1 the pn-junction in equilibrium 2 the i–v characteristics

Introduction to Microelectronics1

PN-JUNCTION

1 The PN-junction in Equilibrium

2 The I–V Characteristics of the PN-Junction

3 Deviations from the Ideal Diode

Introduction to Microelectronics

2

−

⊕ donor atom (positive ion)

− acceptor atom (negative ion)

+ hole

(free) electron


3

PN-junction

“Contactless” pn-junction


1016

1014

1012

1010

106

108

104

102

nn = ND

pp = NA

np = ni2 / NA pn = ni

2/ ND

N A=1016

cm-3

PN D=10

15cm

-3

N

ohmic

contact

ohmic

contact

A B

-N A

N D

x(a) Step junction

(b) “initial” carrier concentrations

ni=1010cm-3

“Before contacting”, the two

regions are electrically neutral

( ρ = 0 ).


5

http://www.pveducation.org/pvcdrom/pn-junction/formation-pn-junction


NA=1016 cm-3 ND=1015 cm-3 P-type Si N-type Si

A B

1016

1012

108

104 n

Space charge region

or depletion region

quasi-neutral regions

ρ = E = 0

Ions (NA-)

Ions (ND+) +

_

VAB = 0

∗ electrons (holes) diffuse from

the n (p) side to the p (n) side,

leaving behind ND+ (NA

-) ionized

donor (acceptor) atoms and,

consequently, a net charge density

ρ ≠ 0, which gives rise to an

electric field ≠ 0

∗ pn = ni2 because VAB = 0

pn-junction in thermal equilibrium


NA=1016 cm-3 ND=1015 cm-3 P-type Si N-type Si A

B

1016

1012

108

104 n

Ions (NA-)

Ions (ND+) +

_


On a linear

scale


A review on Poisson’s equation

E(y1)

ρ(x)

εE(y2)

ρ - charge density ε - permittivity A - area

y1 y2 y

2

1

y

2 1

y

A ε E (y ) A ε E (y ) A ρ (y )d y

d VE (y )

d y

− =

= −

∫ 2

2

d V dE ρ

dy dy ε− = =

Qsd D =∫∫�

�

Gauss’ law:

In one dimension:

Poisson’s equation relates the potential V

to the charge density ρ

10


EE = 0 E = 0

-xp xn

-Emax

-xp xn

-qNA = -1.6 x10-3

c/cm3

x

x

1.6 x10-4

c/cm3qND

ρ (c/cm )

+

-0

x

φJ

xn-xp 0

V

Siερ

dx

dE=

E = 0 xn ≤ x; x ≤ -xp

E = -qNA / εSi ( x + xp )

E = -qND / εSi ( xn - x )

Continuity of electrical field

Naxp = NDxn

dx

dVE −=

V = 0 at x = -xp

( )2

p

Si

A xx2ε

qN+

( )2D

n

Si

qNx-x

2εJφ −

0 ; for x ≤ -xp

V =

φJ; for xn ≤ x


3

P- NNA = 10

16 cm

-3 ND = 1015

cm-3A B


dx

n) d(ln

µ

D- E 0

dx

dnqDEqn µJ

n

nnnn =→=+=

==== ∫

a

bab

b

an

n

nn

ln V-V dx E- q

kT

µ

Dtt φφ

( )( )

−<

>=

p

nJ

xxn

xxnln tφφ

V = 0 and n = ni2 / NA ; for x < -xp

=

2

i

DAJ

n

NNln tφφ

=

20

31

J10

10ln tφφ

2 5 )

J 6 3 2 m V

( t m V

φ

φ

=

=

NA=10 16 cm -3 N D=10 15 cm -3P-type Si N-type Si

φ J

xn0-xpx

V(x)

A B

The Built-in Voltage

Example: ni=1010cm-3

NA=1016cm-3 , ND=1015cm-3

12


Depletion-layer width and maximum

electric field

Si

2

nD2

p

SI

A0)(x

2ε

xqNx

ε 2

N qV −=== Jφ

2d o

A D

1 1X

N N

S i

Jq

εφ

= +

don

D

A

Xx

N1

N

=

+

dop

A

D

Xx

N1

N

=

+

nDpA xNxN =

do n pX x x= +

equilibrium

εSi = 1.04 x10-12 F/cm Xdo = 0.951 µm

NA = 1016cm-3 xn = 0.864 µm

ND = 1015cm-3 xp = 0.0864 µm

2max

do

EX

Jφ=

µm 0.951

V 0.6322Emax

×= kV/cm 13.3E max =

13


P- NNA = 10

16 cm

-3 ND = 1016

cm-3A B

14


P- NNA = 1016 cm-3 ND = 10

17cm

-3A B

15


Biased pn-junction


Biased pn-junction

� Voltage drop at the ohmic contacts remain the same;

� Voltage drops across the quasi-neutral regions is zero (not valid for

high currents);

→ All applied voltage drops across the space charge region

→Electrostatics of the SCR under bias is unchanged from

thermal equilibrium


Biased pn-junction

Electrostatics of the SCR under bias is unchanged from thermal equilibrium

Depletion approximation

J

J

J

J

J

J

J

J

J JJ


PN-Junction in Thermal Equilibrium

Jn diff. + Jn drift = 0

Jp diff. + Jp drift = 0

20

Thermal equilibrium: energy band diagram and carrier flux

qφJ

qφJ


PN-Junction Under Reverse Bias

Energy band diagram for reverse bias (– · – · –)

and at thermal equilibrium ( ______ )

Jn = Jn diff. + Jn drift ≈≈≈≈ Jn drift

Jp = Jp diff. + Jp drift ≈≈≈≈ Jp drift

21

qφJ

φJ

φJ


Jn = Jn diff. + Jn drift ≈≈≈≈ Jn diff.

Jp = Jp diff. + Jp drift ≈≈≈≈ Jp diff.

J = Jn + Jp

VA > 0

The voltage drop across the

(quasi)-neutral regions

is ≈ zero for low-level

injection

PN-Junction Under Forward Bias

22

φJ

φJ

Energy band diagram for forward bias (– · – · –) and

at thermal equilibrium ( )


I

VA

IDO – current downhill α p ( N-side )

P N

φJ

-xp xn

Equilibrium

(VA=0)

M. Born, Atomic Physics, Dover, p. 305

R. Feynman et al., The Feynman Lectures on Physics, Addison

Wesley, vol. 3, p. 14.8.

Hole current

A few holes on the P-side approach the

barrier with enough energy to carry over it

and reach the N-side, where they

recombine

IUP – current uphill α NA exp(-φJ / φt )

(same reasoning for electrons)

IUP is balanced by a continual generation of pairs

by thermal fluctuations near the junction on the

N-side and some of the holes produced fall

down the energy gradient into the P-side giving a current IDO

Equilibrium: IUP = IDO = IS α NA exp(-φJ / φt )

Boltzmann

First order model: IDO is independent of VA – the rate of thermal generation of pairs will not

change for VA ≠≠≠≠0 since it depends only on local properties of the crystal near the junction.


I

VA

IDO =IS

Forward bias

(VA>0)

M. Born, Atomic Physics, Dover, p. 305

R. Feynman et al., The Feynman Lectures on Physics, Addison

Wesley, vol. 3, p. 14.8.

Hole current

IUP – current uphill α NA exp(VA-φJ)/ φt )

(same reasoning for electrons)

Boltzmann

First order model is independent of VA – the rate of thermal generation of pairs will not

change for VA ≠≠≠≠0 since it depends only on local properties of the crystal near the junction.

P N

φJφ

J -VA

-xp

VA=0

xn

IUP = IS exp (VA / φt)

I D= IUP – IDO = IS [exp (VA / φt) -1]


25

At the edges of the depletion region, -xp and

xn, equilibrium conditions do not prevail so

we must use the "law of the junction".

Development of analytical dc model (I-V characteristics)

of the diode I

VA

( )( )

( ) ( )

( )( )

( ) ( ) ( )exp

exp exp

exp

p np A J

n

p np A

n

t

J

t

q x xp x

q x x

V

p x

n x V

n x k

T

T

k

− −− − − = − =

− −− − = =

φ φφ φφ φφ φ φφφφ

φφφφ

φ φφ φφ φφ φ φφφφ

φφφφ

A B

NAP ND

-xp xn

QNRQNR

0

N

φJφ

J - VA

-xp

VA=0

xn

φJ

φJ


Law of the junctionP N

φJ

-xp xn

-VA

( )

( )

;

− ≅

≅

A

n D

pp x

n N

N

x

Boundary conditions:

3. Recalling that we find that 2

exp =J A D

t i

N N

n

φ

φ

( )

( )2

2

e

xp

x

e

pi A

p

A t

i A

n

D t

n Vn

n Vp x

N

N

x

φφφφ

φφφφ− =

=Law of the junction

( )( )

( )( )

( )exp

exp

p

p A J

n t

A J

n t

p x V

p x

n x V

n x

φφφφ

φφφφ

φφφφ

φφφφ

−

− −=

− −=

1. Neutrality at -xp and xn (limits of QNR)

→p(-xp)=n(-xp)+NA n(xn) =ND+p(xn) (III);

2. Low-level injection p(-xp)>>n(-xp) & n(xn) >>p(xn) (IV)

(I)

(II)


( )

( )

;

− ≅

≅

A

n D

pp x

n N

N

x

Boundary conditions for minority carriers

( )2

expi A

p

A t

n Vn x

N− =

φφφφ

Law of the junction

A B

NAP ND

-xp xn

QNRQNR

0

N

-Wp Wn

( )2

i

p

A

nn W

N− = ( )

2

i

n

D

np W

N=

( )2

expi An

D t

n Vp x

N=

φφφφ

Ohmic contacts:

Thermal equilibrium


Calculate the current-voltage characteristic of a “short” P+N junction diode

I

VA

1. Holes are the main carriers;

2. Recombination is negligible in the N region;

3. Diffusion current is dominant

( )2

expi A

n

D t

n Vp x

N φ=

NAND

0 xn Wn

( )2

i

n

D

np W

N=

Law of the

junction

Ohmic

contact

( ) ( )

( )

2

exp 1

n n

p p p

n n

i A

p p p

n Dn t

p W p xdpJ qD qD

dx W x

n VI AJ qAD

W Nx φ

−= − = −

−

= = −

−

2

1 ;φ

≅ = −

≅

A

t

V

p S

i

S p

n D

I I I e

nI qAD

W N


+ vD -

iD

−

= 1

vexpIi D

SD

tnφφt = kT/q


0.10.0-0.1-0.2-2.00e-15

0.00e+0

2.00e-15

4.00e-15

6.00e-15

8.00e-15

1.00e-14

1.20e-14

Diode Voltage (V)

Dio

de C

urr

ent

(A)

IS

Diode behavior near the origin with IS=10-15 A and n=1

φt = 25 mV


Diode Voltage (V)

1.00.90.80.70.60.50.40.30.20.10.010 -15

10 -14

10 -13

10 -12

10 -11

10 -10

10 -9

10 -8

10 -7

10 -6

10 -5

10 -4

10 -3

10 -2

10 -1

Dio

de C

urr

ent

(A)

φt = 0.025 V

Slope - 1 decade/60 mV

I = 10 AS

-15

Diode i -v characteristic on semilog scale

IS : saturation current

n : ideality factor ( 1 to 2 )

φt : kT/q

k : 1.38 x10-23 J/K

Typical values of IS:

10-18 A ≤ IS ≤ 10-9 A

−

= 1

vexpIi D

SD

tnφ

+ vD -

iD


32

References

– EEL 7061 Eletrônica Básicahttp://www.lci.ufsc.br/electronics/index7061.htm

– Reid R. Harrison, “Analog Integrated Circuit Design” ECE/CS 5720/6720 Department of Electrical and Computer Engineering University of Utah

– Charles Sodini, “6.012 Microelectronic Devices and Circuits”,OpenCourseWarehttp://ocw.mit.edu

– Sze & Ng, “Physics of semiconductor devices”, 3rd edn. Wiley

– Pierret, “Semiconductor device fundamentals,” Addison-Wesley

1 the pn-junction in equilibrium 2 the i–v characteristics

Documents