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An Improved Virtual-Source-Based Transport Model for Quasi-Ballistic Transistors—Part I: Capturing Effects of Carrier Degeneracy, Drain-Bias Dependence of Gate Capacitance, and Nonlinear Channel-Access Resistance Shaloo Rakheja, Member, IEEE, Mark S. Lundstrom, Fellow, IEEE, and Dimitri A. Antoniadis, Life Fellow, IEEE Abstract— In this paper, an improved physics-based virtual-source (VS) model to describe transport in quasiballistic transistors is discussed. The model is based on the Landauer scattering theory, and incorporates the effects of: 1) degeneracy on thermal velocity and mean free path of carriers in the channel; 2) drain-bias dependence of gate capacitance and VS charge, including the effects of band nonparabolicity; and 3) nonlinear resistance of the extrinsic device region on gm -degradation at high drain currents in the channel. The improved charge model captures the phenomenon of reduction in VS charge under nonequilibrium transport conditions in a quasi-ballistic transistor. Index Terms— III–V HEMTs, carrier degeneracy, nonlinear channel-access resistance, quantum capacitance, quasi-ballistic transport, Si ETSOI, virtual source (VS).
I. I NTRODUCTION
T
HE basic MIT virtual-source (MVS) model provides a simple, physical description of transistors that operate in a quasi-ballistic regime [1], [2]. With only a few fitting parameters, most of which have a physical significance, the model has well served for technology benchmarking [3]. The charge-based compact model describes very well drain–source I –V and all terminal Q–V characteristics in bulk and ETSOI silicon devices [3], [4], III–V transistors [5], and by extension to ambipolar transport to graphene RF transistors [6]–[8].
Manuscript received May 20, 2015; revised July 12, 2015; accepted July 14, 2015. Date of publication August 3, 2015; date of current version August 19, 2015. This work was supported by the National Science Foundation and Semiconductor Research Corporation through the National Communications Network–Nano-Engineered Electronic Devices Simulation Program within the Division of Electrical, Communications and Cyber Systems under Contract 1227020-EEC. The review of this paper was arranged by Editor K. J. Chen. S. Rakheja is with the Department of Electrical and Computer Engineering, New York University, Brooklyn, NY 11201 USA (e-mail:
[email protected]). M. S. Lundstrom is with Network for Computational Nanotechnology, Purdue University, West Lafayette, IN 47907 USA (e-mail: lundstro@ecn. purdue.edu). D. A. Antoniadis is with Microsystems Technology Laboratories, Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TED.2015.2457781
In the first part of this two-part paper, we will revisit and revise the underlying assumptions and the corresponding model equations in the basic MVS model. There are three important components in the basic MVS model formulation that deserve more scrutiny. First, it is assumed in MVS that the virtual-source (VS) saturation injection velocity of carriers is independent of carrier concentration. When a constant VS saturation injection velocity was used in MVS to describe the I –V characteristics of In0.7 Ga0.3 As channel HEMT devices [9], a good match with experimental data was obtained only for an extremely low effective carrier mass (m ∗ = 0.022m 0 ) corresponding to the fitted VS velocity. However, such low effective carrier mass is not reasonable for In0.7 Ga0.3 As material, and a value of m ∗ = 0.035m 0 is more appropriate for bulk In0.7 Ga0.3 As, as shown in [10], with carrier confinement leading to a further increase in the effective carrier mass [11]. Since the effect of carrier degeneracy on thermal velocity is not modeled in the basic MVS model, only a very low effective carrier mass could produce the experimentally measured ON-current of the device. Second, in the MVS model, the VS charge is not influenced by nonequilibrium transport conditions in the channel, and essentially, the gate capacitance of the device is assumed independent of the drain bias. This assumption is too simplistic for quasi-ballistic devices, where the negative momenta of the VS charge distribution are primarily supplied by the drain contact in near-equilibrium transport (Vds ≈ 0 V) and are missing in nonequilibrium transport (high Vds ). Last, the nonlinearity of the resistances of the extrinsic device regions that are responsible for the reduction in the transconductance gm = ∂ I D /∂ Vg of the III–V HEMT devices for high drain currents is also not modeled in MVS. In this paper, we present a revised VS model for quasi-ballistic transistors that overcomes the above-noted limitations of the MVS model, while still retaining the philosophy of a small number of physically meaningful parameters of the original model. We will refer to this model as MVS-2 throughout this paper to denote the latest stage of evolution. The remainder of this paper is organized as follows. In Section II, the transport formulation of MVS-2 is presented.
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Detailed discussions on: 1) the accurate calculation of VS charge; 2) the dependence of VS injection velocity on carrier concentration; and 3) the modeling of extrinsic device regions as nonlinear current-dependent resistances to capture the reduction in device transconductance under high drain currents are presented in Section III. The key findings of this paper are summarized in Section IV. The derivation of the saturation voltage in MVS-2 is presented in the Appendix. In a companion paper, the MVS-2 model is tested by fitting the experimental dc I –V data of InGaAs quantum-well HEMT and Si ETSOI devices. II. T RANSPORT F ORMULATION In the MVS-2 model, the transistor current I D normalized to the device width W is given as ID = Q x0 v x0 Fsat (1) W where Q x0 is the charge at the top of the barrier (ToB) or commonly known as the VS point, v x0 is the VS saturation injection velocity of carriers, and Fsat is an empirical function that achieves transition from saturation to linear regimes of transport. The effect of the resistances of extrinsic device regions (Rs for the source terminal and Rd for the drain terminal) is to lower the intrinsic drain–source voltage Vds and the gate-source voltage Vgs according to = Vds − I D (Rs + Rd ) Vds
Vgs = Vgs − I D (Rs )
(2a) (2b)
where Vds and Vgs are the extrinsic drain–source and gate– source voltages, respectively. In this paper, Rs and Rd are modeled as voltage-dependent nonlinear resistances as explained in Section III-C. The saturation function Fsat in MVS-2 is given as [1] Fsat =
/V Vds dsat
/V 1 + Vds dsat
β 1/β
(3)
where Vdsat is the saturation voltage and β is an empirical parameter obtained from the experimental calibration. β controls the slope of the transition of current from the linear to the saturation transport regime. A. Saturation Voltage The saturation voltage Vdsat in (3) is given as 1 Tsat vT λ Vdsat = × f1 μeff Tlin 2 − Tsat Q x0 highVds f1 = Q x0 Vds = 0 V
(4a) (4b)
where v T is the thermal velocity of carriers in the channel, μeff is the long-channel mobility of carriers, and λ is the mean free path (MFP) of carriers associated with backscattering. Tlin and Tsat are the Landauer transmission coefficients of the channel in the linear and saturation operation regimes, respectively. The function f1 accounts for the difference in (non-equilibrium) and low V the VS charge Q x0 in high Vds ds (near-equilibrium) conditions in the channel as discussed
Fig. 1. Critical length normalized to the channel length as a function of Vds for various values of θ and β in (7).
in Section III. The derivation of Vdsat for the MVS-2 model is presented in the Appendix. It must be noted that the MFP for backscattering is different from the MFP for momentum relaxation. In the case of 2-D transport, λ = (2/π)λm , where λm is the MFP for momentum relaxation [12]. The Landauer transmission coefficient T , of the transistor channel is given as T =
λ λ + L crit
(5)
where L crit is the critical backscattering length (called critical length hereafter) of the low-field region near the source-end of the channel that governs the steady-state ON-current of the device. For ballistic transport conditions, T is unity, but for quasi-ballistic transport in the channel, T < 1. Assuming the Boltzmann statistics, the critical length L crit can be computed as [13] L eff V (x) − V (0) dx (6) L crit = exp − kB T 0 where L eff is the effective channel length (assumed the same as the gate length here), V (x) is the position-dependent channel potential, and V (0) is the potential at the beginning of the channel. To obtain L crit , one needs to know the exact potential profile V (x) in the channel, which requires numerical techniques as discussed in [14], [15]. For compact modeling purposes, L crit is given semiempirically as L crit = L eff × (1 − f2 ) + ξ L eff × f 2 /(θ φ ) Vds t f2 = β 1/β 1 + Vds /(θ φt )
(7a) (7b)
where φt = k B T /q is the thermal voltage and ξ < 1 is the ratio of the critical length in saturation and the channel length. ξ is independent of the channel length and is universally fitted for a given device technology. The parameter θ is also θ φ , the critical length is equal universally fitted. For Vds t to ξ L eff . Fig. 1 shows L crit normalized to L eff as a function of Vds for various values of θ . For a given β, the value of θ determines the knee drain–source voltage in Fig. 1 at which the critical length begins to saturate to ξ L eff . The typical value of θ is ≈2–3 as discussed in [15]. The parameter β controls
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the sharpness of the transition of L crit from L eff (linear region) to ξ L eff (saturation region). While, in principle, the value of the empirical parameter β in (3) and (7b) can be adjusted independently, in this paper, we assume the same value in the two equations. This allows us to limit the number of empirically adjusted parameters in the MVS-2 model without sacrificing the quality of model fits as discussed in the second part of this paper. Using the definition of L crit in (5), the transmission coefficient in the linear and saturation regimes of transport is given as λ λ + L eff λ = . λ + ξ L eff
Tlin =
(8a)
Tsat
(8b)
It must be noted that in the case of λ L eff , the effect of L crit on the channel transport is not very significant. Hence, there is only a minimal sensitivity of model fits on the choice of the parameters θ and ξ for λ L eff . This will be particularly important in the case of III–V HEMT devices presented in the companion paper. Using the Fermi–Dirac (FD) statistics and single subband occupancy, the MFP of carriers λ for backscattering is given as [16] F0 (ηfs ) (9a) F−1/2 (ηfs ) E fs − 10 + qψs ηfs = (9b) kB T where q is the elementary charge, E fs is the Fermi level in the source contact, 10 is the first subband energy in the channel, ψs is the potential at the charge centroid in the channel, F j (η) is the Blakemore FD integral of the j th order [17], and λ0 is the low-field MFP in the bulk material of the channel and is given as μeff (10) λ0 = 2φt vT 0 where φt is the thermal voltage, μeff is the long-channel effective carrier mobility in the channel material, and v T 0 is the thermal velocity of carriers in the nondegenerate regime given as [18] 2k B T m C (ηfs ) vT 0 = (11) πm D (ηfs )2 λ = λ0
where m C (ηfs ) is the energy-averaged conductivity effective mass of carriers, and m D (ηfs ) is the energy-averaged densityof-states (DOS) effective mass of carriers in the first subband of the channel by accounting for the nonparabolicity of the conduction band as explained in Section III-A. Furthermore, m D (ηfs ) = m C (ηfs ) for no mass anisotropy in the plane of the quantum well for III–V HEMT devices. III. F ORMULATION OF VS C HARGE , VS V ELOCITY, AND C HANNEL -ACCESS R ESISTANCES This section provides models for the VS charge and the VS velocity in MVS-2. Furthermore, the nonlinearity of the
access-region resistances in III–V HEMT devices to explain the gm -reduction as shown in the experimental measurements in [5] is also discussed. A. Virtual Source Charge Q x0 Under low drain–bias conditions (near equilibrium), 1-D electrostatics may be used to obtain the charge at the ToB. For single subband occupancy,1 Q x0 is given as [18] ∞ D2-D (E) f FD (E)d E (12a) Q x0 = −q
10 −qψs
m D (E) D2-D (E) = π h¯ 2 f FD (E) =
1 + exp
(12b) 1
E−E f kB T
(12c)
where the Fermi level E f is the same in the source and the drain contacts in equilibrium, and m D (E) is the energy-dependent DOS effective mass of carriers due to the nonparabolicity of conduction band in the channel. As discussed in [19]–[21], the DOS effective mass of carriers in InGaAs/InAs channel materials increases with the kinetic energy of the electrons according to m D (E) = m D,0 (1 + κm E)
(13)
where m D,0 is the effective DOS mass of electrons when their kinetic energy is zero, and κm is the rate of increase in 1/J of the DOS effective mass with the kinetic energy. Using (13) in (12) and carrying out the integration, the VS charge is given as Q x0 = −q N2-D (ηfs ) ln(1 + eηfs )
(14)
where N2-D (ηfs ) is the energy-averaged 2-D DOS in the channel and is given as kBT (15) N2-D (ηfs ) = m D (ηfs ) 2 π h¯ η Li 2 (−e fs ) m D (ηfs ) = m D,0 1 − κm k B T (16) ln(1 + eηfs ) where the function Li s (.) is the polylogarithm function of order s (also called the dilogarithm function for s = 2). At the VS point and ignoring 2-D electrostatic coupling to the source and the drain contacts, the gate voltage Vg must balance the surface potential and the voltage drop across the insulator according to Q x0 Vg = ψs − (17) Cgc where Cgc is the per-unit-area gate-channel capacitance. Incorporating quantum-mechanical correction due to the QM finite separation x av of the channel charge centroid from the semiconductor–insulator interface, Cgc is given as
1 1 1 (18) = + QM Cgc Cins CQM x av 1 Multiple subband contribution to current conduction as is the case in Si ETSOI devices [4] can be easily incorporated in the VS charge model. This point will be revisited in the companion paper.
RAKHEJA et al.: IMPROVED VS-BASED TRANSPORT MODEL FOR QUASI-BALLISTIC TRANSISTORS
Fig. 2. In the nonequilibrium condition, states at the ToB will be filled differently from the source and the drain contacts. The source contact fills the states according to its Fermi level E fs and the drain contact fills the states according to its Fermi level E fd = E fs − q Vds . The transmission coefficient T accounts for elastic scatterings in the channel in quasi-ballistic transport conditions.
where Cins is given as ins /tins. Here, ins is the insulator dielectric constant and tins is the insulator thickness or effective QM ) is thickness in the case of composite dielectrics. CQM (x av given as QM
ch CQM x av = QM (19) x av where ch is the dielectric constant of the channel region and QM the quantum-mechanical correction x av is given semiempirically as [22] QM = x av
B (Q B + 11/32Q x0)1/3
(20)
where B and Q B are the empirical parameters that are determined from the calibration of the model with experimental C g –Vg data. The parameter Q B can be expressed in terms of B according to
3 B (21) QB = QM x av,0 QM
where x av,0 is the average separation of the charge centroid from the insulator interface when Q x0 is negligible (under low-Vgs conditions). Equations (12)–(20) must be solved selfconsistently to obtain Q x0 versus Vg under low drain–bias conditions (also referred to as near-equilibrium conditions in the channel). For the experimental calibration of the charge model, C g = ∂(−Q x0 )/∂ Vg is obtained by differentiating (17). For near-equilibrium conditions in the channel, C g is independent of the drain bias. is nonzero and When the intrinsic drain–source voltage Vds the device is under nonequilibrium conditions, the correct procedure to obtain Q x0 is by properly accounting for all the ways in which the source and the drain contacts can fill up the energy states at the ToB. As shown in Fig. 2, the contribution to Q x0 comes from the flux injected into the channel by both the source and the drain contacts. While the flux injected from the source contact depends on the source Fermi level E fs , the flux injected from the drain contact depends on the drain . The Landauer transmission Fermi level E fd = E fs − q Vds coefficient T in Fig. 2 accounts for elastic scatterings in
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Fig. 3. 2-D circuit model for quasi-ballisic transistors. The surface potential ψs at the VS point is controlled by (a) terminal voltages through the three capacitors shown and (b) charge at the VS point. The charge at the VS point, in turn, is determined by the source and the drain Fermi levels and the transmission coefficient in the channel.
the channel in quasi-ballistic transport conditions. Under one , Q subband approximation and for a general Vds x0 is given as q (22) Q x0 = − [(2 − T )Fs + T Fd ] vT where v T is the thermal velocity of carriers, and Fs and Fd are the source and the drain fluxes, respectively, and are given as N2-D (ηfs ) Fs = v T ln(1 + eηfs ) (23a) 2 N2-D (ηfd ) ln(1 + eηfd ) (23b) Fd = v T 2 ))/k T . Note that (22) where ηfd = (E fs − 10 + q(ψs − Vds B becomes the same as (12) for Vds = 0 V. For T = 1 and k T /q, the drain flux is cutoff, and for a given value Vds B of ψs , Q x0 in (22) is lower than its value at Vds = 0 V in (12). Hence, ignoring the drain-bias dependence of gate capacitance, as is the case in the basic MVS model, is not physically sound for quasi-ballistic transistors. Including the 2-D electrostatics, the surface potential ψs at the VS point can be related to the various terminal voltages through the capacitors, as shown in Fig. 3 [18]. From Fig. 3, it can be seen that C g−VS Cd−VS Cs−VS Q x0 Vg + Vd + V + (24) ψs = C C C s C where C = Cs−VS + C g−VS + Cd−VS . Furthermore, it must be noted that C =n (25a) C g−VS Cd−VS =δ (25b) C g−VS (25c) C g−VS = Cgc is the nonideality factor and is related where n = n 0 + n d Vds to subthreshold swing (SS) according to n = SS/(2.3φt ), n d is the punchthrough factor, and δ denotes the drain-induced barrier lowering (DIBL). For a well-tempered MOSFET, it is expected that the gate controls the channel, such that C g−VS /C ≈ 1, while Cd−VS /C and Cs−VS /C are both 1. Equations (22)–(25) completely describe the
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Fig. 4. VS charge Q x0 versus Vgs for Vds = 0 and 0.5 V obtained by solving (22)–(25). Here, n = 1.2, Cgc = Cins = 2 μF/cm2 , m D,0 = 0.035m 0 , κm = 0, and T = 1. The effect of DIBL on Q x0 only occurs when Vds = 0 V. Likewise, for Vds = 0 V, there is no effect of T on Q x0 . Inset: zoomed-in view of Q x0 in the high-Vgs regime.
Fig. 5. VS charge normalized to the VS charge for T → 0 (completely diffusive transport) as a function of the Landauer transmission coefficient T in the channel. Here, n = 1.2 and Cgc = Cins = 2 μF/cm2 , and m D is the DOS effective mass of electrons in the channel for a parabolic subband.
relationship between Q x0 and the various terminal voltages if the values of n, δ, and Cgc are known. In this paper, we assume that the parameters n 0 and δ are independent of the magnitude of the channel charge, that is, Vgs does not affect the value of these parameters in the MVS-2 formulation. This assumption is confirmed by our 2-D simulations of the ToB charge density in short-channel FETs with quantum-mechanical effects using the software tool nextnano [23]. Experimentally, it has also been repeatedly observed by us that, for well-behaved transistors, one can describe below and above threshold operation with a single DIBL parameter. Finally, maintaining n 0 and δ constants is a reasonable compromise between simplicity and accuracy. Fig. 4 shows Q x0 versus Vgs for Vds = 0 V (channel in equilibrium) and Vds = 0.5 V (channel out of equilibrium) for m D,0 = 0.035m 0 obtained by solving (22)–(25). Other simulation parameters are noted in Fig. 4. In the absence of DIBL, there is a reduction in Q x0 at high Vds for the entire Vgs range for this simulation. However, a finite DIBL will increase Q x0 at high Vds . The increase in Q x0 due to DIBL and the reduction in Q x0 because of the reduction in flux from the drain at high Vds compete against each other to determine the net behavior of Q x0 with various terminal voltages. As shown in Fig. 5, the reduction in Q x0 at high Vds is more prominent when T is high, that is, when the channel is more ballistic and when N2-D is low.
v x0 in terms of v T according to Tsat vT . v x0 = 2 − Tsat
B. Carrier-Concentration-Dependent VS Saturation Injection Velocity v x0 In the saturation operation regime, the width-normalized device current in the MVS-2 model is given as Q x0 v x0 as the saturation function Fsat in (1) approaches unity when V Vds dsat . According to the Landauer transmission theory, the saturation current I D,sat is given as [9], [24], [25] Tsat I D,sat = (26) Q x0 v T . W 2 − Tsat Equating the saturation current in the MVS-2 model with that from the Landauer transmission theory allows us to express
(27)
Using Tsat (Landauer transmisison coefficient of the channel in the saturation regime) from (8b), the relationship between v x0 and v T can be further simplified as λ . (28) v x0 = v T λ + 2ξ L eff Since both the MFP λ and the thermal velocity v T of carriers vary with carrier concentration, the VS injection velocity v x0 , as defined by (28), will also vary with carrier concentration. Using FD statistics and assuming single subband occupancy, the generalized expression for thermal velocity v T of carriers is given as [14], [15] F1/2 (ηfs ) (29) vT = vT 0 ln(1 + eηfs ) where the nondegenerate thermal velocity v T 0 is given in (11) and ηfs is defined in (9b). Note that in the MVS-2 model formulation, the biasdependent transmission coefficient T in the channel is used for the calculation of the critical length L crit and VS charge Q x0 . However, for the calculation of the VS saturation injection velocity v x0 , only Tsat is used. C. Nonlinear Channel-Access Resistance In III–V HEMT devices, the device transconductance gm degrades under nonequilibrium transport conditions as shown experimentally in [26] and discussed in the companion paper. However, in the basic MVS model, gm saturates at high drain and gate voltages according to gm ≈ WCgc v x0 / (1 + WCgc Rs v x0 ). Here, Rs is the voltage-independent source resistance and v x0 is the fitted VS saturation injection velocity in the basic MVS model. To model the reduction in device transconductance at high Vds , the source and the drain resistances are modeled
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TABLE I D ESCRIPTION OF M ODEL PARAMETERS IN THE MVS AND MVS-2 M ODELS . PARAMETERS IN C OLORED C ELLS A RE U NIVERSALLY F ITTED , T HAT I S , T HEY A RE I NDEPENDENT OF G ATE L ENGTH FOR A G IVEN D EVICE T ECHNOLOGY. E VEN T HOUGH THE MVS-2 M ODEL H AS A F EW M ORE PARAMETERS T HAN THE MVS M ODEL , M OST OF THE PARAMETERS A RE U NIVERSALLY F ITTED IN THE MVS-2 M ODEL . I N A DDITION , THE MVS-2 M ODEL I S AN E FFECTIVE -M ASS -BASED M ODEL AND C APTURES THE P HYSICS OF C ARRIER T RANSPORT IN N ANOSCALE T RANSISTORS M ORE A CCURATELY
as current-dependent nonlinear resistances according to the emprical expression given as [27] Rc Rs,d = β 1/β ID 1 − Iext,sat
(30)
where Rc is the resistance of the extrinsic device region under low I D and Iext,sat is the maximum current supported by the extrinsic device region. The parameter β in (30) is the same
as that in the empirical function Fsat in (3). The maximum current Iext,sat in the extrinsic region is given as Iext,sat = q W n ext v T 0
(31)
where n ext is the charge concentration in the extrinsic device region. According to the formulation in (30), Rs,d will tend to ∞ when I D = Iext,sat. An appropriate value of n ext ensures that Iext,sat > I D for all values of terminal voltages, and that no source starvation [28] occurs.
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A complete list of all the model parameters with a brief description is given in Table I. In the basic MVS model, there are a total of 10 fitting parameters, while there a total of 15 fitting parameters in the MVS-2 model. However, out of the 15 fitting parameters, 11 parameters are universally fitted, which means that they are independent of gate length for a given device technology. The universally fitted parameters are shown in shaded cells in Table I. The only parameters that vary from device-to-device are: n 0 , n d , δ, and (E fs − 10 ). These four parameters are determined by the electrostatic integrity of the particular device, and can be easily extracted from the experimental transfer curves of the device in subthreshold conduction. While with the exception of the gm degradation, which could be added to it, the basic MVS model can fit the experimental data equally well as MVS-2, and with fewer fitting parameters, it does not correctly capture the essential physics of the nanotransistor in the presence of carrier density degeneracy. Furthermore, the model for Q x0 as currently implemented in the MVS model cannot explain the reduction in the VS charge at high drain bias in quasi-ballistic transport conditions. In addition, the effect of band nonparabolicity on the effective mass of carriers is not included in the basic MVS model. In the companion paper, we discuss why the basic MVS model can fit the experimental data equally well with a fewer number of fitting parameters. IV. C ONCLUSION In this paper, an improved VS-based I –V model for quasi-ballistic transistors is presented. A physically sound model for the VS charge is presented that accounts for the reduction in the VS charge due to the reduction in the drain flux at high drain bias in a quasi-ballistic transistor. The quantum correction to the gate-channel capacitance in strong inversion is also included in the charge model. The model also captures the effects of nondegeneracy in carrier concentration on the thermal velocity and the MFP of the carriers in the channel. Finally, the model is able to describe the reduction in the device transconductance under high drain currents by formulating the extrinsic device regions as nonlinear current-dependent resistances. ACKNOWLEDGMENT The authors would like to thank Dr. A. Majumdar from the IBM Thomas J. Watson Research Center, Dr. D.-H. Kim from Global Foundries, and Prof. M. Luisier from ETH Zürich for the insightful discussions. A PPENDIX C ALCULATION OF S ATURATION VOLTAGE IN THE MVS2 M ODEL The saturation voltage Vdsat is defined as the drain–source voltage at which the currents in the linear and saturation regimes of transport become equal at a given gate voltage. That is (32) = Vdsat = I D,sat @Vds = Vdsat I D,lin @Vds
where I D,lin and I D,sat are the transistor drain–source currents in the linear and saturation regimes of transport, respectively. According to the Landauer transmission model in equilibrium [25] μeff I D,lin = Vdsat = Q x0 Vds =0 V @Vds Tlin Vdsat W λ (33) In saturation, the current is simply given as I D,sat @Vds = Vdsat = Q x0 highVds v x0 W Tsat = Q x0 highVds v T . 2 − Tsat Equating (33) and (34), Vdsat is given as vT λ 1 Tsat Vdsat = f1 μeff Tlin 2 − Tsat
Q x0 highVds f1 = Q x0 Vds =0V Using Tlin and Tsat from (8), Vdsat is simplified to vT λ λ + L eff Vdsat = f1 . μeff λ + 2ξ L eff
(34)
(35a) (35b)
(36)
The first term in parentheses in (36) is equal to 2φt for nondegenerate carrier statistics. In the more general case, this term is given as F1/2 (ηfs ) vT λ = 2φt . (37) μeff F−1/2 (ηfs ) in (35b) is The ratio of the VS charge for high and low Vds given as Q x0 highVds (2 − T )Fs + T Fd = (38) f1 = =0 V 2Fs Q x0 Vds
where Fs and Fd are given in (23). Substituting (37) and (38) in (35b), we obtain F1/2 (ηfs ) λ + L eff Vdsat = 2φt F−1/2 (ηfs ) λ + 2ξ L eff
(2 − T )Fs + T Fd . × 2Fs
(39)
= 0 V, f is equal to unity. Hence, the output At Vds = Vds 1 conductance of the device at Vds → 0 V is equal to μapp 1 ∂ ID GD = = Q x0 (40) W Vds →0 W ∂ Vds Vds →0 L eff
where μapp is the apparent mobility of the device. Apparent device mobility is given by the Mattheissen’s sum of the longchannel effective mobility μeff and the ballistic mobility μ B , which scales with the channel length [25]. That is 1 1 1 = + μapp μeff μB L eff μ B = μeff λ L eff . μapp = μeff L eff + λ
(41a) (41b) (41c)
RAKHEJA et al.: IMPROVED VS-BASED TRANSPORT MODEL FOR QUASI-BALLISTIC TRANSISTORS
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Shaloo Rakheja was a Post-Doctoral Research Associate with Microsystems Technology Laboratories, Massachusetts Institute of Technology, Cambridge, MA, USA. She is currently an Assistant Professor of Electrical and Computer Engineering with New York University, Brooklyn, NY, USA, where she is involved in nanoelectronic devices and circuits.
Mark S. Lundstrom (S’72–M’74–SM’80–F’94) is currently the Don and Carol Scifres Professor of Electrical Engineering with Purdue University, West Lafayette, IN, USA, where he is involved in electronic devices. Prof. Lundstrom is a member of the National Academy of Engineering.
Dimitri A. Antoniadis (M’79–SM’83–F’90–LF’14) is currently the Ray and Maria Stata Professor of Electrical Engineering with the Massachusetts Institute of Technology, Cambridge, MA, USA, where he is involved in nanoelectronics. Prof. Antoniadis is a member of the National Academy of Engineering.
