Impact of Surface Orientation on the Sensitivity of
FinFETs to Process Variations—An Assessment
Based on the Analytical Solution of the
Schrödinger Equation
Yu-Sheng Wu, Student Member, IEEE, and Pin Su, Member, IEEE
Abstract—This paper investigates the impact of surface
ori-entation on Vth sensitivity to process variations for Si and Ge
fin-shaped field-effect transistors (FinFETs) using an analytical solution of the Schrödinger equation. Our theoretical model con-siders the parabolic potential well due to short-channel effects and, therefore, can be used to assess the quantum-confinement effect in short-channel FinFETs. Our study indicates that, for ultrascaled FinFETs, the importance of channel thickness (tch) variations
increases due to the quantum-confinement effect. The Si-(100) and Ge-(111) surfaces show lower Vthsensitivity to the tchvariation as
compared with other orientations. On the contrary, the quantum-confinement effect reduces the Vthsensitivity to the Leffvariation,
and Si-(111) and Ge-(100) surfaces show lower Vthsensitivity as
compared with other orientations. Our study may provide insights for device design and circuit optimization using advanced FinFET technologies.
Index Terms—Fin-shaped field-effect transistor (FinFET),
quantum effects, surface orientation, variation. I. INTRODUCTION
A
S THE carrier mobility of a MOSFET depends on surface orientation [1]–[3], it has been proposed that with an optimized surface orientation, the circuit performance of a fin-shaped field-effect transistor (FinFET) structure can be en-hanced [3], [4]. However, with the scaling of device geometry, process variation has become a crucial issue. The immunity of a FinFET structure with various surface orientations to process variations is an important issue. In our previous work [5], we have investigated the threshold voltage (Vth) sensitivity toprocess variations for lightly doped FinFETs using an analytical solution of Poisson’s equation. As the channel thickness of the FinFETs scales down, nevertheless, the quantum-confinement effect may become significant. This 1-D confinement effect may result in a Vth shift and impact the Vth sensitivity to
Manuscript received July 15, 2010; revised September 15, 2010; accepted September 16, 2010. Date of publication October 18, 2010; date of current version November 19, 2010. This work was supported in part by the National Science Council (NSC) of Taiwan under Contract NSC 98-2221-E-009-178 and in part by the Ministry of Education in Taiwan under the Aim for the Top University Program. The review of this paper was arranged by Editor H. S. Momose.
The authors are with the Department of Electronics Engineering and Insti-tute of Electronics, National Chiao Tung University, Hsinchu 30013, Taiwan (e-mail: pinsu@faculty.nctu.edu.tw).
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.2010.2080682
Fig. 1. Schematic of the FinFET structure investigated in this paper. Leff is
the channel length, tchis the channel thickness, and tinis the gate-insulator
thickness.
process variations. Moreover, the impact of quantum confine-ment may show surface-orientation dependence.
In this paper, we investigate the Vth sensitivity to process
variations for Si- and Ge-channel FinFETs with various channel orientations using an analytical solution of the Schrödinger equation. For long-channel undoped devices, the quantum-confinement effect is often considered to be independent of the carrier-flow direction (i.e., channel-length direction), and the potential well in the Schrödinger equation is usually assumed to be flat [6]. For short-channel devices, however, the center of the potential well is altered by source/drain coupling due to short-channel effects, and flat-well approximation is no longer valid. Therefore, an accurate solution of the Schrödinger equation that considers the short-channel effect is crucial to the determination of Vthfor ultrascaled FinFETs.
This paper is organized as follows. In Section II, we ana-lytically derive the solution of the Schrödinger equation for the short-channel FinFETs under the subthreshold region, and the model results are verified with technology computer-aided design (TCAD) simulation. In Section III, we calculate the electron density and Vthusing our theoretical model. Then, we
investigate the Vth sensitivity to process variations for Si- and
Ge-channel FinFETs with various surface orientations. Finally, the conclusion is drawn in Section IV.
II. ANALYTICALSOLUTION OF THESCHRÖDINGER
EQUATION ANDVERIFICATIONWITHTCAD Fig. 1 shows a schematic sketch of a FinFET structure. To consider the quantum-confinement effect along the fin-width (i.e., x) direction, the Schrödinger equation can be expressed as
− 2 2mx d2Ψ j(x) dx2 + EC(x)Ψj(x) = EjΨj(x) (1) 0018-9383/$26.00 © 2010 IEEE
where Ej is the jth eigen-energy, Ψj(x) is the corresponding wave function, is the reduced Planck constant, and mx is the carrier-quantization effective mass. For electrons in the Si and Ge channels, mx for various surface orientations are listed in Table I. If the conduction band edge EC(x) is treated as a flat well with a potential energy β, the solution of (1) is Ψj(x) = (2/tch)1/2sin[(j + 1)π(x + tch/2)/tch] and
Ej= β + (j + 1)2π22/(2mxt2ch) [6]. However, to account
for the source/drain coupling due to short-channel effects, the conduction band edge EC(x) in (1) should be treated as a parabolic well with potential energy EC(x) = αx2+ β [7]. α and β are length-dependent coefficients and can be obtained from the channel potential solution of Poisson’s equation under the subthreshold region.
In our previous work [5], we have derived the 3-D channel potential solution φ(x, y, z) for multigate MOSFETs in the subthreshold region. For the FinFET structure in this study, the potential solution can still be applied after neglecting the top-gate potential coupling along the fin-height direction. In other
α = (−q) i=1 cisinh Weff y + cisinh iπ Weff (Leff − y) × −1 2 iπ Weff 2 sin iπ 2 (3a) β = (−q) b + ∞ i=1 cisinh iπ Weff y + cisinh iπ Weff (Leff − y) × sin iπ 2 − 1 2 Eg q + 1 2 kT q ln Nc Nv (3b) where kT /q is the thermal voltage, Eg is the bandgap of the channel material, and Nc and Nv are the effective density of
φ1(x) = − qNa 2εch x + 1 2tch 2 + a x + 1 2tch + b (2a) a =(qNa/2εch) t2ch+ 2 (εch/εin) t2ch+ 2 (εch/εin) tintch Weff (2b) b = (VGS− Vfb) + εch εin tina (2c) φ2(x, y) = ∞ i=1 cisinh iπ Weff y + cisinh iπ Weff (Leff− y) sin iπ Weff x +1 2tch+ εch εin tin (2d) ci= 1
sinh [iπ (Leff/Weff)]
2(−φms+ VDS− b) 1− (−1)i iπ + 2a tin iπ + (Weff − tin)(−1)i iπ +qNa εch t2in iπ − (Weff− tin)2(−1)i iπ + 2W 2 eff (−1)i− 1 (iπ)3 (2e) ci= 1
sinh [iπ (Leff/Weff)]
2(−φms− b) 1− (−1)i iπ + 2a tin iπ + (Weff− tin)(−1)i iπ +qNa εch t2 in iπ − (Weff − tin)2(−1)i iπ + 2W 2 eff (−1)i− 1 (iπ)3 (2f)
Fig. 2. Conduction-band edge and quantized eigen-energies of a short-channel lightly doped FinFET.
states for conduction and valence bands, respectively. Using the parabolic-well approximation, the solution of (1) can be expressed as [9] Ψj(x) = ∞ n=0 dnxn (4a)
with the coefficient dn being determined by the following recurrence relationship: d2= − mx(Ej− β) 2 d0 d3=− mx(Ej− β) 32 d1 dn+2= − 2mx(Ej− β)/2 (n + 1)(n + 2) × dn+ 2mxα/2 (n + 1)(n + 2)dn−2, n≥ 2. (4b) It should be noted that as α = 0 (i.e., ECis spatially constant), Ψj(x) will return to the form of sinusoidal functions, which is the solution for the flat-well approximation [6]. The jth eigen-energy Ejcan be determined by the boundary condition Ψj(x = tch/2) = 0. Thus, the eigen-energy and the
eigenfunc-tion of the short-channel FinFET under the subthreshold region can be derived.
To validate the accuracy of our model, we compare the calculation results with the TCAD simulation that numerically solves the self-consistent solution of the 2-D Poisson’s and 1-D Schrödinger equations [10]. The Schrödinger equation is solved along the fin-width x-direction to consider the quantum-confinement effect. The effective masses used for various surface orientations in the TCAD simulations are listed in Table I. We assume that the barrier height across the gate insulator/channel is infinite, and the wave functions vanish at the interface. In this study, we focus on FinFETs with lightly doped channel Na= 1015cm−3. The equivalent oxide thickness is 0.5 nm to sustain the electrostatic integrity, and a midgap-gate work function (4.5 eV) is used. Fig. 2 shows that for a short-channel lightly doped FinFET, the conduction band edge EC is bent from a flat well to a parabolic-like well due to the source/drain coupling. It can be seen that the
eigen-Fig. 3. Channel-length dependence of E0 for lightly doped FinFETs with various tchshowing the accuracy of our model.
Fig. 4. Comparison of the square of Ψ0for long- and short-channel FinFETs.
energy calculated by our model considering the parabolic-well approximation agrees well with the TCAD simulation. Since EC is not spatially constant along the x-direction for short-channel devices, we choose EC at the channel center (i.e.,
x = 0) as the reference energy. Fig. 3 shows the channel-length (Leff) dependence of the energy difference of E0(ground-state
energy in a fourfold valley) and the bottom of well EC(x = 0). In contrast to the constant E0 − EC(x = 0) calculated from the flat-well approximation, both the TCAD simulation and our model show that E0 − EC(x = 0) increases with decreasing
Leff. In addition to eigen-energy, the bent potential well due
to the short-channel effect also affects the shape of the wave function. Fig. 4 shows that |Ψ0|2 for lightly doped FinFETs
with shorter Leff (i.e., Leff = 15 nm) is more centralized to the
channel center. This is because the EC barrier at the channel center (x = 0) is lower than that near the insulator/channel interface (x = 0.5tch), and, thus, the electron density becomes
larger at x = 0.
III. RESULTS ANDDISCUSSION
To assess the impact of the quantum confinement on the threshold voltage Vth, Vth is defined as the VGS at which the average electron density of the cross section at y = Leff/2
Fig. 5. Comparison of the electron density calculated from our model and the model using the flat-well approximation.
(the highest potential barrier for low VDS) exceeds the critical concentration 1× 1016 cm−3 [11]. The electron density is determined by the eigen-energy and eigenfunction as
n(x, y) = NC,QMexp −EC− EF kT (5a) NC,QM = kT π2 v,j gvmvd|Ψv,j(x, y)|2 exp −Ev,j− EC kT (5b) where gv is the valley degeneracy, and mvd is the density-of-state effective mass of valley v. gv and mvd for Si and Ge channels are listed in Table I. It can be seen from (5b) that the flat-well approximation may overestimate the electron density for short-channel devices because it underestimates the eigen-energy Ej (as shown in Fig. 3). Fig. 5 compares the electron density distribution calculated from the flat-well approximation and our model. The electron density predicted by our model agrees well with the TCAD simulation, whereas the flat-well approximation shows higher electron density in both sides of the channel.
For the FinFET structure, different surface orientations such as (100), (110), and (111) can be achieved by rotating the device layout in the wafer plane [3]. Fig. 6 shows that for Si FinFETs with a small tch, Vth and its sensitivity to channel
thickness (tch) variations considering the quantum-confinement
effect is larger than that predicted by the classical model (CL). Moreover, the Vth of the (111) and (110) surfaces increase
more rapidly than that of the (100) surface with decreasing tch. This is because the quantum-confinement effect depends
on the surface orientation, as indicated by the inset in Fig. 6. For a FinFET with a small tch, Vth is mainly determined by
E0. In addition, as mx and, thus, the ground-state energy of twofold and fourfold valleys for (100) and (110) surfaces are different (see Table I), the overall lowest state occurs for the valley with larger mx because (to the first order) the eigen-energy is inversely proportional to mx. Therefore, the mx of the twofold valley determines the E0for the (100) surface, and
Fig. 6. Comparison of the tchdependence of Vthfor Si FinFETs with various
surface orientations and the CL. The Vthshift due to quantum confinement is
mainly determined by E0, as indicated by the inset.
Fig. 7. Comparison of the tch dependence of Vth for Ge FinFETs with
various surface orientations and the CL. (Inset) Comparison of E0for various
surface orientations.
the mxof the fourfold valley determines the E0for the (110)
surface. Since the dominant mxof various surface orientations for the Si channel is (111) < (110) < (100), the E0 and, thus,
Vthis (111) > (110) > (100), as shown in Fig. 6.
For a high-mobility channel such as Ge FinFETs, Vth
dis-persion due to quantum confinement becomes more significant. Fig. 7 shows that the Vth of the (100) surface increases more
rapidly than the (110) and (111) surfaces with reducing tch.
This is because the quantum-confinement effect of the (100) surface is larger than that of the (110) and (111) surfaces, as indicated by the inset in Fig. 7. Since the dominant mx of the various surface orientations for the Ge channel is (111) > (110) > (100), the E0and Vthis (100) > (110) > (111).
Besides Vth sensitivity to tch variation, the
quantum-confinement effect also affects Vth sensitivity to Leff
varia-tion. Fig. 8 shows that for Ge FinFETs, the degree of Vth
rolloff predicted by our quantum-confinement model is (100) < (110) < (111) < CL, which is opposite to Vth sensitivity to
tch variations (Fig. 7). In other words, while the
Fig. 8. Comparison of the Leff dependence of Vth (Vth rolloff) for Ge
FinFETs with various surface orientations and the CL. The Vth rolloff is
defined as Vth (Leff)− Vth(Leff= 100 nm). (Inset) Devices with smaller
Leff show larger Vth shift due to the quantum-confinement effect than the
devices with larger Leff.
reduces Vthsensitivity to Leff variation. This can be explained
as follows. The Vthshift due to the quantum-confinement effect
can be expressed as ΔVthQM= S/(ln 10 kT /q)ΔΨQM
s , with S being the subthreshold swing and ΔΨQM
s being the equivalent surface potential shift [12]. The S for a short-channel device is larger than that for a long-channel device because of the enhanced drain coupling with decreasing Leff. Therefore, for
devices with a given surface orientation, the ΔVthQM (which increases Vth) of the short-channel device is larger than that
of the long-channel one, as indicated by the inset in Fig. 8. The discrepancy in ΔVthQM between short- and long-channel devices reduces the Vth rolloff, and the Vth rolloff considering
the quantum-confinement effect becomes smaller than the CL. In addition, as ΔΨQM
s is determined by E0, a larger E0(and,
thus, ΔΨQMs ) results in a larger ΔVthQMand, hence, smaller Vth
rolloff. This explains why the degree of Vth rolloff is (100) <
(110) < (111) for Ge FinFETs.
In addition to the eigen-energies (Fig. 3) and the electron density (Fig. 5), Vthcalculated by our model is physically more
accurate than that calculated by the flat-well approximation. Fig. 9 shows that Vthcalculated using our model and the
flat-well approximation are fairly close for devices with a small tch.
However, the discrepancy between the two models increases with tchbecause the impact of short-channel effects becomes
more significant for devices with a larger tch. Compared with
the flat-well approximation, Vth calculated by our model is
more physical because it returns to the classical one for devices with larger tch, in which the quantum-confinement effect is
negligible.
IV. CONCLUSION
We have investigated the impact of surface orientation on Vth sensitivity to process variations for Si and Ge FinFETs
using an analytical solution of the Schrödinger equation. Our theoretical model considers the parabolic potential well due to short-channel effects and, therefore, can be used to assess the quantum-confinement effect in short-channel FinFETs. Our
Fig. 9. Comparison of the tch dependence of Vth of the short-channel
FinFETs calculated from the CL, our model, and the model using flat-well approximation.
study indicates that, for ultrascaled FinFETs, the importance of channel thickness variations increases due to the quantum-confinement effect. The Si-(100) and Ge-(111) surfaces show lower Vth sensitivity to tch variation as compared with other
orientations. On the contrary, the quantum-confinement effect reduces Vth sensitivity to Leff variation, and Si-(111) and
Ge-(100) surfaces show lower Vth sensitivity as compared
with other orientations. Our study may provide insights for device design and circuit optimization using advanced FinFET technologies.
REFERENCES
[1] M. Yang, E. P. Gusev, M. Ieong, O. Gluschenkov, D. C. Boyd, K. K. Chan, P. M. Kozlowski, C. P. D’Emic, R. M. Sicina, P. C. Jamison, and A. I. Chou, “Performance dependence of CMOS on silicon substrate orientation for ultrathin oxynitride and HfO2gate dielectrics,” IEEE
Elec-tron Device Lett., vol. 24, no. 5, pp. 339–341, May 2003.
[2] E. Landgraf, W. Rösner, M. Städele, L. Dreeskornfeld, J. Hartwich, F. Hofmann, J. Kretz, T. Lutz, R. J. Luyken, T. Schulz, M. Specht, and L. Risch, “Influence of crystal orientation and body doping on trigate transistor performance,” Solid State Electron., vol. 50, no. 1, pp. 38–43, Jan. 2006.
[3] L. Chang, M. Ieong, and M. Yang, “CMOS circuit performance enhance-ment by surface orientation optimization,” IEEE Trans. Electron Devices, vol. 51, no. 10, pp. 1621–1627, Oct. 2004.
[4] S. Gangwal, S. Mukhopadhyay, and K. Roy, “Optimization of surface orientation for high-performance, low-power and robust FinFET SRAM,” in Proc. IEEE CICC, 2006, pp. 433–436.
[5] Y.-S. Wu and P. Su, “Sensitivity of multigate MOSFETs to process variations—An assessment based on analytical solutions of 3-D Poisson’s equation,” IEEE Trans. Nanotechnol., vol. 7, no. 3, pp. 299–304, May 2008.
[6] H. Ananthan and K. Roy, “A compact physical model for yield under gate length and body thickness variations in nanoscale double-gate CMOS,”
IEEE Trans. Electron Devices, vol. 53, no. 9, pp. 2151–2159, Sep. 2006.
[7] Y.-S. Wu and P. Su, “Quantum confinement effect in short-channel gate-all-around MOSFETs and its impact on the sensitivity of threshold voltage to process variations,” in Proc. IEEE Int. SOI Conf., 2009, pp. 1–2. [8] F. Stern and W. E. Howard, “Properties of semiconductor surface
inver-sion layers in the electric quantum limit,” Phys. Rev., vol. 163, no. 3, pp. 816–835, Nov. 1967.
[9] D. G. Zill and M. R. Cullen, Differential Equations With Boundary Value
Problems, 5th ed. Pacific Grove, CA: Brooks/Cole, 2001.
[10] ATLAS User’s Manual, SILVACO, Santa Clara, CA, 2008.
[11] C.-T. Lee and K. K. Young, “Submicrometer near-intrinsic thin-film SOI complementary MOSFETs,” IEEE Trans. Electron Devices, vol. 36, no. 11, pp. 2537–2547, Nov. 1989.
[12] Y. Taur and T. H. Ning, Fundamentals of Modern VLSI Devices. Cambridge, U.K.: Cambridge Univ. Press, 1998.
