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
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
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
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
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