P-N Junction Depletion Region

p-n junction energy band diagram

P-n junction in equilibrium

EF

EF

p n∆E = qVbi

p-n junction energy band diagram

Note that Vr is negative: Vr < 0

EF

EF

p

n

∆E = q(Vbi – Vr)

P-n junction at reverse bias Vr

Electron (n), hole (p) and net charge (ρ) distribution in the p-n junction (in equilibrium)

p << NA;np<< n << NA;p×n = ni

2;ρ ≈ -q NA;

n << ND;pn<< p << ND;p×n = ni

2;ρ ≈ +q ND;

Log

(Con

cent

ratio

n)

x

ND

np

NA

pp0 ≈ NA;np0 << NA;pp0×np0 = ni

2;ρ = 0;

nn0 ≈ ND;pn0 << ND;pn0×nn0 = ni

2;ρ = 0;

Carrier and charge distribution in p-n junction

ϕ(x) varies from 0 (n-side) to 1.1 eV for Si (p-side). Note, kT = 0.026 eV at 300 K;

Ec

Ev

EF

X

ϕ(x)

n p

nn

n(x)

kTenϕ

−~

;

kTx

n enxn)(

)(ϕ

−=

ϕ(x), eV n(x)/nn

0 1

0.1 0.021

0.3 9.74787Ε−06

0.5 4.44819Ε−09

0.75 2.96671Ε−13

1 1.97864Ε−17

Under reverse bias the depletion is even stronger: ϕmax = q(Vbi-Vr)

pn ≈ 0;np ≈ 0;

ρ ≈ -q NA;

nn ≈ 0;pn ≈ 0;

ρ ≈ +q ND;

Con

cent

ratio

n

x

ND

np

NA

ρ = 0; ρ = 0;

Charge distribution in the depletion region.Total charge density in the semiconductor: ρ = q×(Nd + p - Na - n);

Depletion region (Space-Charge) in the p-n junction

Total charge density in the semiconductor: ρ = q×(Nd + p - Na - n);

The gradient of electric field (the slope of the E(x) curve):

0

dEdx

ρε ε

=

(Poisson's equation)

xn0, xp0 = the depletion region width on the n- and p-sides

On the p-side of the junction, 00

Am p

qNE x

εε=

On the n-side of the junction, 00

Dm n

qNE x

εε=

At the junction interface: Em(p-side) = Em (n-side)

0 00 0

0 0

D Am n p

p n D A

qN qNE x x

x x N N( / )εε εε

= =

=

Maximum Field in the p-n junction

Em

0

dEdx

ρε ε

=From

0 00 0

D Am n p

qN qNE x x

εε εε= =

The voltage drop across the p-n junction, p nV E x dx( )− = − ∫= -{AREA UNDER the E(x) curve}Since E(x) curve forms a triangle,

WxNqWEV ndmbi 002

121

εε==

Depletion region width in the p-n junction

In equilibrium:

At reverse bias:

00

1 12 2bi r m d n

qV V E W N x Wε ε

− = =

DA

AD

AD

Dn

Dbi NN

NNWqNN

WqNWxqNV+

=+

=⎟⎟⎠

⎞⎜⎜⎝

⎛= 2

0

2

00 2/11

22/

εεεεεε

The total width of the depletion (space-charge) region,

W = xn0 + xp0 = xn0 (1 + ND/NA);The built-in voltage (the area of the triangle): Vbi = Em×W/2;

212A D bi D

qN N , V N Wε

>> =

If the p-n junction is strongly asymmetrical, for example:

Depletion region width in the p-n junction (cont.)

0 0p n D Ax x N N( / )=From

Under reverse bias, Vbi has to be replaced with Vbi - Vr

1 Depletion region (a.k.a space-charge region) width

Depletion region of the p-n junction summary

2

02D A

bi rA D

N NqV V WN Nεε

− =+

2

0

12A D bi r D

qN N , V V N Wε ε

>> − =

For strongly asymmetrical p-n junction:

2

0

12D A bi r A

qN N , V V N Wε ε

>> − =

The space-charge region is extended mainly to the low-doped side of the p-n junction

Depletion region width sharing between n- and p-sides:

0 0p n D Ax x N N( / )=W = xn0 + xp0

W ≈ xn0

W ≈ xp0

Summary: main characteristics of the p-n junction

2. Maximum electric field

12bi r mV V F W− =

( )0

2 bi rm

q N V VF

'ε ε

−=

20

2m

bi rF

V Vq N '

ε ε− =

20

2BDF

Vq Nmax '

ε ε=