Results

To examine the validity of the channel backscattering theory, we have compared the calculated results with those from 2-D Monte Carlo particle simulation [9]. The calculation values in case 2 are considerably consistent with the Monte Carlo particle simulation results as exhibited in Figure 2-9.

Furthermore, corroborating evidence in terms of the height of the source-channel junction barrier is given in Figure 2-10. However, as the

mobility μ or scattering time τ is increased by a factor of 5, implying that the channel length is effectively reduced from 25nm down to 5nm, the calculated drain currents is also consistent with the Monte Carlo particle simulation results as shown in Figure 2-11. Consequently, the channel backscattering theory remains valid in this study.

Chapter 3 Low-Field Mobility of Electrons in Bulk Silicon

3.1 Channel Backscattering Coefficient

In this study, if the channel is under low electric field conditions, the width of the kBT layer l calculated according to its definition is wide enough to be larger than the channel length L. Therefore, the backscattering coefficient can be estimated from

_Co L

r ⁼ L+λ (3.1)

where λ is the mean-free-path, L is the channel length which is smaller than the k_BT layer width l, implying that the scattering effect is only assumed to occur in channel. When a strong channel electric field is present, i.e. l < L, the backscattering coefficient can be accordingly estimated from

C

r l

l

λ

= + (3.2)

3.2 Electron Transport Simulation

In order to explore the backscattering coefficient rC , we have simulated electron transport through one-dimensional silicon devices by the Monte Carlo technique [8]. A model “channel” with length L = 80 nm or 20 nm is divided into 100 grids to analyze the forward and backward flux in

each grid, and a constant electric field is applied. Figure 3-1 shows the schematic structure of the simulation in the model “channel”. The doping concentration is set to be 8×10¹⁷ /cm³ and the temperature is 300 K. From the definition of backscattering coefficient, we have

( )

where n± is the electron density for forward/backward direction, and v± is the average velocity for forward/backward direction. Fig. 3-2 shows an example case of the schematic velocity distribution. Here the backscattering coefficient rC is just equal to the area ratio of negative to positive. Figure 3-3 shows 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.

It is obvious that the backscattering coefficient is nearly constant at low electric field as a linear region. In contrast to low electric field, the backscattering coefficient for the higher electric field is lower in the saturation region, which is close to the ballistic limit.

3.3 Low-Field Mobility of Electrons

From the channel backscattering theory, since the backscattering coefficient rC is functionally linked to both the quasi-thermal-equilibrium mean-free-path λ for backscattering and the width l of the kBT layer, the mean-free-path λ can be obtained by fitting method. With the Monte Carlo simulation at different temperatures, the mean-free-path λ is extracted as shown in Figure 3-4. The mean-free-path is physically increased with the decrease of the temperature. Furthermore, the low-field mobility of electrons is calculated as shown in Figure 3-5. As a result, it is considerably reasonable [6] to keep the channel backscattering theory remaining valid in this study.

Chapter 4 Physically Based Analytic Model for 1-D case

4.1 Device Under Study

Figure 4-1 describes the diagram of the Si nanowire transistor structure under study: a cylindrical SNWT with <100> oriented channel length equal to 10 nm. The gate length L is equal to the channel length. The silicon body thickness TSi is 3 nm, and the oxide thickness is 1 nm. The source/drain doping concentration is 2×10²⁰ /cm³ and the channel region is undoped. The low-field mobility is assumed to be 200 cm²/V-sec, and the work function of the gate all around is 4.05 eV. All the simulations are conducted at room temperature (T=300 K) with the same voltage of 0.4V applied to both the gate and the drain.

4.2 Model Establishment

Similar to Chapter 2 of this thesis, we have also established a 1-D model for the nanowire transistor system with the concepts of the elementary scattering theory [11].

For 1-D case, the density-of-states is exhibited as follows:

1

with a factor 2 to indicate the two carrier transportation directions in the

nanowire. Under steady-state, non-equilibrium conditions,

( )

dN E 0

dt = (4.2)

and we can solve it for the steady-state number of electrons in the device.

Consequently, the total number of electrons is obtained via integration over the energy in the device as

(1 ) 2 1 2( )

where n_vⁱ is the valley degeneracy, mⁱ_d is the density-of-states effective mass for subband i, and ℑ_{-1/ 2} is the Fermi-Dirac integral of order -1/2 as study, it is assumed that almost all of the electrons occupy the lowest subband (i.e. i=1, n¹_S =n_S), which is the first subband of the four-fold valley (i.e. nvⁱ=4) [13], [14].

With the drain current contributed by subband, i,

the effective thermal injection velocity at the top of source-junction barrier is as follows [4].

The mean-free-path λ for backscattering is exhibited as follows [10].

^{1 2}

where μ is the quasi-equilibrium mobility. The backscattering coefficient r_C is

then the drain current can be obtained as follows.

1

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:

qn_S =C_eff(V_G −V_tho) (4.10)

The k_BT layer widths are extracted from the potential profiles of Fig. 3 in [12]. The density-of-states effective mass is 0.28m_o 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 k_BT/q drop is developed, and r_C 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

_BF

F

_forward

)

l x 0

K

_B

T Conduction band

F

_forward

K

_x

E

Source Channel Drain

F

_backward

(= r

_BF

F

_forward

)

l x 0

K

_B

T Conduction band

F

_forward

K

_x

E

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 r_BF=0 Extract l And let initial r_BF=0

r_c=r_BF & Iteration

Fig. 2-5 Flowchart of our analysis.

X

_R-Scatt

(nm) k

B

T lay er w id th l (nm )

τ_scatt

= 10 fs

Source Scattering Drain Region

k

B

T lay er w id th l (nm )

τ_scatt

= 10 fs

Source Scattering Drain Region

0 XR-Scatt X Non-Scattering Region

X

_R-Scatt

(nm) k

B

T lay er w id th l (nm )

τ_scatt

= 10 fs

Source Scattering Drain Region

0 XR-Scatt X Non-Scattering Region

Source Scattering Drain Region inset shows the definition of X_R-Scatt.

(λ_elastic)

Fig. 2-7 Schematic flux profile in the kBT layer when the factor Q’ is considered.

r_BFQ_invV_inj

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 X_R-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 : k_BT Layer

Source

Drain Incident Beam

L : Channel Length l : k_BT 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

¹

10

²

10

³

10

⁴

10

⁵

10

⁶

0.0

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

d

at

Γ

point (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 r_C is consistent in a whole range of V_GS.

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