MSE-630

Dopant Diffusion

Topics:

•Doping methods

•Resistivity and Resistivity/square

•Dopant Diffusion Calculations

-Gaussian solutions

-Error function solutions

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As devices shrink, controlling diffusion profiles with processing and annealing is critical in acquiring features down to 10-20 nm

Schematic of a MOS device cross section, showing various resistances. Xj is the junction

depth in the table above

As devices shrink, controlling the depth of the gate channel

becomes critical

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Deposition Methods•Chemical Vapor Deposition

•Evaporation

-Physical Vapor Deposition

-Sputtering

•Ion Beam Implantation

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Vapor Deposition: Chemical (CVD)

In Chemical Vapor Deposition (CVD) a

reactive gas is passed over the substrate to be

coated, inside of a heated, environmentally

controlled reaction chamber.

In this case (right) CH4 gas is introduced to

create a diamond-like coating

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Vapor Deposition: Physical (PVD)

Physical Vapor Deposition (PVD) may be from evaporation or

sputtering.

Sometimes a plasma is used to create high energy species that

collide with target (right)

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Sputtering

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Ion beam implantation gives excellent control

over the predeposition dose

and is the most widely used doping

method

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Ion beam implantationIt can cause surface damage in the form of sputtering of surface atoms, surface roughness and changes in

the crystal structure.

Though these defects can be removed by annealing, annealing also results in a

high degree of dopant diffusion.

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Resistivity and Sheet Resistance

From Ohm’s Law: J =

Where J = current density (A/cm2) = electric field strength (V/m)=resistivity (cm)

Thus

= /J

In semiconductors, the doped regions have higher conductivity than the sheet as a whole. We

are interested in the depth of the junction, xj. The resistance we measure is that of a square of any dimension with depth xj, or

R = /xj /square ≡ s

for uniform doping.

For variable doping: dxxnNxnqx jx

Bjs

)()(

11

0

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Solid solubility

Sometimes dopants cluster around vacancies and other

point defects, as above, becoming electrically neutral. As a result, effective level of doping may be lower than equilibrium values in the

adjacent figure

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Diffusion Models

Fick’s 1st law: F = -D dC/dx

Fick’s 2nd law:

C/t = F/x = (Fin – Fout)/x

dC/dt = D d2C/dx2

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Diffusion in SiliconIn general, diffusivity is

given by:

D = Doexp(-Ea/kT)

Where Ea = activation energy ~ 3.5 – 4.5 eV/atom

k = 8.61x10-5 eV/atom-KThis applies to intrinsic

conditions. Dopant levels (ND, NA) need to be less than the intrinsic carrier density, ni

as shown in the graph

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DtxtC

Dtx

DtQtxC

4exp),0(

4exp

2),(

22

QdxtxC

andxfortasCxfortasC

),(

00000

Gaussian Solution in an Infinite Medium

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Gaussian Diffusion near a Surface

DtxtC

Dtx

DTQtxC

4exp),0(

4exp),(

22

DtQtC

),0(

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Error-Function solution in an Infinite Medium

00000

xfortatCCxfortatC

DtxerfCtxC

21

2),(

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Error-Function solution near a Surface

)(1)(

2),(

21),(

xerfxerfcwhere

DtxerfcCtxC

orDtxerfCtxC

s

s

This solution assumes the concentration C is at the solid solubility limit and is infinite

DtCDtxerfCQ s

s 2

21

0

The dose, Q, is calculated by summing the concentration:

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Effect of successive diffusion steps

If diffusion occurs at constant temperature, where the diffusivity is constant, then the effective thermal budget, Dt is:

(Dt)eff = D1t1+D1t2+…D1tn

If D is not constant, then time is increased by the ratio of D2/D1, or

(Dt)eff = D1t1+D1t2(D2/D1)+…D1tn(Dn/D1)

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