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MOS Transistor Theory

1.2.3 MOSFET Operation Modes

Figure 1.9: Basic MOSFET channel formation

n-channel MOSFET:

Source electrode (n+ region) is at the lowest potential

Source potential is the reference potential for all voltages:VDS=VD VS, VGS=VG VS, VSB = (VS VB) (1.61)

VSB >0 because VB must be more negative than VSto make sure that the pn-junctionfrom bulk to source is reverse biased.

VLSI Design

Course 1-13 Darmstadt University of Technology

Institute of Microelectronic Systems

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MOSFET operation Modes: Cutoff, Nonsaturation, Saturation

Cutoff: VGS< VT

Figure 1.10: MOSFET in cutoff mode

Nonsaturation: VGS VT and VDS (VGS VT)Saturation: VGS VT and VDS (VGS VT)

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Figure 1.11: MOSFET in nonsaturation mode

Figure 1.12: MOSFET in saturation mode

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1.2.4 MOSFET current characteristic

The Gradual Channel Approximation

analysis with the gradual channel approximation=reduction of the three-dimensionalproblem to a one-dimensional current flow problem

approximation describes very well large devices analysis first done for VS = 0 assumption for derivation of GCA equations: depletion charge is supported entirely by

the vertical electric fieldEx(y); (assume VT0(QB0) indep. ofV(y))

Figure 1.13: MOSFET geometry used in GCA (MOSFET in linear/nonsaturated region)

The channel electric field Ey(y) is established by the drain source voltageVDS is

Ey(y) = dV(y)dy

(1.62)

with V(y= 0) =VS= 0, V(y = L) =VDS.The depletion depth has its maximum at the drain electrode because V(y) has a maximum aty= L:

Xdm(y)

2SiqNa

[2|F|+ V(y)] (1.63)

The inversion charge density as a function of the position y is given by

QI(y = 0) =

Cox[VGS

VT] (1.64)

QI(y) = Cox[VGS VT V(y)] (1.65)The resistance for a differential channel increment dy is

dR= dynW QI(y)

= dy

A [] (1.66)

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Figure 1.14: Geometry for GCA current analysis

with A : channel cross sectionn : electron surface mobilityW : channel width : conductivity

Rearranging

dV = IDdR= IDdynW QI(y)

(1.67)

IDL0

dy = nWVDS0

QI(V)dV (1.68)

and Integration yields

ID = nCoxW

L

VDS0

(VGS VT V)dV (1.69)

= kW

L

(VGS VT)VDS 1

2V2DS

(1.70)

with the process transconductance parameter k = nCox AV2 and the device transconduc-tance parameter = kWL[A/V

2].

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MOSFET Current Equations

The resulting equation from the GCA for the nonsaturated current in a conveniant form is

ID =

2[2(VGS VT)VDS V2DS] (1.71)

At the onset of saturation the current ID reaches a peak value and remains constant in the

Figure 1.15: Nonsaturated MOS current

saturation region:ID

VDS= 0 = (VGS VT VDS) (1.72)

Evaluation of the derivation yields

VDS,SAT = VGS VT (1.73)

ID,SAT = ID(VDS=VDS,SAT) = 2

(VGS VT)2 (1.74)

parabolic border between saturation and nonsaturation.

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Figure 1.16: Basic MOSFET characteristics

Figure 1.17: Start of Saturation in a MOSFET

Channel length modulation in saturation

The effective channel lenght in saturation is L = L L.From GCA:

QI(L) = 0 (1.75) V(L) VDS,SAT (1.76)

(VDS,SAT =VGS VT0no inversion charge is induced).L may be approximated as a depletion region for a one-sided pn junction with a voltageVDS VDS,SATacross it.

L

2SiqNa

[VDS VDS,SAT] (1.77)

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Figure 1.18: Channel length modulation

The saturated current is modified to

ID k

2

W

L(VGS VT)2 (1.78)

ID01 LL

(1.79)

with

ID0 =

2(VGS VT0)2.

Using the empirical relation

1L

L 1 VDS (1.80)with [V1] the channel length modulation factor and assuming that VDS 1 the currentcan be represented by

ID = ID01 VDS

= ID01 VDS

1 + VDS1 + VDS

= ID0(1 + VDS)

1 (VDS)2

1

(1.81)

ID ID0(1 + VDS) = 2

(VGS VT0)2(1 + VDS) (1.82)

has typical values from 0.1 to 0.01V1 and represents the influence of VDS on ID insaturation. is important in small geometrie devices. In the following exercises we willneglect.

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Figure 1.19: MOSFET characteristics with channel length modulation

1.2.5 Biased MOSFET Current Equations

Figure 1.20: General MOSFET bias

VT = VT0+ (

2|F|+ VSB

2|F|) (1.83)ID 0 (VGS< VT) (1.84)ID =

2

2(VGS VT)VDS V2DS

(VGS> VT, VDS< VDS,sat) (1.85)

VDS,sat = VGS VT (1.86)ID =

2(VGS VT)2(1 + VDS) (VGS> VT, VDS VDS,sat) (1.87)

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Figure 1.21: Body bias effects

1.2.6 Measurement of device parameters

Figure 1.22: Device parameter measurement (a)

Get

(1) VT0from intercept

(2) k = k W

L from slope:

k =

2ID

VGS VT(3) =

VT(VSB) VT02|F|+ VSB

2|F|

and from

(4) ID2

ID1=

1 + VD21 + VD1

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Figure 1.23: Device parameter measurement (b)

1.2.7 The Complete MOSFET GCA Analysis

includes additional depletion charge created by the channel voltage V(y), which is reversebias across the n+p junction at the channel-substrate boundary

assumeVS = 0 = VB calculation for nonsaturated MOSFET

VT0(V) =VFB+ 2|F|+q DICox

+ 1

Cox

2qSiNa(2|F|+ V) (1.88)

The basic GCA integral

ID=

VDS0

[VGS VT0(V) V] dV (1.89)

is modified to (now: VT0 not constant and dependent ofQB0)

ID =

VDS

0

VGS VFB 2|F| qDI

Cox V 1

Cox

2qSiNa(2|F|+ V)

dV (1.90)

which gives for the nonsaturated drain current

ID =

VGS VFB 2|F| qDI

Cox

VDS1

2V2DS

23Cox

2qSiNa[(2|F|+ VDS)3/2 (2|F|)3/2]

. (1.91)

Introduction of a reduction factor M

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Figure 1.24: Comparision of circuit equations with the complete GCA model

Figure 1.25: Comparision of modified circuit equations with the complete GCA model

1.2.8 Depletion mode nchannel MOSFET

only used in NMOS as load device.

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With the donator implant dose DI, VT is modified to

VT =VFB+ 2|F|+ 1Cox

2qSiNa(2|F|+ VSB) qDI

Cox(1.94)

so thatVT of a depletion MOSFET is negative. The n-type layer resulting from donor doping

Figure 1.26: Depletion-mode MOSFET

is modeled by(Nd Na)> 0. (1.95)

The currentID can be modeled by

ID= n

W

L

VDS0

QC(V)dV (1.96)

with QC(V) the channel charge density

QC(V) = Qn+ QS(V) + Qj(V) (1.97)

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Figure 1.27: Simplified depletion-mode MOSFET model

Qn: total charge density of electrons in then-type layer

QS: MOS surface charge density (VFB gives the voltage necessary to create a charge-neutralflatband state at the surface of the semiconductor)

Qj: amount of depletion charge on then-side of the pn junction n-type layersubstrate

Qn = q(Nd Na)a (1.98)QS(V) = Cox[VGS VFB V] (1.99)Qj(V) =

2qSiN(0+ V) (1.100)

0 kTq

ln(Nd Na)Na

N2i

(built-in voltage) (1.101)

N = (Nd Na)Na(Nd Na) + Na

= NaNd

(Nd Na) (1.102)

Using these charge densities gives

ID = n

W

L

VDS0

[q(Nd Na)a + Cox(VGS VFB V)

2qSiN(0+ V)]dV

=

q(Nd Na)a

CoxVDS+

(VGS VFB)VDS 1

2V2DS

23Cox

2qSiN[(0+ VDS)

3/2 (0)3/2]

. (1.103)

This equation is too complicate for hand-calculations, so usually the D-mode MOSFET isdescribed by

ID =

2[2(VGS VT0)VDS V2DS], (1.104)

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whereVT0< 0. Saturation current:

ID,sat=

2(VGS+ |VT0|)2 (1.105)

Application of D-mode MOSFETs often as Depletion load (saturation region):

Figure 1.28: Depletion-mode MOSFET characteristics

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Figure 1.29: Square root of saturated depletion-mode MOSFET current

1.2.9 pchannel MOSFET

Figure 1.30: p-channel MOSFET

The source electrode is connected to VDD.

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Threshold voltage:

VTP=GS 2Fn 1Cox (QSS+ Qox) 1Cox QBnFn=

kTq ln

Ndni

> 0 Nd: n type substrate doping

QBn=2qSiNd[2Fn+ VBSp]VTp=VTOp p

VBSp+ 2Fn

2Fn

with p=

2qNdSiCox

(1.106)

VTp is negative for enhancement p-channel MOSFET. Current equations are similar to n-channel MOSFET but all the signs are opposite.

1.2.10 Conclusions

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n channel transistor p channel transistorFermipotential

Fp = kT

q lnniNa

< 0 Fn = kT

q ln Ndni

> 0

Threshold Voltagepositive negative

VTn =VT0n+ n

(|2Fp| VBS)|2Fp |

VTp =VT0p p

(VBSp+ 2Fn)

2Fn

n =

2qNaSi/Cox p =

2qNdSi/Cox

Current Equations

Cutoff :VGS< VTn |VGS| < |VTp|ID = 0 ID = 0

Nonsaturation

VGS> VTn and VDS

(VGS

VTn)

|VGSp

|>

|VTp

| and

|VDSp

| |VGSp

VTp

|ID =

n2

2(VGSn VTn)VDSn V2DSn

IDp =

p2

2(VSGp+ VTp)VSDp V2SDp

Saturation

VGS> VTn and VDS (VGS VTn) |VGSp | > |VTp| and |VDSp| |VGSp VTp|

ID = n

2(VGS VTn)2 ID = p2(VSGp+ VTp)2

1.2.11 Modelling the MOS Transistor for Circuit simulation

MOSFET SPICE Parameters

SPICE=(Simulation Program with IC Emphasis)

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Symbol Name Parameter Units Default Example

LEVEL Model Index 1VT0 VTO Zero-bias threshold voltage V 0.0 1.0k KP Transconductance parameter A/V2 2.0E-5 3.1E-5

GAMMA Bulk threshold parameter V1/2 0.0 0.372|F| PHI surface potential V 0.6 0.65 LAMBDA Channel-length modulation 1/V 0.0 0.02rd RD Drain ohmic resistance 0.0 1.0rs RS Source ohmic resistance 0.0 1.0Cbd CBD Zero-bias B-D junction capacitance F 0.0 2.0E-14Cbs CBS Zero-bias B-S junction capacitance F 0.0 2.0E-14Is IS Bulk junction saturation current A 1.0E-14 1.0E-150 PB Bulk junction potential V 0.8 0.87

CGSO Gate-source overlap capacitanceper meter channel width F/m 0.0 4.0E-11

CGDO Gate-drain overlap capacitanceper meter channel width F/m 0.0 4.0E-11

CGBO Gate-bulk overlap capacitanceper meter channel length F/m 0.0 2.0E-10

RSH Drain and source diffusionsheet resistance / 0.0 10.0

Cj0

CJ Zero-bias bulk junction bottom capacitanceper square meter of junction area F/m2 0.0 2.0E-4m MJ Bulk junction bottom grading coefficient 0.0 0.5

CJSW Zero-bias bulk junction sidewall capacitanceper meter of junction perimeter F/m 0.0 1.0E-9

m MJSW Bulk junction sidewall grading co efficient 0.33JS Bulk junction saturation current

per square meter of junction area A/m2 1.0E-8tox TOX Oxide thickness m 1.0E-7 1.0E-7NA or ND NSUB Substrate doping 1/cm

3 0.0 4.0E15QSS/q NSS Surface state density 1/cm

2 0.0 1.0E10NFS Fast surface state density 1/cm2 0.0 1.0E10TPG Type of gate material 1.0

+1 opposite to substrate-1 same as substrate

0 Al gateXj XJ Metallurgical junction depth m 0.0 1.0E-6LD LD Lateral diffusion m 0.0 0.8E-6 UO Surface mobility cm2/Vs 600 700

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