i
µ n ZC
V D ( sat .) ≈ V G − V T 1 2 I D ( sat .) ≈ k N (V G − V T ) 2 ∂ I D ( sat .) g m ( sat .) = ≈ k N (V G − V T ) ∂VG
1 2 I D = k N (VG − VT )V D − V D ≈ k N (VG − VT )V D 2 ID ≈ 1 k N (VG − VT )2 ; 2 ID =
Without the consideration of the direction of the current flow
(V G − V T )V D L 1 2 = k N (V G − V T )V D − V D 2 ID = kN =
µ n ZC i
1 2 − VD 2
gm
∂I D = ∂VG
Fig.6-28
1 2 I D = k N [(VG − VT )VD − VD ] 2
gm depending on VD
∂I D gm = ≈ kNVD ∂VG VD (sat.) ≈ VG −VT 1 2 I D (sat.) ≈ kN (VG −VT ) 2 ∂I D (sat.) gm (sat.) = ≈ kN (VG −VT ) ∂VG
φ s ( x ) = V x + 2φ F
Qd (x) Qn (x) = −Ci VG −VFB +Vx + 2φF − Ci
2ε sφs 12 W =[ ] , qN a Qd ( x) = − qN aWm = − 2qε s N a (2φF + Vx ) Qd = − 2qε s N a (2φF )
6.5 The MOS Field Effect Transistor
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µ n ZC
i
L When pinch –off, Qn(x=L)=0 Qn ( x) = −Ci (VG − VT − Vx )
( x = L, Vx = VD , Qn = 0) VD ( sat.) ≈ VG − VT 1 1 2 2 I D ( sat.) ≈ k N (VG − VT ) = k NVD ( sat.) 2 2
ξ p ξ sat ξ f ξ sat
V D − V D ( sat .) ξ max = ∆L ห้องสมุดไป่ตู้he Maximum Electric field near the drain end ∆ L ≈ 3 dx j
Pinch off region
6.5.4 Short channel MOSFET I-V characteristics For short channel devices, very high longitudinal electric fields in the pinchedoff region cause the carrier velocity to saturate. ID does not increaseεcquadratically εx >
µn ZCi
6.5.2 Transfer Characteristics(转 转 移特性, 移特性 ID~VG)
1 2 ID = V (V G − V T ) D − 2 V D L 1 2 = k N (V G − V T ) D − V VD 2 ∂I D gm = ≈ k NVD ∂VG
⌧⌧⌧⌧ ⌧⌧⌧⌧
The mobility of A higher gate bias draws than in bulk channel is lower the carriers closer to the •scattered by surface semiconductor. interface, resulting in
6.5.3 Mobility model
Additional scattering mechanisms for carriers in the channel: •Coulombic interaction with carriers in the fixed charge roughness
⌧⌧⌧⌧
6.5.1 Output characteristics(输出特性 D~VD) 输出特性,I 输出特性
Some assumptions s εy
D
εx
•The voltage drop between the source(or drain) electrode and the end of channel near the source(or drain) is neglected. •The mobility in the inversion layer is assumed to be constant.
dx 1 dx dR( x) = ρ ( x) = Zδ n ( x) σ ( x) Zδ n ( x)
σ ( x) = n( x)qµn
1 dx dx dR( x) = = n( x)qµn Zδ n ( x) Qn ( x)Z µn Qn ( x) = qn( x)δ n ( x)
δn(x)
dVx dVx ID = − = −Qn ( x) Z µ n dR( x) dx I D dx = − Z µ nQn ( x)dVx
ID =
µ n ZCi
L
1 2 (VG − VT )VD − 2 VD
1 2 = k N (VG − VT )VD − VD 2 kN =
µ n ZCi
L
∂I D g= = k N (VG − VT ) (VD << VG − VT ) ∂VD VD ( sat.) ≈ VG − VT 1 1 2 2 I D ( sat.) ≈ k N (VG − VT ) = k NVD ( sat.) 2 2
VG = V FB + Vi + φ s ( x ) Qs Vi = − ; φ s ( x ) = V x + 2φ F Ci VG = V FB Qs − + V x + 2φ F Ci
If we neglect the variation of Qd(x) with bias Vx, Qn(x) can be simplified to
1）flat band voltage ）
VFB
εi
Qi = φms − Ci
−14
(3.9) × (8.85×10 ) −8 2 Ci = = = 6.9 ×10 F / cm −6 d 5 ×10
Qi (5 ×1010 ) × (1.6 ×10−19 ) VFB = φms − = −0.89 − −8 Ci 6.9 ×10 = −0.89 − 0.0116 = −1.01V
Qi Q ; Vi = − s φs will be the same along the Ci Ci surface parallel to channel when the gate voltage is only applied. It VG = VFB + Vi + φs equals 2φF at the onset of strong φ inversion.
Qd ( x) ≈ Qd Qd + 2φ F VT = VFB − Ci
Q s ( x ) = Qd ( x ) + Qn ( x )
Qn ( x) = −Ci (VG − VT − Vx )
∫
L
0
I D dx = − Z µn ∫ Qn ( x)dVx = − Z µ nCi ∫
0
VD
VD
0
(VG − VT − Vx )dVx
θ is called mobility degradation parameter
ID increases sub-linearly with gate bias for high gate voltage
• Effect of longitudinal electric field :
v = µξ v = Vs for for
larger degradation of mobility.
Effect of Transverse electric field
1 1 ξeff = Qd + Qn (electrons / n − channel) 2 εs 1 1 ξeff = Qd + Qn (holes / p − channel) εs 3 µ n ZCi 1 2 ID = (VG − VT )VD − 2 VD L{1 + θ (VG − VT )}
J D with VG-VT, but rather shows a = nqvs I D linear A = nqvs Zδ n = Qn Zvs = nqvs dependence Qn ≈ −Ci (VG − VT ) I D ≈ ZCi (VG − VT )vs
I D ≈ ZCi (VG −VT )vs
With a voltage VD applied, there is a voltage rise Vx from the source to each point x in the channel. So, the potential will become φs(x) dependent of the position along the channel. The potential φs(x) required to achieve strong inversion is written as