Exam Support:: Diode Equations

Dr Ayhan Ozturk I s t a n b u l T e c h n i c a l U n i v e r s i t y

1

Phsical Constants Constant Symbol Magnitude Avagadro’s number NA 6.023 × 1023 [molecules /mol] Boltzmann’s constant kB 1.38 × 10-23 [J/K] = 8.62 × 10-5 [eV/K] Electronic charge q 1.6 × 10-19 [C] Electronvolt eV 1.6 × 10-19 [J] Free-electron mass mo 9.1 × 10-31 [kg] Permittivity of free space Єo 8.854 × 10-14 [F/cm] Permeability of free space µo 1.257 × 10-8 [H/cm] Planck’s constant h 6.625 × 10-34 [J.s] Reduced Planck’s constant ħ 1.054 × 10-34 [J.s] Thermal voltage at 300 °K φT 25.8 [mV] Velocity of light c 3 × 1010 [cm/s] Properties of Material at 300 °K Properity Ge Si GaAs SiO2 Atoms or Molecules/cm3 4.42 × 1022 5.0 × 1022 4.42 × 1022 2.3 × 1022 Atomic or molecular weight

72.6 28.08 144.63 60.08

Density, [g/cm3] 5.32 2.33 5.32 2.27 Breakdown field, [V/cm] ~105 ~3 105 ~4 105 ~107 Crystal structure Diamond Diamond Zinc blend Amorphous Dielectric constant 16 11.8 13.1 3.9 Efective density of states/cm3

Nc = 1.0 × 1019

Nv = 6.1 × 1018 Nc = 2.8 × 1019

Nv = 1.0 × 1019 Nc = 4.7 × 1017

Nv = 7.0 × 1018

Electron affinity, qχ, [eV] 4.0 4.05 4.07 0.9 Energy gap, Eg 0.68 1.12 1.43 9 Intrinsic carrier conc, ni , [1/cm3]

2.4 × 1013 1.5 × 1010 1.8 × 106

Lattice constant, [nm] 0.5646 0.5431 0.5653 Electron effective mass

mn=0.22mo mn*=0.12mo

mn=0.33mo mn*=0.26mo

mn=0.063mo

Hole effective mass mp=0.31mo mp*=0.23mo

mp=0.5mo mp*=0.16mo

mp=0.5mo

Intrinsic electron mobility, [cm2/Vs]

3900 1500 8600

Intrinsic hole mobility, [cm2/Vs]

1900 450 400

Tempreture coefficient of expantion, [1/°C]

5.8 × 10-6 2.6 × 10-6 6.8 × 10-6 5 × 10-7

Termal conductivity, [W/cm °C]

0.6 1.5 0.46 0.01

Exam Support:: Diode Equations

Dr Ayhan Ozturk I s t a n b u l T e c h n i c a l U n i v e r s i t y

2

pn Junction equations:: Equilibrium (T = 300 °K) Entity Formula Comments (Silicon) Mass action law

2innp = T = constant

Carrier concentration in Intrinsic semiconductor [1/cm3]

TkE

iB

g

eTBn−

= 32 ni(Si) = 1.5 1010 [1/cm3] B = 5.4 1031 [1/(K3cm6)] Eg(Si)=1.12 eV

Carrier concentrations in n-type semiconductor [1/cm3]

do Nn ≅ d

io N

np2

≅ Nd: Donor concentration

Carrier concentrations in p-type semiconductor [1/cm3] a

io N

nn2

≅ ao Np ≅

Na: Acceptor concentration

Conductivity of electrons [S/cm]

nn nq µσ ⋅⋅=

µn: electron mobility

Conductivity of holes [S/cm]

pp pq µσ ⋅⋅=

µp: hole mobility

Total Conductivity [S/cm]

pn σσσ +=

Resistance of electrons [Ω·cm]

nn σ

ρ 1=

Resistance of holes [Ω·cm]

pp σ

ρ 1=

Total resistance [Ω·cm]

pn σσρ

+=

1

Square resistance [Ω/]

≡

tRsq

ρ R = Rsq× s S []

Einstein relation

TB

p

p

n

n

qTkDD φ

µµ===

Dn: Diffusion constant for electrons Dp: Diffusion constant for holes

Diffusion constant for electrons [cm2/s]

TnnD φµ ⋅=

Diffusion constant for holes [cm2/s]

TppD φµ ⋅=

WLs ≡

Exam Support:: Diode Equations

Dr Ayhan Ozturk I s t a n b u l T e c h n i c a l U n i v e r s i t y

3

pn Junction Equations:: Equilibrium (T = 300 °K) Entity Formula Comments Potential barrier (Built-in voltage) [V]

= 2ln

i

daTo n

NNφψ oo V=ψ

n-region depletion width [cm] )1(

2

a

dd

oosn

NNqN

Kx+

∈=

ψ Ks: relative dielectric constant

o∈ : Vacuum dielectric constant

p-region depletion width [cm] )1(

2

d

aa

oosp

NNqN

Kx+

∈=

ψ

Total depletion width [cm]

pnd xxx +=

Junction law

d

a

p

n

NN

xx

=

Electron diffusion length [cm] nnn DL τ⋅= Dn: Electron diffusion

constant τn: Mean electron lifetime

Hole diffusion length [cm] ppp DL τ⋅= Dp: Hole diffusion

constant τp: Mean hole lifetime

Max Junction Field [V]

os

ndm K

xNq∈⋅⋅⋅

−=ε

Charge balance

nNpN ad +=+

Exam Support:: Diode Equations

Dr Ayhan Ozturk I s t a n b u l T e c h n i c a l U n i v e r s i t y

4

pn Junction Equations:: Forward Mode Entity Formula Comments n-region depletion width [cm] )1(

)(2

a

dd

foosn

NNqN

VKx

+

−∈=

ψ

Ks: relative dielectric constant

o∈ : Vacuum dielectric constant

p-region depletion width [cm] )1(

)(2

d

aa

foosp

NNqN

VKx

+

−∈=

ψ

Vf : Forward Voltage

Electron drift current [A/cm2]

εµ ⋅⋅⋅= nn nqdriftJ )(

ε: Electrical field q: electronic unit charge

Hole drift current [A/cm2]

εµ ⋅⋅⋅= pp pqdriftJ )(

Electron diffusion current [A/cm2]

⋅⋅=

dxdnDqdiffJ nn )(

Hole diffusion current [A/cm2]

⋅⋅−=

dxdpDqdiffJ pp )(

Scale (saturation) current [A]

⋅

+⋅

⋅⋅=

an

n

dp

piDS NL

DNL

DnqAI 112

Short p diode pn WL → Replace in IS formula

Short n diode np WL → Replace in IS formula

Diode current T

fV

SD eII φ⋅=

Dynamic resistance [Ω]

D

Td I

r φ=

Diffusion capacitance [F]

T

DTD

ICφ

τ ⋅=

τT: transient time

Electron component of current [A]

T

DV

an

niDn e

NLDnqAI φ

⋅

⋅⋅=

12

Hole component of current [A] T

DV

dp

piDp e

NLD

nqAI φ

⋅

⋅⋅=

12

Junction capacitance (for forward bias)

joj CC ⋅= 2

Power [W] DD IVP ⋅= Instantaneous power

Exam Support:: Diode Equations

Dr Ayhan Ozturk I s t a n b u l T e c h n i c a l U n i v e r s i t y

5

pn Junction Equations:: Reverse Mode Entity Formula Comments n-region depletion width [cm] )1(

)(2

a

dd

roosn

NNqN

VKx+

+∈=

ψ Ks: relative dielectric constant

o∈ : Vacuum dielectric constant

p-region depletion width [cm] )1(

)(2

d

aa

roosp

NNqN

VKx+

+∈=

ψ Vr : Reverse Voltage

Diode current SD II −=

Diffusion capacitance [F]

joD CC ⋅= 2

Junction (Depletion) capacitance [F] d

osDjo x

KAC ∈= m

o

R

joj

V

CC

+

=

ψ1

AD: device crossection xd: Equilibrium depletion width m: (1/3) to (1/2)

Breakdown Voltage [V] (BV >> ψo)

d

cos

NqKBV

⋅⋅∈⋅

=2

2ε

Power [W] ZZ IVP ⋅= Instantaneous power for Zener

Prefixes k = 103 m = 10-3 1µm = 10-4 cm 1cm = 104 µm M = 106 µ = 10-6 1nm = 10-7 cm 1nm = 103 µm G = 109 n = 10-9 1mil = 10-3 inch 1mil = 25.4 µm T = 1012 p = 10-12 1mil = 25.4 10-4 cm