Results

This new compact kBT-layer’s width model, which links the width of thermal energy k_BT layer to the geometrical and bias parameters of the devices, is physically derived on the basis of a parabolic potential profile around the source-channel junction barrier of nanoscale-MOSFETs. Moreover, this proposed model is supported not only by experimental data, but also by various simulation works presented in the open literature. Only one fitting parameter remains constant in a wide range of channel length (10 to 65 nm), gate voltage (0.4 to 1.2 V), drain voltage (0.2 to 1.2 V), and temperature (100 to 500 K), which means that the capability of this compact model is universal.

Chapter 4

Re-examination of Mean-Free-path for Backscattering

When applying (1) to nanoscale MOSFETs to predict relevant current-voltage characteristic, we need to quantify the key backscattering coefficient rc. However, there are two important parameters needed in rc calculation. We have discussed the l on the basis of a parabolic potential profile around the source-channel junction barrier and obtained a compact model of l with the channel length, gate overdrive, drain voltage and temperature as input parameters. On the other hand, another parameter λ0 has not been discussed yet. In Section 2.2.3, we extracted this information under near-equilibrium low field condition. When comparing (10) to compare with Monte Carlo results, it seems to have some differences between calculated and simulated rc

in the linear potential profile. In parabolic case, Figure 2-6 shows good agreement between calculation and simulation results. In this chapter, we will discuss the near-equilibrium mean-free-path for backscattering.

Section 4.1 Apparent Mean-Free-Path

At the end of Chapter 2, we pointed out that in the citation [12], the Monte Carlo simulations at room temperature in case of non-degenerate statistics on a linear channel potential profile have exhibited certain relationship: λ'=λ γ₀/ with γ =1.5 to 2.0. Here λ is the apparent mean-free-path that constitutes the ' following expression in the high field case:

c

'

r ⁼ + l λ

l

⁽²⁴⁾

Thus, some clarifications are demanded.

In order to achieve this goal, we perform Monte Carlo particle-based simulation on a silicon bulk conductor like what we have done in Chapter 2, but pay more attention on the velocity distribution on the top of the barrier and flux ration at the end of the kBT layer.

Section 4.1.1 Mean-Free-Path Extraction in Linear Potential

First of all, we extract the mean free path from the simulated rc using (25), proven in Chapter 2. By substituting the simulated rc into (25), the underlying λ ' can be obtained as depicted in Figure 4-1 versus applied voltage. From Figure 4-1, we can see that: (i) λ falls below ' λ₀, and decreases with increasing applied voltage, leading to λ'=λ γ₀/ with γ = 1.5 to 2.5; and (ii) on average, λ decreases with '

decreasing conductor length. However, it is supposed that the upper limit of λ0 can be recovered only with increasing conductor length or decreasing applied voltage. On the other hand, the ratio of λ to ' λ₀ appears to be a weak function of the lattice temperature. The same argument has been produced by the recent Monte Carlo simulations [12] devoted to a linear potential profile at lattice temperature of 300 K as we mentioned at the end of Chapter 2.

Section 4.1.2 Mean-Free-Path Extraction in Parabolic Potential

For a parabolic potential profile, the corresponding simulated

r

c is displayed in Figure. 2-6. Here, the lines are calculated from (25) with λ of 56, ' 105, and 155 nm for 300, 200, and 150 K, respectively. Good reproduction of the data for different conductor lengths, different temperatures, and, especially, different applied voltages can be gotten. Moreover, the reproduction can all be achieved with simply λ'=λ₀, without adjusting any parameters. This means no matter how the quasi-ballistic transport prevails in the kBT layer (λ'> A), the quasi-equilibrium conditions still govern the backscattered carriers.

Section 4.2 Evidence for Carrier Heating

In Section 4.1, we see that the discrepancy in the mean-free-path in linear potential profile. This inconsistency can be contributed to the presence of the carrier heating. In citation [10], all the derivations are carried on the basis of the “relaxation length” approximation, which means that the mean-free-path is a constant, and is independent of the carrier energy or carrier temperature. However, as carriers transport under applied electric field, their energy should be changed, especially in the linear potential profile. Because of the lack of weak field regime near the injection point, the carriers experience a larger electric field in the linear one, and thus the deviation between the lattice temperature and carrier temperature would be possible.

On the other hand, for a parabolic potential profile, as showing Figure 2-2, there is significant fraction of the kBT layer, which can be determined as the weak field regime. Take this into consideration, although the deviations from the lattice temperature would be possible as entering into the remainder, the overall carrier heating in the kBT layer should be weakened. Consequently, when the carriers are injected into the channel at the beginning of the k_BT layer, they immediately undergo the strong acceleration. Also, owing to the quasi-ballistic transport, the carrier

mean-free-path is therefore no longer independent of the carrier energy.

Section 4.2.1 Velocity Distribution at the Injection point

Concerning the difference between the lattice temperature and carrier temperature, the carrier velocity at the injection point is helpful as demonstrated in Figure 4-2 and 4-3. Figure 4-2 shows the velocity distribution of L=25nm at Va=0.8 V and 300 K for two potential profiles. Figure 4-2 clearly shows that two significant differences between the potential profiles. First, the injected single hemi-Maxwellian velocity distribution is retained in the positively-directed carriers with the parabolic one, but it is split into two distinct components in the linear one: One of the longitudinal effective mass and one of the transverse effective mass. Second, the distribution of the negatively-directed or backscattered carriers appears to be wider in the linear potential profile than the parabolic one. The same result can be seen in Figure 4-3 for L=50nm at V_a=1.0 V and 150 K. This is evidence of carrier heating.

Section 4.2.2 Flux Ratio at the End of k_BT Layer also can be cited in the literature [10], which was derived on the basis of a Maxwillien shape distribution in both the forward and backward directions with in the context of the relaxation length approximation. In order to get the ratio of negatively-directed flux to positively-directed flux, we substitute (10) into (26) for r_c=a(0) / (0)b . Then,

accounting for the conservation of the current, the ratio ( ) / ( )b l a l can be straightforwardly calculated. Monte Carlo simulation program can provide the velocity distribution at the end of the kBT layer. For example, Figure 4-4 shows velocity distribution of L=50 nm, T=200 K, and V_a=0.2 V at x= A. The velocity distribution also is split into two components, and velocity distribution of backscattered carrier is wider. The ( ) / ( )b l a l is the ratio of deep grey area to light grey area, like what we did to determine

r

c at the injection point in Section 2.2.2. The calculated lines from (26) are shown in Figure 4-4 for linear potential profile. Also plotted in the figure are those of the Monte Carlo simulation. However, the calculated values are seen to fall below the simulation ones. This reveals the fact that the reflected flux at the end of k_BT layer can be enhanced in the presence of carrier heating. Only with decreasing applied voltage can the deviation between the simulation and model calculation be shortened.

Section 4.3 Results

We executed Monte Carlo simulations on a silicon bulk conductor to re-examine the channel backscattering theory in bulk nano-MOSFETs. Through these simulations, some important points can be addressed again:

(i) The near-equilibrium mean-free-path for backscattering λ₀, is independent of the potential profile.

(ii) λ in a linear potential profile is lower than ' λ₀ of parabolic one due to the presence of the carrier heating. Evidence is highlighted by both the velocity distribution at the injection point and the flux ratio at the end of the k_BT layer.

Chapter 5 Conclusion

The backscattering coefficient is derived through the scattering matrix approach and verified by Monte Carlo simulations. Two important parameters, λ₀ and A , constituting the channel backscattering coefficient have been taken into account. A parabolic barrier oriented compact model has been physically derived for A . The validity of this compact model has been corroborated experimentally and by Monte Carlo simulation results. As forλ₀, the carrier heating as the origin of reduced mean-free-path is inferred on the basis of the simulated carrier velocity distribution at the injection point. Strikingly, for the parabolic potential case, the mean-free-paths remain consistent: λ’= λo. This indicates the absence or weakening of the carrier heating in the layer of interest, valid only for a parabolic potential barrier

References

[1] M. S. Lundstrom, “Elementary scattering theory of the Si MOSFET,” IEEE Electron Device Letters, vol. 18, pp. 361-363, July 1997.

[2] 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.

[3] 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, Sept. 2004.

[4] M.-J. Chen, H.-T. Huang, K.-C. Huang, P.-N. Chen, C.-S. Chang, and C. H. Diaz,

“Temperature dependent channel backscattering coefficients in nanoscale MOSFETs,” in IEDM Tech. Dig., pp. 39-42, Dec. 2002.

[5] M. J. Chen, S. G. Yan, R. T. Chen, C. Y. Hsieh, P. W. Huang, and H. P. Chen,

“Temperature-oriented experiment and simulation as corroborating evidence of MOSFET backscattering theory,” IEEE Electron Device Lett., vol. 28, pp.177-179, Feb. 2007.

[6] A. Rahman and M. S. Lundstrom, “A compact scattering model for the nanoscale double-gate MOSFET,” IEEE Trans. Electron Devices, vol. 49, pp. 481-489, March 2002.

[7] M. J. Chen, R. T. Chen, and Y. S. Lin, “Decoupling channel backscattering coefficients in nanoscale MOSFETs to establish near-source channel conduction-band profiles,” in Silicon Nanoelectronics Workshop, pp. 50-51, June 2005.

[8] P. Palestri, D. Esseni, S. Eminente, C. Fiegna, E. Sangiorgi, and L. Selmi,

“ Understanding quasi-ballistic transport in nano-MOSFETs : Part I – Scattering in the channel and in the drain,” IEEE Trans. Electron Devices, vol. 52, pp.

2727-2735, Dec. 2005.

[9] Mark Lundstrom, Fundamentals of Carrier Transport, second edition, School of Electrical and Computer Engineering Purdue University, West Lafayette, Indiana, USA: Cambridge University Press, 2000.

[10] R. Clerc, P. Palestri, and L. Selmi, “On the physical understanding of the kT-layer concept in quasi-ballistic regime of transport in nanoscale devices,”

IEEE Trans. Electron Devices, vol. 53, pp. 1634-1640, July 2006.

[11] S. Datta, Electronic Transport in Mesoscopic System, Cambridge, U.K.:

Cambridge Univ. Press, 1995.

[12] P. Palestri R. Clerc, and D. Esseni, L. Lucci, and L. Selmi,

“Multi-subband-Monte-Carlo investigation of the mean free path and of the kT layer in degenerated quasi ballistic nanoMOSFETs,” in IEDM Tech. Dig., , pp.945-948, Dec. 2006.

[13] http://www.nanohub.org

[14] M. J. Chen and L. F. Lu, “A parabolic potential barrier oriented compact model for the kBT-layer’s width in nano-MOSFETs,” IEEE Trans. Electron Devices, vol. 53, pp. 1265-1268, May 2008.

[15] R. Clerc, P. Palestri, and L. Selmi, “On the physical understanding of the kT-layer concept in quasi-ballistic regime of transport in nanoscale devices,”

IEEE Trans. Electron Devices, vol. 53, pp. 1634-1640, July 2006.

[16] E. Fuchs, P. Dollfus, G. L. Carval, S. Barraud, D. Villanueva, F. Salvetti, H.

Jaouen, and T. Skotnicki, “ A new backscattering model giving a description of the quasi-ballistic transport in nano-MOSFET,” IEEE Trans. Electron Devices, vol. 52, pp. 2280-2289, Oct. 2005.

[17] J. Saint-Martin, A. Bournel, and P. Dollfus, “On the ballistic transport in nanometer-scaled DG MOSFETs,” IEEE Trans. Electron Devices, vol. 51, pp.

1148-1155, July 2004.

[18]D. Querlioz, J. Saint-Martin, K. Huet, A. Bournel, V. Aubry-Fortuna, C. Chassat, S. Galdin-Retailleau, and P. Dollfus, “On the ability of the particle Monte Carlo technique to include quantum effects in nano-MOSFET simulation,” IEEE Trans.

Electron Devices, vol. 54, pp. 2232-2242, Sept. 2007.

[19] S. Eminente, D. Esseni, P. Palestri, C. Fiegna, L. Selmi and E. Sangiorgi,,

“ Understanding quasi-ballistic transport in nano-MOSFETs : Part II – Technology scaling along the IRTS,” IEEE Trans. Electron Devices, vol. 52, pp.

2736-2743, Dec. 2005.

[20] K. N. Yang, H. T. Huang, M. C. Chang, C. M. Chu, Y. S. Chen, M. J. Chen, Y.

M. Lin, M. C. Yu, S. M. Jang, C. H. Yu, and M. S. Liang, “A physical model for hole direct tunneling current in p⁺ poly-gate pMOSFETs with ultrathin gate oxides,” IEEE Trans. Electron Devices, vol. 47, pp. 2161–2166, Nov. 2000.

L (nm) Vtho (V) DIBL (mV/V) V_D (V) V_G (V) T (K)

M. S. Lundstrom, et al. [2] 10 0.33 140 0.6 0.6 300

A. Rahman, et al. [6] 20 0.33 25 0.2 0.55 300

P. Palestri, et al. [8] 14~37 0.3 110 1 1 100~500

E. Fuchs, et al. [16] 25 0.3 100 0.8 0.5~1.0 300

J. Saint-Martin, et al. [17] 15 0.2 120 0.7 0.7 300

D. Querlioz, et al. [18] 15 0.3 77 0.7 0.7 300

S. Eminente, et al. [19] 14~65 0.3 11~230 1.0~1.2 1.0~1.2 300

Table. 3-1

( ) Ec x

l

F

r Fc

(1− r_c)F

B / k T q

0 L

Fig.1-1

Fig.2-1

linear

l

parabolic

l

Fig.2-2

-5x10⁷-4x10⁷-3x10⁷-2x10⁷-1x10⁷ 0 1x10⁷ 2x10⁷ 3x10⁷ 4x10⁷ 5x10⁷

0.0 0.2 0.4 0.6 0.8 1.0

Doping =10

¹²

#/cm

³

0 x =

L=100nm,T=300K V

_a

=0.6V

Dis tribut ion

Velocity(cm/s)

Linear Potential

Fig.2-3

40 60 80 100 120 140 160 180 0.35

0.40 0.45 0.50 0.55 0.60 0.65

0 c

r L

L λ

= +

T=150K T=200K

r c

Linear Potential T=300K

L=100nm

Parabolic Potential

λ ₀ ^(nm)

Fig.2-4

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

( )

G th

V > V

V

D

≈

B

/ k T q

( ) V x

l ^L%

0

V D

Fig. 3-1

0.4 0.6 0.8 1.0 1.2

0.4 0.6 0.8 1.0 1.2 1.4

0 1 2 3 4 5

0.0 0.2 0.4 0.6 0.8 1.0

-3x10⁷ -2x10⁷ -1x10⁷ 0 1x10⁷ 2x10⁷ 3x10⁷ 4x10⁷ 5x10⁷

-3x10⁷ -2x10⁷ -1x10⁷ 0 1x10⁷ 2x10⁷ 3x10⁷ 4x10⁷ 5x10⁷

Fig.4-4

-5x10⁷-4x10⁷-3x10⁷-2x10⁷-1x10⁷ 0 1x10⁷ 2x10⁷ 3x10⁷ 4x10⁷ 5x10⁷

0.0 0.2 0.4 0.6 0.8 1.0

x = l

L=50nm,T=200K V

_a

=0.2V

Dis tribut ion

Velocity(cm/s)

Linear Potential

a( ) l

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