Extraction of Ballistic Ratio (BR)

In this section, we will directly extract BR from the drain current and compare it with the temperature coefficient method. As shown in Eq.

(3-23), we calculate BR at saturation region and there are two terms,

and , that should be calculated first:

(3-23) In the experiment part, we can measure C-V characteristics to get the corresponding inversion carrier density . Traditionally, the measured result is the macroscopic average. But in the channel backscattering theory it is not. It represents the carrier density at the barrier high in channel. Therefore, in the following we will extract BR at X=Xvs. As shown in Fig. 3-15-1, BR is extracted at the virtual source; it seems to increase with channel length shrinkage. This result shows the reflection rate decreasing with channel length decreasing. Considering the

20

physical limit on saturation velocity, it must saturate at : BR is about 0.75 and rc is about 0.15 at room temperature. Replacing BR with and L_kt, we have higher temperature and BR increases with channel length shrinkage for the two methods. Although they have the same characteristics, the difference between each temperature is not the same. Therefore, we can predict BR extracted from temperature coefficient method. As shown in Eq. (2-16), we can use the five temperature coefficients to extract . Result is shown in Fig. 3-15-3 and it shows a nearly constant characteristic for each channel length. Furthermore, using the temperature coefficient set A assumed in [3] even shows a negative trend at LG=15nm/

20nm. The reason is caused by the error in the five coefficients. In the five coefficients, we could simply assume the error would occur on critical scattering length. Other coefficients have been defined clearly.

Therefore, substituting Eq. (3-24) into (3-23), we have

. (3-25) The calculated critical scattering length is shown in Fig. 3-16-1. And according to the new , we can extract the new BR as shown in Fig.

3-16-2. As our anticipation, it shows similar characteristics as the result of direct calculation.

21

Chapter 4

Conclusion

In the study, we used TCAD simulator to predict the characteristic of DG nMOSFETs with channel length scaling down to nanoscale. Here, the most important issue is the high field velocity saturation. It causes the carrier mobility degradation for the even general applied measurement drain bias of 0.025V, and generates DIBL effect for channel length is smaller than 45nm. Therefore, we should measure the mobility at small enough drain bias like 1mV. But the mobility degradation effect still exists for the operation voltage. For this reason, Prof. Lundstrom, et al. at Purdue University brings up the channel backscattering theory. They used another concept to explain the velocity saturation effect. In the theory, they think the critical scattering length stems from the virtual source point to the point with kT potential drop, rather than the virtual source point.

The result between the DD model and channel backscattering model has not too much difference. But as we used the temperature coefficient method to calculate BR, the slight difference in the critical scattering length causes a strong variation on the temperature coefficient LkT. We could infer carriers would be scattered in a range larger than kT-layer for higher temperature and in smaller than kT-layer for lower temperature.

However, the channel backscattering theory is a useful theory to explain why we would face the awkward situation with the channel length scaling down.

22

References

[1] A. Pirovano, A. Lacaita and A. Spinelli, ―Two-dimensional quantum effects in nanoscale MOSFETs,‖ IEEE Trans. Electron Devices, vol. 49, no. 1, pp25-31, Jan.

2002.

[2] M. S. Lundstrom, ―Elementary scattering theory of the Si MOSFET,‖ IEEE Electron Device Letters, vol. 18, no. 7, pp. 361-363, July 1997.

[3] 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 IEEE IEDM Tech. Dig., pp. 39-42, 2002.

[4]D. M. Caughey and R. E. Thomas, ―Carrier mobilities in Silicon empirically related to doping and field,‖ Proc. IEEE, pp. 2192–2193, Dec. 1967.

[5] M. J. Chen and L. F. Lu, ―A parabolic potential barrier-oriented compact model for the k_bT layer’s width in Nano-MOSFETs,‖ IEEE Trans. Electron Devices, vol. 55, no. 5, pp. 1265-1268, May 2008.

[6] A. Rahman and M. S. Lundstrom, ―A compact scattering model for the nanoscale Double-Gate MOSFET,‖ IEEE Trans. Electron Devices, vol. 49, no. 3, pp. 481-489, March 2002.

[7] D. Vasileska, D. K. Schroder and D. K. Ferry, ―Scaled silicon MOSFET’s: Part II - Degradation of the total gate capacitance,‖ IEEE Trans. Electron Devices, vol. 44, no. 4, pp. 584-587, April 1997.

[8] F. Assad, Z. Ren, D. Vasileska, S. Datta and M. Lundstrom, ―On the performance limits for Si MOSFETs: a theoretical study,‖ IEEE Trans. Electron Devices, vol. 47, pp. 232-240, Jan. 2000.

[9] M.Zilli, P.Palestri, D.Esseni and L.Selmi, ―On the experimental determination of channel back-scattering in nanoMOSFETs,‖ in IEEE IEDM Tech. Dig., pp. 105-108,

23

2007.

[10] V. Barral, T. Poiroux, J. Saint-Martin, D. Munteanu, J. Autran and S. Deleonibus,

―Experimental investigation on the Quasi-ballistic transport: Part I—determination of a new backscattering coefficient extraction methodology,‖ IEEE Trans. Electron Devices, vol. 56, no. 3, pp. 408-419, March 2009.

[11] V. Barral, T. Poiroux, D. Munteanu, J. Autran and S. Deleonibus, ―Experimental investigation on the Quasi-ballistic transport: Part II—backscattering coefficient extraction and link with the mobility,‖ IEEE Trans. Electron Devices, vol. 56, no. 3, pp. 420-430, March 2009.

[12] K. Natori, ―Ballistic metal-oxide-semiconductor field effect transistor,‖ J. Appl.

Phys., vol. 76, pp. 4879–4890, 1994.

[13] D. J. Frank, S. E. Laux and M. V. Fischetti, ―Monte Carlo simulation of a 30 nm Dual-Gate MOSFET: how short can Si go?,‖ in IEEE IEDM Tech. Dig., pp. 553-556, 1992.

[14] K. Banoo and M. S. Lundstrom, ―Electron transport in a model Si transitor,‖

Solid- State Electronics, vol. 44, issue 9, pp. 1689-1695, Sep. 2002.

24

Fig. 1-1 Schematic illustration of DG MOSFET.

25

Fig. 1-2 Schematic illustration of channel backscattering theory in terms of the conduction band profile. F⁺: the incident flux from the source is located at the peak of the source-channel barrier. Fb

-: the incident flux from the drain. T: the transmission coefficient for the flux cross the barrier for both directions.

26

-20 -15 -10 -5 0 5 10 15 20 -1.0

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

C ond uction ba nd ene rgy E

C

( eV )

Y-position (nm)

ConductionBand V_G=0.8V V_G=-0.2V

at X=0=mid of channel

Fig. 3-1 The conduction energy band diagram along Y-sirection.

T=300K, V

_D

=0V, V

_G

=-0.2V/ 0.8V and L

_G

=90nm.

27

-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 -1.2

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

At Y=0 V_D=1V V_G=0.8V

C ond uction ba nd ene rg y E

C

( eV )

X-position (nm)

L_G=90nm L_G=45nm L_G=20nm L_G=15nm

Fig. 3-2 The conduction energy band diagram along the channel.

T=300K, V

_D

=1V, V

_G

=1V and L

_G

=15/ 20/ 45/ 90nm.

28

-45 -40 -35 -30 -25 -20 -15 -10 -5 0 0.005

0.010 0.015 0.020 0.025 0.030 0.035 0.040

L_G=15nm L_G=90nm

T=250K T=300K

T he low est su bba nd ene rgy E

11

( eV )

X-position (nm)

T=350K Dash line: V_D= 0 Straight line: V_D=1 mV

mid of channel

Fig. 3-3 The lowest subband E

₁₁

along the channel. T=250/ 300/

350K, L

_G

=15/ 90nm and V

_D

=0/1mV. The difference between

two different V

_D

is due to the variation of Fermi-level.

29

Fig. 3-4 Schematic diagram of sheet carriers.

30

Inve rsion ca rr ier de nsity N

inv

( cm

-2

)

X-position (nm)

Inve rsio n ca rr ier de nsity N

inv

( cm

-2

)

V_d= 1V

DIBL gives arise in the carrier density increasing at L

_G

=15nm.

31

Inversion carrier density N inv (cm-2 )

V_G (V)

TCAD L_G=90nm TCAD L_G=15nm Schred

Fig. 3-6-1 The inversion carrier density N

inv

for V

G

=-0.2~1V at L

_G

=15/ 90nm, T=300K and V

_D

=0V of TCAD and Schred.

Fig. 3-6-2 The inversion carrier density N

inv

for V

G

=-0.5~1V at L

G

=90nm, T=300K and V

D

=0V/ 50mV.

Inversion carrier density N inv (cm-2 )

V_G (V)

32

-60 -40 -20 0 20 40 60

0.0 0.2 0.4 0.6 0.8 1.0

E lectr on Qua si-F er mi le vel E

f

( me V )

X-position (nm)

L_S=L_D=20nm L_G=15/20/45/90nm

at Y=0 V_D=1mV V_G=0.8V T=250K

T=300K T=350K

Fig.3-7 Electron Quasi-Fermi Level alone the channel for

L

_G

=15/ 20/ 45/ 90nm, V

_D

=1mV and V

_G

=0.8V at Y=0.

33

34

0 10 20 30 40 50 60 70 80 90 100 0

50 100 150 200 250

R

Channel

= W V

Channel

/I

D

(



m)

R_tot Solid: V_D=25mV Open: V_D=1mV Square: T=350K

Circle: T=300K Triangle: T=250K Line: Linear fitting

Channel Length (nm)

Fig.3-8-3 Channel resistance for L

_G

=15/ 20/ 45/ 90nm at

V

_D

=1mV/ 25mV and V

_G

=0.8V.

35

240 260 280 300 320 340 360

400

240 260 280 300 320 340 360

300

36

260 280 300 320 340 360

300

240 260 280 300 320 340 360

400

37

Table 1 Caughey-Thomas formula parameters

Symbol Parameter Value Unit

vsat0 1.07x10⁷ cm/s

vsatexp 0.87 1

beta0 1.109 1

betaexp 0.66 1

38

The lowest subband energy E 11 (meV)

X-X_VS position (nm)

L_G=15nm

L_G=90nm

350K

Fig 3-10-1. Profile of the lowest subband energy E

11

at V

G

=0.8V and V

D

=1V.

Critical scattering length L kT (nm)

Temperature (K)

Lg=15nm _LkT=1.609 Lg=20nm _LkT=1.277 Lg=45nm _LkT=1.585 Lg=90nm _LkT=1.456

Fig 3-10-2. kT-layer extension as a function of the temperature

at V

_G

=0.8V and V

_D

=1V.

39

Injection velocity v inj (107 cm/s)

N_inv (cm^-2)

Cal: using E_ij from Schred T=250K

T=300K T=350K

Fig 3-11-1. Injection velocity against the inversion carrier density for three temperatures.

Fig 3-11-2. Injection velocity as a function of the temperature at

V

_G

=0.8V and V

_D

=1V.

40

Inversion carrier density N inv (cm-2 )

V_G (V)

Inversion carrier density N inv (cm-2 )

V_G (V)

41

Inversion carrier density N inv (cm-2 )

V_G (V)

42

Fig 3-13-1. Drain current vs. gate voltage.

10 20 30 40 50 60 70 80 90 100

Solid: T=350K Open: T=300K Center: T=250K V_D=1V

43

10 20 30 40 50 60 70 80 90 100 -2.5

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

A lpha



( 10

-3

K

-1

)

Channel Length (nm)

_th @ V_D=1V

_th @ V_D=25mV

_th @ V_D=1mV

_ID @ V_D=1V

Fig 3-13-3. Alpha α against the channel length.

44

10 20 30 40 50 60 70 80 90 100 0.5

1 1.5 2

_e

X=X

_vs

_e

X=0

_th

X=0

(  - 

satexp

/ T ) / 

Channel Length (nm)

Fig 3-14. Ratio=

.

45

BR dir = I D/Q inv(X vs)V inj = (1-r c)/(1+r c)

Channel Length (nm)

and V

_D

=1V. BR increases with channel length scaling down.

10 20 30 40 50 60 70 80 90 100

and V

_D

=1V. BR increases with channel length scaling down.

46

10 20 30 40 50 60 70 80 90 100 0.0

0.2 0.4 0.6 0.8 1.0

B R ( T C m ode l)

Channel Length (nm)

Solid: BR_dir Open: BR_ T=350K

T=300K T=250K

TC model set A [3]

Fig. 3-15-3 BR (TC model) calculation.

Set A [3]:

.

47

240 260 280 300 320 340 360

0

Fig. 3-16-1 The calculated L

kTcal

versus temperature for different

L

G

, along with the extracted L

kT

.

different temperature. BR

_TC

is also shown in the comparison of

calculated L

_kT

.

在文檔中 利用通道背向散射理論及溫度係數模型分析奈米級金氧半電晶體電子遷移率的劣化之研究 (頁 30-0)