Semiconductor DevicesSemiconductor DevicesSemiconductor DevicesSemiconductor Devices

P fP f R b t A T lR b t A T lProf. Prof. Robert A. TaylorRobert A. Taylor

The aim of the course is to give an introduction to semiconductor device physics. The syllabus for the course is:

Simple treatment of p-n junction, p-n and p-i-n structures as photodetectors light emitting diodes and lasers (excludingphotodetectors, light-emitting diodes and lasers (excluding optical gain and cavity properties). Semiconductor heterojunctions, quantum wells and nanostructures. Low dimensional semiconductor qdevices, e.g. quantum well laser. (Non-examinable) transistors and their uses.

Semiconductor devicesSemiconductor devicesSemiconductor devices

* pn junction: (i) at equilibrium, (ii) under bias* photodiodes* solar cells (photovoltaics)* light emitting diodes* semiconductor lasers* bipolar junction transistors* field effect transistors* heterojunctions* quantum wells and dots, quantum well lasers* single electron transistors

Electron drift velocity:

where τ is the mean time between scattering events, and μn is the electron mobility. There is a similar relation for holes. The mobility depends on field at high field due to inelastic scattering processes, and the drift velocity saturates; the maximum value of vn is ~105ms-1 in Si. The net current density is:

*n nn

qv E E

m

τ μ= − = −

( )0 0n p n pq n pμ μ= + = +J J J E

Carrier drift in applied fieldCarrier drift in applied field

When there is a spatial variation in the electron density, there will be a diffusion current. In one dimension:

where Dn is the electron diffusivity. Dn and μn are related via the Einstein relation:

The hole mobility and diffusivity are usually much smaller than the electron values because of the hole’s heavier mass.

n n

dnJ qD

dx=

B Bn n *

n

k T k TτD

q mμ= =

Carrier diffusionCarrier diffusion

1E14 1E16 1E18 1E20

DENSITY (cm-3)

Density dependence of D,μDensity dependence of D,μ

The recombination rate R is proportional to the number of electrons and the number of holes :

In thermal equilibrium the recombination rate is equal to the generation rate:

If there is an excess of carriers of a particular type, e.g. caused by illumination of a doped semiconductor, the excess carriers will recombine. If Δp holes are injected into n-type material:

(βn0)-1 is the minority carrier lifetime τp.

R npβ=

0 0Rth thG n pβ= =

00 0 ( ) n tdp

n p p t p pedt

ββ −= − ⇒ = + Δ

Carrier recombinationCarrier recombination

The continuity equations for electrons and holes are:

Under steady state conditions:

Using the boundary condition p(x→∞) = p0 yields:

where Lp =√(Dpτp) is the diffusion length, and (p(0)-p0) is the excess hole population at x = 0.

1 1 pn

n n p p

JJn pG R G R

t q x t q x

∂∂∂ ∂= + − = − + −

∂ ∂ ∂ ∂

20

20 p

p

p pp pD

t x τ−∂ ∂

= = −∂ ∂

( ) [ ]0 0(0) p

xLp x p p p e

−= + −

ContinuityContinuity

p-n junction at equilibriump-n junction at equilibrium

np

nn

pp

-|E|

Depletion Region or Space Charge Region

The electrostatics of the junction at equilibrium are described by the Poisson equation:

where we have assumed that all donors and acceptors are ionized. Far away from the junction we have:

Assume the junction is abrupt. In the depletion region the free carriers are totally depleted, so that :

( )2

20 0

D Ar r

d V dE qN N p n

dx dx

ρε ε ε ε

= − = − = − − + −

( )2

20 0D A

d VN N p n

dx= ⇒ − + − =

2

20

2

20

0

0

AA

r

DD

r

qNd Vd x

dx

qNd Vx d

dx

ε ε

ε ε

= − ≤ <

= − < ≤

p-n junction at equilibriump-n junction at equilibrium

The electric field is then:

The potential difference across the junction is:

and W is the width of the junction:

0

0 00

2 2

0

( ) ( )( )d d d

q 1 ( ) (i.e. the area under )

2 2

D D

A A

d d

A A D DC

r rd d

A A D D Mr

qN x d qN x dV E x x x x

N d N d E W E

ε ε ε ε

ε ε

− −

+ −= − = −

= + =

∫ ∫ ∫

02 ( )r A D CA D

A D

N N VW d d

qN N

ε ε += + =

0

0

0 0

( )( ) 0

( ) 0

where

A AA

r

DM D

r

D D A AM

r r

qN x dE x d x

qN xE x - E x d

qN d qN dE

ε ε

ε ε

ε ε ε ε

+= − − ≤ <

= < ≤

= =

Depletion layerDepletion layer

VC can be obtained from the condition that the drift current balances the diffusion current:

Integrate this equation from -dA to dD to obtain:

where nn is the electron density in the n-region, and np is the electron density in the p-region.N.B. , where ni is the intrinsic carrier density.

d d ( )( ) 0

d dn n

V n xn x q qD

x xμ− + =

ln( )nBC n p

p

nk TV V V

q n= − =

2 /p i An n N=

contact potentialcontact potential

p-n junction under biasp-n junction under bias

The ideal diode is assumed to operate under the following conditions:(i) the depletion layer is abrupt, and there is charge neutrality outside of the layer,(ii) the charge densities at the boundaries are given by the electrostatic potential,(iii) the minority carrier injection is weak, much less than the majority densities,(iv) there are no generation or recombination currents in the depletion layer.

voltage-current relation for a p-n diodevoltage-current relation for a p-n diode

The relation for VC can be re-written:

We expect these relations to hold for non-equilibrium carrier densities when a bias V is applied:

Since we then have :

In the neutral n- and p-regions there is no electric field:

/ / and C B C BqV k T qV k Tn p p nn n e p p e= =

( ) / ( ) /ˆ ˆ ˆ ˆ and C B C Bq V V k T q V V k Tn p p nn n e p p e− −= =

/

/

ˆ ( 1) at

ˆ ( 1) at

B

B

qV k Tp p p A

qV k Tn n n D

n n n e x d

p p p e x d

− = − = −

− = − =

2/ ( ) /

2

2/ ( ) /

2

ˆ ˆˆ ( 1)

ˆˆˆ ( 1)

B D D

B A A

qV k T x d Ln n nn n n

p p

p p qV k T x d Lnp p p

n n

d p p pp p p e e

dx D

n nd nn n n e e

dx D

τ

τ

− −

+

−= → − = −

−= → − = −

ˆn nn n≈

p-n junction under biasp-n junction under bias

The total current flowing is:

The current increases exponentially under forward bias, but saturates at –JS for negative bias.

/

2

( ) ( )

ˆˆ

( 1),

where

D A

B

p D n A

pnp n

d d

qV k TS

p n n pS

p e

p ni

p D e A

J J d J d

dndpqD qD

dx dx

J e

qD p qD nJ

L L

D Dqn

L N L N

−

= + −

⎡ ⎤⎡ ⎤= − + ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

= −

= +

⎛ ⎞= +⎜ ⎟⎜ ⎟

⎝ ⎠

Forward Si

Reverse Si

Theory forward (Ge)

Theory reverse (Ge)

Junctionbreakdown

current-voltage characteristicscurrent-voltage characteristics

-5 -4 -3 -2 -1 0 1 2 3 4 5

-2

0

2

4

6

qV/kT

J/Js

current-voltage characteristicscurrent-voltage characteristics

( )/e 1BqV k TsJ J= −

Web resourcesWeb resources

http://jas.eng.buffalo.edu/

http://ece-www.colorado.edu/~bart/book/book/title.htm

http://cnx.org/content/m1004/latest/

http://people.deas.harvard.edu/~jones/ap216/images/pn_junction/pn_junction.html

http://www.nanohub.org/simulation_tools/pnjunction_tool_information

http://www.mtmi.vu.lt/pfk/funkc_dariniai/diod/index.html