Long Channel MOSFET

1. Theory

Definition: A MOSFET is a four-terminal field-effect transistor in which a gate voltage, applied across a thin oxide, induces a conducting inversion channel between source and drain.
Reference device: This page uses an n-channel enhancement-mode MOSFET on a uniformly doped p-type substrate; a PMOS follows with $N_{A} \to N_{D}$ , $μ_{n} \to μ_{p}$ , $ϕ_{F} \to - ϕ_{F}$ and all polarities reversed.
Long-channel assumptions: Channel length $L$ is much larger than the source/drain depletion widths, mobility is field-independent and the gradual-channel approximation holds, so short-channel effects are negligible.
MOS capacitor regimes: As gate voltage rises, the surface goes from accumulation to depletion, then weak inversion, then strong inversion at $ψ_{s} = 2 ϕ_{F}$ .
Threshold voltage: $V_{T H} = V_{F B} + 2 ϕ_{F} - Q_{d m} / C_{o x}$ (with $Q_{d m}$ the depletion charge), set by the work-function difference, oxide charge, substrate doping and oxide thickness.
Body effect: A source-substrate reverse bias widens the depletion region and raises $V_{T H}$ ; the substrate acts as a weak second gate.
Unipolar, voltage-controlled: Only electrons carry the channel current, and the insulated gate draws no DC current.

2. Working Principle

Cut-off and subthreshold: Below $V_{T H}$ a small diffusion current flows that rises exponentially with $V_{GS}$ , with a swing of at least about 60 mV/dec at 300 K.
Linear (triode) region: For small $V_{D S}$ the channel acts as a gate-controlled resistor, $R_{o n} = 1/ [μ_{n} C_{o x} (Z / L) (V_{GS} - V_{T H})]$ .
Pinch-off and saturation: When $V_{D S}$ reaches $V_{GS} - V_{T H}$ the inversion charge vanishes at the drain end; extra drain voltage drops across the pinched-off region and current saturates.
Square-law saturation current: $I_{D s a t} = \frac{1}{2} μ_{n} C_{o x} (Z / L) (V_{GS} - V_{T H})^{2}$ , with a bulk-charge factor $m$ correcting for depletion charge along the channel.
Channel-length modulation: The pinched-off region grows with $V_{D S}$ , shortening the channel and giving a finite output resistance (set by $λ$ ).
Speed: $f_{T}$ scales as $1/ L^{2}$ , so long channels are slow but have high intrinsic gain.

3. Applications

CMOS logic and memory: Complementary NMOS/PMOS pairs are the basis of essentially all digital ICs.
Precision analog circuits: Long channels give high output resistance, good matching and lower $1/ f$ noise for current mirrors, bias networks and op-amp input pairs.
Analog switches and sample-and-hold circuits: Near-zero gate current and bidirectional conduction make an ideal voltage-controlled switch.
Mature-node mixed-signal ICs: Power-management, sensor-interface and automotive ICs built at 180 nm and above, where long-channel behaviour applies.

Name	Formula	Related Formulae	Variables/Constants
Gate oxide capacitance	$C_{o x} = \frac{ε _{o x}}{t _{o x}}$	$ε_{o x} = 3.9 ε_{0} (SiO_{2})$	$C_{o x} : g a t e o x i d e c a p a c i t an ce p er u ni t a r e a$ $ε_{o x} : o x i d e p er mi tt i v i t y$ $t_{o x} : o x i d e t hi c k n ess$
Fermi potential	$ϕ_{F} = \frac{k _{B} T}{q} ln (\frac{N _{A}}{n _{i}})$	$p_{0} = N_{A} = n_{i} e^{q ϕ_{F} / k_{B} T}$	$ϕ_{F} : p o t e n t ia l d i f f er e n ce b e tw ee n t h e in t r in s i c l e v e l an d t h e F er mi l e v e l in t h e n e u t r a l p - t y p e s u b s t r a t e$ $k_{B} : B o l t z mann co n s t an t$ $T : ab so l u t e t e m p er a t u r e$ $n_{i} : in t r in s i c c a r r i er co n ce n t r a t i o n$
Surface potential regimes	$⎩ ⎨ ⎧ ψ_{s} < 0 0 < ψ_{s} < ϕ_{F} ϕ_{F} < ψ_{s} < 2 ϕ_{F} ψ_{s} > 2 ϕ_{F} accumulation depletion weak inversion strong inversion$	$ψ_{s} = 2 ϕ_{F} + V_{S B} at the onset of strong inversion$	$ψ_{s} : ban d b e n d in g a t t h e o x i d e - se mi co n d u c t or in t er f a ce r e l a t i v e t o t h e n e u t r a l s u b s t r a t e$ $V_{S B} : so u r ce - s u b s t r a t e v o l t a g e$
Maximum depletion width	$W_{d m} = \frac{2 ε _{s} ( 2 ϕ _{F} )}{q N _{A}} = \frac{4 ε _{s} ϕ _{F}}{q N _{A}}$	$Q_{d m} = - q N_{A} W_{d m} = - 4 q ε_{s} N_{A} ϕ_{F}$	$W_{d m} : ma x im u m s u r f a ce d e pl e t i o n w i d t h a t t h e o n se t o f s t r o n g in v er s i o n$ $Q_{d m} : d e pl e t i o n c ha r g e p er u ni t a r e a a t t ha t p o in t$ $ε_{s} : se mi co n d u c t or p er mi tt i v i t y$
Flat-band voltage	$V_{F B} = ϕ_{m s} - \frac{Q _{f}}{C _{o x}}$	$ϕ_{m s} = ϕ_{m} - (χ + \frac{E _{g}}{2 q} + ϕ_{F})$	$V_{F B} : g a t e v o l t a g e a t w hi c h t h e se mi co n d u c t or ban d s a r e f l a t$ $ϕ_{m s} : m e t a l - se mi co n d u c t or w or k f u n c t i o n d i f f er e n ce$ $Q_{f} : e f f ec t i v e f i x e d o x i d e c ha r g e p er u ni t a r e a (o x i d e an d in t er f a ce c ha r g es l u m p e d a t t h e in t er f a ce)$ $ϕ_{m} : g a t e w or k f u n c t i o n$ $χ : se mi co n d u c t or e l ec t r o n a f f ini t y$ $E_{g} : se mi co n d u c t or ban d g a p$
Threshold voltage	$V_{T H} = V_{F B} + 2 ϕ_{F} + \frac{4 q ε _{s} N _{A} ϕ _{F}}{C _{o x}}$	$V_{T H} = V_{F B} + 2 ϕ_{F} - \frac{Q _{d m}}{C _{o x}}$	$V_{T H} : g a t e - so u r ce v o l t a g e a t w hi c h s t r o n g in v er s i o n b e g in s (ψ_{s} = 2 ϕ_{F}) w i t h V_{S B} = 0$
Body effect	$V_{T H} = V_{T H 0} + γ (2 ϕ_{F} + V_{S B} - 2 ϕ_{F})$	$γ = \frac{2 q ε _{s} N _{A}}{C _{o x}}$	$V_{T H 0} : t h r es h o l d v o l t a g e a t V_{S B} = 0$ $γ : b o d y e f f ec t coe f f i c i e n t$ $V_{S B} : so u r ce - s u b s t r a t e r e v er se bia s (p os i t i v e f or N M O S)$
Inversion charge (charge-sheet)	$Q_{n} (y) = - [C_{o x} (V_{GS} - V_{F B} - 2 ϕ_{F} - V (y)) - 2 q ε_{s} N_{A} (2 ϕ_{F} + V (y))]$	$Q_{n} (y) \approx - C_{o x} (V_{GS} - V_{T H} - V (y))$	$Q_{n} (y) : m o bi l e in v er s i o n c ha r g e p er u ni t a r e a a t p os i t i o n y a l o n g t h e c hann e l$ $V (y) : c hann e l p o t e n t ia l r e l a t i v e t o t h e so u r ce (0 a t t h e so u r ce, V_{D S} a t t h e d r ain)$ $(the approximate form is the simple square-law model, which neglects the change in depletion charge along the channel)$
Linear-region drain current (square-law)	$I_{D} = μ_{n} C_{o x} \frac{Z}{L} [(V_{GS} - V_{T H}) V_{D S} - \frac{V _{D S}^{2}}{2}]$	$I_{D} = μ_{n} C_{o x} \frac{Z}{L} {(V_{GS} - V_{F B} - 2 ϕ_{F} - \frac{V _{D S}}{2}) V_{D S} - \frac{2}{3} γ [(V_{D S} + 2 ϕ_{F})^{3/2} - (2 ϕ_{F})^{3/2}]} (bulk-charge, V_{S B} = 0)$	$I_{D} : d r ain c u r r e n t$ $V_{D S}, V_{GS} : d r ain - so u r ce an d g a t e - so u r ce v o l t a g es$ $μ_{n} : e l ec t r o n c hann e l m o bi l i t y$ $Z, L : c hann e l w i d t h an d l e n g t h$
Channel conductance and on-resistance	$g_{d} = \frac{\partial I _{D}}{\partial V _{D S}}_{V_{D S} \to 0} = μ_{n} C_{o x} \frac{Z}{L} (V_{GS} - V_{T H})$	$R_{o n} = \frac{1}{g _{d}}$	$g_{d} : d r ain (c hann e l) co n d u c t an ce a t s ma l l V_{D S}$ $R_{o n} : o n - r es i s t an ce o f t h e d e v i ce in t h e l in e a r r e g i o n$
Saturation drain voltage	$V_{D s a t} = V_{GS} - V_{T H}$	$V_{D s a t} = V_{GS} - V_{F B} - 2 ϕ_{F} + \frac{γ ^{2}}{2} - γ V_{GS} - V_{F B} + \frac{γ ^{2}}{4} (bulk-charge, V_{S B} = 0)$	$V_{D s a t} : d r ain v o l t a g e a t w hi c h t h e in v er s i o n c ha r g e a t t h e d r ain e n d g oes t o z er o (c hann e l p in c h - o f f)$ $(the square-law value is the same as the overdrive voltage V_{o v} = V_{GS} - V_{T H})$
Saturation drain current (square-law)	$I_{D s a t} = \frac{μ _{n} C _{o x}}{2} \frac{Z}{L} (V_{GS} - V_{T H})^{2}$	$I_{D s a t} \approx \frac{μ _{n} C _{o x}}{2 m} \frac{Z}{L} (V_{GS} - V_{T H})^{2}, m = 1 + \frac{C _{d}}{C _{o x}} = 1 + \frac{γ}{2 2 ϕ _{F} + V _{S B}}$	$I_{D s a t} : d r ain c u r r e n t in t h e s a t u r a t i o n r e g i o n$ $m : b o d y f a c t or (b u l k - c ha r g e cor r ec t i o n)$ $C_{d} = ε_{s} / W_{d m} : d e pl e t i o n c a p a c i t an ce p er u ni t a r e a$ $(the m form approximates the bulk charge result; the square-law form is recovered for m = 1)$
Transconductance	$g_{m} = μ_{n} C_{o x} \frac{Z}{L} (V_{GS} - V_{T H}) (sat.), g_{m} = μ_{n} C_{o x} \frac{Z}{L} V_{D S} (lin.)$	$g_{m} = 2 μ_{n} C_{o x} \frac{Z}{L} I_{D s a t} = \frac{2 I _{D s a t}}{V _{GS} - V _{T H}}$	$g_{m} : t r an sco n d u c t an ce \partial I_{D} / \partial V_{GS} a t co n s t an t V_{D S}$
Body transconductance	$g_{mb} = \frac{\partial I _{D}}{\partial V _{B S}} = η g_{m}$	$η = \frac{γ}{2 2 ϕ _{F} + V _{S B}} = \frac{C _{d}}{C _{o x}}$	$g_{mb} : c han g e in d r ain c u r r e n t p er u ni t c han g e in s u b s t r a t e v o l t a g e$ $η : b o d y e f f ec t r a t i o$ $V_{B S} : s u b s t r a t e - so u r ce v o l t a g e$
Output conductance (channel-length modulation)	$g_{d s} = λ I_{D s a t}$	$I_{D} = I_{D s a t} (1 + λ V_{D S}), r_{o} = \frac{1}{g _{d s}}$	$g_{d s} : s ma l l - s i g na l o u tp u t co n d u c t an ce in s a t u r a t i o n$ $λ : c hann e l - l e n g t h m o d u l a t i o n p a r am e t er, d escr ibin g t h e s h or t e nin g o f t h e in v er s i o n c hann e l a s V_{D S} e x cee d s V_{D s a t}$ $(same definition as in the JFET page)$
Channel-length modulation	$Δ L = \frac{2 ε _{s} ( V _{D S} - V _{D s a t} )}{q N _{A}}$	$I_{D} = I_{D s a t} \frac{L}{L - Δ L} \approx I_{D s a t} (1 + \frac{Δ L}{L})$	$Δ L : l e n g t h o f t h e p in c h e d - o f f (d e pl e t e d) r e g i o n b e tw ee n t h e e n d o f t h e in v er s i o n c hann e l an d t h e d r ain$ $(first order abrupt junction estimate; assumes the full excess drain voltage drops across the pinched-off region)$
Subthreshold drain current	$I_{D} \approx μ_{n} C_{d} \frac{Z}{L} (\frac{k _{B} T}{q})^{2} exp (\frac{q ( V _{GS} - V _{T H} )}{m k _{B} T}) [1 - exp (- \frac{q V _{D S}}{k _{B} T})]$		$(weak inversion, V_{GS} < V_{T H}; diffusion current dominates; for V_{D S} > 3 k_{B} T / q the current is independent of V_{D S})$ $m : b o d y f a c t or d e f in e d in t h e s a t u r a t i o n c u r r e n t r o w$ $k_{B} : B o l t z mann co n s t an t$
Subthreshold swing	$S = \frac{d V _{GS}}{d ( l o g _{10} I _{D} )} = ln (10) \frac{k _{B} T}{q} (1 + \frac{C _{d}}{C _{o x}})$	$S_{min} = ln (10) \frac{k _{B} T}{q} \approx 60 mV/dec at 300 K$	$S : g a t e v o l t a g e c han g e n ee d e d t o c han g e t h e s u b t h r es h o l d c u r r e n t b y o n e d ec a d e$ $S_{min} : i d e a l l imi t, r e a c h e d f or C_{d} / C_{o x} \to 0$ $k_{B} : B o l t z mann co n s t an t$
Gate capacitance (strong inversion)	$C_{g s} = \frac{2}{3} C_{o x} Z L, C_{g d} \approx 0 (saturation)$	$C_{G} = C_{o x} Z L (linear region, V_{D S} \to 0)$	$C_{g s}, C_{g d} : g a t e - so u r ce an d g a t e - d r ain c a p a c i t an ces (in t r in s i c)$ $C_{G} : t o t a l in t r in s i c g a t e c a p a c i t an ce$ $(Meyer model; neglects overlap and fringing capacitances)$
Cutoff frequency	$f_{T} = \frac{g _{m}}{2 π C _{g s}} = \frac{3 μ _{n} ( V _{GS} - V _{T H} )}{4 π L ^{2}}$	$f_{T} \propto \frac{1}{L ^{2}}$	$f_{T} : f r e q u e n cy a t w hi c h t h e s h or t - c i r c u i t c u r r e n t g ain f a l l s t o u ni t y$ $(neglects overlap capacitance and velocity saturation, which limit short-channel devices)$
Intrinsic voltage gain (saturation)	$A_{0} = g_{m} r_{o} = \frac{g _{m}}{g _{d s}} = \frac{2}{λ ( V _{GS} - V _{T H} )}$		$A_{0} : ma x im u m s ma l l - s i g na l v o l t a g e g ain o f a s in g l e t r an s i s t or s t a g e w i t h i d e a l c u r r e n t so u r ce l o a d$

Important Questions

Ques: Why does strong inversion begin at a surface potential of $2 ϕ_{F}$ ?

In a p-type substrate the hole concentration far from the surface is $N_{A}$ , which sits $ϕ_{F}$ above the intrinsic level in the Fermi-potential sense. When the surface potential reaches $ϕ_{F}$ , the electron concentration at the surface equals the bulk hole concentration (the intrinsic level at the surface crosses the Fermi level). Because carrier concentrations depend exponentially on band bending, the surface electron concentration only becomes comparable to $N_{A}$ , and a substantial mobile electron layer builds up, when the bands bend by a further $ϕ_{F}$ , giving the usual threshold criterion $ψ_{s} = 2 ϕ_{F}$ . Beyond this point nearly all additional gate charge goes into mobile inversion charge, so the depletion width stays at its maximum value $W_{d m}$ . Reference: S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007.

Ques: What does “pinch-off” mean in a MOSFET, and why does the drain current saturate?

The channel potential $V (y)$ rises from $0$ at the source to $V_{D S}$ at the drain, so the voltage between the gate and the channel, and therefore the inversion charge $Q_{n} (y)$ , is smallest at the drain end. When $V_{D S}$ reaches $V_{D s a t}$ , $Q_{n}$ goes to zero at the drain end (pinch-off). Current does not stop, because carriers reaching the pinch-off point are injected into the high-field depleted region next to the drain and swept across it. Beyond $V_{D s a t}$ the extra drain voltage is dropped across this short region, while the voltage across the inverted channel stays at about $V_{D s a t}$ , so the current stays nearly constant. The small remaining slope is the channel-length modulation described by $λ$ . The mechanism is similar to JFET pinch-off (see the JFET page), but in a JFET the channel is closed by the gate depletion regions, whereas in a MOSFET the inversion layer itself vanishes. Reference: S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007.

Ques: What is the body effect, and why does it change the threshold voltage?

A positive source-substrate bias $V_{S B}$ on an NMOS device widens the depletion region under the channel, because the inversion layer is now separated from the substrate by a larger reverse bias. The extra depletion charge must be balanced by additional gate charge, so a larger gate voltage is needed to reach inversion, and $V_{T H}$ increases. The size of the shift is set by $γ$ , which grows with substrate doping and with oxide thickness. The same dependence means that the substrate behaves as a second, weaker gate, which gives the body transconductance $g_{mb} = η g_{m}$ . Reference: S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007.

Ques: When should the square-law model be used versus the bulk-charge model?

The square-law model assumes that the depletion charge is constant along the channel, which makes $I_{D}$ a simple quadratic function of $V_{GS} - V_{T H}$ and needs only $μ_{n} C_{o x} Z / L$ and $V_{T H}$ . The bulk-charge model keeps the increase in depletion charge toward the drain, so it predicts a lower drain current and a larger $V_{D s a t}$ -dependence on doping and oxide thickness. The square-law form is adequate for hand analysis and first-pass circuit design, while the bulk-charge form (or the factor $m$ in the saturation current) should be used when predicting behavior from doping and oxide thickness. Both are long-channel models and break down once velocity saturation and other short-channel effects become important. Reference: S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007.

Ques: Why does the MOSFET still conduct below threshold, and what limits the subthreshold swing?

Below $V_{T H}$ the surface is in weak inversion: there are few mobile electrons, but the source-channel barrier is still lowered by the gate voltage. The resulting current is dominated by diffusion, not drift, and depends exponentially on $V_{GS}$ . Only the fraction $C_{o x} / (C_{o x} + C_{d})$ of a change in gate voltage appears as a change in surface potential, since the rest is dropped across the oxide and depletion capacitances in series. This capacitive divider sets the swing $S = ln (10) (k_{B} T / q) (1 + C_{d} / C_{o x})$ , whose thermal lower limit of about 60 mV/dec at room temperature cannot be beaten by reducing $C_{d} / C_{o x}$ alone. Reference: S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007.

Quartz 5

Explorer

Long Channel MOSFET

1. Theory

2. Working Principle

3. Applications

Important Questions

Ques: Why does strong inversion begin at a surface potential of $2 ϕ_{F}$ ?

Ques: What does “pinch-off” mean in a MOSFET, and why does the drain current saturate?

Ques: What is the body effect, and why does it change the threshold voltage?

Ques: When should the square-law model be used versus the bulk-charge model?

Ques: Why does the MOSFET still conduct below threshold, and what limits the subthreshold swing?

Graph View

Table of Contents

Backlinks

Quartz 5

Explorer

Long Channel MOSFET

1. Theory

2. Working Principle

3. Applications

Important Questions

Ques: Why does strong inversion begin at a surface potential of 2ϕF​?

Ques: What does “pinch-off” mean in a MOSFET, and why does the drain current saturate?

Ques: What is the body effect, and why does it change the threshold voltage?

Ques: When should the square-law model be used versus the bulk-charge model?

Ques: Why does the MOSFET still conduct below threshold, and what limits the subthreshold swing?

Graph View

Table of Contents

Backlinks

Ques: Why does strong inversion begin at a surface potential of $2 ϕ_{F}$ ?