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An Improved Model for Quasi-Ballistic Transport in MOSFETs Avirup Dasgupta, Graduate Student Member, IEEE , Amit Agarwal, and Yogesh Singh Chauhan, Senior Member, IEEE
Abstract — We have already presented a compact model for FETs operating in the quasi-ballistic regime [1]. However, this model suffers from two important problems: 1) the profile for charge density along the channel is not correctly accounted for and 2) current is not conserved throughout the channel. In this brief, we propose improvement, which does away with these inaccuracies. Index Terms — Ballistic, drift diffusion, FinFET, HEMT, mean free path, nanowire, quasi-ballistic, surface potential.
I. I NTRODUCTION
T
HE problems with the existing model [1] are clearly shown in Figs. 1 and 2. Fig. 1 shows the inaccuracy in the charge density profile. If the boundary conditions, i.e., the charge densities at the source and drain sides, are used to calibrate the existing model, it overestimates the charge density throughout the channel. Fig. 2 shows the nonzero error in current conservation along the channel, for the existing model [1]. These problems arise, because the model [1] does not consider charges flowing in from the drain and does not take current conservation into account while calculating the electrostatic parameters. In this brief, we present an improved model that is more physical and is free from these issues. This brief is arranged as follows. Section II describes the new model for charge profile, which is used for the derivation of the potential profile while taking current conservation into account. Section III shows the use of these potential and charge profiles in the calculation of the drain current. The improved model is compared with the existing model highlighting its superiority in Section IV. We conclude in Section V.
Fig. 1. Results from the model and Monte Carlo simulations [3] for charge density of a quasi-ballistic double-gate FET (Lg = 10 nm) at drain-tosource voltage V ds = 0.5 V and gate-to-source voltage Vgs = 1 V. The improved model captures the charge density accurately. The figure also shows the charge profile predicted by the existing model. Since the existing model does not account for the charges flowing in from the drain, it fails to account for the increasing trend in the charge density at the drain edge.
II. D ERIVATION OF P OTENTIAL P ROFILE Poisson’s equation along the channel (x) is given as [1] Vgfb − ψc 1 ∂ 2 ψc q = n(x) + κ ∂x2 ξ s
(1)
Manuscript received February 16, 2017; revised April 1, 2017; accepted May 15, 2017. Date of publication May 29, 2017; date of current version June 19, 2017. This work was supported in part by Ramanujan Fellowship Research Grant, in part by the Council of Scientific and Industrial Research, India, in part by the Department of Science and Technology, India, and in part by Semiconductor Research Corporation. The review of this brief was arranged by Editor B. Iñiguez. (Corresponding author: Avirup Dasgupta.) A. Dasgupta and Y. S. Chauhan are with the Nanolab, Department of Electrical Engineering, IIT Kanpur, Kanpur 208016, India (e-mail:
[email protected];
[email protected]). A. Agarwal is with the Department of Physics, IIT Kanpur, Kanpur 208016, India. 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.2017.2706301
Fig. 2. Relative error in the current density along the channel length from the source (x = 0) to the drain (x = L) calculated using the existing model [1] as well as the improved model proposed in this brief. The simulation has been performed for a quasi-ballistic device with gate length, L = 50 nm and S = 0.006, at V gs = Vds = 1 V. The existing model [1] does not conserve current along the channel, but the new electrostatics model has current conservation built into it through (2) and results in zero relative error.
where n(x) represents the total charge density, while all other symbols used have been defined in Table I. Current
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A. Source-Side Injection
TABLE I D ESCRIPTION OF THE C OMMON S YMBOLS U SED IN T HIS B RIEF
Using (4) in (3), we get α
∂ 2 ψc1 ∂ψc1 + + φt α 2 = 0 ∂x ∂x2
(6)
where α = ln(γ )/λ and ψc1 is the potential due to the source2 /∂ x 2 ) from (1) can be substituted in (6) injected charge. (∂ψc1 to yield a first-order linear differential equation as κ ∂ψc1 + ψc1 = M1 − N1 γ x/λ ∂x αξ where M and N are given as κ qκ Vgfb − n b-source − φt α; M1 = αξ αs
N1 =
(7)
qκ n 0s . αs
(8)
Equation (7) can be solved analytically to give κ κ x/λ e αξ x αξ x −κ αξ M N e γ 1 1 x − + C1 ψc1 = e αξ κ κ α + αξ
(9)
where C1 is a constant of integration, which can be calculated using the suitable boundary conditions. conservation dictates that at any point in the channel ∂ψc ∂n dd n b v b + μn dd + μφt = constant ∂ x ∂ x
Ballistic
Drift
B. Drain-Side Injection (2)
where n b and v b represent the ballistic charge density and the ballistic velocity, respectively, while n dd is the drift–diffusive charge density. Differentiating (2) with respect to x yields (3)
In [1], it has been shown that the drift–diffusive charge density injected from the source as a function of x is of the form n dd-source(x) = n s
S γ (x/λ) = n 0s γ (x/λ) 1−S
(4)
where n 0s = n s [S/(1 − S)] and γ is a material-dependent parameter. From [1], γ = (1 − S), where S is the fraction of incoming charge carriers that scatter at every λ interval. The ballistic charge density is given as n b-source = (1 − S) L/λ n s . However, [1] only considers charges being injected from the source and not the charges being injected from the drain [2]. Since the scattering model presented in [1] is not dependent on the direction of charge flow, we can use the same model to write the charge density injected from the drain as n dd-drain(x) = n d
S γ L−x/λ = n 0d γ −x/λ 1−S
∂ 2 ψc2 ∂ψc2 − − φt α 2 = 0 (10) ∂x ∂x2 where ψc2 is the potential due to the drain-injected charge. Equation (10) combined with (3) can be written as α
Diffusion
∂ 2 ψc ∂ 2 n dd ∂n dd ∂ψc + μn dd μ + μφ = 0. t ∂x ∂x ∂x2 ∂x2
Similarly, using (5) in (3), we get
(5)
where n 0d = n d (S/(1 − S))γ L/λ . The ballistic charge injected from the drain is given as n b-drain = (1 − S) L/λ n d . Since (1) and (3) are both linear equations (first degree), we can use the principle of superposition to say that the final potential profile is a sum of the individual potential profiles due to n s−d (x) = n b-source + n dd-source and n d−s (x) = n b-drain + n dd-drain.
∂ψc2 κ − ψc2 = M2 + N2 γ −x/λ ∂x αξ where M2 and N2 are given as qκ κ qκ Vgfb ; N2 = n b-drain + φt α − n 0d . M2 = αs αξ αs
(11)
(12)
Equation (11) can be solved to get the analytical expression for ψc2 as κ κ − αξ x κ N2 γ −x/λ e− αξ x x −αξ M2 e αξ ψc2 = e − + C2 (13) κ κ α + αξ where C2 is a constant of integration.
C. Final Potential Profile The final potential profile, (ψc ), due to all charges can be obtained through superposition as ψc = ψc1 + ψc2 .
(14)
The potential at the surface, (ψ), is therefore given as [1] ψc + (κ − 1)Vgfb . κ Using the boundary conditions ψ=
ψ(x = 0) = 0; ψ(x = L) = Vds
(15)
(16)
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Fig. 3. Plots show the validation of the improved model presented in this brief with measurement for an In0.85 Ga0.15 AS nanowire FET [4] with varying gate and drain voltages in parts (a) and (b), respectively. The model matches the measurement with good accuracy. Parameters used: S = 0.005, A = 0.4, U0 = 500 cm2 /Vs, UA = 40 V/m, ETA0 = 70 × 10-3 , THETASAT = 1, NFACTOR = 0.01, and CDSCD = 0. Parameters not specified in this brief are from [5].
where Vds is the drain-to-source voltage, and defining K 1 and K 2 as K1 =
N2 + N1 αξ(M2 − M1 ) + κ − (κ − 1)Vgfb κ α + αξ
(17)
N2 γ −L/λ + N1 γ L/λ αξ(M2 − M1 ) + κ κ α + αξ − (κ − 1)Vgfb (18)
K 2 = κ Vds +
Idd =
we can express the integration constants as κL
C2 =
K 2 e αξ − K 1 e
2κ L αξ
−1
; C1 = K 1 − C2.
B. Drift–Diffusive Current The drift–diffusive current can be calculated as in [1] while using n dd,eff (x) = n dd-source(x) − n dd-drain(x) as the drift–diffusive charge density. The second term comes from the current flowing from the drain to source, which was not considered in [1]. This yields the total drift–diffusive current as
(19)
III. D RAIN C URRENT
qμeff W Ts L N2 n 0s + N1 n 0d × αL α + κ N1 n 0s − (γ −2L/λ − 1) + κ n 0s (κ + α −1 ) 2(α + κ ) ×{γ L/λ (C1 e−κ L + C2 eκ L ) − (C1 + C2 )} − κ n 0d
The drain current can be written as the sum of the ballistic current (Ib ) and the drift–diffusive current (Idd ). Care must be taken to consider the flow of electrons both from the source to the drain as well as from the drain to the source.
×(κ +α −1 ){γ −L/λ (C1 e−κ L +C2 eκ L )−(C1 +C2 )} (22) +φt {n 0s (γ L/λ − 1) − n 0d (γ −L/λ − 1)} . where κ = κ/(αξ ) and μeff is the effective mobility including degradation due to vertical field, as described in [1].
A. Ballistic Current IV. D ISCUSSION
The ballistic current can be written as Ib = Ib,s−d − Ib,d−s
(20)
where Ib,s−d and Ib,d−s denote the magnitudes of the current due to electrons flowing from the source to the drain and from the drain to the source, respectively. The drain-to-source current, Ib,d−s , has not been taken into account in [1]. Borrowing the ballistic current model from our previous work [1], Ib can be given as Ib = q Ts W (n b-source v b (x = 0) − n b-drain v b (x = L)) 1 − e−A·Vds /φt × (21) 1 + e−A·Vds /φt where v b is calculated as in [1].
Fig. 1 shows the charge density obtained from the improved model presented here and from the existing model [1], along with the results from Monte Carlo simulations [3], for a 10-nm quasi-ballistic device. Fig. 1 shows the improvement in accuracy. The accurate match with the numerical simulation also demonstrates the physicality of the proposed electrostatics model. Fig. 2 shows the comparison of the relative error in the current density along the channel. The existing model has issues with current conservation as illustrated by the nonzero error. Also, the error decreases at the terminals, as Ids is calculated using the terminal values of charges and potentials. However, as we move away from the source and drain edges, the error increases due to the deviation of the model from the actual value of the charges and potentials. This is
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Fig. 4. (a) and (b) Comparison of the model presented in this brief with the existing model [1] for the same parameter set as in Fig. 3. The significant difference between the results from the two models is because: 1) the value of ballisticity in the improved model is bias-dependent and different for the flow of electrons from the source and from the drain resulting in the different effective ballisticity of this model as compared with the constant ballisticity of the existing model [1] and 2) the drain-to-source electron flow is not taken into account in the existing model [1]. If the effective ballisticity of the improved model is made equal to the ballisticity of the existing model [1] and the drain-to-source electron flow is turned OFF, the currents predicted by the two models are nearly equal, as shown in (c).
because: 1) the model for the charge density does not consider the charges flowing in from the drain and 2) current conservation is not considered in the electrostatics model while calculating the charge and potential profiles. Fig. 3(a) and (b) shows the validation of the model presented in this brief with measurements for a quasi-ballistic In0.85 Ga0.15 As nanowire FET [4] with L = 75 nm. Short channel effects have been included as in [1] and [5]. We have also used the same parameter set to simulate the existing model [1] and the results are shown in Fig. 4(a) and (b). As can be seen, the existing model overestimates the drain current. Note that, with suitable changes to the parameter set, the existing model [1] can also be made to fit the measurement reasonably well. This flexibility of the existing model has already been demonstrated in [1]. However, the improvements suggested in this brief make it more physical. The difference between the drain current obtained from the existing and improved models (with the same parameter set), as shown in Fig. 4, is because of the following reasons:
A. Effective Ballisticity For the existing model [1], since there is no charge flowing from the drain to the source, the ballisticity is simply defined as n b /n s , that is, the ratio of the carrier density reaching the drain ballistically to the total carrier density at the source. However, in the improved model, there are charges flowing in both the directions, source to drain and drain to source, quasi-ballistically depending on the source-side and drainside charge densities, respectively. Although we follow the same scattering picture for each of these currents, the effective ballisticity (keeping the same definition as in [1]) for each transport direction can be given as: n b-source βeff,s−d = n s + n dd-drain(x = 0) + n b-drain (x = 0) n b-drain (23) βeff,d−s = n d + n dd-source(x = L) + n b-source (x = L) where βeff,s−d is the effective ballisticity for charges flowing from the source to the drain while βeff,d−s is the effective
ballisticity for the charges flowing the other way. Since the amount of current flowing from source to drain and drain to source is proportional to n s and n d , respectively, we can write the effective ballisticity for the total current flow as a weighted average given by βeff =
n s βeff,s−d + n d βeff,d−s . ns + nd
(24)
Note that the same value of S will result in different values of β (for the existing model [1]) and βeff (for the improved model). Also, it is evident from the formulation of βeff that it is bias-dependent, as opposed to the constant β used in the existing model [1].
B. Drain-to-Source Charge Flow The improved model presented in this brief also considers electrons flowing from the drain to the source in addition to the ones flowing from the source to the drain. This additional component reduces the net current flowing from the drain to the source, as compared with the existing model [1], which did not take it into account. To explicitly prove that these two factors contribute to the significant difference between the two models, as shown in Fig. 4, we have: 1) changed the value of S in the existing model [1] so that the ballisticity, β, is the same as mean(βeff ) = ( N βeff )/N, where N is the number of points in the gate voltage sweep and 2) turned the flow of electrons from the drain to the source OFF in the improved model. Taking the mean was necessary to get a constant value for β, as the existing model [1] does not allow for bias-dependent ballisticity. The result obtained by making these changes is shown in Fig. 4(c). As can be seen, both the models predict nearly equal values of the drain current. The slight difference seen in Fig. 4(c) is due to the improved model having a biasdependent ballisticity. The bias point where the two models give exactly the same current values represents the gate voltage at which βeff (Vg ) = mean(βeff ) = β.
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V. C ONCLUSION We have presented an improved electrostatics model, as compared with our existing work [1], which captures the charge density profile along the channel accurately and also takes current conservation into account. The charge density profile has been validated with Monte Carlo simulation for a 10-nm quasi-ballistic device and has been shown to be more accurate than the existing model [1]. We have also explicitly shown that the error in current conservation along the channel is zero for the improved model, unlike the nonzero error of the existing work [1]. This improvement has been attained with no significant increase in computational cost and can be easily incorporated in our existing compact model [1] to make it highly accurate. Moreover, having validated the improved model with measured data, we have compared it with the prediction from the existing model [1] and have elaborately discussed the causes for the observed differences. R EFERENCES [1] A. Dasgupta, A. Agarwal, and Y. S. Chauhan, “Compact modeling of surface potential, charge, and current in nanoscale transistors under quasi-ballistic regime,” IEEE Trans. Electron Devices, vol. 63, no. 11, pp. 4151–4159, Nov. 2016. [2] A. Rahman, J. Guo, S. Datta, and M. S. Lundstrom, “Theory of ballistic nanotransistors,” IEEE Trans. Electron Devices, vol. 50, no. 9, pp. 1853–1864, Sep. 2003. [3] A. Mangla, J.-M. Sallese, C. Sampedro, F. Gamiz, and C. Enz, “Modeling the channel charge and potential in quasi-ballistic nanoscale double-gate MOSFETs,” IEEE Trans. Electron Devices, vol. 61, no. 8, pp. 2640–2646, Aug. 2014. [4] C. B. Zota, L.-E. Wernersson, and E. Lind, “High-performance lateral nanowire InGaAs MOSFETs with improved on-current,” IEEE Electron Device Lett., vol. 37, no. 10, pp. 1264–1267, Oct. 2016. [5] Y. S. Chauhan et al., “FinFET modeling for IC simulation and design,” Using the BSIM-CMG Standard. San Diego, CA, USA: Academic, 2015.
Avirup Dasgupta (GS’14) is currently pursuing the Ph.D. degree with IIT Kanpur, Kanpur, India. He has multiple international journal and conference publications to his name. He is a CoDeveloper of the ASM-HEMT model for GaN HEMTs, which is under industry standardization at the Compact Model Coalition. His current research interests include the physics and modeling of nanoscale semiconductor devices.
Amit Agarwal is currently an Assistant Professor with IIT Kanpur, Kanpur, India. He is also an Adjunct Faculty with the Harish Chandra Research Institute, Allahabad, India, and a Junior Associate with the International Center for Theoretical Physics, Trieste, Italy. His current research interests include theoretical condensed matter theory, particularly low dimensional systems, and nanoscale device modeling.
Yogesh Singh Chauhan (SM’12) is currently an Associate Professor with IIT Kanpur, Kanpur, India. He is a Co-Developer of the ASM-HEMT model for GaN HEMTs, which is under industry standardization at the Compact Model Coalition. Dr. Chauhan received the Ramanujan Fellowship in 2012, the IBM Faculty Award in 2013, and the P. K. Kelkar Fellowship in 2015.
