Chapter 4 Physically Based Analytic Model for 1-D case
4.3 Analysis with 1-D model
Similar to Chapter 2, by using the Schrödinger-Poisson solver, the effective gate capacitance Ceff and quasi-equilibrium threshold voltage Vtho
can be obtained with the relationship as follows:
qnS =Ceff(VG −Vtho) (4.10)
The kBT layer widths are extracted from the potential profiles of Fig. 3 in [12]. The density-of-states effective mass is 0.28mo as shown in Figure 4-2 for wire diameter TSi equal to 3 nm [14]. Following the case 1 of the flow chart in Chapter 2, the drain current at VG=VD=0.4V can be obtained.
4.4 Results
To examine the validity of the channel backscattering theory on 1-D case, we have compared the calculated results with those from another model called Büttiker probes [12]. The calculated drain currents appear to lie a little above the Büttiker probes ones as shown in Fig. 4-3. It suggests that the importance of the source-to-drain tunneling is increased for channel length down to below 10 nm. The existing channel backscattering formula might be improved with consideration of the source-to-drain tunneling effect.
Chapter 5 Conclusion
The physically based analytic models of the ultra-thin film double-gate MOSFETs and silicon nanowire transistors have been established. The validity of the models has been confirmed using sophisticated simulations such as 1-D Schrödinger-Poisson solving, 2-D and 1-D ballistic I-V simulations, and 1-D Monte Carlo particle simulations with the scattering in the channel. The issues of concern have been focused on the effect of backward to forward flux ratio on the thermal injection velocity at the top of the source-channel junction barrier. It is argued that the backward to forward flux ratio can determine the channel backscattering coefficient in the framework of the channel backscattering theory.
References
[1] S. Datta, Electronic Transport in Mesoscopic Systems. Cambridge, U.K.:
Cambridge Univ. Press, 1995.
[2] M. S. Lundstrom, “Elementary scattering theory of the Si MOSFET,”
IEEE Electron Device Letters, vol. 18, pp. 361-363, July 1997.
[3] F. Assad, Z. Ren, S. Datta, and M. S. Lundstrom, “Performance limits of silicon MOSFET’s,” in IEDM Tech. Dig., Dec. 1999, pp. 547-550.
[4] F. Assad, Z. Ren, D. Vasileska, S. Datta, and M. S. Lundstrom, “On the performance limits for Si MOSFET’s: A theoretical study,” IEEE Trans.
Electron Devices, vol. 47, pp. 232-240, Jan. 2000.
[5] M. S. Lundstrom and Z. Ren, “Essential physics of carrier transport in nanoscale MOSFETs,” IEEE Trans. Electron Devices, vol. 49, pp.
133-141, Jan. 2002.
[6] Mark Lundstrom, Fundamentals of Carrier Transport, second edition, School of Electrical and Computer Engineering Purdue University, West Lafayette, Indiana, USA: Cambridge University Press, 2000.
[7] M. J. Chen, H. T. Huang, Y. C. Chou, R. T. Chen, Y. T. Tseng, P. N. Chen, and C. H. Diaz, “Separation of Channel Backscattering Coefficients in Nanoscale MOSFETs,” IEEE Trans. Electron Devices, vol. 51, pp.
1409-1415, September 2004.
[8] http://www.nanohub.org/
[9] A. Svizhenko and M. P. Anatram, “Role of scattering in nanotransistors,”
IEEE TED, p. 1459, 2003.
[10] A. Rahman and M. S. Lundstrom, “A compact scattering model for the nanoscale double-gate MOSFET,” IEEE TED, p. 481, 2002.
[11] M. S. Lundstrom, “Notes on Ballistic MOSFET,” Network for Computational Nanotechnology and Purdue University.
[12] J. Wang, E. Polizzi, M. S. Lundstrom, “A three-dimensional quantum simulation of silicon nanowire transistors with the effective-mass approximation,” Journal of Applied Physics, vol. 96, p. 2192, 2004.
[13] M. Bescond, N. Cavassilas, K. Kalna, K. Nehari, L. Raymond, J.L.
Autran, M. Lannoo, A. Asenov, “Ballistic transport in Si, Ge, and GaAs nanowire MOSFETs,” in Proc. IEEE Int. Electron Devices Meeting, Dec. 5, 2005, pp. 526-529.
[14] Jing Wang, Anisur Rahman, Gerhard Klimeck and Mark Lundstrom,
“Bandstructure and Orientation Effects in Ballistic Si and Ge Nanowire FETs” in Proc. IEEE Int. Electron Devices Meeting, Dec. 5, 2005, pp.
530-533.
Fig. 1-1 Schematic diagram of channel backscattering theory. F is the incident flux from the source, l is the critical length over which a kBT/q drop is developed, and rC is the channel backscattering coefficient. The channel length Leff is the physical gate length minus the source/drain extensions.
L = 25nm
Work function = 4.25eV
Metal Contact
Work function = 4.25eV
Y
Work function = 4.25eV
Metal Contact
Work function = 4.25eV
Y
X
Fig. 2-1 Schematic cross section of the device under study.
Fig. 2-2 Schematic conduction-band profile from source to drain. An E-k diagram is plotted showing forward and backward flux at the peak of the source-channel barrier.
F
backward(= r
BFF
forward)
l x 0
K
BT Conduction band
F
forwardK
xE
Source Channel Drain
F
backward(= r
BFF
forward)
l x 0
K
BT Conduction band
F
forwardK
xE
Source Channel Drain
0.0 0.2 0.4 0.6 0.8 1.0 0.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5
Subba nd_ Energy Le ve l [ e V ]
V
G[V]
Ef E
0 E
1 E
2
E3 E'0 E'1
Fig. 2-3 Channel subband levels and Fermi level versus gate voltage obtained from 1-D self-consistent Schrödinger-Poisson simulation.
0.0 0.2 0.4 0.6 0.8 1.0 0
20 40 60 80 100
Subband O c cupancy [%]
V
G[V]
E0 E1 E2 E3 E'0 E'1
Fig. 2-4 Channel subband level occupancy versus gate voltage from 1-D self-consistent Schrödinger-Poisson simulation.
Extract l And let initial rBF=0 Extract l And let initial rBF=0
rc=rBF & Iteration
Fig. 2-5 Flowchart of our analysis.
X
R-Scatt(nm) k
BT lay er w id th l (nm )
τscatt
= 10 fs
Source Scattering Drain Region
k
BT lay er w id th l (nm )
τscatt
= 10 fs
Source Scattering Drain Region
0 XR-Scatt X Non-Scattering Region
X
R-Scatt(nm) k
BT lay er w id th l (nm )
τscatt
= 10 fs
Source Scattering Drain Region
0 XR-Scatt X Non-Scattering Region
Source Scattering Drain Region inset shows the definition of XR-Scatt.
(λelastic)
Fig. 2-7 Schematic flux profile in the kBT layer when the factor Q’ is considered.
rBFQinvVinj
Fig. 2-8 In order to analyze, we must transform the flux profile to adapt our model.
-15 -10 -5 0 5 10 15
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Monte Carlo Simulation Value without Q'
Calculation value with Q'
τ
scatt= 10fs
I
D(mA/ µ m)
X
R-Scatt(nm)
Fig. 2-9 Comparison of calculated drain current versus XR-Scatt with that from Monte Carlo particle simulation.
-15 -10 -5 0 5 10 15 0
20 40 60 80 100
Monte Carlo simulation Calculation value with Q'
τ
scatt= 10fs
Barrier Hei g h t (meV)
X
R-Scatt(nm)
Fig. 2-10 Comparison of calculated barrier height versus XR-Scatt with that from Monte Carlo particle simulation.
-15 -10 -5 0 5 10 15
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Monte Carlo Simulation Value without Q'
Calculation value with Q'
τ
scatt= 50fs I
D(mA/ µ m)
X
R-Scatt(nm)
Fig. 2-11 Comparison of calculated drain current versus XR-Scatt with that from Monte Carlo particle simulation.
Source
Drain Incident Beam
L : Channel Length l : kBT Layer
Source
Drain Incident Beam
L : Channel Length l : kBT Layer
Fig. 3-1 Schematic structure of the simulation in the model “channel”.
-5.00x107 -2.50x107 0.00 2.50x107 5.00x107 0.0
0.2 0.4 0.6 0.8 1.0
Distribution Function
velocity(cm/s)
Fig. 3-2 An example case of the schematic velocity distribution. The backscattering coefficient rC is just equal to the area ratio of negative to positive
10
110
210
310
410
510
60.0
0.2 0.4 0.6 0.8 rc
Electric Field (V/cm) L=80nm
L=20nm
Fig. 3-3 The simulated backscattering coefficient of Monte Carlo evaluation under electric field from 10 V/cm to 106 V/cm for L = 80 nm and L
= 20 nm.
100 101 102 103 104 105 106
Fig. 3-4 The mean-free-path λ extracted from the rC-E relation.
100 101 102 103
Fig. 3-5 The low-field mobility of electrons
Temperature (K)
(a)
Fig. 4-1 (a) A schematic diagram of the simulated nanowire FETs.
(b) Schematic cross section of the Si nanowire under study.
1 2 3 4 5 0.2
0.3 0.4 0.5 0.6
Diameter (nm) m
dat
Γpoint (in m
o)
Fig. 4-2 The density-of-states effective mass at Γ point in the wire conduction band versus wire diameter D for a [100] oriented Si nanowire.
(a)
ID(scatt.)(Buttiker probes)
IDS (A)
ID(scatt.)(Buttiker probes)
IDS (A)
VGS (V)
Fig. 4-3 Comparison of calculated drain current by (a) linear and (b) logarithmic scale versus VGS with that from Büttiker probes model.
Here we assumed that rC is consistent in a whole range of VGS.