GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2018 1

Cryogenic MOS Transistor ModelArnout Beckers, Farzan Jazaeri, and Christian Enz

Abstract— This paper presents a physics-based analyt-ical model for the MOS transistor operating continuouslyfrom room temperature down to liquid-helium temperature(4.2 K) from depletion to strong inversion and in the linearand saturation regimes. The model is developed relying onthe 1D Poisson equation and the drift-diffusion transportmechanism. The validity of the Maxwell-Boltzmann approx-imation is demonstrated in the limit to zero Kelvin as a re-sult of dopant freeze-out in cryogenic equilibrium. ExplicitMOS transistor expressions are then derived includingincomplete dopant-ionization, bandgap widening, mobilityreduction, and interface charge traps. The temperature-dependency of the interface trapping process explains thediscrepancy between the measured value of the subthresh-old swing and the thermal limit at deep-cryogenic temper-atures. The accuracy of the developed model is validatedby experimental results on a commercially available 28nm bulk CMOS process. The proposed model providesthe core expressions for the development of physically-accurate compact models dedicated to low-temperatureCMOS circuit simulation.

Index Terms— cryogenic MOSFET, cryo-CMOS, freeze-out, incomplete ionization, interface traps, low temperature,MOS transistor, physical modeling

I. INTRODUCTION

ADVANCED CMOS processes perform increasingly wellfrom room temperature down to deep-cryogenic tem-

peratures (< 10 K) [1]–[4]. At these temperatures the idealswitch with a step-like subthreshold slope comes within reach[5]. Furthermore, cryo-electronics [6]–[8] can provide an in-terface with superconducting devices on the quest for exascalesupercomputing [9]. Ultimately, quantum-engineered devicescontrolled by cryo-CMOS circuits can bring new functionalityto existing computing technologies [10], [11].

Large-scale integration of silicon spin qubits [12], [13] andcryo-CMOS control circuits is envisioned to take solid-statequantum computing to the next level [14]. Digital, analog,and RF CMOS circuits [15]–[17] are then required to operateat millikelvin temperatures for initialization, manipulation,and read-out of the qubits, as well as error correction [18],[19]. Since the cooling power at millikelvin temperatures isreduced, the system could feature a cryogenic temperaturegradient, where the control circuits operate at a higher cryo-genic temperature than the qubits, e.g. 4.2 K [15]. However, the

This project has received funding from the European Union’s Horizon2020 Research & Innovation Programme under grant agreement No.688539 MOS-Quito (MOS-based Quantum Information Technology)which aims to bring quantum computing to a CMOS platform.

The authors are with the Integrated Circuits Laboratory (ICLAB) at theEcole Polytechnique Federale de Lausanne (EPFL), 2000 Neuchatel,Switzerland (e-mail: arnout.beckers@epfl.ch).

optimal design of power-hungry and thermal-noise dissipatingcircuits operating in close proximity to the qubits is yet to beexplored. In this context, the main hurdle to overcome is thelack of compact MOS transistor models in circuit simulators,remaining physically accurate below 10 K [15], [17].

II. CRYO-MOS TRANSISTOR MODELING

The low-temperature circuits developed for spacecraft [20],[21], scientific equipment [22], ultra-low-noise detectors [23],cryobiology [24], and others, have been custom-designedrelying on a semi-empirical approach. This approach re-quires laborious and expensive low-temperature measurementsto extract model parameters for tuning room-temperaturecompact models to the target low-temperature [23], [25],[26]. Empirical temperature-scaling laws have been added tothe room temperature physics-based MOS transistor model[27], [28] to capture cryogenic operation down to 4.2 K[29]–[31]. However, the discrepancy between the measuredvalue of the subthreshold swing for a long device at 4.2 K(≈ 10 mV/decade) [3], [4], [32], and the theoretical thermallimit, UT ln 10 (≈ 0.8 mV/decade) reveals that something morefundamental is missing. As we will demonstrate along thispaper, important physical phenomena at low temperatures suchas interface trapping [28], [33] and incomplete ionization [34],[35] have not been properly included to date. Furthermore,the intrinsic carrier concentration, ni, takes on extremelysmall values below 10 K, causing arithmetic underflow inimplemented analytical expressions or convergence problemsin computer-aided-design simulations [36]–[38]. Therefore,standard references on semiconductor devices treat only thecryogenic equilibrium condition in bulk semiconductors above10 K [27], [28], [39]. Analytical device-physics models, start-ing from the Poisson equation at low temperature, leave agap unfilled between the zero-Kelvin approximation and 77 K[40]–[43].

In this work, we develop a MOS transistor model valid fromroom temperature (RT) down to deep-cryogenic temperatures,entirely based on physics principles and validated with exper-imental results. We start by verifying the continued validity ofthe Boltzmann statistics down to the deep-cryogenic regime.

III. MOS ELECTROSTATICS FROM RT TO 4.2 K

We model a long, planar n-channel MOS field-effect tran-sistor in silicon, depicted in Fig. 1. Uniform operation acrossthe width of the transistor is assumed and the gradual channelapproximation is adopted. The electrostatics can then bedescribed by the 1D Poisson equation [27], [28].

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2 GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2018

A. Poisson-Fermi equationMerging the 1D Poisson equation with the mobile carrier

concentrations, n and p, given by Fermi-Dirac statistics, gives

∂2ψ(y)

∂y2= − q

εsi

(−n+ p−N−A

), (1)

where q is the elementary charge, εsi the silicon permittivity,and ψ , (EF −Ei)/q the potential, with EF the Fermi-leveland Ei the intrinsic energy level. The first term on the RHS ofequation (1) represents the electron contribution, n, the secondterm the hole contribution, p, and the third term the ionizeddopant-contribution, N−A .

1) Incompletely-ionized dopants: Under thermal equilib-rium, both at room and cryogenic temperatures, the majoritycarrier concentration can defer from the implanted dopingvalue, NA, due to incomplete ionization of the dopants. Incryogenic equilibrium, incomplete ionization is strong andknown as freeze-out, since thermal dopant-ionization is verylow [35]. However, during MOS operation, also field-assistedionization comes into play. Fermi-Dirac statistics provides afundamental way to model incomplete ionization which in-cludes both dopant-ionization mechanisms. The concentrationof ionized dopants, N−A , is then equal to the total concentra-tion of implanted dopants times the Fermi-Dirac occupationprobability of the acceptor energy EA, i.e.NA × f(EA), or

N−A =NA

1 + gAeEA−EF,n

kT

=NA

1 + gAeψA−(ψ−Vch)

UT

, (2)

where the electron quasi-Fermi-level is given by EF,n =EF−qVch. The RHS of (2) is obtained by replacing EA−EF,nwith EA − Ei + Ei − EF,n in the exponential term, andby defining an acceptor potential, ψA , (EA − Ei)/q, asindicated in Fig. 1. The channel voltage, Vch, denotes theshift of the quasi-Fermi-potential due to the drain-to-sourcevoltage, VDS. The second expression in (2) highlights thetwo dopant-ionization contributions, i.e. the potential (field-assisted ionization [35]) and temperature (thermal ionization).The acceptor-site degeneracy factor, gA, is set to four dueto fourfold degeneracy (heavy-light hole, spin up-down) [28],[39]. Note that setting gA to zero is equivalent to assumingcomplete ionization.

2) Mobile carrier concentrations: Since n and p given byFermi-Dirac statistics in (1) require numerical integration overenergy, this inhibits explicit solutions for the charge densitiesand current in the MOS transistor. Expressing n and p usingBoltzmann statistics allows to obtain such relations. However,the validity of the Maxwell-Boltzmann approximation downto deep-cryogenic temperatures is questionable. It has beenreported [38], [42] that semiconductors become strongly de-generate at deep-cryogenic temperatures, preventing its use.This is however inconsistent with the zero-Kelvin limits ofthe Fermi-level position in the bandgap derived by Pierret[39]. Therefore, in the next subsection we aim to verify theMaxwell-Boltzmann approximation down to deep-cryogenictemperatures.

3) Verification of Boltzmann statistics: We numerically cal-culate the position of the equilibrium Fermi-level, EF , down to100 mK relying on Fermi-Dirac statistics in an extrinsic bulk

E

B

S

G

D

-q

-q

q

4.2 K

Fig. 1. Schematic representation of a long nMOS transistor withannotated band diagram. The drift-diffusion and Poisson equations aresolved along the x and y-directions respectively. At 4.2 K, and forNA = 1018 cm−3, EF lies below EA in the bulk (see Fig. 2a), leadingto bulk freeze-out according to (2). WhenEA bends underEF near thesurface, the acceptor dopants become rapidly completely ionized dueto field-assisted ionization. The quasi-Fermi potential is not taken intoaccount in this figure.

semiconductor, e.g. p-type silicon. In this case, the Poissonequation imposes the charge neutrality, pp = N−A , wherepp is expressed by Fermi-Dirac statistics [28], [39] and N−Aby (2). This yields an implicit equation for EF , which issolved numerically at each temperature and doping valueusing an extension of the arithmetic precision. As illustratedin Fig. 2a, below 120 K, EF remains off the valence bandedge with an offset larger than 3kT for doping values belowthe degenerate limit (i.e.NA = 4× 1018 cm−3 in Si:B)[28], [39], [45]. Note that this is predicted correctly onlywhen incomplete ionization is taken into account. Completeionization (gA = 0) would predict an offset smaller than3kT for NA = 1018 cm−3, and hence a degenerate semi-conductor. It should therefore be emphasized that incompleteionization maintains the non-degeneracy of a highly-dopedsemiconductor at temperatures down to 100 mK. Furthermore,near zero Kelvin, EF tends to saturate at (EA − Ev)/2 forall considered doping values. This corresponds to the zero-Kelvin limit in Pierret [39] assuming Boltzmann statistics.Using the now validated Maxwell-Boltzmann description forpp, i.e.Nv exp[(Ev − EF )/kT ], in pp = N−A , leads to aquadratic equation in exp [(Ev − EF )/kT ] with as solution,

EF − Ev = kT lnNvNA

+ kT ln1 +

√1 + 4gA

NANveEA−EvkT

2.

(3)

Considering the temperature dependency of Nv [28], [39],while taking the limit of (3) to 0 K, leads to limT→0K EF =Ev + (EA − Ev)/2.

Performing the same numerical EF -calculation for an in-trinsic semiconductor, the extremely small value of ni can beverified relying on Fermi-Dirac statistics. The Poisson equa-

BECKERS et al.: CRYOGENIC MOS TRANSISTOR MODEL 3

0.05

0

EF -

Ev [eV

]

120110100908070605040302010

T [K]

incomplete ionization

NA = 1018

cm-3

0

Ev

3kT

77 K4.2 K

NA = 1012

cm-3

NA = 1018

cm-3

complete ionization

NA = 1012

cm-3

EA

EA - Ev

2

0.045

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

EF -

Ei [

eV

]

450400350300250200150100500

T [K]

Ec

ED

EA

Ev

Ei

n-type Si

p-type Si

1014

1016

1018

gD = 2

gA = 4

Conduction band

Valence band

RT

NA [cm-3

]

ND [cm-3

]

1012

1012

1014

1016

1018

(a) (b)

Fig. 2. Thermal equilibrium in extrinsic bulk silicon. Right: position of the Fermi-level, EF , in the bandgap as a function of doping and temperature.TheEF -position is calculated from RT down to 100 mK using an extension of the arithmetic precision and anEF -resolution of 1 meV. Left: magnifiedview of the cryogenic regime (below 120 K). When incomplete ionization is taken into account, the distance of EF to the valence-band edge, Ev ,stays larger than 3kT , validating the use of the Maxwell-Boltzmann approximation down to millikelvin temperatures. This figure applies to the bulkof the MOS transistor in all regions of operation, and to the whole body of the MOS transistor in the flatband condition. Bandgap temperaturedependency is taken from Varshni [44] and a standard, temperature-independent value of EA − Ev = 0.045eV in Si:B is assumed.

tion then imposes the charge neutrality, n = p = ni, wheren and p are given by Fermi-Dirac statistics. As illustrated inFig. 3, this yields ni-values lying outside the range of IEEEdouble-precision arithmetic (10−308 − 10308), e.g. at 4.2 K,≈ 10−678 cm−3. Therefore, an extension of the arithmeticprecision will also be used in the remainder of this work basedon Boltzmann statistics, since the carrier concentrations arethen expressed through ni.

B. Poisson-Boltzmann equation

Using the Maxwell-Boltzmann approximation of n and p,validated down to deep-cryogenic temperatures in the previoussection, we combine the 1D Poisson equation with Boltzmannstatistics, which leads to

∂2ψ(y)

∂y2= − q

εsi

(−nie

ψ−VchUT + nie

− ψUT −N−A

), (4)

where UT , kT/q is the thermal voltage. The first term on theRHS of equation (4) represents the electron contribution, n,and the second term the hole contribution, p. The intrinsic car-rier concentration is given by ni =

√NcNv exp(−Eg/2kT ),

with Eg the bandgap, and Nc and Nv the effective density-of-states in the conduction band and valence band respectively.The temperature-dependency of Eg as described by Varshni[44] is used. The extremely small, but finite value of ni atdeep-cryogenic temperatures cannot be assumed zero—whichwould be equivalent to the zero-Kelvin approximation [40]or considering f(E) as a step function—since this leadsto zero mobile carrier concentrations independently of thepotential. This is irreconcilable with the observed field-effectand correct functioning of the MOS transistor at 4.2 K [3], [5].For UT small, the exponential factor has a very big dynamicrange when ψ changes during MOS transistor operation, largeenough to overrule ni in the multiplication.

1) Derivation of the electric field at the surface: Introducing(2) for N−A in Eq. (4), and then multiplying (4) on both sideswith 2(∂ψ/∂y) gives

∂

∂y

[(∂ψ(y)

∂y

)2]=

2q

εsi

(nie

ψ−VchUT − nie−

ψUT

+NA

1 + gAeψA−(ψ−Vch)

UT

)∂ψ

∂y.

(5)

Integrating (5) from bulk to surface with E = −∂ψ/∂y andEb = 0 yields

E2s =

2q

εsi

∫ ψs

ψb

(nie

ψ−VchUT − nie−

ψUT

+NA

1 + gAeψA−(ψ−Vch)

UT

)dψ.

(6)

In (6) the additional potential dependence due to field-assistedionization of the dopants can be straightforwardly integratedas well, i.e. by replacing NA with NA{1+gA exp[(ψA−(ψ−Vch))/UT ]−gA exp[(ψA− (ψ−Vch))/UT ]} in the numeratorof the third term and splitting the resulting integral. This givesan expression for the square of the electric field at the surface,

E2s =

2qniUTεsi

(eψs−VchUT − e

ψb−VchUT + e

− ΨsUT − e−

ΨbUT

)+

2qNAεsi

[ψs − ψb − UT ln

fs(EA)

fb(EA)

],

(7)

where ψb , (EF,b − Ei)/q is the bulk potential and ψs ,(EF,s−Ei)/q the surface potential, as indicated in Fig.1. EF,sdenotes the Fermi-level at the surface, and EF,b the Fermi-level in the bulk. The logarithmic term in (7) is the contribution

4 GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2018

of incomplete ionization, where

fs(EA) ,1

1 + gAeEA−EF,s

kT

=1

1 + gAeψA−(ψs−Vch)

UT

(8)

the Fermi-Dirac ionization probability at the surface, and

fb(EA) ,1

1 + gAeEA−EF,b

kT

=1

1 + gAeψA−ψbUT

, (9)

the Fermi-Dirac ionization probability in the bulk, assumingthat Vch is zero in the bulk. Both ionization probabilitiesare qualitatively shown in Fig. 1. If complete ionization isassumed, then fs(EA) = fb(EA) = 1 and the incompleteionization term cancels in (7), leading to the expressionwidely-used at RT [27], [28]. The surface-ionization proba-bility fs(EA) is plotted in Fig. 4 as a function of thermal andfield-assisted ionization. Immediately evident is that freeze-out at the surface (arbitrarily defined when fs(EA) < 0.2)is only present when the temperature is below ≈ 50 K andthe potential is close to the flatband condition (ψs ≈ ψb).Above ψb, the ionization probability rapidly transitions to onedue to field-assisted ionization. This transition correspondsto the bending of EA under EF at the surface in Fig. 1.Therefore, complete ionization is a valid approximation evenat deep-cryogenic temperatures, although the shift in EFdue to incomplete ionization (Fig. 2a) should be taken intoaccount since it affects the threshold voltage. This EF -shiftcan be quantified by using fb(EA) from (9) in the bulkcharge neutrality condition, pp = N−A , which leads to thequadratic equation exp(2ψb/UT ) − (ni/NA) exp(ψb/UT ) −(gA/NA) exp(ψA/UT ) with as solution,

ψb = UT lnniNA

+ UT ln1 +

√1 + 4NAni gAe

ψAUT

2. (10)

The second term in (10) is the shift of EF by includingincomplete ionization, which is only dependent on temperatureand doping. Assuming complete ionization, i.e. gA = 0, thewell-known UT ln(ni/NA) is obtained.

2) Derivation of the charge densities: Applying the Gaussianlaw over the semiconductor body in Fig. 1, the total semi-conductor charge density per unit area, Qsc, is obtained byQsc = −εsiEs, with Es given by (7). The obtained Qsc isplotted in Fig. 5a at RT, 77 K, and 4.2 K. For 77 K and 4.2 K,small kinks are noticeable close to ψb due to the transitionfrom incomplete to complete ionization when EA bends underEF at the surface (EF,s), or equivalently, ψs becomes lessnegative than ψA. Above this transition, fs(EA) ≈ 1 accordingto (2). At RT, EF lies above EA in the flatband condition(see Fig.2b) and hence no transitional kink is noticeable.There is however a ψb-shift also at RT due to incompleteionization according to (10). Note that for complete ionization(dashed lines) no kinks are observed since the logarithmic termcancels in (7). Assuming the charge-sheet and fully-depletionapproximations [27], the fixed charge density per unit area,Qf , is given by

Qf = −εsi

√2qNAεsi

(ψs − ψb

)− 2qNAUT

εsilnfs(EA)

fb(EA). (11)

Fig. 3. The intrinsic carrier concentration reaches extremely smallvalues at 4.2 K, left: Fermi-Dirac distribution function approaching astep function at 4.2 K, middle: density-of-states in the conduction band,right: overlap between the density-of-states in the conduction band andthe Fermi-Dirac distribution function at 4.2 K, 77 K and room temper-ature (RT). The overlap function gc(E) × f(E) becomes extremelysmall in magnitude and very peaked at 4.2K. The area under theoverlap function is equal to the intrinsic carrier concentration. Bandgaptemperature-dependency used from Varshni [44] and effective massvalues from Pierret [39].

Relying on the charge neutrality, the mobile charge densityper unit area, Qm, can be obtained from Qm = Qsc − Qf ,resulting in Eq.(12). Qm is plotted in Fig. 6a for RT, 77 K, and4.2 K. As can be observed in this figure, incomplete ionizationdoes not affect the turn-on rate of Qm, but contributes a smalldecrease in the charge threshold voltage, due to the shift ofthe EF -position closer towards the conduction-band edge asshown in Fig. 2a and derived in (10).

Therefore, from this section we conclude that incompleteionization cannot explain the offset between the measuredsubthreshold swing at 4.2 K and the thermal limit. As we willshow in the next section, the temperature-dependent occupa-tion of interface charge traps can degrade the subthresholdswing down to 4.2 K.

3) Interface charge traps: Defects and lattice breaking atthe oxide-semiconductor interface introduce trap energy levels,Et, in the bandgap which degrade the control of the gate-to-bulk voltage, VGB, over the channel. In what follows, theFermi-Dirac occupation of interface traps, f(Et), is includedin the surface-boundary condition and the effect on the Qmturn-on rate is analyzed at 4.2 K. The surface-boundarycondition, i.e. the link between VGB and ψs, is given byVGB = VFB + εsiEs/Cox + (ψs−ψb) where Cox is the oxidecapacitance per unit area, and VFB is the flatband voltage,given by VFB , φms − Qit/Cox [27], [28]. Here Qit isthe interface trap charge density per unit area. We considerthe summation of the discrete acceptor trap energy levels (alldonor states are occupied and neutral during turn-on in nMOS[28]). Each discrete trap-energy-level, Et,j , at position j in thebandgap has its particular N it,j-value assigned to it, whereN it is the density-of-interface-traps per unit area. Qit can thenbe expressed as Qit = −q

∑Nj N it,jfs(Et,j) where N is the

number of interface traps, and

fs(Et,j) =1

1 + gteEt,j−EF,s

kT

=1

1 + gteψt,j−(ψs−Vch)

UT

, (13)

BECKERS et al.: CRYOGENIC MOS TRANSISTOR MODEL 5

Qm = −εsi

√√√√2qniUT

εsi

(eψs−VchUT − e

ψb−VchUT

)+

2qNA

εsi

[ψs − ψb − UT ln

fs(EA)

fb(EA)

]+ εsi

√√√√2qNA

εsi

[(ψs − ψb

)− UT ln

fs(EA)

fb(EA)

](12)

is the Fermi-Dirac occupation probability of the trap energylevel Et,j . The RHS of Eq. (13) is obtained by defining thetrap potentials, ψt,j , (Et,j − Ei)/q [46]–[48]. This leads tothe flatband voltage

VFB =φms +q

Cox

N∑j

Nit,j1 + gt exp{[ψt,j−(ψs−Vch)] /UT }

.

(14)Plotting Qm from (12) versus VGB at 4.2 K in Fig. 6b, includ-ing four interface traps close to the conduction band, revealshow each interface trap degrades the turn-on of Qm separately,as well as the combined effect of the sum of the interface traps.

IV. CURRENT DERIVATION

To derive the current in the linear regime, this core-modelassumes drift-diffusion transport, and does not include ballisticnor quantum transport. To verify the drift-diffusion transport-mechanism at cryogenic temperatures, the proposed model forthe drain-to-source current will be experimentally validated inSection V. Neglecting the hole-contribution to the current, theexpression for the total drain-source current is given by IDS =

−µn(W/L)∫ VDB

VSBQm(Vch)dVch, where the electron mobility

µn is assumed constant along the channel, and W/L is thedevice aspect-ratio, as indicated in Fig. 1. In the linear regime,Qm can be assumed independent of Vch. In this case, the totaldrain-source current is given by IDS = −µn(W/L)QmVDS.In saturation, the integral over Vch cannot be readily solved.

Fig. 4. Dopant ionization at the surface is an interplay between thermalionization (T ) and field-assisted ionization (ψs − ψb). Freeze-out isassumed when 20 % of the dopants are ionized. This happens onlywhen T is below ≈ 50K and close to the flatband condition (ψs ≈ ψb).Whenψs increases, a rapid transition takes place to complete ionizationfor all temperatures. In the flatband condition (ψs = ψb) the ionizationprobability at the surface is only due to thermal ionization and equalsthe ionization probability in the bulk, fb(EA).

Therefore, starting from the drift-diffusion equation gives

IDS = −WL

∫ ψs,D

ψs,S

µnQmdψ+W

L

∫ Qm,D

Qm,S

µnUT dQm. (15)

Assuming a linearization of the mobile charge density withrespect to the surface potential at constant gate voltage [49] in(15), i.e., Qm = mCox(ψs−ψP), with m , ∂(Qm/Cox)/∂ψsand ψP the pinch-off potential, and integrating, results in anexpression for the total drain-source current in saturation,

IDS =W

Lµn

[−Q2m,D −Q2

m,S

2mCox+UT (Qm,D −Qm,S)

]. (16)

Qm,S and Qm,D are obtained from (12), setting Vch to zero andVDS respectively. At cryogenic temperature an improvement inthe low-field mobility, µ0, is observed due to a reduction ofthe phonon scattering [1], [6]. In addition, the mobility reducesat higher gate voltage due to surface roughness scattering athigh vertical electric field [1]. This mobility reduction can bemodeled by µn = µ0/(1 + θVGB) where θ is the mobilityreduction factor.

V. EXPERIMENTAL RESULTS AND DISCUSSION

Room temperature and cryogenic measurements were per-formed on devices fabricated in a 28-nm bulk CMOS pro-cess. The full set of measurements, measurement set-up, andcharacterization were previously reported in [3], [4]. Aftermeasuring at RT, the samples were immersed into liquidhelium (4.2 K) and liquid nitrogen (77 K) baths with a dipstick.Fig. 7a favorably compares the model with the linear transfercharacteristics (VDB = 20 mV) measured at RT and 4.2 Kon a long nMOS device with W/L = 3µm / 1µm, in linearand logarithmic scales. The extracted µ0-values from themodel are in accordance with the characterization performedin [3]. Furthermore Fig. 7b analyzes the effect of incompleteionization, interface traps, and mobility, on the current at 4.2 K.Note that incomplete ionization reduces the threshold voltage,and interface traps can degrade the subthreshold swing (SS) to≈ 10 mV/decade. The strong increase in the mobility increasesthe on-state current at 4.2 K. Fig. 8 validates the model for thecurrent in saturation (|VDB| = 0.9 V) using the measurementsperformed at RT, 77, and 4.2 K on a long pMOS devicewith W/L = 3µm / 1µm, in linear and logarithmic scales. Themetal-semiconductor work function difference, φms, increasesin absolute value at lower temperatures according to thechange in EF -position (Fig. 2).

VI. SUBTHRESHOLD-SWING DERIVATION

In this section a SS-expression including incomplete ioniza-tion and temperature-dependent interface trapping is derived.Incomplete ionization is included to prove the minimal influ-ence on SS shown in Fig.7b. The temperature-dependency

6 GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2018

1x10-6

0.8

0.6

0.4

0.2

0

|Qsc

| [

C c

m-

2]

0.80.60.40.20-0.2-0.4-0.6

ys - Vch [V]

Incomplete ionization

Complete ionization

Vch = 0

NA = 1018

cm-3

T [K]: 4.2, 77, 300

77

300

4.2

7x10-7

6

5

4

3

2

1

0

|Qf |

[C

cm

-2]

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

ys - Vch [V]

Incomplete ionization

Complete ionizationNA = 10

18 cm

-3

4.2

77

77 K

4.2 K

RT

(a) (b)

Fig. 5. (a) Total semiconductor charge density, Qsc, and (b) fixed charge density, Qf , at room temperature (RT, red), liquid-nitrogen temperature(77K, green), and liquid-helium temperature (4.2K, blue) including incomplete ionization (solid lines) or assuming complete ionization (dashedlines). The potential is swept starting from the bulk potential, ψb, calculated at a given temperature and doping according to (10). Horizontal arrowsshow the shifts in ψb by including incomplete ionization at a given temperature. Skewed arrows in the insets indicate the kinks at 77 and 4.2 K dueto the transition from incomplete to complete ionization when EA bends under EF (Fig. 1).

10-12

10-11

10-10

10-9

10-8

10-7

10-6

|Qm |

= |

Qsc

| -

|Q

f | [

C c

m-

2]

1.210.80.60.40.2

VGB [V]

1x10-6

0.8

0.6

0.4

0.2

0

300 77

4.2

fms = -0.7 V

Nit, j = 0

NA = 1018

cm-3

Cox =20 mF m-2

T [K]

IncompleteComplete

Vch = 0

10-13

10-12

10-11

10-10

10-9

10-8

10-7

|Qm

| [C

cm

-2]

0.760.750.740.730.72

VGB [V]

4.2

fms = -0.7 V

Nit, j = 1011

cm-2

NA = 1018

cm-3

Cox = 20 mF m-2

gA = 4

gt = 4

without traps

yt, j = 0.58 V + 3UT

yt, j = 0.58 V + 2UT

yt, j = 0.58 V + UT

yt, j = 0.58 V

sum4.2 K

Vch = 0

(a) (b)

Fig. 6. Mobile charge density, (a) without interface traps at RT, 77K and 4.2K including incomplete ionization (solid lines), and assuming completeionization (dashed lines). Incomplete ionization yields a small decrease in the charge-threshold voltage, (b) Influence of four single interface trapsclose to the conduction band, and their combined effect on the turn-on rate of Qm at 4.2K.

of interface-trap occupation, fs(Et), allows to obtain the∆SS-offset of ≈ 10 mV/dec above the thermal limit, UT ln 10,previously observed on long-channel devices [3], [4].

The subthreshold swing, SS, is usually expressed asnUT ln 10, where the non-ideality factor or slope factor, n, isgiven by (∂VGB/∂ψs), describing the deviation from the ther-mal limit. Assuming fs(Et) from (13) to be one, (∂VGB/∂ψs)yields 1 +

√2qNAεsi/

[Cox(2

√ψs − ψb)

]+ qN it/Cox [27],

[50]. However, at 4.2 K, and assuming the highest possibledoping value below the degenerate limit, a large N it-value inthe order of 1013 cm−2 is extracted to accommodate for a SSof ≈ 10 mV/decade [33], [51], since N it becomes multipliedwith UT in this expression. However, it should be emphasizedthat in the used expression for SS, the temperature dependencyof interface-trap occupation is not taken into account. Relyingon drift-diffusion transport in the linear regime and assumingµ independent of VGB, the subthreshold slope, SS−1, is givenby

SS−1 =1

ln 10

1

Qm

∂Qm∂ψs

∂ψs∂VGB

. (17)

The factor (1/Qm)(∂Qm/∂ψs) is found from (12) by con-sidering Qm � Qf or Qsc ≈ Qf in the subthreshold region.After some mathematical manipulation, we find

1

Qm

∂Qm∂ψs

= qεsi

[1

QmQfnie

ψs−VchUT

− 1

Q2f

NA

(1− UT

f(EA)

∂f(EA)

∂ψs

)].

(18)

Merging (11) and (18), and inverting, gives

SS =2NA ln 10

[(ψs − ψb

)− UT ln f(EA)

fb(EA)

]QfQm

nieψs−VchUT︸ ︷︷ ︸

(a)

−NA(

1− UTf(EA)

∂f(EA)∂ψs

) ∂VGB

∂ψs. (19)

The following relation can be derived for (a) in the aboveequation (see Appendix),

nieψs−VchUT =

QmQf

2NA

[(ψs − ψb

)− UT ln f(EA)

fb(EA)

]UT

. (20)

BECKERS et al.: CRYOGENIC MOS TRANSISTOR MODEL 7

10-8

10-7

10-6

10-5

10-4

I DS [

A]

0.90.80.70.60.50.40.30.2

VGB [V]

30

25

20

15

10

5

0

ID

S [µA

]

298

4.2

T [K]

m0 = 300 cm2/Vs

Nit,j = 1.2 x 1011

cm-2

f ms = - 0.78 V

m0 = 750 cm2/Vs

Nit,j = 1.2 x 1011

cm-2

f ms = - 0.83 V

Measurement Model

nMOS W/L = 3 µm / 1 µm VDB = 20 mV Cox = 20 mF m-2

q = 0.1

NA = 1018

cm-3

gt = 4, gA = 4

10-8

10-7

10-6

10-5

10-4

I DS [

A]

0.80.750.70.650.60.550.5

VGB [V]

m

4.2 K

Incomplete ionizationComplete ionization

Measurement

gA : 4 0

Nit, j = 0

Nit, j

(a) (b)

Fig. 7. (a) Linear model validation with measurements at RT and 4.2K on a long nMOS device (28-nm bulk CMOS process). In the measurementsthe gate voltage was swept from 0.2V to 0.9V with a step size of 1mV in order to reliably resolve the steep subthreshold slope at cryogenictemperature. The sets of used physical model parameters are shown. Five interface traps are placed at ψt,j = 0.58V − 2UT : UT : 0.58V+2UT , (b) Overview of the phenomena influencing the current at 4.2 K: incomplete ionization (gA = 4) leads to a decrease in the threshold voltage;interface traps (Nit,j ) strongly degrade the subthreshold swing, and mobility (µ) increases the on-state current.

10-8

10-7

10-6

10-5

10-4

I SD

[A

]

0.90.80.70.60.50.40.30.2

VBG [V]

60

50

40

30

20

10

0

IS

D [µ

A]

298

4.2 T [K]

77

m0 = 130 cm2/Vs

Nit,j = 5 x 1010

cm-2

f ms = - 0.81 V

m0 = 325 cm2/Vs

Nit,j = 1.2 x 1011

cm-2

f ms = - 0.87 V

Measurement Model

pMOS W/L = 3 µm / 1 µm VBD = 0.9 V Cox = 20 mF m-2

q = 0.9

NA = 1018

cm-3

gt = 4

gA = 4

Fig. 8. Saturation model validation with measurements at RT, 77Kand 4.2K on a long pMOS device (28-nm bulk CMOS process). Fourinterface traps are placed at ψt,j = 0.58V − 2UT : UT : 0.58V +UT . The used physical model parameters for RT and 4.2K are shownin the figure. For 77K, the model parameters are: φms = −0.87V,Nit,j = 1.1× 1011 cm−2, and µ0 = 300 cm2 V−1 s−1.

Plugging this in (19) one finds

SS = UT ln(10)1

1−UT

(1− UT

fs(EA)

∂fs(EA)

∂ψs

)2

[(ψs−ψb

)−UT ln

fs(EA)

fb(EA)

]∂VGB

∂ψs, (21)

where

∂VGB

∂ψs= 1 +

√2qNAεsiCox

1− UTfs(EA)

∂fs(EA)∂ψs

2√

(ψs − ψb)− UT ln fs(EA)fb(EA)

− q

Cox

∑j

N it,j∂fs(Et,j)

∂ψs.

(22)

follows from the surface-boundary condition derived inSec. III-B.3. In the subthreshold region, far above the flatbandcondition, fs(EA) = 1 can be assumed (Fig. 4), and UT �2(ψs − ψb), leading to SS = UT ln 10(∂VGB/∂ψs) with

∂VGB

∂ψs= 1 +

√2qNAεsiCox

1

2√ψs − ψb

− q

Cox

∑j

N it,j∂fs(Et,j)

∂ψs.

(23)

Taking the derivative of (13), (23) becomes

∂VGB

∂ψs= 1 +

√2qNAεsiCox

1

2√ψs − ψb

+q

Cox

1

UT

∑j

N it,jgt exp[(ψt,j − ψs)/UT ]

{1 + gt exp[(ψt,j − ψs)/UT ]}2.

(24)

Note the appearance of a factor 1/UT in the third term on theRHS of (24). Placing a discrete interface trap, ψt,j , at eachψs-value in the subthreshold region, and assuming a uniformN it,j-value for each trap, leads to

∂VGB

∂ψs= 1 +

√2qNAεsiCox

1

2√ψs − ψb

+qN it

Cox

1

UT

gt(1 + gt)2

.

(25)

The first two terms in (25) yield the non-ideality or slope factorwithout interface traps, n0. The SS-expression becomes

SS = n0UT ln 10 +qN it

Cox

gt(1 + gt)2

ln 10 (26)

where the second term on the RHS is the sought ∆SS-offset.The non-ideality factor n0 has an upper bound of 2 mainlyrelated to doping. Therefore, at 4.2 K, the first term on theRHS of (26) is limited by 1.6 mV/decade. Note that N it doesnot become multiplied with UT in (26). Therefore, assuminga reasonable value for N it = 3× 1011 cm−2, the second termgives ≈ 9 mV/decade (with Cox = 20 mF m−2 and gt =4).

8 GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2018

Together they yield the SS-degradation observed on a longnMOS device at 4.2 K. At 77 K, a similar calculation usingn0 = 1.08 and Nit = 3 × 1011 cm−2 gives 25 mV/dec,corresponding to the subthreshold swing measured [3] on along pMOS device at 77 K (Fig. 8).

VII. CONCLUSION

A theoretical MOS transistor model is developed valid fromroom temperature down to liquid-helium temperature. Themodel relies on Boltzmann statistics, verified in the limitto zero Kelvin, and includes incomplete ionization, interfacetraps, bandgap temperature dependency, and mobility reduc-tion. It is evidenced that incomplete ionization maintains thenon-degeneracy of a semiconductor at deep-cryogenic temper-atures, and leads to a decrease in the threshold voltage on topof the overall increase due to Fermi-Dirac distribution scaling.The Fermi-Dirac temperature-dependency of interface-trap oc-cupation degrades the subthreshold swing down to 4.2 K. Anexpression for the subthreshold swing including incompleteionization and temperature-dependent interface trapping isderived. The proposed model builds the indispensable physicalfoundation for future low-temperature CMOS circuit design.

APPENDIX

Starting from Qm+Qf = Qsc, we can write (Qm+Qf )2 =ε2siE

2s, with Es given by (7). Solving a quadratic equation for

Qm leads to

QmQf

= −1 +

√1 +

2qniUT εsiQ2f

eψs−VchUT (27)

where we neglected the exponential term in ψb/UT . In the sub-threshold region (Qm � Qf ),

√1 + x can be approximated

by 1 + x/2 for x→ 0. Using (11) for Q2f leads to (20).

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