7 nm Model Parameters
Since the frequency response in homework 2 produced a unity gain frequency for the shorter gate length that wasn't higher than the long gate length FinFET, I suspect a problem with the model or the way I'm using it so I investigated the model further. The following model parameters were used for this investigation and for homework 3.
| parameter | Model | Value | units
|
| Fin Length | L | 20.0 | nm
|
| Number of fins | NFIN | 3 |
|
| Number of fingers | NFGR | 1 |
|
| Fin Height | HFIN | 32.0 | nm
|
| Fin Thickness | TFIN | 6.5 | nm
|
| Fin Pitch | FPITCH | 27.0 | nm
|
| Fin Width, 1 fins | W | 70.5 | nm
|
| Fin Width, 3 fins | | 211.5 | nm
|
| W/L for 3 fins | | 10.6 |
|
| Gate oxide thickness | EOT | 1 | nm |
The fin width was computed as NFIN (2 HFIN + TFIN) where HFIN is the fin height and TFIN is the fin thickness using the .model variables. This makes W/L = 10.6 for a FinFet with 3 fins.
Using methods described in the course and in the book, the performance parameters for the 7 nm models were extracted. I haven't found a references to confirm the performance values yet.
| Parameter | NMOS | PMOS | Units
|
| μCOX | 156.6 | 110.2 | μA/V2
|
| Vt | 230 | -210 | mV
|
| λ | 0.1795 | 0.3563 | 1/V
|
| λL | 0.0036 | 0.0071 | μm/V
|
| COX | | | fF/μm2
|
| n | 1.05 | 1.02 |
|
| θ | 2.94 | 1.92 | V
|
| COV/W = LOVCOX | 0.16 | 0.16 | fF/μm
|
| Cdb/W ≈ CSb | 5.5*10-8 | 5.6*10-8 | fF/μm |
The 7 nm models from the PTM website [1] are define with the .model function and have more parameters defined compared to the modified (simplified) model provided for the course. These models use the BSIM-CMG [2] model. The PTM website indicates that the model process maximum supply voltage is 700 mV. The homework, however, specifies V
DS = 800 mV in the problems so was used for the solutions. The solutions are based on the materials presented in the course and the book Analog Integrated Circuit Design [3].
The sections below show how the parameters were determined and how the SIMetrix simulator was setup to make the measurements. Mathematica was used to add the line guides to estimate the projection of the g
m slopes to the x-axis and to the projections of maximum g
m.
NMOS μnCOX, Vt, subthreshold slope, n, and θ
The schematic used for finding μ
nC
OX, V
t, subthreshold slope, n, and θ is shown below.
![figure 1 7nm finfet model schmatic 7nm nmos gm vgs]()
The top plot in the figure below shows g
m/I
D vs V
GS. The minimum was found to be 62.4 mV/decade using
Measure -> 8 Minimum. n=1.05 was found using Ln(10)(nKT/q) = subthreshold slope (i.e n = 62.4 mV/(2.3kT/q)).
The middle plot shows g
m vs V
GS. The linear slope of g
m is μ
nC
OX(W/L) and was used to determine μ
nC
OX at 156.6 μA/V
2 for W/L = 211.5/20. The projection of the slope to the x-axis gives V
t at 230 mV. The voltage V
GS at the intersection of where the projection of the g
m slope to a line that is parallel to where g
m flattens is used to determine V
eff. At that point V
eff = 1/(2θ). Solving for θ, θ = 2.9/V [4].
The bottom plot shows I
D in the active region.
![figure 2 7nm finfet model plot 7nm nmos gm vt theta vgs]()
PMOS μpCOX, Vt, subthreshold slope, n, and θ
The schematic used for finding μ
pC
OX, V
t, subthreshold slope, n, and θ is shown below.
![figure 3 7nm finfet model schmatic 7nm pmos gm vgs]()
The top plot in the figure below shows g
m/I
D vs V
GS. The minimum was found to be 60.8 mV/decade using
Measure -> 8 Minimum. n=1.02 was found using Ln(10)(nKT/q) = subthreshold slope (i.e n = 60.8 mV/(2.3kT/q)).
The middle plot shows g
m vs V
GS. The linear slope of g
m is μ
pC
OX(W/L) and was used to determine μ
pC
OX at 110.2 μA/V
2 for W/L = 211.5/20. The projection of the slope to the x-axis gives V
t at 210 mV. The voltage V
GS at the intersection of where the projection of the g
m slope to a line that is parallel to where g
m flattens is used to determine V
eff. At that point V
eff = 1/(2θ). Solving for θ, θ = 1.9/V
The bottom plot shows I
D in the active region.
![figure 4 7nm finfet model plot 7nm pmos gm vt theta vgs]()
NMOS λL
The schematic below was used determine λ.
![figure 5 7nm finfet model schemactic 7nm nmos gds vds lambda l]()
The top plot in the figure below shows g
ds/I
D and the bottom shows I
D vs V
DS for V
GS = 300 mV, 500 mV, 800 mV, and 1 V. To generate a useful plot g
ds/I
D, the
Choose Analysis V
DS sweep can't start a 0 V so that g
ds/I
D doesn't become infinite.
![figure 6 7nm finfet model plot 7nm nmos gds id vs vds lambda l]()
g
ds/I
D and I
D vs V
DS were plotted again but only for V
GS = 500 mV to determine λ and λL. Recall that
$$g_{ds}=\frac{\partial I_d}{\partial V_{DS}}=\lambda (\frac{\mu_n C_{OX}}{2})(\frac{W}{L})V_{eff}^2=\lambda I_{D-sat}=\lambda I_d$$
In short, g
ds/I
D = λ and was sampled in the middle of the active range at V
DS = 654 mV as the start of the active region occurs at V
DS-sat = V
eff = 500 mV - 230 mV = 270 mV. λ = 0.1795/V and λL = 0.00359 μm/V [5].
![figure 7 7nm finfet model plot 7nm nmos gds id vs vds lambda l gs 500mv]()
PMOS λL
![figure 8 7nm finfet model schemactic 7nm pmos gds vds lambda l]()
g
ds/I
D and I
D vs V
DS were plotted again but only for V
GS = 500 mV to determine λ and λL. Recall that $$g_{ds}=\frac{\partial I_d}{\partial V_{DS}}=\lambda (\frac{\mu_p C_{OX}}{2})(\frac{W}{L})V_{eff}^2=\lambda I_{D-sat}=\lambda I_d$$
In short, g
ds/I
D = λ and was sampled in the middle of the active range at V
DS = 693 mV as the start of the active region occurs at V
DS-sat = V
eff = 500 mV - 210 mV = 290 mV. λ = 0.3563/V and λL = 0.00713 μm/V.
![figure 9 7nm finfet model plot 7nm pmos gds id vs vds lambda l gs 500mv]()
NMOS COV/W and Cdb/W
The schematic below was used to determine C
OV/W.
![figure 10 7nm finfet model schematic 7nm nmos cov]()
The plot below shows the gate current I
g vs frequency. The gate current is function of frequency and I
g = 2πfV
gC
OV. Solving for C
OV, C
OV = I
g/(2πfV
g) with f = 10 MHz and V
g = 2 V because the AC source produces a 1 V peak output. I
g = 4.25 nA at 10 MHz so C
OV/W = 0.16 fF/μm for the 211.5 nm gate width [6].
![figure 11 7nm finfet model plot 7nm nmos cov w]()
The schematic below was used to find C
db/W.
![figure 12 7nm finfet model schemactic 7nm nmos cdb w]()
The plot below shows the body current I
b vs frequency. The body current is function of frequency and I
b = 2πfV
bC
b where C
b = C
db+C
sb. Solving for C
b, C
b = I
b/(2πfV
bs) with f = 10 MHz and V
b = 2 V because the AC source produces a 1 V peak output. I
b = 2.95 fA at 10 MHz so C
b/W = 1.1*10
-7 fF/μm for the 211.5 nm gate width. For C
db = C
sb = C
b/2 = 5.5*10
-8 fF/μm [7].
![figure 13 7nm finfet model plot ac 7nm nmos cdb csb w]()
PMOS COV/W and Cdb/W
The schematic below was used to determine C
OV/W.
![figure 14 7nm finfet model schematic 7nm pmos cov]()
The
Simulate -> Choose Analysis setup for the AC analysis shown below.
![figure 15 7nm finfet model choose analyis 7nm pmos config]()
I
g = 4.25 nA at 10 MHz so C
OV/W = 0.16 fF/μm for the 211.5 nm gate width.
![figure 16 7nm finfet model plot 7nm pmos cdb csb w]()
The schematic below was used to find C
db/W.
![figure 17 7nm finfet model schematic 7nm pmos cdb]()
I
b = 3.099 fA at 10 MHz so C
b/W = 1.13*10
-7 fF/μm for the 211.5 nm gate width. For C
db = C
sb = C
b/2 = 5.57*10
-8 fF/μm.
![figure 18 7nm finfet model plot ac 7nm pmos cdb csb w]()
NMOS ID and Rin vs VGS
The gate and drain for the nmos_lvt and nmos_rvt transistors are connected together to simulated the IV curve of a diode connected transistor that is used as part of a current mirror. The MOSFET connected in this way isn't really a diode but its IV curve behaves like an diode IV curve because the V
GS = V
DS and the transistor operates in the active region. The I
D vs V
DS is
$$I_d=\frac{1}{2}\mu_n C_{OX} \frac{W}{L} (V_{GS}-V_t)^2 = \frac{1}{2} \mu_n C_{OX} \frac{W}{L}(V_{DS}-V_t)^2$$
Solving for V
DS gives [8]
$$V_{DS}=\sqrt{\frac{I_d}{\frac{1}{2} \mu_n C_{OX} \frac{W}{L}}}+V_t$$
Using the model parameters in the table above at 100 μA, V
DS = 577.5 mV for the nmos_lvt transistor, and V
DS = 609.2 mV at 100 μA in the simulation. This represents a 32 mV error between the simulation and the calculation.
| Transistor | VDS | Simulation | Units
|
| nmos_lvt | 577.5 | 609.2 | mV |
The schematic below shows the diode connected nmos_lvt and nmos_rvt FinFETs in single and stacted configurations.
![figure 19 7nm finfet model schematic 7nm nmos common g d single_stacked]()
The bottom plot in the figure below shows the I
D vs V
GS for the nmos_lvt and nmos-rvt transistors. The top two traces are the single gate-drain connected transistors, and the bottom two traces are the stacked transistors. The REF cursor is set at the I
D = 100 μA for the single nmos_lvt transistor. The B cursor is set at the I
D = 100 μA for the stacked nmos_lvt transistors. The middle plot shows g
m for the transistors. The single transistors have the higher g
m compared with the stacked transistors. The top plot shows the output resistance. The cursors was placed at 100 μA for the nmos_lvt device and the V
DS is 609.2 mV and an resistance of 6.093 kOhms. The stacked nmos_lvt transistors have a V
DS of 814.0 mV and a resistance of 8.138 kOhms.
![figure 20 7nm finfet model plot 7nm nmos lvt rvt common gd id gm rout_single_stacked]()
PMOS ID and Rin vs VGS
The simulation was repeated for the pmos_lvt and pmos_rvt transistors for both single and stacked configurations. The schematic for this simulation is shown below.
![figure 21 7nm finfet model schematic 7nm nmos lvt rvt single stacked common g d]()
The bottom plot in the figure below shows the I
D vs V
GS for the nmos_lvt and nmos-rvt transistors. The top two traces are the single gate-drain connected transistors, and the bottom two traces are the stacked transistors. The REF cursor is set at the I
D = 100 μA for the single nmos_lvt transistor. The B cursor is set at the I
D = 100 μA for the stacked nmos_lvt transistors. The middle plot shows g
m for the transistors. The single transistors have the higher g
m compared with the stacked transistors. The top plot shows the output resistance. The cursors was placed at 100 μA for the nmos_lvt device and the V
DS is 650.5 mV and an resistance of 6.505 kOhms. The stacked nmos_lvt transistors have a V
DS of 902.5 mV and a resistance of 9.021 kOhms.
![figure 22 7nm finfet model plot 7nm nmos lvt rvt single stacked common gd id gm rout]()
The figures below shows how the
Choose Analysis and
Define Curve configuration panels were setup to produce the plot above.
![figure 23 7nm finfet model choose_analysis_7nm pmos common gs_config]()
![figure 24 7nm finfet model define_curve_7nm pmos lvt config]()
![figure 25 7nm finfet model define_curve_7nm pmos rvt config]()
Index
References
[1]:
ASU Predictive Technology Model (PTM)
[2]:
BSIM Group Berkeley BSIM-CMG Model
[3]: T. C. Carusone, D. A. Johns, and K. W. Martin, Analog Integrated Circuit Design, 2nd Edition. Wiley, 2011.
[4]: T. C. Carusone, D. A. Johns, and K. W. Martin, Analog Integrated Circuit Design, 2nd Edition. Wiley, 2011. see pg. 28, 30, 47.
[5]: T. C. Carusone, D. A. Johns, and K. W. Martin, Analog Integrated Circuit Design, 2nd Edition. Wiley, 2011. see pg. 29.
[6]:
45nm CMOS process – Learning Microelectronics
[7]:
45nm CMOS process – Learning Microelectronics.
[8]: T. C. Carusone, D. A. Johns, and K. W. Martin, Analog Integrated Circuit Design, 2nd Edition. Wiley, 2011. see pg. 129.
