From Eq. (2-7), we can get BR directly and compare it with the result of TC model by using Eq. (2-16). Generally, the two methods may not get the same BR value for any conditions, but it would suggest that there is another factor needed to add into the simulation or the theory should be corrected. In Chapter 3, we will show the calculated result between the two methods.
10
Chapter 3
Analysis of TCAD simulation result
Section 3.1 Introduction
In this chapter, we focus on: (i) the extraction of five power-law parameters ( , , , , ); (ii) the extraction of BR; (iii) the comparison of BR between the direct extraction and the calculation result by using TC model; and (iv) the difference between the key point, X=Xvs, and the balance point, X=0. In the study, the whole channel status is taken into consideration by using the TCAD simulator. We used the quantum mechanics to accurately deal with the sub-band energy in the y direction along the channel, and drift and diffusion transport model to get a self-consistent solution along the channel.
Section 3.2 Extraction of Five Temperature Coefficients
Section 3.2.1 Schrödinger–Poisson–Drift–Diffusion (SPDD) model
The SPDD model was proposed in [1], [2]. From Schrödinger equation (3-1), a set of quantum energy states {Eij}, , , can be solved:
(3-1) With device parameters and the operation temperature and applied bias as input, TCAD can accurately calculate the sub-band energy and the
11
Fermi-level, thus producing the inversion layer carrier density (2DEG) per sub-band as:
, (3-2) where i=1, 2 (valley), j=1, 2, 3 (sub-band)…; nvi is the degeneracy factor
of ith valley; mdi is the density of states effective mass of ith valley. Ef is the quasi-Fermi level.
And by using the Poisson equation as Eq. (3-3), we can get a self-consistent result of the Schrödinger–Poisson as shown in Fig. 3-1:
(3-3) We get the 1-D Schrödinger–Poisson solution in the y-direction, and we can apply it to 2-D case by appending the DD model to constitute a self-consistent solution along the two directions as shown in Fig. 3-2. The electron and hole continuity equations are written as:
(3-4)
, (3-5) where Rnet is the net electron/ hole recombination rate, is the electron current density, and is the hole current density.
The DD model is widely used as a carrier transport model in semiconductors and is defined by the following equations for the current densities of electrons and holes:
(3-6) (3-7)
12
where μ n and μ p are the electron and hole mobilities, and Φn and Φp
are the electron and hole quasi-Fermi potentials, respectively.
Section 3.2.2 Channel Status at Equilibrium
Usually we thought the highest energy of conduction band would locate at the mid of channel, but it would vary from the mid of channel toward the source and drain with increasing gate voltage. As shown in Fig.
3-3 is a profile of the lowest sub-band, E11, along the channel. We found the highest energy is located 2~5nm away from the junction of source/channel and channel/drain in channel. Fig. 3-4 show the sheet charge. In basic Poisson equation, we considered only forward or backward direction. In reality, we should take both into consideration.
This leads to the result of Fig. 3-3. Apparently, the highest point of E11 is not located at the mid-channel. The difference between VD=0V and 1mV is due to the variation of Fermi-Level.
The local higher potential causes a relative lower carrier density at that region. As shown in Fig. 3-5 is the inversion carrier density Ninv
along the channel. We get almost the same Ninv at VD=0V and 1mV. This shows channel is at equilibrium condition for VD=1mV. Furthermore, we find that the inversion carrier density can be gradually affected as the channel length is scaled down. The effect of drain induced barrier lowering, DIBL, can cause the carrier density increasing for LG=15nm, except at LG=90nm. It shows a similar characteristic as [13]. The short channel effect would appear for channel length is smaller than 40nm or the temperature of 150K.
13
Finally, we want to compare how much difference between each channel length different gate voltages can produce. Here, we used a quite correct simulator, Schred [7], evaluated by Prof. Lundstrom, et al. at Purdue University, as the standard to check and analyze the TCAD’s simulation result. As shown in Fig. 3-6-1, we can get a perfect match between TCAD and Schred at long channel, but source and drain affect the channel as channel scaling to 15nm. Usually, we think DIBL is a constant for sub-threshold and above threshold region. Indeed, the channel resistance is high below the sub-threshold region; and above threshold region, the source and drain resistance are relative higher.
Therefore, DIBL is not a constant for any gate voltage as shown in Fig.
3-6-2.
Section 3.2.3 Mobility Extraction at Equilibrium
After analyzing the status of channel at equilibrium, it provided two ways to calculate the mobility for us. First, according to the experimental concept, using the results of ID against VG and the inversion carrier density calculated by C-V measurement, we should assume that the drain current uniformly flows through the channel and the carrier uniformly distributes in channel. The results are shown in Eq. (3-8). Second, according to the ballistic theory, carriers come from the virtual source and may be reflected with a critical scattering length Lkt and a mean free path λ. Whatever the methods used, it should follow the DD model as Eq. (3-6) and (3-7). As shown in equation (3-8), we can get a relationship between ID per unit width and current density of electrons and holes. We can even reduce Eq. (3-8) to Eq. (3-9), because the majority carriers are electrons:
14
(3-8)
(3-9) Although the drain current is conserved along the channel, we can clearly find that from Fig. 3-5 the carrier density is not the same along the channel. From electron quasi Fermi-level along the channel as shown in Fig. 3-7, we can extract the series resistances of source and drain as shown in Fig. 3-8-1 and Fig. 3-8-2:
. (3-10) Fig. 3-8-1, 3-8-2 and 3-8-3 show Rtot, RSD and RChannel, respectively, for four different channel lengths from 90nm to 15nm and three temperatures. There is a slight difference between two applied VD, which is caused by the mobility degradation at VD=0.025V. That suggests that it is not appropriate to extract mobility at drain bias of 0.025V for nanoscale device. We could predict the trend of mobility by using the characteristic of the channel resistance that is proportional to 1/LG as revealed by Eq. (3-11). However, is not zero at LG=0nm at all for VD=0.025V:
; (3-11)
, (3-12) where is the apparent mobility, is the low field mobility, and is a correction term due to the mobility degradation.
15
In Fig. 3-9-1, 3-9-2 and 3-9-3, we sight on the mid-channel (X=0) and X=Xvs. When VD is 1mV, there is no decrease on mobility with the channel length, but we can find a strong degradation on mobility at VD=25mV. This suggests the decrease of mobility with scaling down is caused by the saturation velocity. According to the Canali model evaluated by Canali et al. [4], it is used to explain the high field saturation phenomenon. The Caughey-Thomas formula is shown in below:
(3-13)
where is the saturation velocity and is a fitting parameter for Caughey-Thomas formula. Parameters’ setting is shown in Table 1. We confirmed the mobility degradation by using the Caughey-Thomas formula. As shown in Fig. 3-9-4, we set the low field mobility the same as the mobility extracted from DD model at VD=1mV and LG=90nm. The result agrees with our prediction. That is the mobility degradation caused by the limit of saturation velocity with channel length shrinkage.
Section 3.2.4 Critical Scattering Length L
kTExtraction in Saturation Region (V
G=0.8V and V
D=1V)
Within the framework of the channel backscattering theory, carrier will be scattered in the k-T layer. As the drain voltage is smaller than the thermal energy, the critical scattering length must be equal to the channel length. That explains why we should extract LkT in the saturation region.
16
LkT is expressed as:
(3-14) where and are taken as parameters that can fit the simulation result. Generally, we extract LkT from the conduction band energy profile . But would vary with x and y direction. Therefore, we used the lowest sub-band energy profile along the channel to substitute . As shown in Fig. 3-10-1, LkT increases with increasing channel length and increasing temperature. Fig. 3-10-2 shows the extracted LkT and the temperature coefficient for each channel length and temperature. It has the similar trend with Zilli, et al. [9]
Section 3.2.5 Injection velocity v
injExtraction in Saturation Region (V
G=0.8V and V
D=1V)
Within the framework of the channel backscattering theory, carriers come from the virtual source. Therefore, the effective thermal injection velocity should be extracted at the source-channel barrier position [8].
17
Fermi-Dirac integral of order one-half. For two-fold valley, and for four-fold valley,
. Here the longitudinal mass =0.916 and the transverse mass =0.19 . The result has been shown in Fig. 3-11-1 and Fig.
3-11-2. It shows an increasing trend with temperature increasing and the temperature coefficient will increase with the channel length increasing.
Section 3.2.6 Inversion Carrier Density N
invExtraction in Above Threshold Region (V
G=0.8 and V
D=1mV/ 1V)
With channel length scaling down to nanoscale, an important question we faced is the short channel effects. The short channel effects are attributed to two physical phenomena: (i) the limitation of the drift characteristic; and (ii) the variation of the threshold voltage. Here, we focus on the phenomenon (ii). This is the variation of the carrier density with channel length scaling down for each temperature. As shown in Fig.
3-12-1 and Fig. 3-12-2, the applied drain voltage will cause the channel carrier density decreasing at LG=90nm and 45nm; and for LG=20nm and 15nm, the channel carrier density increases due to the DIBL effect.
Therefore, the temperature coefficient of the inversion carrier density will vary with the channel length change. In the extraction of beta, there are two methods to do:
; (3-17)
(3-18) Furthermore, DIBL is not a constant for any gate voltage as shown
18
in Fig. 3-6-2. It complicates the status of channel with the varying applied gate voltage. Therefore, we focus on the above threshold region and use the maximum transconductance method to extract now. Result is shown in Fig. 3-12-4. Beta appears to slightly increase with channel length decreasing.
Section 3.2.7 Temperature Coefficient of Drain Current in Saturation Region (V
G=0.8V and V
D=1V)
The temperature coefficient of drain current α is the most important factor. It represents the combination of all the effects. Therefore, it should be the same as the result of the combination of each factor. Fig. 3-13-1 shows the same characteristic as Fig. 3-12-2. Here α can be expressed as:
to extract the real variation of drain current between three temperatures.
Owing to the same reason as α , is used to calculate the BR.
Section 3.2.8 Verification of Temperature Coefficient Method
Like the derivation shown in section 2.3, differentiating Eq. (3-9) with respect to temperature is performed:
19
The result of each differential term has been shown in section 3-3, 3-6 and 3-7. At the saturation region, we assume the third of the right-hand side of Eq. (3-11) term is zero and
. Therefore, we can derive Eq. (3-22) as:
. (3-22) Result is shown in Fig. 3-14. It suggests that using the temperature coefficient method is feasible and the accuracy is better than 50%.
Section 3.3 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 Lkt, 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 kbT 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 VG=0.8V VG=-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 VD=1V VG=0.8V
C ond uction ba nd ene rg y E
C( eV )
X-position (nm)
LG=90nm LG=45nm LG=20nm LG=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
LG=15nm LG=90nm
T=250K T=300K
T he low est su bba nd ene rgy E
11( eV )
X-position (nm)
T=350K Dash line: VD= 0 Straight line: VD=1 mV
mid of channel
Fig. 3-3 The lowest subband E
11along the channel. T=250/ 300/
350K, L
G=15/ 90nm and V
D=0/1mV. The difference between
two different V
Dis 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)
Vd= 1V
DIBL gives arise in the carrier density increasing at L
G=15nm.
31
Inversion carrier density N inv (cm-2 )
VG (V)
TCAD LG=90nm TCAD LG=15nm Schred
Fig. 3-6-1 The inversion carrier density N
invfor 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
invfor V
G=-0.5~1V at L
G=90nm, T=300K and V
D=0V/ 50mV.
Inversion carrier density N inv (cm-2 )
VG (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)
LS=LD=20nm LG=15/20/45/90nm
at Y=0 VD=1mV VG=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)
Rtot Solid: VD=25mV Open: VD=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-XVS position (nm)
LG=15nm
LG=90nm
350K
Fig 3-10-1. Profile of the lowest subband energy E
11at 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)
Ninv (cm-2)
Cal: using Eij 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 )
VG (V)
Inversion carrier density N inv (cm-2 )
VG (V)
41
Inversion carrier density N inv (cm-2 )
VG (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 VD=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
-3K
-1)
Channel Length (nm)
th @ VD=1V
th @ VD=25mV
th @ VD=1mV
ID @ VD=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
vse
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: BRdir 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