18 IEEE ELECTRON DEVICE LETTERS, VOL. 32, NO. 1, JANUARY 2011
Impact of Quantum Confinement on
Short-Channel Effects for Ultrathin-Body
Germanium-on-Insulator MOSFETs
Yu-Sheng Wu, Student Member, IEEE, Hsin-Yuan Hsieh,
Vita Pi-Ho Hu, Student Member, IEEE, and Pin Su, Member, IEEE
Abstract—This letter investigates the impact of quantum
con-finement (QC) on the short-channel effect (SCE) of ultrathin-body (UTB) and thin-buried-oxide germanium-on-insulator (GeOI) MOSFETs using an analytical solution of Schrödinger equation verified with TCAD simulation. Our study indicates that, al-though the QC effect increases the threshold voltage (Vth) roll-off when the channel thickness (Tch) is larger than a critical value (Tch,crit), it may decrease the Vth roll-off of GeOI MOSFETs when the Tch is smaller than Tch,crit. Since Ge and Si channels exhibit different degrees of confinement and Tch,crit, the impact of QC must be considered when one-to-one comparisons between UTB GeOI and Si-on-insulator MOSFETs regarding the SCE are made.
Index Terms—Germanium-on-insulator (GeOI), quantum
con-finement (QC), threshold voltage roll-off.
I. INTRODUCTION
G
ERMANIUM as a channel material has been proposed to enable the mobility scaling for CMOS devices. As the higher permittivity makes Ge more susceptible to short-channel effects (SCEs), an ultrathin-body (UTB) germanium-on-insulator (GeOI) structure with thin buried oxide (BOX) has been suggested to improve the electrostatic integrity [1], [2]. With the scaling of channel thickness, the quantum-confinement (QC) effect may become significant and impact the SCE of scaled UTB devices. Using the density gradient model [3], Omura et al. [4] have observed increased threshold voltage (Vth) roll-off due to QC in UTB Si-on-insulator (SOI)devices. Whether there exists any difference between GeOI and SOI devices regarding the impact of QC on SCEs is not clearly known and merits investigation. In this letter, we tackle the problem using an analytically derived solution of Schrödinger equation verified with TCAD simulation. We report our new findings for UTB GeOI MOSFETs with thin BOX.
Manuscript received October 4, 2010; revised October 13, 2010; accepted October 14, 2010. Date of publication December 3, 2010; date of current version December 27, 2010. This work was supported in part by the National Science Council of Taiwan under Contract NSC 99-2221-E-009-174 and in part by the Ministry of Education in Taiwan under the ATU Program. The review of this letter was arranged by Editor L. Selmi.
The authors are with the Department of Electronics Engineering and the Institute of Electronics, National Chiao Tung University, Hsinchu 30013, Taiwan (e-mail: [email protected]).
Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/LED.2010.2089425
II. ANALYTICALSOLUTION OFSCHRÖDINGEREQUATION
To consider the QC effect along the channel thickness (i.e.,
x-) direction, the Schrödinger equation can be express as − 2
2mx ·
d2Ψ
j(x)
dx2 + EC(x)· Ψj(x) = Ej· Ψj(x) (1)
where Ej is the jth eigenenergy, Ψj(x) is the corresponding
wavefunction, and mxis the carrier quantization effective mass.
For long-channel undoped UTB MOSFETs, the conduction band edge EC(x) was usually treated as a triangular well [5].
However, to account for the source/drain coupling due to SCEs, the conduction band edge EC(x) in (1) should be treated as a
parabolic well [6] with potential energy EC(x) = αx2+ βx +
γ, where α, β, and γ are channel-length-dependent coefficients
and can be obtained from the channel potential solution of Poisson’s equation under the subthreshold region [7]. Using the parabolic-well approximation, the solution of (1) can be expressed as Ψj(x) =
dn· xnwith the coefficients dn’s
d2=− mx 2 (Ej− γ) · d0 d3=− mx 32[(Ej− γ) · d1− β · d0] dn=− 2mx n(n− 1)2 × [(Ej− γ) · dn−2− β · dn−3− α · dn−4] , n≥ 4. (2) The jth eigenenergy Ej can be determined by the boundary
condition Ψj(x = 0) = Ψj(x = Tch) = 0, where x = 0 and
x = Tch (channel thickness) are defined as the interface
po-sitions of BOX/channel and channel/gate oxide, respectively. Thus, the eigenenergy and eigenfunction of short-channel UTB MOSFETs under the subthreshold region can be derived. We have verified our model using the TCAD simulation that nu-merically solves the self-consistent solution of 2-D Poisson and 1-D Schrödinger equations [8]. Fig. 1(a) and (b) shows that, for both the triangular potential well of long-channel devices and the parabolic well (due to SCEs) of short-channel ones, the
Ej’s calculated by our model are fairly accurate. It should be
noted that a scalable QC model with accurate channel length dependence is crucial to this work.
WU et al.: IMPACT OF QUANTUM CONFINEMENT ON SHORT-CHANNEL EFFECTS FOR UTB GeOI 19
Fig. 1. Conduction band edge and quantized eigenenergies of lightly doped GeOI MOSFETs. (a) Long-channel device with triangular well. (b) Short-channel device with parabolic well.
Fig. 2. Comparison of the electron density distributions with and without considering the QC effect. The electron density is calculated from 2-D density of states, eigenenergies, and wavefunctions.
III. IMPACT OFQCONVthROLL-OFF
To assess the impact of QC effect on Vth, the Vthis defined
as the VGSat which the average electron density of the cross
section at y = ymin (the minimum potential along the carrier
flow direction) exceeds a critical concentration that is equal to the channel doping. Note that the choice of other critical concentrations for determining Vth[9], [10] will result in a shift
in Vth but will not affect the results of Vthcomparisons in this
study. Using the calculated eigenenergies and wavefunctions, the electron density can be derived [11]. Fig. 2 shows that the peak of electron density calculated by the classical (CL) model is not located at the channel/BOX interface (x = 0) because the use of thin BOX (10 nm) instead of thick BOX suppresses the buried-insulator-induced-barrier lowering (BIIBL) [4]. Al-though the peak of electron density calculated by the QC model is shifted toward the channel center, the main current flow paths predicted by both models are quite similar for the UTB structure with thin BOX.
Fig. 3 shows that, for GeOI MOSFETs with channel thick-ness (Tch) = 10 nm, the Vth roll-off [defined as Vth(L)−
Vth(L = 100 nm)] predicted by the QC model is larger than
that predicted by the CL model. This is consistent with the result reported for SOI MOSFET [4] and can be explained as follows. The Vth shift due to the QC effect can be
ex-pressed as ΔVthQM∼= S/(ln 10· kT/q) · ΔψsQM, with S being
Fig. 3. Comparison of the Vth roll-offs between QC and CL models for
Tch= 10 nm. The QC effect alters Lmin(where the Vthroll-off =−0.2 V [12]) by about +2 nm. The inset indicates that, for GeOI MOSFETs with larger Tch, the difference in E0’s of long-channel (E0,long) and short-channel (E0,short) devices is significant due to electrical confinement.
Fig. 4. Comparison of the Vth roll-offs between QC and CL models for
Tch= 5 nm. The QC effect alters Lminby about−1 nm. The inset indicates that, for GeOI MOSFETs with smaller Tch, the difference in E0’s of long-channel (E0,long) and short-long-channel (E0,short) devices is small because the degree of structural confinement is similar.
the subthreshold swing and ΔψQM
s being the equivalent surface
potential shift due to the QC effect [5], [12]. The inset of Fig. 3 shows that, for GeOI devices with larger Tch (10 nm), the
“electrical confinement” [5] dominates the carrier quantization. The E0 (ground-state energy) of the triangular well (for
long-channel devices) is much larger than that of the parabolic well (for short-channel devices) because of the larger electric field in the triangular one. As ΔψsQMis mainly determined by E0, the
ΔψQM
s and, thus, ΔV
QM
th for the long-channel device are larger
than that of the short-channel one. Therefore, the Vth roll-off
considering the QC effect is larger.
As the Tch scales down, however, a different trend can be
observed. Fig. 4 shows that, for GeOI MOSFETs with Tch=
5 nm, the Vth roll-off predicted by the QC model becomes
smaller than that predicted by the CL model, which is opposite to the larger Tch case and [4]. This cannot be explained by
the reduction of BIIBL due to the QC effect [4], because in this study, thin BOX (TBOX= 10 nm) is used and the impact
of BIIBL is not significant (see Fig. 2). Since the “structural confinement” [5] dominates the carrier quantization for GeOI devices with smaller Tch (5 nm), the inset of Fig. 4 shows
20 IEEE ELECTRON DEVICE LETTERS, VOL. 32, NO. 1, JANUARY 2011
Fig. 5. Vthroll-off comparison between SOI and GeOI devices. As the QC effect is considered, a crossover near Tch= 4 nm can be seen.
Fig. 6. Difference in Vthroll-offs between the QC and CL models depends on TBOXand channel material. The filled region denotes that the QC effect enhances the Vth roll-off, while the blank region denotes that the QC effect suppresses the Vthroll-off.
that the E0(and, hence, ΔψsQM) of the long-channel device is
close to that of the short-channel one. Nevertheless, due to the SCE, the subthreshold swing S of the short-channel device is larger than that of the long-channel one. Therefore, the ΔVthQM of the short-channel device is larger than that of the long-channel device, and the Vthroll-off considering the QC effect
is smaller. This mechanism is important because it may alter the comparison result for Vth roll-off between SOI and GeOI
devices. Fig. 5 shows that, contrary to the prediction by the CL model, the Vth roll-off for GeOI devices with smaller Tchcan
be smaller than that of the SOI counterparts as the QC effect is considered.
In summary, depending on Tch, the QC effect may increase
or decrease the SCE of UTB devices. The critical channel thick-ness (Tch,crit) determining whether the QC effect enhances
or decreases the Vth roll-off depends on the BOX thickness
(TBOX) and the channel material. Fig. 6 shows that the Tch,crit
of GeOI MOSFETs increases with TBOX. In addition, for a
given TBOX, the Tch,critof SOI MOSFETs is smaller than that
of the GeOI MOSFETs. This may explain why the suppression
of Vthroll-off by the QC effect was not observed for the UTB
SOI devices (with Tch= 10 nm) in [4].
IV. CONCLUSION
We have investigated the impact of QC on the SCE of UTB GeOI MOSFETs using a derived analytical solution of Schrödinger equation verified with TCAD simulation. Our study indicates that the impact of QC effect on the Vthroll-off
of UTB GeOI MOSFETs shows two distinct trends. For GeOI devices with Tch larger than Tch,crit, the QC effect increases
the Vth roll-off, as previously observed in SOI devices [4].
However, for GeOI devices with Tchsmaller than Tch,crit, QC
decreases the Vth roll-off. Since Ge and Si channels exhibit
different degrees of confinement (because of the discrepancy in effective mass) and Tch,crit, the impact of QC must be
considered when one-to-one comparisons [13], [14] between UTB GeOI and SOI MOSFETs regarding the SCE are made.
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