Diffusion

Impurity Diffusion • Fundamental process step for

microelectronics

– Controls majority carrier type

– Controls semiconductor resistivity

• We want Substitutional diffusion

– Needed to provide carriers

Silicon Dopant Types • N-type (electron donor)

– P, As, Sb

• P-type (hole donor)

– B

– (Al+Ga have high diffusion constants/don’t mask well)

III IV V

Sb

Methods for Doping Silicon

• Diffusion

• Ion-Implantation

• Combinations of the above

Diffusion Fick’s First Law

Particle flux J is proportional to the negative

of the gradient of the particle concentration

J DN

x

D = diffusion coefficient

• Same mathematical “model” as oxidation model

Diffusion Fick’s Second Law

Continuity Equation for Particle Flux :

Rate of increase of concentration is equal to the

negative of the divergence of the particle flux

N

t

J

x

(in one dimension)

Fick's Second Law of Diffusion :

Combine First Law with Continuity Eqn.

N

t D

2N

x 2

D assumed to be independent of concentration!

• We use this because we are in a non-steady state situation, dopants continually diffuse

• Dose (Q) = Impurities/cm^2

Constant Source Diffusion Complementary Error Function Profiles

FunctionError ary Complement=erfc

tCoefficienDiffusion

ionConcentrat Surface

2, :Dose Total

2, :ionConcentrat

0

0

0

0

D

N

DtNdttxNQ

Dt

xerfcNtxN

erfc z 1 erf z

erf z 2

exp x 2 dx

0

z

• Solve PDE with boundary conditions (No=const)

• Dose changes over time

• Furnace/chamber/etc

Limited Source Diffusion Gaussian Profiles

Concentration :

N x,t N0 exp x

2 Dt

2

Q

Dtexp

x

2 Dt

2

N0 Surface Concentration N0 Q

Dt

D Diffusion Coefficient

Gaussian Profile

Initial Impulse with Dose Q

• Solve PDE with boundary condition (Impulse dose at surf) • Source never is

replenished • Area under each curve

(dose) is constant

Diffusion Profile Comparison

Complementary Error Function and Gaussian Profiles are Similar in Shape

erfc z 1 erf z

erf z 2

exp x 2 dx

0

z

Diffusion Coefficients

Substitutional Diffusers Interstitial

Diffusers

Diffusion Coefficients

D DO exp EAkT

Arrhenius Relationship

E A activation energy

k = Boltzmann' s constant =1.38 x10 -23 J/K

T = absolute temperature

• Dt product is the measure of driving force in the diffusion – D is proportional to Temp – Time (t) – Increase either of these or both and you will change

the diffusion parameters

• At high concentrations (~ni) diffusion constant becomes dependant on concentration

Two-step Diffusion Process • Short, high concentration

constant source pre-diffusion approximates impulse dose at surface

• Longer “drive in” step diffuses impurities into lattice

• If Dt for drive in >> Dt for predeposition – Final profile will be Gaussian - -

- MOST CASES

• If Dt for drive in << Dt for predeposition – Final profile will be Erfc fn.

Successive Diffusions

• Successive diffusions using different times and temperatures

• Any process which involves high temperatures also affect this

• Final result depends upon the total Dt product

• This (Dt)tot is plugged into the equation to determine final distribution

Dt tot Di

i

ti

Diffusion Solid Solubility Limits

• There is a limit to the amount of a given impurity that can be “dissolved” in silicon (the Solid Solubility Limit)

• At high concentrations, all of the impurities introduced into silicon will not be electrically active

Diffusion p-n Junction Formation

B

01-

B

0

N

Nerfc 2 :profileFunction Error

N

Nln 2 :ProfileGaussian

DepthJunction calMetallurgi

Dtx

Dtx

x

j

j

j

• P-n junction occurs where the net

impurity concentration is = 0 • P doping cancels n doping/ etc.

• Set N(xj)=0 • Solve equations for xj

Lateral Diffusion Under Mask Edge

Original Mask

Concentration Dependent Diffusion

Second Law of Diffusion

N

t

xD x

N

x

Profiles More Abrupt at High Concentrations

Concentration Dependent Diffusion

• Phosphorus diffusion is more complex, includes a “Kink” which makes it harder to use in actual devices

• Arsenic used instead

Diffusion Resistivity vs. Doping

1 q nn p p 1

n type : qn ND NA 1

p type : qp NA ND 1

Resistors Sheet Resistance

A W t

R

t

L

W

RS

L

W

RS

t= Sheet Resistance [Ohms per Square]

L

W

Number of Squares of Material

Resistors Counting Squares

Figure 4.14

• Top and Side Views of Two Resistors of Different Size

• Resistors Have Same Value of Resistance

• Each Resistor is 7 sq in Length

• Each End Contributes Approximately 0.65 sq

• Total for Each is 8.3 sq

Resistors Contact and Corner Contributions

• Effective Square Contributions of Various Resistor End and Corner Configurations

Figure 4.15

Sheet Resistance

Irvin’s Curves

• Irvin Evaluated this Integral and Published a Set of Normalized Curves Plot Surface Concentration Versus Average Resistivity

• Four Sets of Curves – n-type and p-type

– Gaussian and erfc

1

1

1

x j x dx

0

x j

RS

x j

1

x dx0

x j

RS qN x dx0

x j

1

RSx j

Two Step Diffusion Sheet Resistance - Predep Step

Initial Profile

No 1.1x1020 /cm 3

NB 3x1016 /cm 3

x j 0.0587 m

p type erfc profile

RSx j 50 -m

RS 32 -m

0.0587 m 850 /Square

Two Step Diffusion Sheet Resistance - Drive-in Step

Final Profile

No 1.1x1018 /cm 3

NB 3x1016 /cm 3

x j 2.73 m

p type Gaussian profile

RSx j 700 -m

RS 700 -m

2.73 m 260 /Square

Doping Systems • Spin on

– Glass containing the dopant impurity • Not as uniform of a doping

• Furnaces (3 zone) – Source material

• Liquid, Solid, Gas

– Boron • Gas/solids react to supply impurities on surf

– Phosphorus • Gas/solids react to supply impurities on surf

– Arsenic • Hard to make high concentrations with furnace methods –

Use ion implantation