emrani | On the understanding of electron and hole mobility models from room to liquid helium temperatures[END_REF], [START_REF] Bennamane | Mobility coupling effects due to remote Coulomb scattering in thin-film FD-SOI CMOS devices[END_REF]).

This yields a normalized drain current noise given by [START_REF] Liu | Ultra-thin-body and BOX (UTBB) Fully Depleted (FD) device integration for 22nm node and beyond[END_REF]:

  2 2 2 / 1 m d ox eff d m Vfb d Id g I C µ I g S I S            (2)
and to an input-referred gate voltage noise S Vg , at strong inversion as,

𝑆 𝑉𝑔 = 𝑆 𝐼𝑑 𝑔 𝑚 2 = 𝑆 𝑉𝑓𝑏1 [1 + 𝛼𝜇 0 𝐶 𝑜𝑥 (𝑉 𝑔 -𝑉 𝑡 )] 2 (3)
where µ 0 is the low field mobility, V t the threshold voltage and

2 ox Qit Vfb WLC S S  (4)
the flat band voltage power spectral density (PSD), with S Qit being the PSD of the dynamic interface charge per unit area. Note that S Vg is different from S Vfb , especially at strong inversion where CMF prevail.

The PSD of the interface charge is intimately related to the physical trapping mechanism into the gate dielectric. For a tunnelling process (McWorther scheme [START_REF] Mcwhorter | McWorther Proc Physics of Semiconductor Surface[END_REF]), the trapping probability decreases exponentially with dielectric depth, so that the PSD of the flat band voltage reads for a uniform trap depth profile [START_REF] Mcwhorter | McWorther Proc Physics of Semiconductor Surface[END_REF], [START_REF] Christensson | Low frequency noise in MOS transistors-I Theory[END_REF]:

f WLC N kT q S ox t Vfb 2 2   ( 5
)
where f is the frequency, λ is the tunnel attenuation distance (0.1nm in SiO 2 ), kT is the thermal energy and N t is the volumetric trap density (/eVcm 3 ). However, some experiments [START_REF] Nagumo | Statistical characterization of trap position, energy, amplitude and time constants by RTN measurement of multiple individual traps[END_REF], [START_REF] Grasser | On the microscopic structure of hole traps in pMOSFETs[END_REF] show that the time constants are either weakly or not at all correlated with depth. In that case, one has to consider that the trapping probability decreases exponentially with the activation energy of the defect cross section E a , as in thermally assisted trapping processes (Dutta scheme [START_REF] Dutta | Low-frequency fluctuations in solids: 1/f noise[END_REF]). This is also consistent with Random Telegraph Noise data from [START_REF] Nagumo | Statistical characterization of trap position, energy, amplitude and time constants by RTN measurement of multiple individual traps[END_REF], [START_REF] Grasser | On the microscopic structure of hole traps in pMOSFETs[END_REF], [START_REF] Scholten | Noise modeling for RF CMOS circuit simulation[END_REF]. Accounting for such a dependence of τ and E a , the PSD of the flat band voltage takes the form [START_REF] Surya | A thermal activation model for 1/f^γ noise in Si-MOSFETs[END_REF]:

a ox st Vfb E f WLC N T k q S   2 2 2 2 (6)
where E a is the spread of the activation energy and N st is the areal surface trap density (/eVcm 2 ).

The CNF-CMF derivation of Eq. ( 2) can be generalized within the gradual channel approximation to the non-linear MOSFET operation region after integration of the local sheet conductivity  fluctuations along the channel, yielding [START_REF] Boutchacha | Improved modeling of low-frequency noise in MOSFETs -Focus on surface roughness effect and saturation region[END_REF]:

dUc dUc C µ V S Id S d V d V ox eff fb Vfb Id                        0 0 2 2 1 (7)
where =qµ eff .Q i with Q i being the inversion charge and U c is the channel potential.

As illustrated in Figure 1(a), the variation rate of the input gate voltage noise S Vg (V d ) strongly depends on the CMF amplitude factor () and shows clear presence of CMF in the experimental results as in Figure 1(b).

b. Hooge Mobility fluctuations (HMF)

According to the Hooge mobility model [START_REF] Hooge | 1/f noise[END_REF], the drain current noise comes from the fluctuations of the carrier mobility stemming from variations in the scattering probability due to phonon number fluctuations [START_REF] Jindal | Model for mobility fluctuation 1/f noise.pdf[END_REF], [START_REF] Van Vliet | Noise in Semiconductors and Photoconductors[END_REF]. This results in a flicker noise with amplitude inversely proportional to the total number of carriers in the device. The normalized drain current noise in linear operation then reads [START_REF] Hafez | A study of flicker noise in MOS transistors operated at room and liquid helium temperatures[END_REF]: where Q i is the inversion charge and  H is the Hooge parameter (10 -4 -10 -7 ).

In the case of non-linear operation region, it can be shown that the normalized drain current noise becomes after integration along the channel [START_REF] Hafez | A study of flicker noise in MOS transistors operated at room and liquid helium temperatures[END_REF]:

f I L V µ q I S d d eff H d Id 2 2     (9)
where <µ eff > stands for the average mobility along the channel.

As the Hooge mobility fluctuations depend only on the phonon scattering rate [START_REF] Hooge | 1/F Noise Sources[END_REF], the Hooge mobility parameter should be modulated by its contribution among other scattering mechanisms limiting the carrier mobility. Therefore, in the case of a MOSFET, the Hooge parameter should be written as:

2 0 1 1 1 1 .                  SR C ph ph H H µ µ µ µ   ( 10
)
where  H0 refers to the intrinsic Hooge parameter, µ ph , µ C and µ SR are respectively the phonon, Coulomb and surface roughness scattering limited mobilities in the MOSFET inversion layer [START_REF] Takagi | Excess noise generation by carrier fluctuation in semiconductor devices[END_REF].

Accounting for the phonon, Coulomb and surface roughness scattering universal mobility law [START_REF] Takagi | Excess noise generation by carrier fluctuation in semiconductor devices[END_REF], [START_REF] Karatsori | Influence of AC signal oscillator level on effective mobility measurement by split C-V technique in MOSFETs[END_REF] against effective electric field, the dependence of the Hooge parameter given by Eq. ( 10) can be evaluated theoretically with the MOSFET inversion charge from weak to strong inversion as illustrated in Figure 2. It clearly reveals that, in MOSFET, the Hooge parameter is far from being independent of inversion charge i.e. gate voltage while varying from subthreshold to strong inversion region. The Hooge parameter is maximized when the phonon scattering contribution prevails with respect to Coulomb and surface roughness scattering rates. ).

Figure 3 shows the impact of the Hooge parameter dependence with inversion charge (Eq. ( 10)) on the associated normalized drain current noise. In this situation, S Id /I d 2 is no longer simply inversely proportional to the inversion charge as it were the case for HMF model with constant mobility. Therefore, even if the Hooge model is empirical, the α H parameter should not be considered constant with gate bias, due to its modulation by the interface charge Coulomb scattering.

c. Impact of Access resistance

The impact of the access resistance R sd on the LF noise can be obtained by adding to the channel drain current noise the LF noise component arising from the resistance of the source and drain access region. To this end, one should note that the sensitivity of the drain current to R sd variation is simply given by,

2 m d sd d g g R I     (11)
where g d =dI d /dVd is the ouput conductance and assuming R sd =2R s =2R d . Therefore, the extrinsic normalized drain current noise including the access region noise can be expressed as:

  Rsd m d channel d Id d Id S g g I S I S 2 2 2 2            (12)
where S Rsd is the PSD associated to the access resistance. In Eq. ( 12) the channel component is either calculated by the CNF/CMF (2)- [START_REF] Doris | Extreme scaling with ultra-thin Si channel MOSFETs[END_REF] or by the HMF (9) model.

d. Random Telegraph Noise (RTN)

The observation of RTN in small area MOS devices in the 80s was attributed to elementary carrier trapping at the channel-gate dielectric interface [START_REF] Ralls | Discrete Resistance Switching in Submicrometer Silicon Inversion Layers: Individual Interface Traps and Low-Frequency (1/f) Noise[END_REF], [START_REF] Uren | 1/F and Random Telegraph Noise in Silicon Metal-Oxide-Semiconductor Field-Effect Transistors[END_REF]. The drain current RTN was commonly interpreted as a conductance modulation due to the fluctuation in the carrier number with associated mobility fluctuations. The two-level RTN signal of a single trap is statistically characterized by three parameters: i) the average amplitude of the drain current jump I d between the low and high levels, ii) the average time passed on the high level normally associated to the capture time  c and iii) the average time passed on the low level normally related to the emission time  e . In this case, the histogram of the drain current amplitude exhibits two peaks at a distance equal to I d . The histogram of the high and low level times are generally exponentially distributed due to the Poisson process governing the dynamic trapping events [START_REF] Kirton | Individual defects at the Si:SiO2 interface[END_REF]. In the case were more than one traps are active under the same gate bias, the result is a multi-level RTN signal (not discussed here).

The drain current RTN amplitude can be evaluated by considering that the trapping of an elementary charge q from the channel into an oxide defect modulates the local conductivity [START_REF] Ghibaudo | Improved Analysis of Low Frequency Noise in Field-Effect MOS Transistors[END_REF], [START_REF] Kirton | Individual defects at the Si:SiO2 interface[END_REF]. In a first order approximation, the relative drain RTN amplitude accounting for both CNF and CMF is given by [START_REF] Ghibaudo | Improved Analysis of Low Frequency Noise in Field-Effect MOS Transistors[END_REF], [START_REF] Dit Buisson | Model for drain current {RTS} amplitude in small-area {MOS} transistors[END_REF]:

  m d ox eff ox t ox d m d d g I C µ t x WLC q I g I I / 1 1              (13)
where x t is trap depth in the gate dielectric and t ox is the gate oxide thickness. Therefore, the drain current RTN relative amplitude is maximized in the subthreshold region where the transistor gain g m /I d is larger. Furthermore, in order to account for abnormally high RTN amplitudes in dramatically scaled devices, a parameter η should be added to [START_REF] Husseini | New numerical low frequency noise model for front and buried oxide trap density characterization in FDSOI MOSFETs[END_REF] [START_REF] Simoen | On the gateand drainvoltage dependence of the RTS amplitude in submicron MOSTs[END_REF]:

1 1 ds m t ds ds ds ox ox m I g x I q I I WLC t g                ( 14
)
where η is unity for a homogeneous channel in the linear region and Ω=αμ eff C ox the CMF factor [START_REF] Ioannidis | Analytical low-frequency noise model in the linear region of lightly doped nanoscale double-gate metal-oxide-semiconductor field-effect transistors[END_REF].

The large deviation of the parameter  from unity [START_REF] Karatsori | Study of Hot-Carrier-Induced Traps in Nanoscale UTBB FD-SOI MOSFETs by Low-Frequency Noise Measurements[END_REF], [START_REF] Campbell | Large random telegraph noise in sub-threshold operation of nano-scale nMOSFETs[END_REF]- [START_REF] Subirats | Impact of single charge trapping on the variability of ultrascaled planar and trigate FDSOI MOSFETs: Experiment versus simulation[END_REF] could be explained by the huge variability of the threshold voltage associated with single charge trapping, which exhibits an exponential distribution due to the impact of trap location within the gate dielectric and facing the channel [START_REF] Subirats | Impact of single charge trapping on the variability of ultrascaled planar and trigate FDSOI MOSFETs: Experiment versus simulation[END_REF], [START_REF] Takeuchi | Single-charge-based modeling of transistor characteristics fluctuations based on statistical measurement of RTN amplitude[END_REF].

Regarding the trap kinetics, in general, the RTN capture and emission times are governed by the Shockley-Read-Hall statistics [START_REF] Kirton | Individual defects at the Si:SiO2 interface[END_REF] and reads:

th s c v n   1  (a) and th e v n 1 1    (b) (15)
where v th is the thermal velocity,  is the trap cross section, n s is the surface carrier concentration and n 1 is the surface carrier concentration when the Fermi level E f crosses the trap energy E t . The trap cross section might depend on the trap depth into the oxide and on temperature as = 0 exp(-E a /kT).exp(-x t /λ) [START_REF] Kirton | Individual defects at the Si:SiO2 interface[END_REF].

However, when the trap is not located right at the oxide-channel interface, but at a depth x t in the oxide, the apparent trap energy E t depends on the band bending in the gate dielectric as [START_REF] Ralls | Discrete Resistance Switching in Submicrometer Silicon Inversion Layers: Individual Interface Traps and Low-Frequency (1/f) Noise[END_REF], [START_REF] Kirton | Individual defects at the Si:SiO2 interface[END_REF]:

  s fb g ox t t t V V V t x q E E     0 (16)
where V s is the surface potential and V g is the gate voltage. Another way to express this difference is through n 1 in Eq.( 15)(b), if we replace it with:

𝑛 1 = 𝑛 0 𝑒 𝜓 𝑠1 𝑘𝑇 = 𝑛 0 𝑒 𝜓 𝑠1,0 -𝛥𝜓 1 𝑘𝑇 (17)
where ψ s1 is the surface potential for which E t coincides with E f , and Δψ 1 =x t ˑQi/ε ox corresponds to the potential drop across the oxide, from the interface to the trap depth.

It should also be noted that the capture and emission times of Eqs [START_REF] Lukyanchikova | Influence of an accumulation back-gate voltage on the low-frequency noise spectra of fabricated on ELTRAN and UNIBOND wafers[END_REF] are evaluated within the classical statistics i.e. using carrier volumetric concentration at the surface. They have to be updated when quantum mechanical effects become important in the MOSFET inversion layer, since n s is cancelled out at the surface. Indeed, the capture probability is proportional to the escape frequency of the electrons in the quantized sub-band and to the barrier tunnelling transparency to reach the trap in the oxide. If in addition we take into consideration Eq. ( 17), the capture and emission time can be expressed in a way that accounts for the trap depth x t and the single sub-band approximation:

𝜏̅ 𝑐 = 𝑞 𝜎 • 𝑓 𝑒 • 𝑄 𝑖 (𝑎) and 𝜏̅ 𝑒 = 𝑞. 𝑒 𝑥 𝑡 •(𝑄 𝑖 +𝑄 𝑑 ) 𝑘𝑇•𝜀 𝑜𝑥 𝜎 • 𝑓 𝑒 • 𝑄 𝑖1 (𝑏) (18)
where f e is the escape frequency (2.10 13 Hz),  ox is the oxide permittivity, Q i1 the inversion charge when the Fermi level E f crosses the trap energy E t and Q d is the depletion charge. In Figure 4 are illustrated typical variations of the capture and emission times calculated with Eqs (18) as a function of gate voltage for a FDSOI structure and showing the usual huge decrease of  c and the slight increase of  e with V g in strong inversion. It should be emphasized that this formulation [START_REF] Ghibaudo | On the theory of carrier number fluctuations in MOS devices[END_REF] of the capture and emission times will also be of great interest for compact modelling applied to circuit simulation (see section 5).

t ox =2nm, x t =0.05.t ox ,  0 =10 -18 /cm 2 , E a =0).
Furthermore, in strong inversion we can consider Q i = C ox (V g -V t ), so Eqs. [START_REF] Ghibaudo | On the theory of carrier number fluctuations in MOS devices[END_REF] It should be also mentioned that more sophisticated capture and emission time models have been proposed for a better description of the trapping-detrapping dynamics using multi-state defects with multiphonon theory (see [START_REF] Grasser | Stochastic charge trapping in oxides: From random telegraph noise to bias temperature instabilities[END_REF]).

Finally, it is worth recalling that the drain current power spectral density of a two-level RTN follows a Lorentzian spectrum given by [START_REF] Machlup | Noise in semiconductors: Spectrum of a two-parameter random signal[END_REF]:

2 2 2 1 . . 4       d Id I A S ( 20
)
where =(1/ c +1/ e ) -1 is the effective time constant, A=/( c + e )=f t (1-f t ) is the space mark ratio, =2f is the angular frequency and f t is the trap occupancy factor, f t =1/{1+exp[(E t -E f )/kT]}.

The RTN phenomenon in MOSFETs and the various characterization methods are discussed in more detail in Chapters 4 [START_REF] Marquez | Systematic Characterization of Random Telegraph Noise and Its Dependence with Magnetic Fields in MOSFET Devices[END_REF],7 [START_REF] Stampfer | Advanced Electrical Characterization of Single Oxide Defects utilizing Noise Signals[END_REF],14 [START_REF] Martin-Martinez | Advanced characterization and analysis of Random Telegraph Noise in CMOS devices[END_REF] and 18 [START_REF] Rouf | Oxide trap-induced RTS in MOSFETs[END_REF] of this book.

Noise model development and challenges

a. Multi-interface CNF approach

In multi-gate devices, trapping and de-trapping of carriers can occur at multiple channel/oxide interfaces. Therefore, if one applied the CNF model as described in 2.1 for this case, the normalized flicker drain current noise would be expressed as:

𝑆 𝐼𝑑 𝐼 𝑑 2 (𝑚𝑢𝑙𝑡𝑖-𝑔𝑎𝑡𝑒) = ∑ 𝑆 𝑉𝑓𝑏𝑖 ( 𝑔 𝑚𝑖 𝐼 𝑑 ) 2 𝑛 𝑖=1 (21)
where the subscript i refers to each interface, n is the total number of interfaces, and S Vfb is defined by [START_REF] Andrieu | Low leakage and low variability Ultra-Thin Body and Buried Oxide (UT2B) SOI technology for 20nm low power CMOS and beyond[END_REF]. In SOI devices, there are only two flicker noise sources: the interfaces between silicon / front gate oxide (t ox ) and silicon / buried oxide (t BOX ), thus for SOI can be expressed as:

𝑆 𝐼𝑑 𝐼 𝑑 2 (𝑆𝑂𝐼) = 𝑆 𝑉𝑓𝑏1 ( 𝑔 𝑚1 𝐼 𝑑 ) 2 + 𝑆 𝑉𝑓𝑏2 ( 𝑔 𝑚2 𝐼 𝑑 ) 2 (22)
where subscripts 1 and 2 refer to the top and back interface, respectively. It follows that g m1 = dI d /dV G1 and g m2 = dI d /dV G2 are the respective front and back gate transconductance. Then, taking into account that only the oxide capacitance and trap densities differ between S Vfb1 and S Vfb2 , the front input referred noise S VG1 = S Id /g m1 2 can be expressed through:

𝑆 𝑉𝑔1 (𝑆𝑂𝐼) = 𝑆 𝑉𝑓𝑏1 [1 + 𝐶 21 2 𝑁 𝑡2 𝑁 𝑡1 ( 𝐶 𝑜𝑥1 𝐶 𝑜𝑥2 ) 2 ] ( 23
)
where C 21 =g m2 /g m1. If the two oxides are made from the same dielectric material and with the same process, then the approximation N t1 = N t2 can be considered. However, the high-k dielectric used in the front gate to reduce the leakage current degrades the interface quality compared to buried thermal oxide, resulting in N t1 values typically one decade higher than N t2 [START_REF] Theodorou | Impact of front-back gate coupling on low frequency noise in 28 nm FDSOI MOSFETs[END_REF]. Nevertheless, from [START_REF] Mcwhorter | McWorther Proc Physics of Semiconductor Surface[END_REF], it becomes clear that the defining factor for the coupling effect between the two noise sources is the product (C 21 ) 2 (N t2 /N t1 )(C ox1 /C ox2 ) 2 , revealing that the contribution of the back interface to the total drain current noise is not just depending on the back/front trap density ratio, but also on the back/front transconductance and oxide thickness ratios. From [START_REF] Mcwhorter | McWorther Proc Physics of Semiconductor Surface[END_REF] we also conclude that in strong inversion, where g m1 ≈g m2 [START_REF] Theodorou | Low-Frequency Noise Sources in Advanced UTBB FD-SOI MOSFETs[END_REF], the front/back noise coupling only depends on the quality ratio of the two interfaces (N t2 /N t1 ) and the ratio between the buried oxide and channel thicknesses.

To illustrate these effects, we run a series of TCAD simulations in FlexPDE (a fast and reliable partial differential equations solver), for FD-SOI MOSFETs with t Si = 7 nm, t ox1 = 1.5 nm and t ox2 = 25 nm. Some results are shown in Figure 5, where we varied the trap density ratio and in Figure 6, where the simulations were repeated for different back-gate voltage V G2 . As can be seen in Figure 5, when the volumetric trap density values N t1 and N t2 are close to each other, both noise sources -front and bottom-contribute to the total noise level, while if one oxide has a much worse interface quality, its noise becomes dominant. This makes it very difficult to extract N t1 and N t2 from a single set of measurements in a reliable way. However, as shown in Figure 6, the noise intensity of the bottom interface can be controlled by V G2 , resulting in a different total noise level for each back-bias value. Thus, one can extract the characteristic parameter values of each interface combining simultaneously noise measurement data at various bias conditions: front-gate mode, back-gate mode, back-accumulation mode, constant current etc. [START_REF] Zafari | Low frequency noise in multi-gate SOI CMOS devices[END_REF], [START_REF] Theodorou | Low-frequency noise behavior of n-channel UTBB FD-SOI MOSFETs[END_REF].

b. Two-interface CNF/CMF modeling

Now, if we also take into account the contribution from remote Coulomb scattering, we can add the CMF terms in [START_REF] Bennamane | Mobility coupling effects due to remote Coulomb scattering in thin-film FD-SOI CMOS devices[END_REF] and obtain:

𝑆 𝐼𝑑 𝐼 𝑑 2 (𝑆𝑂𝐼) = 𝑆 𝑉𝑓𝑏1 ( 𝑔 𝑚1 𝐼 𝑑 ) 2 (1 + 𝛺 1 𝐼 𝑑 𝑔 𝑚1 ) 2 + 𝑆 𝑉𝑓𝑏2 ( 𝑔 𝑚2 𝐼 𝑑 ) 2 (1 + 𝛺 2 𝐼 𝑑 𝑔 𝑚2 ) 2 (24)
or, expressed as input-referred gate voltage power spectral density [START_REF] Theodorou | Low-Frequency Noise Sources in Advanced UTBB FD-SOI MOSFETs[END_REF]:

𝑆 𝑉𝑔1 (𝑆𝑂𝐼) = 𝑆 𝑉𝑓𝑏1 [(1 + 𝛺 1 𝐼 𝑑 𝑔 𝑚1 ) 2 + 𝐶 21 2 𝑁 𝑡2 𝑁 𝑡1 ( 𝐶 𝑜𝑥1 𝐶 𝑜𝑥2 ) 2 (1 + 𝛺 2 𝐼 𝑑 𝑔 𝑚2 ) 2 ] (25)
It becomes clear from [START_REF] Nagumo | Statistical characterization of trap position, energy, amplitude and time constants by RTN measurement of multiple individual traps[END_REF] that the last parameters determining the total drain current noise are the CMF factors Ω 1 = α sc1 μ eff C ox1 and Ω 2 = α sc2 μ eff C ox2 . Regarding these two factors, a relation between them can be found using the dependence of α sc on the carriers' position. According to [START_REF] Nazarov | Semiconductor-On-Insulator Materials for Nanoelectronics Applications[END_REF] (Chapter 3), α sc depends on the distance x between the inversion charge centroid and the interface through:

𝛼 𝑠𝑐 = 𝛼 0 (1 + x 𝜆 𝑐 ) 2 ( 26
)
where α 0 = 10 5 Vs/C approximately in ultra-thin body SOI devices [START_REF] Theodorou | Impact of front-back gate coupling on low frequency noise in 28 nm FDSOI MOSFETs[END_REF] and λ c =1.2 nm. As obtained by the numerical simulations shown in Figure 7 (after [START_REF] Theodorou | Impact of front-back gate coupling on low frequency noise in 28 nm FDSOI MOSFETs[END_REF]) for a constant drain current, the strong dependence of the charge centroid / interface distance on the back-bias voltage V G2 results in a significant increase of the RCS coefficient  sc -through ( 26)-for negative values of V G2 . This occurs because as the bottom interface goes towards accumulation region, for the same total charge, the centroid moves much closer to the front interface. Considering a distance x from the front interface and t Si -x from the back interface, an expression for the ratio Ω 1 /Ω 2 can be derived as [START_REF] Theodorou | Low-Frequency Noise Sources in Advanced UTBB FD-SOI MOSFETs[END_REF]:

𝛺 1 𝛺 2 = 𝐶 𝑜𝑥1 (1 + 𝑥 𝜆 𝑐 ) 2 𝐶 𝑜𝑥2 (1 + 𝑡 𝑆𝑖 -𝑥 𝜆 𝑐 ) 2 = (1 + 𝑡 𝑆𝑖 -𝑥 𝜆 𝑐 ) 2 (1 + 𝑥 𝜆 𝑐 ) 2 𝑡 𝑜𝑥2 𝜀 𝑜𝑥 𝜀 𝑜𝑥 𝑡 𝑜𝑥1 = ( 𝜆 𝑐 + 𝑡 𝑆𝑖 -𝑥 𝜆 𝑐 + 𝑥 ) 2 𝑡 𝑜𝑥2 𝑡 𝑜𝑥1 (27)
Figure 8 (after [START_REF] Celler | Frontiers of silicon-on-insulator[END_REF]) shows a calculation example of ( 27) for devices with t Si = 8.7 nm, t ox1 = 1.2 nm and t ox2 = 10 nm. This figure gives a rough estimation of Ω 1 and Ω 2 , demonstrating that when the charge is mainly concentrated near the front interface, then Ω 1 ≫Ω 2 . Thus, Ω 2 can be considered as negligible when the device is biased in front-gate (FG) mode without any back-bias. [START_REF] Theodorou | Low-Frequency Noise Sources in Advanced UTBB FD-SOI MOSFETs[END_REF] Therefore, for the FG mode and V G2 = 0 V, ( 25) can be approximated as:

Figure 8. CMF factors ratio Ω 1 /Ω 2 versus carriers distance from front interface

𝑆 𝑉𝑔1 (𝑆𝑂𝐼) = 𝑆 𝑉𝑓𝑏1 [(1 + 𝛺 1 𝐼 𝑑 𝑔 𝑚1 ) 2 + 𝐶 21 2 𝑁 𝑡2 𝑁 𝑡1 ( 𝐶 𝑜𝑥1 𝐶 𝑜𝑥2 ) 2 ] (28)
which, for fully depleted channel (ultra-thin and lightly doped), reduces to the final CNF/CMF model approach for FD-SOI MOSFETs for the FG mode:

𝑆 𝑉𝑔1 (𝐹𝐷-𝑆𝑂𝐼) = 𝑆 𝑉𝑓𝑏1 [(1 + 𝛺 1 𝐼 𝑑 𝑔 𝑚1 ) 2 + 𝑁 𝑡2 𝑁 𝑡1 ( 1 1 + 𝐶 𝑜𝑥2 /𝐶 𝑆𝑖 ) 2 ] (29)
Conversely, for back-gate (BG) mode and V G1 = 0 V, we obtain:

𝑆 𝑉𝑔2 (𝐹𝐷-𝑆𝑂𝐼) = 𝑆 𝑉𝑓𝑏2 [(1 + 𝛺 2 𝐼 𝑑 𝑔 𝑚2 ) 2 + 𝑁 𝑡1 𝑁 𝑡2 ( 1 1 + 𝐶 𝑜𝑥1 /𝐶 𝑆𝑖 ) 2 ] (30)
The above equations ( 29) and ( 30) can be used to fit experimental data and extract the noise parameters N t1 , N t2 and Ω 1 and Ω 2 . The need of combining the results from both the FG and BG modes is demonstrated through TCAD simulations that illustrate the carrier concentration distribution in the silicon film channel [START_REF] Theodorou | Low-Frequency Noise Sources in Advanced UTBB FD-SOI MOSFETs[END_REF]. As shown in Figure 9(a), while the total charge is exactly the same, one can observe a lack of symmetry in the distribution of the charge, depending on FG or BG operation mode. In FG mode it is almost uniformly distributed across the channel, whereas for BG mode most of the charge is concentrated near the bottom interface. This means that measuring in BG mode we have a better de-coupling, leading to an easier extraction of N t2 and Ω 2 , which then can be used in [START_REF] Scholten | Noise modeling for RF CMOS circuit simulation[END_REF] to extract N t1 and Ω 1 using the FG mode results. An experimental confirmation of this effect is shown in Figure 9(b) [START_REF] Theodorou | Low-Frequency Noise Sources in Advanced UTBB FD-SOI MOSFETs[END_REF], where a reduction of the coupling factor (t ox1 g m1 /t ox2 g m2 ) 2 was measured in BG mode, compared to FG mode.

For practical reasons and simplicity of use in SPICE simulation, it is very helpful to transfer all the noise sources to the front gate, including the bottom interface component, as it was implemented in [START_REF] Ioannidis | Drain-Current Flicker Noise Modeling in nMOSFETs From a 14-nm FDSOI Technology[END_REF]. To this end, one could define the normalized flicker noise using a classic CNF/CMF model equation as in Eq. ( 2), but with front gate input-referred effective bias-dependent values of S Vfb,eff and Ω eff :

2 m1 d 2 1 , 2 ) g . 1 .( .             eff d m eff Vfb d Id I g S I S ( 31
)
S Vfb,eff can be obtained from weak inversion region limits where CNF dominates as [START_REF] Ioannidis | Drain-Current Flicker Noise Modeling in nMOSFETs From a 14-nm FDSOI Technology[END_REF]:

2 1 , , 2
ox eff t eff Vfb fWLC kTN q S   (32)
where N t,eff is the effective slow oxide trap volumetric density, related the front and back interface trap densities via the coupling factor as 𝑁 𝑡,𝑒𝑓𝑓 = 𝑁 𝑡1 + 𝐶 [START_REF] Emrani | On the understanding of electron and hole mobility models from room to liquid helium temperatures[END_REF] 2 𝑁 𝑡2 Ω 𝑒𝑓𝑓 on the other hand can be obtained from strong inversion region limits where CMF prevails as [START_REF] Ioannidis | Drain-Current Flicker Noise Modeling in nMOSFETs From a 14-nm FDSOI Technology[END_REF]: [START_REF] Jindal | Model for mobility fluctuation 1/f noise.pdf[END_REF] where S Vfb2 /S Vfb1 =(t 2 ox2 .N t2 / t 2 ox1 .N t1 ) revealing the effect of the front and back gate oxide thickness and volumetric trap density ratio on the total noise level. These equations indicate that the effective parameters S Vfb,eff and Ω eff are univocally related to both front and back interface parameters and coupling factor.

                      1 2 2 21 2 2 1 2 2 1 . 1 . Vfb Vfb Vfb Vfb eff S S C S S
The model in [START_REF] Boutchacha | Improved modeling of low-frequency noise in MOSFETs -Focus on surface roughness effect and saturation region[END_REF], combined with [START_REF] Hooge | 1/f noise[END_REF] and [START_REF] Jindal | Model for mobility fluctuation 1/f noise.pdf[END_REF], can be easily used to fit drain current flicker noise data from various FD-SOI technology nodes, extract Ω eff and N t,eff, and use these parameters to perform accurate circuit noise simulations.

Noise characterization in FD-SOI MOSFETs

Flicker noise in FD-SOI MOSFETs

a. Front/back coupling effects

As explained in Section 2, a very important parameter for the flicker noise analysis is the squared transistor gain (g m1,2 /I D ) 2 . In front gate mode, g m1 can be directly measured from the transfer characteristic at constant V G2 as g m1 = dI d /dV G1 , whereas a supplementary static measurement under a nearby back-gate bias (i.e. V G2 = 50 mV for 10 nm BOX) is needed to extract g m2 = dI d /dV G2 at each value of V FG . For the back-gate mode, the inverse procedure has to be followed. In Figure 10, (g m1,2 /I D ) 2 are plotted for various back-bias values as a function of drain current in 28 nm FD-SOI MOSFETs [START_REF] Theodorou | Impact of front-back gate coupling on low frequency noise in 28 nm FDSOI MOSFETs[END_REF]. As usual, (g m /I D ) 2 exhibits a plateau in weak inversion, before dropping above threshold in strong inversion. The top gate gain factor seems to be affected by V G2 only for high positive values, where the position of the inversion channel is modified. However, (g m2 /I d ) 2 depends strongly on V G2 , becoming negligible for very negative back-gate bias, where accumulation is achieved. The ratio (coupling factor) between the two gain factors is plotted in Figure 11. As can be seen, it lies around 100 when the bottom interface is in depletion and weak inversion regimes (V G2 > 0 V), whereas it increases up to almost 10 4 for strong accumulation at the bottom (V G2 ≪ 0 V). This feature indicates that the bottom interface LF noise contribution could be eliminated in the latter situation, allowing for an easier extraction of the front interface noise parameters. One can also notice that c 1 is almost independent of I D in the depletion and weak inversion region. Nevertheless, when comparing the drain current PSD at different back-bias voltages (Figure 12), we conclude that a high negative V G2 value can cause an increase of the flicker noise level, whereas a positive one the opposite. The influence of the back interface coupling effect on the front gate operation LF noise can be better analysed by plotting the normalized drain current noise, measured at a constant drain current, as a function of back gate voltage as shown in Figure 13(a). As can be seen, the normalized drain current noise is significantly higher when V G2 takes high negative value. It should also be noted, by comparing Figure 11 and Figure 13(a), that the normalized drain current noise has a similar behaviour with V G2 as that of the noise coupling factor c 1 . This type of noise dependence on the back-bias voltage was also confirmed in [START_REF] Pirro | RTN and LFN Noise Performance in Advanced FDSOI Technology[END_REF] for the 22 nm FD-SOI node (Figure 13(b)).

Figure 13. a) Comparison of experimental normalized drain current PSD versus V G2 results with the CNF and CNF/CMF model results, for samples from STMicroelectronics (after [17]), b) Percentage change of S Vg median over 30 dies versus back-bias

(after [START_REF] Pirro | RTN and LFN Noise Performance in Advanced FDSOI Technology[END_REF]).

If only the CNF model of Eq. ( 22) is taken into account without CMF, this behavior cannot be explained, because neither (g m1 /I D ) 2 nor (g m2 /I D ) 2 is increasing for negative V G2 . However, the CNF LF noise model applied in the positive V G2 range allows us to solve Eq. ( 22) considering negligible CMF contribution at low drain currents. By combining different bias conditions one can extract constant values for S Vfb1 and S Vfb2 , from which the front and back gate oxide trap densities can be determined using Eq. ( 5) (N t1 = 910 17 cm -3 eV -1 and N t2 =210 17 cm -3 eV -1 for the 28 nm FDSOI in [START_REF] Theodorou | Impact of front-back gate coupling on low frequency noise in 28 nm FDSOI MOSFETs[END_REF]). Note that the front interface shows higher oxide trap density as expected due to the high-k/metal gate stack, whereas the back interface trap density is lower due to the higher quality of a pure thermal oxide.

In order to interpret the abnormal LF noise behavior in the negative V G2 range, one has to account for the CMF dependence on the carriers position, through the influence of V G2 on the remote Coulomb scattering (RCS) parameter  sc , as already explained through Eq. ( 26) and Figure 7 [START_REF] Theodorou | Impact of front-back gate coupling on low frequency noise in 28 nm FDSOI MOSFETs[END_REF]. Using this RCS formulation and choosing values of  sc that correspond to the dependence shown in Figure 7, the variation of noise with the back gate voltage can be well approached (see fitting lines in Figure 12(a) and Figure 13(a)).

This CMF dependence on back-bias has been also confirmed for 14 nm FDSOI node MOSFETs of STMicroelectronics [START_REF] Ioannidis | Drain-Current Flicker Noise Modeling in nMOSFETs From a 14-nm FDSOI Technology[END_REF]. As shown in Figure 14(a), it was found that Ω eff (see Eq. ( 31)) increases for negative back gate bias and tends to very small values for positive ones. This behavior cannot be interpreted using constant RCS coefficients α sc1,2 occurring in Eq. [START_REF] Jindal | Model for mobility fluctuation 1/f noise.pdf[END_REF]. Based on the carrier centroid dependence of α sc1,2 (see Figure 7), it has been proposed to generalize the CMF coefficients under the symmetric forms [START_REF] Ioannidis | Drain-Current Flicker Noise Modeling in nMOSFETs From a 14-nm FDSOI Technology[END_REF]:

  1 2 2 1 1 1 1 1 . . . . . . c ox eff sc c ox eff sc X C µ X C µ       (34) and   2 2 2 2 1 1 2 1 . . . . . . c ox eff sc c ox eff sc X C µ X C µ       (35)
where X c1,2 are the normalized carrier centroid positions for the front and back interface, respectively (X c =1 at the front interface and X c =0 at the back interface). Using these expressions in Eq. ( 33) and extracting the centroid position by solving Poisson's equation with TCAD simulations allows obtaining the simulated trend in Figure 14(a), justifying reasonably well the experimental data behavior. As already explained in 2.2.b, the combination of front and back gate modes of operation can be very useful for achieving different coupling conditions in order to modify the noise contribution of each interface and help the parameter extraction procedure. Indeed, as shown in Figure 15(a) [START_REF] Theodorou | Low-frequency noise behavior of n-channel UTBB FD-SOI MOSFETs[END_REF], in the sub-threshold region (g m1 /I D ) 2 decreases in changing from front-gate (V BG = 0 V) to back-gate (V FG = 0 V) mode of operation, whereas (g m2 /I D ) 2 shows a relative increase between FG and BG mode of operation as shown in Figure 15(b). The above behavior is typical for asymmetrical double-gate devices such as UTTB FDSOI MOSFETs [START_REF] Skotnicki | Competitive SOC with UTBB SOI[END_REF], related to the different channel position in the silicon film under various bias conditions. A more direct way for an experimental representation of the carrier distribution at different bias conditions is the plot of the coupling factor C = (g m1 /g m2 ) 2 as a function of the drain current as described in the previous section. The dramatic reduction of C under the backgate mode of operation (Figure 15 (c)) clearly shows that in that case the carriers have a higher concentration near the Si/BOX interface. This can be taken advantage of and combine measurements from both front and back gate modes of operation to extract a unique set of parameters that fits all data. An example is shown in Figure 16, where the noise parameters S Vfb1 , S Vfb2 , Ω 1 and Ω 2 were obtained by fitting Eq. ( 24) simultaneously with all 1/f noise data for the front gate mode (V G2 = 0 V and -5 V) and back gate mode (V G1 = 0V). Otherwise, fitting the noise data for a specific bias condition (for example V G2 = 0 V), may lead to a numerous number of model parameters. The difference between the 'Measured (f=10Hz)' and '1/f component' seen in Figure 16 means that around 10 Hz there is a strong presence of Lorentzian-like noise, which needs to be removed in order to extract the flicker noise contribution (more on that in 3.2). Furthermore, one can observe the dependence on the bias conditions of the Si/BOX interface contribution to the total noise level. In front-gate mode, there is sufficient de-coupling of the two interfaces in the sub-threshold region, whereas in strong inversion the back interface affects the noise level significantly. In back-gate mode, as expected, there is a strong influence on the noise of both back and front interfaces. Moreover, the involvement of the correlated mobility fluctuation is also bias dependent. In front-gate mode, the CMF takes place only at the front interface. Respectively, when the back gate bias voltage V BG is varying, the CMF appears only at the back interface. This behavior confirms the analysis regarding the Ω 1 and Ω 2 CMF factors, as already explained in Paragraph 11 through Eq. ( 27) and Figure 8. The same technique was also used successfully in [START_REF] Theodorou | Low-Frequency Noise Sources in Advanced UTBB FD-SOI MOSFETs[END_REF] (see Figure 17), for both n-and p-channel FD-SOI MOSFETs.

b. Impact of channel geometry on RCS

In Figure 18(a)-(c) [START_REF] Theodorou | Low-Frequency Noise Sources in Advanced UTBB FD-SOI MOSFETs[END_REF], the plotted model lines have been derived using (29) as a fitting function. It is clear from Figure 18(a)-(b) that for both 20 nm node n-and p-channel FDSOI devices of constant gate length and varying width, there is a strong dependence of Ω 1 on the gate width W. However, this does not seem to be the case for devices of constant gate width. As shown in Figure 18(c), in devices with constant gate width and different gate lengths, Ω 1 is independent on the gate length. This behavior is summarized in Figure 18(d), where the extracted values of Ω 1 are plotted versus the gate width W. It is obvious that for both n-and p-channel devices, the CMF factor Ω 1 is increasing as W decreases. This effect can be attributed to the decrease of the channel cross-section, as W is reduced, which leads to a shorter average distance between carriers and interface. Furthermore, higher values of Ω 1 are clearly observed for the p-channel devices, which are related to the higher values of RCS coefficient α sc observed for holes [START_REF] Bennamane | Mobility coupling effects due to remote Coulomb scattering in thin-film FD-SOI CMOS devices[END_REF]. This lack of dependence of the CMF on the channel length was not, however, confirmed for the 14 nm FDSOI node MOSFETs of STMicroelectronics, studied in [START_REF] Ioannidis | Drain-Current Flicker Noise Modeling in nMOSFETs From a 14-nm FDSOI Technology[END_REF]. As shown in Figure 19(a), the Ω eff parameter of Eq. ( 31) was found to increase in long channel devices. This trend is primarily related to the strong mobility dependence on channel length [START_REF] Shin | Magnetoresistance mobility characterization in advanced FD-SOI n-MOSFETs[END_REF] that also follows a similar tendency, as shown in Figure 19(a). The small deviation can be attributed to the mobility degradation taking place in strong inversion, which was not taken into account. Nevertheless, contrary to the CMF dependence on channel geometry, no such impact was observed for the CNF factors S Vfb1 and S Vfb2 , meaning that the oxide trap density can be considered constant with channel width and length, for both top and bottom interfaces. This is clear in Figure 18(a)-(c), where S VG1 WL is constant with W and L in weak inversion, but also in Figure 14(b) and Figure 19(b), where the extracted S Vfb,eff , normalized by the surface, is proven to stay constant with W and L, respectively. For this effect of L on Ω eff , as well as the impact of V b (V G2 ) a unified empirical equation that can fit to all data (as shown in Figure 20) was proposed [START_REF] Ioannidis | Drain-Current Flicker Noise Modeling in nMOSFETs From a 14-nm FDSOI Technology[END_REF]:

) / 1 ( ) , ( 0 / 0 0 L L e L V V V b eff b       ( 36
)
where Ω 0 = 6.5 V -1 , V 0 = 3.42 V -1 and L 0 = 0.12μm are some fitting parameters, with values that were extracted for the 14nm FDSOI technology node devices in [START_REF] Ioannidis | Drain-Current Flicker Noise Modeling in nMOSFETs From a 14-nm FDSOI Technology[END_REF]. In another study [START_REF] Theodorou | Low-Frequency Noise Sources in Advanced UTBB FD-SOI MOSFETs[END_REF], a comparative parameter extraction was done for three different FDSOI wafers. Table I presents the characteristic manufacturing details of the wafers, while Table II the extracted noise parameters. As shown, N t1 has almost the same value for all wafers, whereas N t2 depends slightly on t BOX . In all wafers, the same front oxide high-k metal gate technology is used in terms of materials and thicknesses, explaining the constant N t1 values. However, the thickness and the quality of the buried oxide change from wafer to wafer, thus affecting the value of N t2 . Moreover, the CMF factor Ω 1 =α sc1 μ eff C ox was found to decrease with the thickness of the silicon body, due to the increase in the average distance between interface-inversion charge centroid, as explained in 2.2.b.

Generation-recombination noise

The physical phenomenon that results in a Lorentzian type of spectrum is the carrier generationrecombination occurring when:

a free electron and a free hole recombine, or a pair of a free electron and a free hole are generated, or a free electron is trapped at an empty trap, or a free hole is trapped at a filled trap.

The Random Telegraph Noise (explained in 2.1.d) is a case of g-r noise, for which there is only one discrete trap and not a group of defects.

In general, when the noise spectra are composed of 1/f and Lorentzian noise components due to several distinct and uncorrelated trap levels, the total current noise power spectral density S Id can be written as:

  2 1 . 1/ d N f i I i ci K A S f ff     ( 37
)
where K f is the amplitude of the flicker noise, A i is the Lorentzian plateau value and f ci is the corner frequency of each Lorentzian noise spectrum. Such an example can be seen in Figure 21, where F corresponds to K f , L i to A i and T+B to the sum of thermal and background noise.

The origin of the g-r noise can be clarified by determining the dependence of the Lorentzian time constant (τ = 1/2πf C ) on the gate bias. When the time constant remains unchanged with gate voltage, the g-r centers are uniformly distributed within the depletion region of the silicon body, whereas when it passes through a maximum, the g-r noise centers are defects located at the interface [START_REF] Lukyanchikova | Non-trivial GR and 1/f noise generated in the p-Si layer of SOI and SOS MOSFETs near the inverted front or buried p-Si/SiO2 interface[END_REF] or inside the oxide region. A recent study has shown that in fully-depleted devices the g-r traps positioned in the Si film can have a misleading time constant behavior, due to its dependence on the carrier concentration above a gate voltage level [START_REF] Rodríguez | Dependence of generation-recombination noise with gate voltage in FD SOI MOSFETs[END_REF]: the time constant maintains the same value as long as the Fermi level is below the trap energy level and τ starts decreasing with increasing gate voltage when the Fermi level exceeds the trap energy level.

Figure 21. Example of spectrum decomposition for a MOSFET drain current PSD measurement

In Figure 22(a) [START_REF] Theodorou | Low-frequency noise behavior of n-channel UTBB FD-SOI MOSFETs[END_REF], one can clearly notice that when V BG = 0 V, the total noise is composed of a flicker noise and a Lorentzian component. A direct way to ascertain if the g-r component is current dependent is to plot noise spectra, obtained at a constant drain current under different bias conditions, as shown in Figure 22(b). It is evident that the g-r component is mainly dependent on the carriers' position and not on the total channel charge, whereas the flicker noise component seems to change slightly. In addition, from Figure 22(b) one can safely conclude that the specific Lorentziantype noise is not related to the front gate interface.

After decomposing all spectra into their noise components, the Lorentzian noise parameters (time constants and plateau values) were extracted. In Figure 23 (after [START_REF] Theodorou | Low-frequency noise behavior of n-channel UTBB FD-SOI MOSFETs[END_REF]) these parameters are plotted as a function of the front and back gate voltages under back and front gate modes of operation, respectively, adjusted to correspond to the same drain current range. One can notice the different trend of the two sets of data. The time constant is not drain current related, but it is defined by the applied bias voltages. This finding leads to the conclusion that the time constant is related to the distribution of the carriers in the channel and not with the total charge. As shown in Figure 23(a), for V BG = 0 V there is no voltage dependence of the time constant and the g-r plateau reaches a constant value above threshold. However, in back-gate mode (Figure 23 (b)) and in the voltage range above the maximum value of S g-r (0), the time constant decreases exponentially with increasing V BG . The results of Figure 23 can be explained considering that in FDSOI MOSFETs there is almost no depletion layer in the Si film when the inversion channel is created, as simulations have shown in [START_REF] Zafari | Low frequency noise in multi-gate SOI CMOS devices[END_REF], [START_REF] Rodríguez | Dependence of generation-recombination noise with gate voltage in FD SOI MOSFETs[END_REF]. If the electron density n 1 (see Eq. ( 15)(b)) at the position where the Fermi level is coinciding with the trap energy level in the Si film is higher than the density n of carriers all across the channel, the time constant of the g-r noise is dependent only on n 1 [START_REF] Rodríguez | Dependence of generation-recombination noise with gate voltage in FD SOI MOSFETs[END_REF], thus it remains unchanged as shown in Figure 23(a). When n becomes comparable or larger than n t , the time constant becomes inversely proportional to n, leading to the behavior of Figure 23(b). At low back gate voltages τ is almost constant, but when S g-r (0) reaches a maximum value the Fermi level approaches the trap energy level. After this point, the time constant decreases following the decrease of the g-r noise plateau. This observation leads to the conclusion that these defects are located inside the silicon channel, but near the bottom silicon/buried oxide interface. This explains why τ stays constant with the front gate voltage.

Noise variability a. General properties of LFN variability

The reduction of the device area in advanced CMOS technologies, such as the FD-SOI structure, can lead to important issues regarding the device performance variability [START_REF] Frank | Design considerations for CMOS near the limits of scaling[END_REF]. As far as the low frequency noise (LFN) is concerned, one of its main properties is the scaling with the reciprocal of the device area [START_REF] Mcwhorter | McWorther Proc Physics of Semiconductor Surface[END_REF] (Figure 24), which has become a major concern for both analog and digital circuits operation, such as oscillators, RAM cells, inverters and other mixed signal circuits [START_REF] Takeuchi | Single-charge-based modeling of transistor characteristics fluctuations based on statistical measurement of RTN amplitude[END_REF]. Furthermore, the nanoscale-induced uncertainty in number of traps, the random telegraph noise (RTN) presence, common in small-area MOSFETs (<1μm 2 ) [START_REF] Dit Buisson | Model for drain current {RTS} amplitude in small-area {MOS} transistors[END_REF], as well as generation-recombination (GR) noise centers in the Si film [START_REF] Simoen | Towards single-trap spectroscopy: Generation-recombination noise in UTBOX SOI nMOSFETs[END_REF] directly lead to enormous levels of LFN variability (Figure 24) that further limit the device performance and reliability and introduce a high inaccuracy in the prediction of circuit noise levels. Thus, a detailed study of the advanced FD-SOI LFN variability is crucial in order to better understand its origin and by turn its prediction and/or reduction. An example of noise variability experimental results is shown in Figure 25 (after [START_REF] Theodorou | Low frequency noise variability in ultra scaled FD-SOI n-MOSFETs: Dependence on gate bias, frequency and temperature[END_REF]), where S Vg is plotted versus frequency for all measured devices across a 300 mm wafer biased under the same gate voltage bias V G = 0.4 V. Notice that few spectra are 1/f-like, but there is also a significant number of Lorentzian-like spectra, commonly observed in such small area (W.L = 0.0012 μm 2 ) devices, due to Random Telegraph Noise (RTN), or due to generation-recombination (GR) centers located in the Si film [START_REF] Rodríguez | Dependence of generation-recombination noise with gate voltage in FD SOI MOSFETs[END_REF], [START_REF] Santos | Low-frequency noise assessment in advanced UTBOX SOI nMOSFETs with different gate dielectrics[END_REF]. From Figure 25 (after [START_REF] Theodorou | Low frequency noise variability in ultra scaled FD-SOI n-MOSFETs: Dependence on gate bias, frequency and temperature[END_REF]), it is clear that the noise variability is strongly enhanced when Lorentzian noise is present. [START_REF] Theodorou | Low frequency noise variability in ultra scaled FD-SOI n-MOSFETs: Dependence on gate bias, frequency and temperature[END_REF]).

Figure 25. Input-referred gate voltage noise spectra at V G = 0.4 V for 85 dies (colored lines) and geometric mean spectrum (black line) for 14 nm FD-SOI MOSFETs with L = 20 nm and W = 60 nm (after

Based on Eqs. ( 13) and ( 20), without taking into account the CMF component, this dispersion in the noise spectra can be expressed mathematically through [START_REF] Ghibaudo | Impact of Scaling Down on Low Frequency Noise in Silicon MOS Transistors[END_REF], [START_REF] Ioannidis | Low frequency noise variability in high-k/metal gate stack 28nm bulk and FD-SOI CMOS transistors[END_REF]: [START_REF] Karatsori | Influence of AC signal oscillator level on effective mobility measurement by split C-V technique in MOSFETs[END_REF] where τ refers to the effective trap time constant, A=f t (1-f t ) is the space mark ratio, x t the trap depth and N is the number of traps in the gate oxide within the energy range swept by the Fermi level (N=WˑLˑt ox ˑN t ˑΔE f , N t being the volumetric oxide trap density and ΔE f the Fermi level excursion). Therefore, since each device has a different number of traps N (i.e. RTN fluctuators) and each trap its own x t , τ and A, an induced variability increase is expected as N decreases dramatically, which is the case in nano-scale devices. This effect is further demonstrated in Figure 26 (after [START_REF] Ioannidis | Low frequency noise statistical characterization of 14nm FDSOI technology node[END_REF]), where it is shown that the noise level dispersion is enhanced by 2-3 decades going from the large (Figure 26(a)) to the small (Figure 26(b)) area device. Furthermore, the shape of the spectrum is noticeably changing from 1/f to more Lorentzian like behavior, indicating the impact of RTN in LFN variability. Furthermore, as already shown in [START_REF] Ioannidis | Impact of low-frequency noise variability on statistical parameter extraction in ultra-scaled CMOS devices[END_REF]- [START_REF] Morshed | Compact modeling of flicker noise variability in small size MOSFETs[END_REF], the drain current noise data of ultra-scaled devices follow a log-normal distribution, thus the geometric mean (or, alternatively, the log-mean) spectrum can better represent the average noise:

𝑆 𝑉𝑔,𝑖 = 𝑞 2 (𝑊𝐿𝐶 𝑜𝑥 ) 2 ∑ (1 - 𝑥 𝑡,𝑘 𝑡 𝑜𝑥 ) 4𝐴 𝑘 𝜏 𝑘 1 + 𝜔 2 𝜏 𝑘 2 𝑁 𝑘=1
〈𝑆〉 𝑔𝑒𝑜𝑚. = √𝑆 1 𝑆 2 𝑆 3 ⋯ 𝑆 𝑛 𝑛 (39)
Indeed, as can be seen in Figure 25, the geometric mean spectrum has a clear 1/f behavior, that can be used for both interface characterization and noise modeling after the correction given in [START_REF] Ioannidis | Impact of low-frequency noise variability on statistical parameter extraction in ultra-scaled CMOS devices[END_REF]. Thus, in this section we always consider a geometric mean average in the noise variability analysis.

Finally, in order to quantify the LFN variability effect and be able to make a variability parameter comparison between different technologies, one needs to calculate the standard deviation of the surface-normalized noise at a certain extraction frequency (usually at 10 Hz) [START_REF] Ioannidis | Low frequency noise variability in high-k/metal gate stack 28nm bulk and FD-SOI CMOS transistors[END_REF]:

𝐷𝑟𝑎𝑖𝑛 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝐿𝐹𝑁 𝑣𝑎𝑟𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦: 𝜎 (log (𝑊𝐿 𝑆 𝐼𝑑 𝐼 𝑑 2 )) (40)
or, even better, in order to eliminate the influence of the subthreshold swing or threshold voltage variability in the LFN variability estimation, one can use [START_REF] Theodorou | Low frequency noise variability in ultra scaled FD-SOI n-MOSFETs: Dependence on gate bias, frequency and temperature[END_REF]:

𝐼𝑛𝑝𝑢𝑡 -𝑟𝑒𝑓𝑒𝑟𝑟𝑒𝑑 𝑔𝑎𝑡𝑒 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝐿𝐹𝑁 𝑣𝑎𝑟𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦: 𝜎 (log (𝑊𝐿S 𝑉 𝑔 )) [START_REF] Kirton | Individual defects at the Si:SiO2 interface[END_REF] where S Vg =S Id /g m 2 is the input-referred gate voltage noise.

b. Variability comparison between different technology nodes

Due to the fact that the trap number variation is a local effect, if the normalized LFN standard deviation (Eq. ( 40) or ( 41)) is plotted versus 1/√𝑊𝐿, a Pelgrom's law [START_REF] Pelgrom | Matching Properties of MOS Transistors[END_REF] trend is observed, from which one can extract a noise matching parameter A. An example is presented in Figure 27 (after [START_REF] Ioannidis | Low frequency noise variability in high-k/metal gate stack 28nm bulk and FD-SOI CMOS transistors[END_REF]), where the slope A seems to be changing, depending on the technology node.

Following this methodology, the noise variability in of the FDSOI technology [START_REF] Weber | 14nm FDSOI technology for high speed and energy efficient applications[END_REF] was found to be improved as compared to advanced bulk technology nodes, as shown in Figure 27 (after [START_REF] Ioannidis | Low frequency noise variability in high-k/metal gate stack 28nm bulk and FD-SOI CMOS transistors[END_REF] and [START_REF] Ioannidis | Low frequency noise statistical characterization of 14nm FDSOI technology node[END_REF]). This was further verified in [START_REF] Ioannidis | Low frequency noise statistical characterization of 14nm FDSOI technology node[END_REF], where the 14nm-node FDSOI technology [START_REF] Weber | 14nm FDSOI technology for high speed and energy efficient applications[END_REF] was found to be improved as compared to both 20 nm Bulk and 28 nm FDSOI nodes, as shown in Figure 28 (after [START_REF] Ioannidis | Low frequency noise statistical characterization of 14nm FDSOI technology node[END_REF]). Interestingly, the noise matching parameter A is following the same trend as the product of the extracted oxide trap density and the squared equivalent oxide thickness, N t ×EOT 2 , indicating that the use of a thinner oxide and better quality interface are the main reasons for this noise variability improvement [START_REF] Ioannidis | Evolution of low frequency noise and noise variability through CMOS bulk technology nodes from 0.5 ??m down to 20 nm[END_REF].

c. Statistical LFN/RTN characterization methods

As shown in Figure 26, when going from large to smaller area devices, not only we induce an increase in noise variability, but there is a change in the spectral shape, from 1/f-like to more Lorentzian-like, because the average number of active traps is decreased. Thus, the CNF/CMF flicker noise model, needed for the extraction of N t , is very difficult to be applied in nano-scale MOSFETs, due to the lack of uniformity in the oxide trap distribution, which results in non-flicker noise spectra. However, if the statistical sample is large enough, the average noise spectrum will be 1/f-like (see Figure 25) [START_REF] Scholten | Noise modeling for RF CMOS circuit simulation[END_REF], [START_REF] Theodorou | Low frequency noise variability in ultra scaled FD-SOI n-MOSFETs: Dependence on gate bias, frequency and temperature[END_REF], making it suitable for extracting the average trap density N t [START_REF] Ioannidis | Low frequency noise statistical characterization of 14nm FDSOI technology node[END_REF]. Indeed, if we plot the extracted density values at 10 Hz from the spectra of Figure 26 versus the drain current (see Figure 29), the average LFN level follows the CNF model [START_REF] Ioannidis | Low frequency noise statistical characterization of 14nm FDSOI technology node[END_REF]. Therefore, with a sufficient number of measured dies, one can extract the average N t for all gate oxide areas. An example of such an extraction is shown in Figure 30(a), where an almost constant state-of-the-art volumetric oxide trap density is extracted for both n-and p-channel 14 nm FDSOI MOSFETs for all measured device areas. Besides, Figure 30(b) shows the full wafer mean trap areal density D st and corresponding average absolute number per device, N st =D st .(WL), values, as calculated using Eq. ( 42). The noiseinduced threshold voltage standard deviation can be calculated using the spectra integrals (Eq. ( 43)). This way, a clear gate voltage dependence is revealed, due to specific trap-induced RTN.

Figure 30. a) N t versus device area for n-and p-MOS from 14nm technology node and b) Full wafer mean trap areal density

D st and average absolute number per device N st (W=60nm, L=20 nm) (after [START_REF] Ioannidis | Low frequency noise statistical characterization of 14nm FDSOI technology node[END_REF]).

kT q C L W D Vt ox st 2 2 2 . . .   ( 42
)
where

   max min f f f Vg Vt df S  ( 43
)
Therefore, following this method, one can obtain a more representative and realistic illustration of the oxide trap density in the full studied frequency range i.e. the whole time constant space, as compared to the standard analysis that extracts the volume trap density N t from a pure flicker noise spectrum in a restricted frequency range e.g. around 10 Hz.

For certain studies, which want to examine the trap activity and to detect the presence of RTN signals, it is not mandatory to extract the exact parameters (ΔI d , τ c , τ e ) of every RTN-inducing trap for the whole wafer and every gate voltage bias, which would be also very time-consuming. Alternatively, in order to probe the RTN impact on the spectrum at each gate voltage and measured die, a new RTN strength indicator is proposed [START_REF] Ioannidis | Low frequency noise statistical characterization of 14nm FDSOI technology node[END_REF]: the standard deviation of log(S VG .f) vs frequency, where S VG = S Id /g m 2 . As one can see in Figure 31, when a single RTN source is active, which occurs in a certain bias range, log(S VG .f) strongly deviates from the flat line that corresponds to 1/f-like noise, giving rise to σ(log(S VG .f)). [START_REF] Ioannidis | Low frequency noise statistical characterization of 14nm FDSOI technology node[END_REF]). [START_REF] Ioannidis | Low frequency noise statistical characterization of 14nm FDSOI technology node[END_REF]).

Figure 31. log(S Vg .f) for 1/f-like (red) and Lorentzian-like (black) spectra (n-MOS 14 nm FD-SOI, W=60nm, L=20 nm) (after

Figure 32. Colored wafer maps based on the values of σ(log(S Vg .f)) (dec) for various gate voltage values (n-MOS 14nm FD-SOI, W=60nm, L=20nm) (after

It can be seen from the full wafer cartography of Figure 32 that σ(log(S VG .f)) is bias dependent, due to the RTN trap activity, and randomly distributed over the wafer map, revealing few RTNdominated dies.

Considering the cumulative distribution function (CDF) for an arbitrarily selected gate voltage, Figure 33(a) shows that the statistical distribution of σ(log(S VG .f)) can be easily modeled by a specific number (in this case 4) of randomly distributed additional RTN fluctuators over 6 frequency decades, superimposed to a 1/f background spectrum. Note also that σ(log(S VG .f)) is very sensitive to the number of RTN per decade, demonstrating the RTN statistical probing capability of this quantity. Figure 33(b) shows the full wafer cartography of the mean values of σ(log(S VG .f)) for all gate voltages. It is clear that they exhibit a smoother wafer distribution, meaning that σ(log(S VG .f)) follows well the convergence of many RTN-like spectra to a 1/f-spectrum when averaged.

d. Dependence on frequency, gate bias and temperature

In Figure 25, a dispersion of up to 4 decades of magnitude can be observed at high frequencies, while almost 2 decades around f = 1 Hz, revealing a possible frequency dependence of the standard deviation. Indeed, as shown in Figure 34(a) (after [START_REF] Theodorou | Low frequency noise variability in ultra scaled FD-SOI n-MOSFETs: Dependence on gate bias, frequency and temperature[END_REF]), where the measured spectra of 85 devices at V G = 0.4 V are plotted along with the noise standard deviation, a clear dependence of LFN variability with frequency is visible. In fact, in the frequency regions where σ(log(S Vg )) is increased, there are Lorentzian-like spectra with significantly higher density levels than the remaining spectra. It becomes thus obvious that the dependence of the noise variability on frequency is directly related to the RTN or GR noise presence in many of the devices due to the random distribution of trap energy and trap position in the gate oxide or Si film for each device. A more representative method for validating and analyzing the log-normal distribution is the use of the inverse error function (InvErf) of (2•CDF-1), where CDF is the cumulative distribution function. If the data distribution is indeed log-normal, the InvErf(2•CDF-1) versus the logarithm of the data should follow a straight line, and the inverse of the slope can provide the standard deviation. So, in order to further demonstrate the frequency dependence of the noise variability, the InvErf(2•CDF-1) is shown versus S Vg (in log-scale) at V G = 0.26 V for 4 different extraction frequencies in Figure 34(b). The log-normal distribution in all extraction frequencies is evident from the linear fit of InvErf(2•CDF-1) with S Vg in log-scale. Furthermore, from the difference observed between the linear fit slopes, it seems that at this specific gate voltage bias, the noise variability has its maximum value at low frequencies around 10 Hz, then it decreases with frequency reaching a minimum around 10 kHz, whereas it again increases at high frequencies around 250 kHz. This trend is different than the one shown in Figure 34(a) for V G = 0.4 V, thus the way that the noise variability depends on the frequency changes with the device bias. Following the same data analysis methodology, a variability dependence on gate voltage and temperature was also found [START_REF] Theodorou | Low frequency noise variability in ultra scaled FD-SOI n-MOSFETs: Dependence on gate bias, frequency and temperature[END_REF] (see Figure 35). To further probe the gate bias dependence of σ(log(S Vg )), a more intuitive plot is shown in Figure 36 (after [START_REF] Theodorou | Low frequency noise variability in ultra scaled FD-SOI n-MOSFETs: Dependence on gate bias, frequency and temperature[END_REF]), where σ(log(S Vg )) is plotted versus frequency and V G at the same time. For all bias conditions, a spectral peak of variability at a specific frequency is observed, the position of which is shifted to higher frequencies with increasing V G . The noise variability dependence on V G could be explained by the voltage dependence of the trap time constants (see Figure 4), as verified in Figure 37(a). A similar time constant dependence is observed in Figure 37(b), where the frequencynormalized noise is plotted versus frequency at V G = 0.26 V and different temperatures. The cut-off frequency of the Lorentzian spectrum corresponds to the maximum value of f•S Vg , which evidently increases with increasing the temperature. This behavior is well known in RTN analysis, where the trapping process is thermally activated, resulting in a decrease of the trap time constant with temperature [START_REF] Kirton | Capture and emission kinetics of individual Si:SiO2interface states[END_REF]. This translates into an augmentation of the Lorentzian cut-off frequency as the temperature rises, since f c =1/2πτ. Regarding the impact of temperature on the noise variability, we can relate the behavior shown in Figure 37(b) to the results of Figure 35(b). At f = 10 Hz, the noise is dominated by RTN at T = 300 K, whereas as the temperature increases the noise becomes more and more 1/f-like, resulting in a flat f•S Vg around f = 10 Hz for T = 400 K. This can explain the reduction of σ(log(S Vg )) with increasing temperature at a given frequency for this particular wafer.

From noise modeling to circuit simulations

Noise model implementation a. Using Verilog-A for noise modeling

Verilog-A [START_REF] Depeyrot | Verilog-A Compact Model Coding Whitepaper[END_REF] is a behavioral description language which is used by Cadence Spectre circuit simulator. It is a very powerful and useful tool, since it provides the potential for full description of a circuit netlist, as well as the behavior of a single device and its interface connections. In this section, two different ways to model a device behavior are presented. When the current-voltage characteristics are described by analytical and compact mathematical equations valid in all regions of operation, these equations can be written in the Verilog-A code and implemented as an analytical model. However, when the device behavior is known only through experimental measurements or TCAD simulations, one has to recall these data so that Spectre can predict all the in-between values through interpolation.

Provided that there is an analytical expression for the drain current as a function of both the gate and drain to source voltages, one can easily include a white or flicker noise source in the Verilog-A code. If the input-referred power spectral density S Vg is constant, as in the case of the 1/f noise originating from carrier number fluctuations (CNF), then the device noise can be modeled by incorporating a voltage noise source at the transistor gate. This can be done by the following line of code, using the contributor symbol '<+' which adds a voltage signal on the declared node:

V(g) <+ flicker_noise(Svfb, 1, "Svg");
where Svfb is the flat-band voltage power spectral density, the number 1 represents the value of the exponent γ and "Svg" is the name of the noise source. This voltage source will result to a drain current noise density equal to S Vg *g m 2 .

On the other hand, Verilog-A can also be used for time-dependent modules, since it gives access to the running time value of a transient simulation, as well as control over parameters such as permitted time-step etc. This is very useful for the simulation of RTN, but also LFN (through postsimulation FFT analysis) and BTI (time-dependent degradation) through defect-aware transient simulations, as will be shown in (c).

b. Implementing the two-interface CNF/CMF model

If the values of the drain current or transconductance are needed for the noise expression, as in the case of correlated mobility fluctuations or thermal noise, the transconductance has to be calculated first. If the analytical drain current model provides a simple expression for the Id-Vg dependence, it is easy to obtain the derivative in the Verilog-A code as below:

gm = ddx (Ids,V(g));
where gm is the variable name for the transconductance, ddx is the operator used for non-transient derivatives and Ids is the variable name for the drain current. Thus, the improved 1/f noise model that takes into account the correlated mobility fluctuations can be described as: V(g) <+ flicker_noise(Svfb*pow((1+alpha*ueff*Cox*Ids/gm),2), 1, "Svg"); where alpha, ueff and Cox are the variable names of α, μ eff and C ox respectively. Using this method, a voltage noise source is being connected to the transistor gate, as illustrated in Figure 38. As can be seen, both S Vg source and the calculation of g m regard the interior of the module. The FDSOI noise model presented in paragraph 2.2.b of this chapter takes into consideration the contribution of both front and back interfaces to the total noise level. Following the two-interface approach, the gate voltage noise sources can be declared as: In this way, two noise sources are implemented in the module as illustrated in Figure 39: The above method is very efficient for circuit simulations in the frequency domain accounting for 1/f noise. A good example of such case is the phase noise, because in frequencies close to the oscillation frequency it is directly proportional to the LFN amplitude [START_REF] Hajimiri | A general theory of phase noise in electrical oscillators[END_REF]. In order to demonstrate the importance of accurate LFN modelling in FDSOI circuits, we took the example of a 3-stage ring oscillator circuit. Figure 41 shows three examples: one case where only the front interface noise is considered (N t2 =0), one where N t2 =N t1 /2 and finally a case with equally defective front and back oxides (N t2 = N t1 ). From the figure becomes clear that if the N t2 contribution is not taken into account, both the phase noise level and the 1/f corner frequency are underestimated by 2-3 times, which may lead to false design decisions.

c. Defect-aware time-domain module

Following the RTN modeling approach described in 2.1.d, one can build a self-contained defectaware Verilog-A module for time-domain simulations [START_REF] Theodorou | A Self-contained Defect-aware Module for Realistic Simulations of LFN, RTN and Time-dependent Variability in FD-SOI Devices and Circuits[END_REF], [START_REF] Kaczer | Atomistic approach to variability of bias-temperature instability in circuit simulations[END_REF]. For the device static response, we can use the model presented in [START_REF] Karatsori | Analytical Compact Model for Lightly Doped Nanoscale Ultrathin-Body and Box SOI MOSFETs With Back-Gate Control[END_REF], because it is threshold-voltage-based, allowing for a much more straightforward electrostatic trap impact declaration through an approximate relation:

ΔV 𝑡 = q (1 - x t t ox ) WLC 𝑜𝑥 (44)
For simplification purposes, the CMF impact on the RTN amplitude was neglected, focusing on the feasibility of the module. The flowcharts for the initialization and the transient processes are shown in Figure 42(a) and (b) respectively, while the module parameters are shown in Table I As can be seen in Figure 42(a), right before the transient simulation starts, the number of active traps, is chosen from a Poisson distribution with an average value N T =WLt ox N t ΔE f based on the inputs (W,L,C ox and N t ). In [START_REF] Kaczer | Atomistic approach to variability of bias-temperature instability in circuit simulations[END_REF], Kaczer et al. approached the 𝜏̅ 𝑐 and 𝜏̅ 𝑒 estimation picking values for 𝜏̅ 𝑒,𝐻 (high bias state of a digital circuit) uniformly distributed on the log scale between 10 -9 and 10 9 and 𝜏̅ 𝑒,𝐿 , 𝜏̅ 𝑐,𝐻 , 𝜏̅ 𝑐,𝐿 calculated accordingly through different correlation levels. Regarding the intermediate voltages conditions, 𝜏̅ 𝑐 and 𝜏̅ 𝑒 were interpolated. In our work [START_REF] Theodorou | A Self-contained Defect-aware Module for Realistic Simulations of LFN, RTN and Time-dependent Variability in FD-SOI Devices and Circuits[END_REF], a random vertical and lateral position is chosen for each trap assuming a uniform distribution in the oxide. Moreover, for the module to be functional under both stationary and non-stationary conditions, 𝜏̅ 𝑐 and 𝜏̅ 𝑒 are initialized and then re-calculated at every change of voltage bias or temperature (see Figure 42(b)), using Equations [START_REF] Ghibaudo | On the theory of carrier number fluctuations in MOS devices[END_REF] and the local inversion charge density Q iyt calculated by the model [START_REF] Karatsori | Analytical Compact Model for Lightly Doped Nanoscale Ultrathin-Body and Box SOI MOSFETs With Back-Gate Control[END_REF]. This way, this module can be also used for simulating the recoverable part of the Bias Temperature Instabilities (BTI) degradation (as shown in [START_REF] Theodorou | A Self-contained Defect-aware Module for Realistic Simulations of LFN, RTN and Time-dependent Variability in FD-SOI Devices and Circuits[END_REF]).

A similar Verilog-A based implementation approach is also presented in Chapter 4 [START_REF] Marquez | Systematic Characterization of Random Telegraph Noise and Its Dependence with Magnetic Fields in MOSFET Devices[END_REF] of this book. In order to verify that our module can generate an RTN signal with a behavior analogous to the number of active traps N T , we simulated three different cases (see Figure 43). When only 1 trap is active, we obtain the typical two-level RTN pulse, whereas when N T = 2 it becomes four-level and for N T =10 the number of levels is 2 10 , thus becoming indistinguishable. In the frequency domain, these cases translate into one Lorentzian-like LFN spectrum per trap or a 1/f-like spectrum for N T = 10, exactly as expected and calculated using the traps input parameters. In order to make the use of our module completely generic and current model independent, we have to make it suitable to be used aside already existing PDK device instances. To do so, the simplest way would be to create a sub-circuit instance that contains the PDK transistor, together with a RTN voltage source in series with the transistor gate, as shown in Figure 44(a), in order that V g ΄= V g + ΔVg, where ΔV g = -ΔV t_RTN . However, since the industrial PDK device models usually do not provide access to the inversion charge values, we need to obtain the drain current, I d , values during a transient so that we can calculate τ c and τ e for each trap. If we neglect, for simplicity reasons, the mobility degradation effects, one can express the inversion charge as a function of I d as following [START_REF] Theodorou | Low-frequency noise and random telegraph noise in nanoscale devices: modeling and impact on circuit operation[END_REF]: There is a serious issue, however, with the above implementation approach: I d (t) is the device current containing the defect activity through the RTN-induced ΔV t that by turn causes a ΔI d shift. Therefore, the capture time calculation during a transient simulation is affected by the trap occupancy itself, creating continuity errors. The solution we proposed in [START_REF] Theodorou | Low-frequency noise and random telegraph noise in nanoscale devices: modeling and impact on circuit operation[END_REF] is to utilize an ideal "dummy" transistor with no RTN inside the sub-circuit. Its role is to always provide the theoretical I d (t) values, when no traps are occupied, in all moments of the transient simulation. This method is illustrated schematically in Figure 44(b). Finally, regarding the energy levels, since we cannot express a charge-voltage relation as done in [START_REF] Theodorou | A Self-contained Defect-aware Module for Realistic Simulations of LFN, RTN and Time-dependent Variability in FD-SOI Devices and Circuits[END_REF], they can be declared through a characteristic I dt , for which E t coincides with E f and calculate Q it through (45)(b), to use it in Eq. ( 18)(b) for the calculation of τ e .

𝑄 𝑖 (𝑡) = 𝐿 𝑊 𝐼 𝑑 (𝑡) 𝜇 0 𝑉 𝑑 (𝑎) , 𝑄 𝑖𝑡 = 𝐿 𝑊 𝐼 𝑑𝑡 𝜇 0 𝑉 𝑑 (𝑏) (45)

Impact of LFN/RTN on circuit operation a. Phase noise in FD-SOI ring oscillator circuits

The frequency domain implementation of noise models, such as the one described in paragraph 4.1.b of this chapter, can be very useful when it comes to performing circuit simulations in the frequency domain accounting for 1/f noise. A good example for the latter is the phase noise, because the 1/f noise is up-converted to phase noise around the oscillation frequency [START_REF] Hajimiri | A general theory of phase noise in electrical oscillators[END_REF]. In order to demonstrate the importance of accounting for both interfaces contributions in FD-SOI circuits, we took the example of a 3-stage ring oscillator circuit (schematic shown in Figure 45). Here we present three cases: one where only the front interface noise is taken into account (N t2 =0), one where N t2 =N t1 /2 and finally an example where the front and back oxide are equally defective (N t2 = N t1 ). One ) scematic (left) and phase noise versus frequency for three different N t2 /N t1 ratios (right) (after [START_REF] Theodorou | Low-frequency noise and random telegraph noise in nanoscale devices: modeling and impact on circuit operation[END_REF]), using the model of ( 9). The flicker/thermal corner frequency points are noted with a circle.

b. The SRAM cell as a circuit reference

In order to examine the impact of LFN or RTN (or both) on a circuit operation, or verify any model approaches, we need a circuit that that has wide application, a well-known standard operation and a noise-sensitive dynamic behavior. The typical 6T SRAM cell (Figure 46) has all these characteristics and is very easy to design and simulate. The letters L and R refer to the left and right symmetrical sides, while BL stands for Bit-Line, PG for Pass-Gate, PD for Pull-Down and PU for Pull-Up. The two bit versions (0 and 1) are written using the Write-Line (WL) and stored at the nodes CH and CL. The SRAM cell's dynamic limitations can be measured through what we call "stability metrics". Some of the most common SRAM stability metrics are: the Read and Write Static Noise Margin (RSNM & WSNM) and the Supply and Wordline Read Retention Voltage (SRRV & WRRV) [START_REF] Guo | Large-scale SRAM variability characterization in 45 nm CMOS[END_REF]. Here, we will focus on the SRRV and RSNM metrics that can provide a representative overview of the SRAM cell dynamic stability.

The SRRV is defined as the maximum allowed reduction of cell supply voltage for which the read stability is not affected. Figure 47(a) and (b) show the time-domain bias setup for properly monitoring the SRRV. During the measurement (or simulation), the word-line (WL) and the right bitline (BLR) are biased at V WL = V BL(R) = 1 V. An initial time is needed for the pre-charge of 0, for example, at the left bit node (CL), and then V BL is constant at 1 V while V CELL is ramped down in order to read the bit-line current I BL . As shown in Figure 47(c), at a certain point the stored bit is flipped and I BL drops suddenly, becoming significantly lower and equal to the Pass-Gate (PG) current. The V CELL value at that point is called V flip and SRRV is extracted as the difference between the power supply V DD and V flip .

The RSNM is an equally important stability metric for SRAM cells [START_REF] Guo | Large-scale SRAM variability characterization in 45 nm CMOS[END_REF]. It represents the maximum tolerable DC noise voltage at each storage node before causing a read upset. Regarding its measurement, it requires a more difficult procedure, compared to the simple SRRV measurement setup, because it needs a high-speed voltage monitoring, as well as access to the internal nodes and it has to be performed in two steps. The RSNM simulation, nevertheless, is far more easy to perform than the SRRV and it can be also used to study the SRAM dynamic variability behavior. Many RTN simulation studies have used these metrics to explore how the RTN amplitude and trap kinetics can affect the SRAM cell stability or even cause read/write errors, arguing this way on the importance of understanding and properly modeling the RTN behavior [START_REF] Takeuchi | Single-charge-based modeling of transistor characteristics fluctuations based on statistical measurement of RTN amplitude[END_REF], [START_REF] Brusamarello | Statistical RTS model for digital circuits[END_REF]- [START_REF] Takeuchi | Direct observation of RTN-induced SRAM failure by accelerated testing and its application to product reliability assessment[END_REF]. Here, we will present some novel approaches, regarding the aspects of the periodic transient noise and the defect-aware time-domain circuit simulations.

c. "Periodic Transient Noise" approach

Cadence Spectre incorporates a transient simulation option called "Transient Noise" (TN). When activated, this method generates time-domain current fluctuations that correspond to the noise power spectral density that has been declared in the model. Details of this simulation process can be found elsewhere [START_REF] Bolcato | A new approach for noise simulation in transient analysis[END_REF], [START_REF] Sung | A transient noise model for frequencydependent noise sources[END_REF]. As a result, when a transient I d -V g simulation with a ramp gate voltage as an input is run repeatedly, it produces a set of curves as in Figure 48(a) (after [START_REF] Theodorou | Dynamic variability in 14nm FD-SOI MOSFETs and transient simulation methodology[END_REF]), where we show an example of 100 multiple runs. The disadvantage of this simulation approach is that the noise bandwidth Δf = f max -f min taken into account by the TN is limited by the rise time, i.e. the duration of each run, because f min = 1/t r , and by the timestep Δt through f max = 1 / 2.Δt. This leads to a constant threshold voltage variation σ Vth for all rise time values (see Figure 48(b)), because the low-frequency part of the spectrum is not considered. To resolve this issue, one can use the Periodic Transient Noise (PTN) approach [START_REF] Theodorou | Dynamic variability in 14nm FD-SOI MOSFETs and transient simulation methodology[END_REF], which means running a periodic transient simulation instead of multiple runs, with the TN module activated. This can enlarge the simulation duration and thus increase the noise bandwidth to lower frequencies as much as necessary. This way, the noise-induced threshold voltage variations are both speed-and duration-sensitive, as expressed in 3.3.c through Eq. ( 43). Also, using the PTN method, the simulation stop time can approach the real duration of operation, obtaining more realistic results.

Using the PTN method, we can easily simulate the noise-induced degradation of the SRRV metric [START_REF] Theodorou | Noise-induced dynamic variability in nano-scale CMOS SRAM cells[END_REF] through a series of I BL -V CELL simulations (setup in Figure 47), with the transistors LFN level being controlled by the flat band voltage fluctuations power spectral density S Vfb (Eq. ( 5)) declared in the flicker noise model (4.1.b). A reference simulation with S Vfb = 10 -8 V 2 /Hz, stop time t stop = 10 ms and cell supply voltage fall time t fall = 0.1 ms is shown in Figure 49(a) (after [START_REF] Theodorou | Noise-induced dynamic variability in nano-scale CMOS SRAM cells[END_REF]), where a small but not significant noise-induced variability of SRRV can be observed. In the simulations of Figure 49(b), (c) and (d), one of these three parameters was changed while keeping the rest constant, to examine the impact of each one separately. In Figure 49(b), the noise level was increased to S Vfb = 10 -6 V 2 /Hz, in accordance with the worst case LFN level of nano-scale FDSOI MOSFETs [START_REF] Ioannidis | Low frequency noise statistical characterization of 14nm FDSOI technology node[END_REF]. In Figure 49(c) the V CELL fall time was reduced down to t fall = 1 μs to increase the speed of operation, and in Figure 49(d) the simulation stop time was increased to t stop = 1 s to include the noise contribution from the lower part of the frequency bandwidth. In all three cases, compared to the reference, a significant increase of the SRRV dynamic variability is observed, concluding that the noise-induced FDSOI SRAM dynamic variability is increasing with the transistor LFN level, the circuit operation speed, and the duration of operation. These results also verify the measured behavior of SRRV shown in Figure 50, further underlining the validity of the PTN simulation method. Regarding the PTN simulation of the LFN impact on the RSNM, some representative results are shown in Figure 51 (after [START_REF] Theodorou | Noise-induced dynamic variability in nano-scale CMOS SRAM cells[END_REF]). In order to capture both local and time-domain RSNM variations, Figure 51(a) shows local variability results through mismatch-related Monte Carlo simulations. One can notice how the RSNM can be reduced by more than 50% from cell to cell, due to the threshold voltage static mismatch. The noise-induced dynamic variability of a single FDSOI SRAM cell, on the other hand, as shown in Figure 51 (b), can lead to a RSNM reduction of up to 20%. For the latter simulation, the device noise level was boosted by two orders of magnitude (S Vfb = 10 -6 V 2 /Hz), in order to consider the worst case scenario, which is to have either huge LFN variability or the presence of high amplitude RTN. This alarming finding means that in FDSOI circuits consisting of nano-scale transistors, the LFN/RTN-induced variations can be comparable to the mismatch variations. Considering that these two sources of variations (topological and temporal) will in reality be added together and result to a variability level significantly higher than what is often taken into account in simulations (only process variations), we conclude that the accurate simulation of LFN and RTN becomes more and more important in circuit design. (after [START_REF] Theodorou | Noise-induced dynamic variability in nano-scale CMOS SRAM cells[END_REF]).

d. Defect-aware time-domain simulations

Using the defect-aware module described in 4.1.c, the impact of RTN-TDV on the Read Static to Noise Margin (RSNM) of a SRAM cell was simulated. Figure 52(a) presents the 6T-cell that was used in our simulations, as well as the number of traps that was chosen by the simulator, considering a uniform trap distribution. The transistor width and length was fixed at W = 80 nm and L = 30 nm, respectively. In order to simulate the phenomenon, a periodic ramp voltage was applied first on the left node (L) and then on the right one (R), while monitoring the voltage at the opposite node. A zoomed-in result is shown in Figure 52(b), where one can also note the presence of RTNrelated fluctuations in the output voltage. A number of 1000 ramp cycles was chosen, with t rise = t fall = 100 μs, in order to have a good statistical sample concerning the trap activity.

The final results are shown in Figure 52(c), where V R is plotted versus V L for both DC and transient simulations. The impact of the trap activity on the reduction of RSNM is visible: RSNM ≈ 0.13V in the DC simulation, while it drops at around 0.1 V in the case of the defect-aware transient, revealing a more than 20% reduction of RSNM, only by trapping/de-trapping events in a very low number of traps, in line with the transient noise results presented in (c), where the same reduction was found by using a high level 1/f noise spectrum with the Transient Noise module in Spectre.

Conclusion

In this chapter, we presented a review of the most important aspects of our research work regarding the low frequency noise behavior of Fully Depleted Silicon-On-Insulator MOSFETs.

First, we found it necessary to give a brief explanation of important terms and concepts regarding the fluctuations mechanisms in MOSFETs, providing some fresh points of view regarding RTN and the Hooge mobility fluctuations. Next, we presented the methodology we followed in order to derive the FDSOI LFN models, which were used on the one hand to extract the noise-related parameters for various FDSOI wafers and on the other hand to perform accurate simulations of noise-induced limitations of a circuit's performance. It was shown that neglecting the impact of the bottom interface on the total noise level or the latter's dependence on back-bias can lead to serious errors in the extraction of trap densities etc.

Regarding the noise characterization of FDSOI MOSFETs, we analyzed the impact of front-back gate coupling on both carrier number and correlated mobility fluctuations, the impact of channel geometry (thickness, width and length) on the strength of the remote Coulomb scattering, and the presence of generation-recombination noise centers in the semiconductor region of such ultra-thin channels. We demonstrated that reducing the channel cross-section (width*thickness) or applying negative back-bias to increase the threshold voltage can lead to enhanced CMF levels.

Afterwards, we presented a thorough study of the LFN variability phenomenon and its properties. A comparison was shown between many different technology nodes, revealing a superiority of the FDSOI technology with respect to the LFN device-to-device variations. Moreover, some new methods of statistical LFN/RTN characterization were demonstrated, as well as the LFN variability dependences on gate bias, frequency and temperature. Finally, our methodology regarding the noise model implementation for circuit simulations was presented in detail. The "Periodic Transient Noise" simulation method was also explained thoroughly, demonstrating its ability to produce realistic LFN-aware time-domain simulations. Also, after showing in which way both frequency and time-domain noise model approaches can be implemented through Verilog-A, we showed how the LFN and RTN of a nano-scale device can affect the stability and thus the performance limitations of a SRAM cell.

We hope that the content of this chapter will contribute to the clarification of any misconceptions regarding the fluctuation mechanisms in FDSOI MOSFETs and help towards a better understanding of the LFN and RTN phenomena in general, as well as the adoption of the noise characterization, modeling and simulation methods presented.

Figure 1 .

 1 Figure 1. a) Theoretical S Vg (V d ) characteristics as obtained from CNF-CMF model of Eq. (7) and b) typical experimental S Vg (V d ) characteristics (after[START_REF] Boutchacha | Improved modeling of low-frequency noise in MOSFETs -Focus on surface roughness effect and saturation region[END_REF]).

Figure 2 .

 2 Figure 2. Theoretical variations of µ eff (a) and Hooge parameter (b) with MOSFET inversion charge Q i for various interface charge Q it levels modulating the Coulomb scattering rate ( H0 =10 -5

Figure 3 .

 3 Figure 3. Variation of normalized drain current noise S Id /I d 2 with inversion charge for HMF model with Hooge parameter of Figure 2.

Figure 4 .

 4 Figure 4. Typical variations of capture and emission with gate voltage for a FDSOI structure (Q d =0) with parameters:

Figure 5 .

 5 Figure 5. TCAD simulations results of normalized noise for different trap density ratio between top and bottom interface. The dashed lines correspond to the two terms of the model expressed in (22) and the continuous line to their sum.

Figure 6 .

 6 Figure 6. TCAD simulations results of normalized drain current noise for different back gate voltage values

Figure 7 .

 7 Figure 7. Calculation results for carriers distribution centroid-front interface distance x and coulomb scattering coefficient versus bias voltage V G2 , keeping the drain current constant[START_REF] Theodorou | Impact of front-back gate coupling on low frequency noise in 28 nm FDSOI MOSFETs[END_REF].

Figure 9 .

 9 Figure 9. Electron density versus silicon film depth at VBG = 0 V (solid line) and VFG = 0 V (dotted line) for the same total drain current Id = 10 μΑ (W = 10 μm , L = 87 nm). [57]

Figure 10 .

 10 Figure 10. (g m1 /I D ) 2 (a) and (g m2 /I D ) 2 (b) versus drain current for various V G2 values (after [17]).

Figure 11 .

 11 Figure 11. Coupling factor c 1 and V G1 dependence on V G2 (inset) for various drain current values (after[START_REF] Theodorou | Impact of front-back gate coupling on low frequency noise in 28 nm FDSOI MOSFETs[END_REF]).

Figure 12

 12 Figure 12. a) Normalized PSD at f=10 Hz versus drain current for different V G2 values with CNF/CMF model and b) Drain curernt PSD for I D = 4.5 A measured at different V G2 (after [17]).

Figure 14 .

 14 Figure 14. Ω eff (a) and W.L.S Vfb,eff (b) dependence for various back gate biases for n-MOS 14 nm FDSOI node MOSFETs (after[START_REF] Ioannidis | Drain-Current Flicker Noise Modeling in nMOSFETs From a 14-nm FDSOI Technology[END_REF]).

Figure 15 .

 15 Figure 15. CNF factors (g m1 /I D ) 2 (a), (g m2 /I D ) 2 (b) and coupling factor (c) versus drain current for n-MOS transistors issued from 20 nm FDSOI CMOS technology with W = 10 µm and L = 87 nm, t ox = 1.2 nm, t Si = 8.7 nm and t ox2 = 10 nm (after [58]).

Figure 16 .

 16 Figure 16. Normalized power spectral densities, flicker noise component values and CNF/CMF model fitting versus drain current for V BG = 0 V (a), V BG = -5 V (b) and V FG = 0 V (c) for n-MOS transistors issued from 20 nm FDSOI CMOS technologywith W = 10 µm and L = 87 nm, t ox = 1.2 nm, t Si = 8.7 nm and t ox2 = 10 nm (after[START_REF] Theodorou | Low-frequency noise behavior of n-channel UTBB FD-SOI MOSFETs[END_REF]).

Figure 17 .

 17 Figure 17. Measured normalized drain current S I /I d 2 at f = 10 Hz, extracted 1/f component and CNF/CMF model versus drain current for n (solid symbols) and p (open symbols) channel devices at front-gate (a) and back-gate mode (b) with W = 10 μm and L = 87 nm (after [57]).

Figure 18 .

 18 Figure 18. Normalized input-referred noise voltage S VG1 WL and noise model fit versus I d /g m1 for n-channel (a, c) and pchannel (b) devices, and extracted values of Ω 1 versus channel width W (d) for transistors issued from 20 nm FDSOI CMOStechnology with t ox = 1.2 nm, t Si = 8.7 nm and t ox2 = 10 nm (after[START_REF] Theodorou | Low-Frequency Noise Sources in Advanced UTBB FD-SOI MOSFETs[END_REF]).

Figure 19 .

 19 Figure[START_REF] Hung | A unified model for the flicker noise in metaloxide-semiconductor field-effect transistors[END_REF]. Ω eff (a) and W.L.S Vfb,eff (b) versus device length for n-MOS, respectively, for 14 nm FDSOI MOSFETs (after[START_REF] Ioannidis | Drain-Current Flicker Noise Modeling in nMOSFETs From a 14-nm FDSOI Technology[END_REF]).

Figure 20 .

 20 Figure 20. Ωeff dependence on back gate bias for various channel lengths.

Figure 22 .

 22 Figure 22. Power spectral density versus frequency for V BG = 0 V (a) and for different bias voltages for a constant drain current I D = 10 μA (b) (after [58]).

Figure 23 .

 23 Figure 23. Extracted values of the Lorentzian plateau (squares) and time constant (triangles) versus V FG for V BG = 0 V (a) and versus V BG for V FG = 0 V (b), plotted at the same drain current region (after [58]).

Figure 24 .

 24 Figure 24. Both LFN level and LFN variability scale reciprocally with the device area.

Figure 26 .

 26 Figure 26. Normalized drain current spectral density S Id /I d 2 versus frequency for large (left) and small (right) area n-channel 14 nm FDSOI MOSFETs (after [72]).

Figure 27 .

 27 Figure 27. Comparison of the drain current noise variability as a function of the inverse square root of the area between 28nm SOI/Bulk and 45nm Bulk (after[START_REF] Ioannidis | Low frequency noise variability in high-k/metal gate stack 28nm bulk and FD-SOI CMOS transistors[END_REF]) (a) and between 28nm/14nm FDSOI and 20nm bulk technology nodes (after[START_REF] Ioannidis | Low frequency noise statistical characterization of 14nm FDSOI technology node[END_REF]) (b).

Figure 28 .

 28 Figure 28. a) Comparison of the drain current noise matching parameter, b) Comparison of the product N t .EOT 2 for n-MOSFETs issued from 28nm & 14nm FDSOI and 20nm bulk technology nodes (after [72]).

Figure 29 .

 29 Figure[START_REF] Scholten | Noise modeling for RF CMOS circuit simulation[END_REF]. Normalized drain current noise (symbols) and CNF model (lines) at 10Hz for large (left) and small (right) area nchannel 14 nm FDSOI MOSFETs (after[START_REF] Ioannidis | Low frequency noise statistical characterization of 14nm FDSOI technology node[END_REF]).

Figure 33 .

 33 Figure 33. a) CDF of σ(log(S VG .f)) (dec) (measurements and model) and b) Colored wafer maps based on the mean values of σ(log(S Vg .f)) (dec) (n-MOS 14 nm FD_SOI, W=60nm, L=20nm) (after [72]).

Figure 34 .

 34 Figure 34. a) Input-referred gate voltage noise (left Y-axis) and standard deviation value (right Y-axis) at V G = 0.4 V versus frequency, and b) Inverse error function versus input-referred gate voltage noise at V G = 0.26V for 4 different frequencies(after[START_REF] Theodorou | Low frequency noise variability in ultra scaled FD-SOI n-MOSFETs: Dependence on gate bias, frequency and temperature[END_REF]).

Figure 35 .

 35 Figure 35. Inverse error function versus input-referred gate voltage noise at f = 10 Hz for a) 4 different gate voltage values under T = 25 o C and b) for three different temperatures T under V G = 0.26V (after [68]).

Figure 36 .

 36 Figure 36. Input-referred gate voltage noise standard deviation versus frequency for all measured gate voltage values (after[START_REF] Theodorou | Low frequency noise variability in ultra scaled FD-SOI n-MOSFETs: Dependence on gate bias, frequency and temperature[END_REF]).

Figure 37 .

 37 Figure 37. Frequency-normalized input-referred gate voltage noise versus frequency a) for 4 different gate voltage bias (1 die example) at T = 300 K and b) for three different temperatures T (1 die example) under V G = 0.26 V (after [68]).

Figure 38 .

 38 Figure 38. Schematic representation of the Verilog-A CNF/CMF noise model implementation (after[START_REF] Theodorou | Low-frequency noise in advanced CMOS/SOI nanoscale multi-gate devices and noise model development for circuit applications[END_REF]).

V

  (g) <+ flicker_noise(Svfb1*pow((1+Omega1*Ids/gm1),2), 1, "Svg1"); V(b) <+ flicker_noise(Svfb2*pow((1+Omega2*Ids/gm2),2), 1, "Svg2");

Figure 39 .

 39 Figure 39. Schematic illustration of the FDSOI Verilog-A noise model implementation (after[START_REF] Theodorou | Low-frequency noise in advanced CMOS/SOI nanoscale multi-gate devices and noise model development for circuit applications[END_REF]).

Figure 40

 40 Figure 40 shows a comparison between Spectre simulation and theoretical calculation of the normalized drain current noise S Id /I d 2 values at 10 Hz versus drain current. The simulation data accurately follow the trend of the noise model, for both CNF and CNF/CMF cases.

Figure 40 .

 40 Figure 40. Normalized drain current noise S Id /I d 2 simulated (symbols) and theoretically calculated (lines) values versus drain current for CNF with N t1 = 5*10 17 cm-3 and N t2 = 10 17 cm -3 and CNF/CMF with Ω 1 = 5 V -1 and Ω 2 = 0.5 V -1 (after[START_REF] Theodorou | Low-frequency noise in advanced CMOS/SOI nanoscale multi-gate devices and noise model development for circuit applications[END_REF]).

Figure 41 .

 41 Figure[START_REF] Kirton | Individual defects at the Si:SiO2 interface[END_REF]. Simulated 3-stage oscillator (W=10μm, L=30nm) phase noise versus frequency for three different N t2 /N t1 ratios, using the model of (9). The flicker/thermal corner frequency points are noted with a circle.

Figure 42 .

 42 Figure 42. Flowcharts of a) trap initialization process and b) trap activation process during transient simulation (after [82]).

Figure 43

 43 Figure 43. a) Simulated RTN signal examples for NT = 1, 2 and 10 and b) corresponding normalized spectra through FFT (after [82]).

Figure 44 .

 44 Figure[START_REF] Ioannidis | Analytical low-frequency noise model in the linear region of lightly doped nanoscale double-gate metal-oxide-semiconductor field-effect transistors[END_REF]. Μodule implementation methods in existing PDKs: without (a) and with (b) noise-less dummy transistor (after[START_REF] Theodorou | Low-frequency noise and random telegraph noise in nanoscale devices: modeling and impact on circuit operation[END_REF]).

  in Figure45that if the bottom interface contribution (N t2 ) is not taken into account, both the phase noise level and the 1/f corner frequency are underestimated by 2-3 times[START_REF] Theodorou | Low-frequency noise and random telegraph noise in nanoscale devices: modeling and impact on circuit operation[END_REF]. This might cause wrong noise yield estimations or/and lead to false design decisions.

Figure 45 .

 45 Figure[START_REF] Karatsori | Study of Hot-Carrier-Induced Traps in Nanoscale UTBB FD-SOI MOSFETs by Low-Frequency Noise Measurements[END_REF]. Simulated 3-stage oscillator (W=10μm, L=30nm) scematic (left) and phase noise versus frequency for three different N t2 /N t1 ratios (right) (after[START_REF] Theodorou | Low-frequency noise and random telegraph noise in nanoscale devices: modeling and impact on circuit operation[END_REF]), using the model of (9). The flicker/thermal corner frequency points are noted with a circle.

Figure 46 .

 46 Figure 46. Typical SRAM cell schematic: R stands for Right, L for left, PU for pull-up, PD for pull-down, PG for pass-gate, BL for bit-line and WL for word-line.

Figure 47 .

 47 Figure 47. Setup for measuring the Supply Read Retention Voltage: (a) Bit-line voltage and (b) Cell supply voltage versus time. (c) Measured bit-line current versus cell supply voltage (after [87])

Figure 48 .V 2 /

 482 Figure 48. a) Transient noise simulation (100 multiple runs) example . b) Dynamic standard deviation of V th versus rise time for periodic and multiple runs transient noise simulation (W = 80 nm, L = 30 nm, S Vg =A/f with A=1.8×10 -8 V 2/Hz) (from[START_REF] Theodorou | Dynamic variability in 14nm FD-SOI MOSFETs and transient simulation methodology[END_REF]).

Figure 49 .

 49 Figure 49. Periodic transient noise simulation results of the bit-line current versus cell supply voltage for four different configurations: (a) Reference simulation, (b) S Vfb ˑ100, (c) t fall /1000, (d) t stop ˑ100 (after [87]).

Figure 50 .

 50 Figure 50. Measured SRAM cell bit-line current versus cell supply voltage for three different cases: (a) Cell #1 with t fall = 5 ms, (b) Cell #2 with t fall = 5 ms, (c) Cell #2 with t fall = 5 μs (after [87]).

Figure 51 .

 51 Figure 51. a) Impact of V t mismatch on RSNM, (MC: Monte Carlo), b) Impact of LFN on RSNM (PTN: Periodic Transient Noise)(after[START_REF] Theodorou | Noise-induced dynamic variability in nano-scale CMOS SRAM cells[END_REF]).

Figure 52 .

 52 Figure 52. a) SRAM cell circuit used in the simulations. The numbers in red show the total number of traps per device, b) Voltage bias setup during transient simulation with RTN and c) Right-node vs left-node voltage for the extraction of RSNM (after [82]).

Table 1 .

 1

	Wafer number	Technology node (nm)	BOX thickness t BOX (nm)	Si body thickness t Si (nm)	Gate oxide trap density N t1 (cm -3 eV -1 )	Buried oxide trap density N t2 (cm -3 eV -1 )	CMF factor Ω 1 (V -1 )
	1	14	10	8.7	8.710 17	3.510 17	1.8
	2	28	25	6.7	9 . 10 17	2.710 17	12.2
	3	28	145	10	80 17	8.510 16	0.8

FDSOI wafer characteristics & Extracted Flicker Noise Parameters (after

[START_REF] Theodorou | Low-Frequency Noise Sources in Advanced UTBB FD-SOI MOSFETs[END_REF]

).

  .

	W gate width	V t0 V t for 0 active traps
	L gate length	Q iyt surface charge at trap position
	t ox gate oxide EOT	Q iyt0 value of Q iyt when E T = E F
	N t trap density (/cm 3 /eV)	τ c capture time
	N T_Avg average # of traps	τ e emission time
	x t trap depth in oxide	t up time on capture event
	y t trap position across channel	t down time on emission event
	V gtrap V gt for which E T = E F	state 0 when empty, 1 when occupied

Table 2 .

 2 Module parameters (after[START_REF] Theodorou | A Self-contained Defect-aware Module for Realistic Simulations of LFN, RTN and Time-dependent Variability in FD-SOI Devices and Circuits[END_REF]).

Abbreviations