p-n junctionscombine p- and n-type semiconductor region

most important characteristics: rectification(current flows “easily“ only in one direction)

important as rectifier but also: Basic building block in bipolar transistors, field-effect transistors, LED‘s, solar cells ….

forward bias

reversebias region

S. M. Sze, “Semiconductor Devices– Physics and Technology“

Formation of a p-n junction:

Conceptionally:

Take a p-type SC and combine it with an n-type SC

The actual process isof course different …

S. M. Sze, “Physics of Semiconductor Devices“

Static conditions – thermodynamic equilibrium

We start with:

S. M. Sze, “Semiconductor Devices– Physics and Technology“

D. A. B. Miller, lecture notes, Stanford University, 1999

Establish contact: e- and h+ concentration gradient !

→ electrons diffuse from the n-side into the p-side and holes diffuse from the p-side into the n-side → DIFFUSION CURRENT !

Acceptor and donor ions are fixed !

→ negative space charge on the p-side of the junction

→ positive space charge on the n-side of the junctionD. A. B. Miller, lecture notes, Stanford University, 1999

→ depletion region

(virtually no freecarriers → charged dopands !)

electric field

→ DRIFT CURRENT !

(In the oppositedirection of thediffusion current)

electrons holes

Diffusion current density

xdndDej ndiff,n =

xdpdDej pdiff,p −=

Dn, Dp … diffusion coefficients given by the Einstein relations

nn ekTD µ= pp e

kTD µ=

µn and µp … mobilities of electrons and holes, respectively

Drift current densitypFej pdrift,p µ=nFej ndrift,n µ=

F … driving electric field

What happens, if we apply an electric field to the semiconductor ?

dxdE

edxdUF 1

=−=S. M. Sze, “Semiconductor Devices– Physics and Technology“

U … electrostatic potential

E … EC, EV, Evac, Ei (intrinsic Fermi level)

Junction in thermodynamic equilibrium

Let‘s look at the hole current density …

xdpdDepFej ppp −= µ

⎟⎟⎠

⎞⎜⎜⎝

⎛−=→

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=→

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

dxdE

xdEd

Tkp

xdpd

dxdE

xdEd

TkTkEEexpn

xdpd

TkEEexpnp

Fi

b

Fi

bB

Fii

B

Fii

1

and

dxdE

eF i1=

D. A. B. Miller, lecture notes, Stanford University, 1999

dxdEpj

dxdE

xdEd

Tkp

eTke

dxdE

epej

Fpp

Fi

bp

Bipp

µ

µµ

=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

1

Thermodynamic equilibrium: The current flow across thejunction has to be zero !

0=dx

dEF

In the thermodynamic equilibrium (i.e., if no charge carriersare injected), the Fermi energy in the device is CONSTANT !

What is the electrostatic potential difference between the p-side and the n-side ?

D. A. B. Miller, lecture notes, Stanford University, 1999

Vbi … built in potentialeVbi= minus

But: How do the bands inside the depletion region reallylook like ?

D. A. B. Miller, lecture notes, Stanford University, 1999

Assume abrupt junction:

S. M. Sze, “Physics of Semiconductor Devices“junction

Red: assumed abrupt junction (abrupt changes of the net chargesbetween the depletion region and the p and n regions)

Potential from Poisson equation

ερ

−=2

2

dxVd

ρ … charge density; ε … dielectric constant

0<≤− xxpfor

nxx ≤<0for

εAeN

dxVd

=2

2

εDeN

dxVd

−=2

2

And as a side-condition:

nDpA xNxN =

Solving these differential equations allows usto calculate:

• The field-distribution in the depletion region

• The maximum field in the depletion region

• The width of the depletion region

• The “shape“ of the band bending• The capacitance of the diode

• …..

Biasing a diode:

Forward bias of VF:• total electrostatic potential across junction decreases by VF

• depletion layer width decreases

Reverse bias of VR:• total electrostatic potential across junction increases by VF

• depletion layer width increases

The system is no longer in thermodynamic equilibrium !

→ current will flow !

S. M. Sze, “Semiconductor Devices– Physics and Technology“

Current through diode

Applying a voltage: disturb balance between diffusioncurrent and drift current

Forward bias: drift current reduced in comparison to diffusion current

→ enhanced diffusion of holes from p-side to n-side

→ enhanced diffusion of electrons from n-side to p-side

minority-carrier injection → current

Reverse bias: increase of electrostatic potential acrossdepletion region

→ reduce minority carrier concentrations

→ reduce also diffusion current

S. M. Sze, “Semiconductor Devices –Physics and Technology“

Ideal current-voltage characteristicsSeveral approximations … including no generation or recombinationof carriers in the depletion region

Using the “semiconductor statistiscs“ and current equations wediscussed before (after a somewhat lengthy derivation):

( )1−= Tk/eVS

Bejj

jS … sturation current (contains diffusion coefficients, diffusionlengths and equilibrium concentrations of the minority carriersin the p and n regions)

• Current grows exponentially with the applied voltage in forward direction• Current density saturates at –jS in reverse direction

S. M. Sze, “Semiconductor Devices –Physics and Technology“

Deviations from the ideal characteristics

Most relevant for our purposes:

Recombination currente- from n-type region and h+ from p-type region recombine in thedepletion region

→ added current in forward direction

can happen through traps (heating of the device)

or through the emission of a photon (LED)

PhotocurrentOptical absorption → extra pair of e- and h+ in depletion region

→ extra reverse current

Heterostructure diodesBand bending of course also happens in heterostructures withdifferent doping levels

But then in addition to the band bending also band offsets

D. A. B. Miller, lecture notes, Stanford University, 1999

D. A. B. Miller, lecture notes, Stanford University, 1999

Metal-semiconductor contacts (Schottky-contacts)

(here we again assume that there are no interface dipoles and weneglect effects like mirror charges and Fermi level pinning …)

S. M. Sze, “Semiconductor Devices –Physics and Technology“

Some similarities to a p-njunction (metal plays the “role“ of the p or n type semiconductor)

But also certain differences …

D. A. B. Miller, lecture notes, Stanford University, 1999n-type semiconductor:

p-type semiconductor:

EF constant !

e- or h+ flow from semiconductor into metal

built-in voltage Vbi and band bending in the SC

→ depletion region

potential barrier Φbn (lowered by Schottky effect)DN

width 1∝

Charging, fields and shape og band bending

S. M. Sze, “Semiconductor Devices –Physics and Technology“

Charge transport through Schottky-junctionsdifferent from p-n junction:

• charge transported by the majority carriers

• thermionic emission of majority carriers from the semiconductorover the potential barrier into the metal

• equilibrium: balanced by flow of electrons from metal intosemiconductor

rectification ! with current density given by: ( )1−= Tk/eVS

Bejj

Ohmic contact:

Definition: metal-semiconductor contact with “negligible“contact resistance

How can this be realized ?

very high doping of the semiconductor

→ depletion region extremely short

→ e- and h+ can “easily” tunnel through resulting very thin barrier

Ohmic contacts

no rectification !

D. A. B. Miller, lecture notes, Stanford University, 1999