APPENDIX The following information may be used where applicable: Elementary charge Permittivity of free space Room temperature Thermal voltage at 300 K Intrinsic carrier concentration of silicon at 300 K Relative permittivity of silicon Relative permittivity of silicon dioxide q ε0 VT ni εr, (Si) εr, (SiO2) = 1.602 x 10-19 C = 8.854 x 10-14 F cm-1 = 300 K = kT/q = 0.025 V = 25 mV = 1.5 x 1010 cm-3 = 11.7 = 3.9

Summary of Equations I. Semiconductor Physics Equation Remarks

Quantity Law of mass action

p0n0 = ni2 p + ND = n + N A

ni = 1.5 × 1010 cm-3 for Si at
T = 300 K

Charge neutrality Carrier concentrations in p-type semiconductor at thermal equilibrium (cm-3) Carrier concentrations in n-type semiconductor at thermal equilibrium (cm-3) Drift current density (A cm-2)

p p0 = N A n p0 = ni2 NA ni2 ND

T = 300 K NA = net p-type doping concentration T = 300 K ND = net n-type doping concentration

nn 0 = N D pn 0 =

J drift = J p,drift + J n,drif
J p,drift = qpμ p E

σ is conductivity, E is electric field

J n,drift = qnμn E

J drift = q pμ p + nμn E = σE
Conductivity (Ω cm)
-1

(

)

σ = q( pμ p + nμn ) =

1

ρ is resistivity

ρ

Diffusion current density (A cm-2)

J diff = J p,diff + J n,diff dp dx dn J n,diff = qDn dx dp dn J diff = − qD p + qDn dx dx D p Dn kT = = = VT q μ p μn J p, diff = − qD p

Einstein Relation

k = 1.381×10-23 J/K = 8.62×10-5 eV/K q = 1.602×10-19 C VT ≈ 0.025 V at T = 300 K.

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 II. pn Junction Equation Remarks

Quantity

Junction built-in voltage (V)

⎛N N ⎞ V0 = VT ln⎜ A 2 D ⎟ ⎜ n ⎟ i ⎝ ⎠

Width of depletion region (cm or μm)

Wdep = x p + xn = xp xn

2ε s q
ND NA

⎡ 1 1 ⎤ ⎢ N + N ⎥(V0 − V ) D⎦ ⎣ A

V0 ≅ 0.6 - 0.8 V for Si at T = 300 K. ND is the doping concentration of the n-region. NA is the doping concentration of the p-region. V is positive for forward bias and negative for reverse bias ε s = 11.7 × ε 0 for Si

ε 0 = 8.854 × 10−14 F cm-1. xp is the depletion width in the pregion. xn is the depletion width in the nregion. = thermal equilibrium minority carrier concentration x eV / VT

=

Minority carrier concentration at the edge of depletion region (cm-3) Minority carrier concentration distributions in the neutral n- and pregions (cm-3)

p-side: n p ( − x p ) = n p0 eV /VT n-side: pn ( xn ) = pn0 eV /VT n-side : pn ( x ) = pn 0 + [ pn ( x n ) − pn 0 ] e
− ( x − xn ) / L p

L p , Ln are the diffusion lengths of

the minority carriers in the n-side and p-side respectively. In the equations, x > xn on the nside, and x < - xp on the p-side.

p-side : n p ( x ) = n p 0 + [n p ( − x p ) − n p 0 ] e

(x + x p )/ Ln

Current (A)

I = I p + I n = I s eV / nVT − 1

(

)
−1

n is a value between 1 and 2.

I p = Aqni2
I n = Aqni2 Saturation current (A)

Dp Lp N D

(e
(

V / nVT

)

I P is the hole current injected into the neutral n-region.

Dn eV / nVT − 1 Ln N A

)

I n is the electron current injected into the neutral p-region.

⎛ Dp Dn ⎞ ⎟ + I S = Aqni2 ⎜ ⎜ L p N D Ln N A ⎟ ⎝ ⎠

Amount of charge stored in (one side) of the depletion region (C) Depletion or junction capacitance (F)

q j = qx p N A A = qxn N D A ⎛ N AN D ⎞ = q⎜ ⎟ ⎜ N + N ⎟Wdep A D⎠ ⎝ A

Cj =

εs A
Wdep

m is a value the ranges between 1/3 to 1/2.

2

m

⎡ qε ⎛ N A N D ⎞ 1 ⎤ C j = A⎢ s ⎜ ⎜ N + N ⎟ (V − V ) ⎥ ⎟ D⎠ 0 ⎦ ⎣ 2 ⎝ A ⎛ V ⎞ C j = A × C j0 / ⎜1 − ⎟ ⎜ V ⎟ 0⎠ ⎝ C j0 Small signal (or incremental) resistance of the pn junction diode (Ω) ⎡ qε =⎢ s ⎣ 2 m ⎛ N AN D ⎞ 1 ⎤ ⎜ ⎜ N + N ⎟V ⎥ ⎟ D⎠ 0⎦ ⎝ A

m

rd =

nVT ID

I D is the (d.c.) diode current at the operating point, n is a value between 1 and 2.

III. Bipolar Junction Transistor (BJT) Quantity Magnitude of the diffusion current of the minority carriers (electrons) in the base of an npn BJT (A) Equation Remarks Dn is the diffusivity of the electrons in the base, wB is the width of the neutral part of the base, np(0) and np(wB) are the concentrations of the electrons on the base side of the edges of the E-B and C-B space charge regions respectively.

iEn = qADn

n p ( 0) − n p ( wB ) 0 − wB

Collector current of an npn BJT (A) Collector saturation current of an npn BJT (A) Base current of an npn BJT (A)

⎛v ⎞ iC = I S exp⎜ BE ⎟ ⎜V ⎟ ⎝ T ⎠

I S = qA

Dn ni2 wB N AB

Note that the symbol IS is also the symbol for the saturation current of a p-n junction diode, but the expression is different. The expression for β here is for an npn BJT with a long emitter.

iB =

IS

⎛v ⎞ exp⎜ BE ⎟ ⎜V ⎟ β ⎝ T ⎠

where β = Common emitter current gain of a BJT (general definition) Common base current gain of a BJT (general definition) Relationship between α and β Collector current of an npn BJT, with Early effect taken into account (A)

Dn L p N DE D p wB N AB

β= α=

iC iB iC iE

β=

α 1−α
VA is the Early voltage.

⎛ v ⎞⎛ v ⎞ iC = I S exp⎜ BE ⎟⎜1 + CE ⎟ ⎜ V ⎟⎜ VA ⎟ ⎠ ⎝ T ⎠⎝

3

Transconductance in the small signal equivalent circuit of a BJT (Ω-1) Input resistance in the small signal equivalent circuit of a BJT (Ω) Output resistance in the small signal equivalent circuit of a BJT (Ω)

gm =

IC VT

at bias point

IC is the value of the collector current at the d.c. operating (bias) point.

rπ =

β gm IC is the value of the collector current at the d.c. operating (bias) point.

ro ≈

VA IC

IV. Metal Oxide Semiconductor Field Effect Transistor (MOSFET) Quantity Equation Drain current of an nW 1 2 MOSFET operating in iD = μn Cox ⎡( vGS − VTH ) vDS − vDS ⎤ ⎢ ⎥ L 2 ⎣ ⎦ the linear region 1 2 ⎤ ⎡ = 2 K n ⎢( vGS − VTH ) vDS − vDS ⎥ 2 ⎣ ⎦ Remarks Valid for

v DS ≤ vGS − VTH

Drain current of an nMOSFET operating in the saturation region Transconductance parameter of an nMOSFET

iD = iDsat =

1 W μn Cox (vGS − VTH )2 2 L

Valid for

= K n ( vGS − VTH )2
Kn = 1 W μn Cox 2 L

vDS ≥ vGS − VTH

Drain current of a pValid for W 1 MOSFET operating in iD = μ p Cox ⎡( vGS − VTH ) vDS − vDS 2 ⎤ ⎢ ⎥ v DS ≤ vGS − VTH L 2 ⎣ ⎦ the linear region
1 2⎤ ⎡ = 2 K p ⎢( vGS − VTH ) vDS − vDS ⎥ 2 ⎣ ⎦

Drain current of a pMOSFET operating in the saturation region Transconductance parameter of a pMOSFET Gate oxide capacitance (per unit gate area) of a MOSFET Transconductance in the small signal equivalent circuit of an n-MOSFET operating in the saturation region

iD = iDsat =

1 W μ p Cox ( vGS − VTH )2 2 L

Valid for

= K p ( vGS − VTH ) 2
Kp = W 1 μ p Cox 2 L

vDS ≥ vGS − VTH

Cox =

ε ox tox ID and VGS are the magnitudes of the drain current and gate-tosource voltage, respectively, at the d.c. operating point.

g m = 2 K n (VGS − VTH ) = 2 K n I D

4

Transconductance in the small signal equivalent circuit of a p-MOSFET operating in the saturation region Drain-to-source resistance in the large signal model of the nMOSFET operating in the linear region Drain-to-source resistance in the large signal model of the pMOSFET operating in the linear region Output resistance in the small signal equivalent circuit of a MOSFET operating in the saturation region Drain current of an nMOSFET operating in the saturation region, including channel length modulation effect Drain current of an pMOSFET operating in the saturation region, including channel length modulation effect

g m = 2 K p (VGS − VTH ) = 2 K p I D

ID and VGS are the magnitudes of the drain current and gate-tosource voltage, respectively, at the d.c. operating point. VGS is the magnitude of the gateto-source voltage, at the d.c. operating point. VGS is the magnitude of the gateto-source voltage, at the d.c. operating point. ID is the magnitude of the drain current at the d.c. operating point.

RDS =

2 K n (VGS − VTH )

1

RDS =

2 K p (VGS − VTH

1

)

ro ≈

VA 1 = I D λI D

iD = K n ( vGS − VTH ) 2 (1 + λv DS )

Valid for

vDS ≥ vGS − VTH

iD = K p ( vGS − VTH ) 2 (1 + λ v DS )

Valid for

vDS ≥ vGS − VTH

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