Chemistry 140a

Lecture #5Jan, 29 2002

Fermi-Level Equilibration• When placing two surfaces in contact, they will equilibrate; just

like the water level in a canal lock.

• The EF of the semi-conductor will always lower to the EF of the metal or the solution. This can be understood by looking at the density of states for each material/soln.

Initial EF

Semi-Conductor Metal/Soln.

Eq. EF Initial EF

Eq. EF

Fermi-Level Equilibration• Charge comes from the easiest thing to ionize, the dopant

atoms. This leads to a large region of (+) charges within the semi-conductor.

• In the metal all of the charge goes to the surface. (Gauss’s Law)

• The more charge transferred the more band bending. E

x

EF

EF

E

x

EF

ECB

EVBEVB

ECB

Vbi

Vbi

E

x

EF

EF

E

x

EFECB

EVB

EVB

ECB

VbiVbi

Depletion Approximation

• All donors are fully ionized to a certain distance, W, from the interface.

• W=W(ND,Vbi)

XW

ND W

Vbi W

++++++++++

-----

-----

Final Picture

E

x

EF

EF

E

x

EF

ECB

EVBEVB

ECB

Vbi

Vbi

EVacEVac

scm

Eg

-

-

- ++ +

Useful Equations

Poisson’s Eqn:

d(x)dx

E x q(x)

x q p x n x NA x ND x

d2(x)dx 2

x K0

E(x)

E(x) = Electric Field (V/cm) (x) = Electric Potential (V)

(x) = Electric Potential Energy (J)

Electric Potential (V)Integrate Poisson’s Eqn.

B.C.’s

Result:

XW

+++++++Q=qNDW

-----

-----

x)

qND

d2(x)dx 2

x K0

d(x)dx

0

x 0

x W

x W

(x) qND2K0

x W 2

-qNDW2

(2K0)

(x)

x

quadratic Vbi=

Depletion Width

• Rearranging for W:

• As expected, W increases w/ Vbi and decreases w/ ND

• If one accounts for the free carrier distribution’s tail around x=W

W 2K0VbiqND

W 2K0 Vbi

kTq

qND

Typical Values

Vbimax (V) ND (cm-3) W (m) Q (C/cm2)

1 1013 11 1010

1 1016 0.36 3*1011

Electric Potential Energy• E(x) = -q(x)

(0) = -Vbi

• qVbi = (EF,SC-EF,M)

B = Vbi + Vn

– Barrier height– Independent of doping– Vbi and Vn are doping dependent

(x)

x

E

x

EF

ECB

EVB

Vbi Be-

h+

Net = 0 @ Eq.

Vn

Electric Field (V/cm)

d(x)dx

E(x)

qNDK0

x W

W

Emax=-qNDW/(K

x

Ex)

I-V Curve

No Band Bending

Low Band Bending

High Band Bending

I

V

Review

• N-type P-typeE

x

EF

ECB

EVB

Vbi

EVac

scm

Eg

-

-

- ++ +

E

x

EF

ECB

EVB

Vbi

EVac

sc

m

Eg

+

+

+

__ -

Solution Contact• 10^17 atoms in 1mL of 1mM solution • D.O.S. argument holds• Difference in exchange current across the

interface

++++++++++++++++

A-A-A-A-A-A-

Li+Li+Li+Li+Li+Li+

5-10 Angstroms

*Significantly less than typical W ~ 10nm

Semiconductor Contacting Phase

• No longer 1-Sided Abrupt Jxn. as the semi-conductor doesn’t have infinite capacity to accept charge

• Assume ND(n-type)=NA(p-type), then Wn=Wp

e-

h+

n-type p-type

Diodedirectionalized current

Degenerate Doping

• Dope p-type degenerately• NA>>ND --> 1-sided Abrupt Jxn.

Wn Wp

P-N HomojunctionN-type

BBB

N-type P+-type

Heterojunctions

• 2 different semiconductors grown w/ the same cyrstal structure (difficult)– Ge/GaAs ao~5.65 angstroms

Normal Staggered Broken

LASERs

• 3 Pieces --> 2 Heterjunctions– p-(Al,Ga)As | GaAs | n-(Al, Ga) As

e-

h+

h

Traps electrons and holes

Fermi-Level Pinning

EF,M

1

•Ideal Case

(only works for very ionic semiconductors like TiO2 and SnO2)

Never works for Si

Fermi-Level Pinning

Sze p. 278

Slope

A-B

1

CdS

TiO2 SnO2

Si GaAs

What’s Missing?• Fermi-Level pinning hurts

– Hinders our ability to fine tune Vbi

Vbi/Ni~Vbi/Pt~Vbi/Au

• Why does this happen?

E

x

EF

ECB

EVB

EVac

B

E

x

EF

ECB

EVB

EVac

B

vs.

*Solution contact for GaAs sees Fermi-level pinning, while the barrierheight correlates well with the electro-chemical potential for solutioncontact to Si

Devious Experimenter• Given a Si sample with a

magic type of metal on the surface X

• Thus the Fermi-level will alwaysequilibrate to the Fermi-level of X

• Thin interface --> e-’s tunnel through it and no additional potential drop is observed

E

x

EF,X

ECB

EVB

E

x

EF,X

ECB

EVB

EF,X-EF,M

What is X?

• Any source or sink for charge at the interface– Dangling bonds– Surface states– etc.

Questions

• Questions– Abrupt 1-sided junction(What is it?)– Sign of Electric P.E. and Electric Potential(Are they correct? I put them as they were in

the notes, but this doesn’t seem to agree with the algebra to me)