Fick’s LawsFick’s Laws Combining the continuity equation with Combining the continuity equation with

the first law, we obtain Fick’s second law:the first law, we obtain Fick’s second law:

c

tD

c

x

2

2

Solutions to Fick’s Laws depend on the Solutions to Fick’s Laws depend on the boundary conditions.boundary conditions.

Assumptions Assumptions – D is independent of concentrationD is independent of concentration– Semiconductor is a semi-infinite slab with Semiconductor is a semi-infinite slab with

eithereither Continuous supply of impurities that can Continuous supply of impurities that can

move into wafermove into wafer Fixed supply of impurities that can be Fixed supply of impurities that can be

depleteddepleted

Solutions To Fick’s Second Solutions To Fick’s Second LawLaw

The simplest solution The simplest solution is at steady state and is at steady state and there is no variation of there is no variation of the concentration with the concentration with timetime– Concentration of Concentration of

diffusing impurities is diffusing impurities is linear over distancelinear over distance

This was the solution This was the solution for the flow of oxygen for the flow of oxygen from the surface to from the surface to the Si/SiOthe Si/SiO22 interface in interface in the last chapterthe last chapter

bxaxc

x

cD

)(

02

2

Solutions To Fick’s Second Solutions To Fick’s Second LawLaw

For a semi-infinite slab with a constant For a semi-infinite slab with a constant (infinite) supply of atoms at the surface(infinite) supply of atoms at the surface

The dose isThe dose is

0 02, DtcdxtxcQ

Dt

xerfcctxc o

2),(

Solutions To Fick’s Second Solutions To Fick’s Second LawLaw

Complimentary error function (erfc) is Complimentary error function (erfc) is defined as erfc(x) = 1 - erf(x)defined as erfc(x) = 1 - erf(x)

The error function is defined asThe error function is defined as

– This is a tabulated function. There are This is a tabulated function. There are several approximations. It can be found as a several approximations. It can be found as a built-in function in MatLab, MathCad, and built-in function in MatLab, MathCad, and MathematicaMathematica

z

dzerf0

2exp2

)(

Solutions To Fick’s Second Solutions To Fick’s Second LawLaw

This solution models short diffusions This solution models short diffusions from a gas-phase or liquid phase from a gas-phase or liquid phase sourcesource

Typical solutions have the following Typical solutions have the following shapeshape

c0

cB

Distance from surface, x

1 2 3

D3t3 > D2t2 > D1t1

Impu

rity

con

cent

rati

on, c

(x)

c ( x, t )

Solutions To Fick’s Second Solutions To Fick’s Second LawLaw

Constant source diffusion Constant source diffusion has a solution of the formhas a solution of the form

Here, Q is the does or the Here, Q is the does or the total number of dopant total number of dopant atoms diffused into the Siatoms diffused into the Si

The surface The surface concentration is given by:concentration is given by:

dxtxcQ

0

),(

Dt

Qtc

),0(

Dt

x

eDt

Qtxc 4

2

),(

Solutions To Fick’s Second Solutions To Fick’s Second LawLaw

Limited source diffusion looks likeLimited source diffusion looks like

c ( x, t )c01

c02

c03

cB

1 2 3

Distance from surface, x

D3t3 > D2t2 > D1t1

Impu

rity

con

cent

rati

on, c

(x)

Comparison of limited Comparison of limited source and constant source source and constant source

modelsmodels1

10-1

10-2

10-3

10-4

10-5

10-6

0 0.5 1 1.5 2 2.5 3 3.5

Val

ue o

f fu

ncti

ons

Normalized distance from surface, x_

xx

Dt

_

2

exp(- )x_

2

erfc( )x_

Predep and DrivePredep and Drive

PredepositionPredeposition– Usually a short diffusion using a Usually a short diffusion using a

constant sourceconstant source DriveDrive

– A limited source diffusionA limited source diffusion The diffusion dose is generally the dopants The diffusion dose is generally the dopants

introduced into the semiconductor during introduced into the semiconductor during the predepthe predep

A DtA Dteffeff is not used in this case. is not used in this case.

Diffusion CoefficientDiffusion Coefficient Probability of a jump isProbability of a jump is

Diffusion coefficient is proportional to Diffusion coefficient is proportional to jump probabilityjump probability

P P Pj v m

e eE kT E kTf m

D D e E kTD 0

Diffusion CoefficientDiffusion Coefficient Typical diffusion coefficients in siliconTypical diffusion coefficients in silicon

ElemenElementt

DDoo (cm (cm22/s)/s) EEDD (eV)(eV)

BB 10.510.5 3.693.69

AlAl 8.008.00 3.473.47

GaGa 3.603.60 3.513.51

InIn 16.516.5 3.903.90

PP 10.510.5 3.693.69

AsAs 0.320.32 3.563.56

SbSb 5.605.60 3.953.95

Diffusion Of Impurities In Diffusion Of Impurities In SiliconSilicon

Arrhenius plots of diffusion in siliconArrhenius plots of diffusion in silicon

1400 1300 1200 1100 1000

Temperature (o C)

10-9

10-10

10-11

10-12

10-13

10-14

0.6 0.65 0.7 0.75 0.8 0.85Temperature, 1000/T (K-1)

Al

Ga

B,P

In

AsSb

Dif

fusi

on c

oeff

icie

nt, D

(cm

2 /se

c)

10-4

10-5

10-6

10-7

10-8

0.6 0.7 0.8 0.9 1.0 1.1Temperature, 1000/T (K-1)

1200 1100 1000 900 800 700

Temperature (o C)

Dif

fusi

on c

oeff

icie

nt, D

(cm

2 /se

c)

Li

Cu

Fe

Au

Diffusion Of Impurities In Diffusion Of Impurities In SiliconSilicon

The intrinsic carrier concentration in The intrinsic carrier concentration in Si is about 7 x 10Si is about 7 x 101818/cm/cm33 at 1000 at 1000 ooCC– If NIf NAA and N and NDD are <n are <nii, the material will , the material will

behave as if it were intrinsic; there are behave as if it were intrinsic; there are many practical situations where this is a many practical situations where this is a good assumptiongood assumption

Diffusion Of Impurities In Diffusion Of Impurities In SiliconSilicon

Dopants cluster into “fast” diffusers Dopants cluster into “fast” diffusers (P, B, In) and “slow” diffusers (As, Sb)(P, B, In) and “slow” diffusers (As, Sb)– As we develop shallow junction devices, As we develop shallow junction devices,

slow diffusers are becoming very slow diffusers are becoming very importantimportant

– B is the only p-type dopant that has a B is the only p-type dopant that has a high solubility; therefore, it is very hard high solubility; therefore, it is very hard to make shallow p-type junctions with to make shallow p-type junctions with this fast diffuserthis fast diffuser

Limitations of TheoryLimitations of Theory

Theories given here break down at high Theories given here break down at high concentrations of dopantsconcentrations of dopants– NNDD or N or NAA >> n >> nii at diffusion temperature at diffusion temperature

If there are different species of the same If there are different species of the same atom diffuse into the semiconductoratom diffuse into the semiconductor– Multiple diffusion frontsMultiple diffusion fronts

Example: P in SiExample: P in Si

– Diffusion mechanism are differentDiffusion mechanism are different Example: Zn in GaAsExample: Zn in GaAs

– Surface pile-up vs. segregationSurface pile-up vs. segregation B and P in SiB and P in Si

Successive DiffusionsSuccessive Diffusions To create devices, successive diffusions To create devices, successive diffusions

of n- and p-type dopantsof n- and p-type dopants– Impurities will move as succeeding dopant Impurities will move as succeeding dopant

or oxidation steps are performedor oxidation steps are performed The effective Dt product isThe effective Dt product is

– No difference between diffusion in one step No difference between diffusion in one step or in several steps at the same temperatureor in several steps at the same temperature

If diffusions are done at different If diffusions are done at different temperaturestemperatures

2111211 )()( tDtDttDDt eff

2211)( tDtDDt eff

Successive DiffusionsSuccessive Diffusions The effective Dt product is given byThe effective Dt product is given by

DDii and t and tii are the diffusion coefficient and are the diffusion coefficient and time for itime for ithth step step– Assuming that the diffusion constant is only a Assuming that the diffusion constant is only a

function of temperature.function of temperature.– The same type of diffusion is conducted The same type of diffusion is conducted

(constant or limited source)(constant or limited source)

i

iieff tDDt

Junction FormationJunction Formation When diffuse n- and p-type materials, When diffuse n- and p-type materials,

we create a pn junctionwe create a pn junction– When NWhen NDD = N = NAA , the semiconductor material , the semiconductor material

is compensated and we create a is compensated and we create a metallurgical junctionmetallurgical junction

– At metallurgical junction the material At metallurgical junction the material behaves intrinsicbehaves intrinsic

– Calculate the position of the metallurgical Calculate the position of the metallurgical junction for those systems for which our junction for those systems for which our analytical model is a good fitanalytical model is a good fit

Junction FormationJunction Formation Formation of a pn junction by diffusionFormation of a pn junction by diffusion

Impurity

concentrationN(x)

N0

NB

(log

sca

le)

xj

p-type Gaussian diffusion(boron)

n-type silicon

background

Distance from surface, x

Net impurityconcentration

|N(x) - NB |

N0 - NB

p-type

region

n-type region

xj

Distance from surface, x

(log

sca

le)

Junction FormationJunction Formation The position of the junction for a The position of the junction for a

limited source diffused impurity in a limited source diffused impurity in a constant background is given byconstant background is given by

The position of the junction for a The position of the junction for a continuous source diffused impurity continuous source diffused impurity is given byis given by

x Dt NNjB

2 0ln

x Dt NNj

B 2 1

0erfc

Junction FormationJunction Formation

Junction Depth Lateral Diffusion

Design and EvaluationDesign and Evaluation

There are three parameters that There are three parameters that define a diffused regiondefine a diffused region– The surface concentrationThe surface concentration– The junction depthThe junction depth– The sheet resistanceThe sheet resistance

These parameters are not independentThese parameters are not independent

Irvin developed a relationship that Irvin developed a relationship that describes these parametersdescribes these parameters

jx

B

jS

dxxnNxnqx

0

)()(

11

Irvin’s CurvesIrvin’s Curves

In designing processes, we need to In designing processes, we need to use all available datause all available data– We need to determine if one of the We need to determine if one of the

analytic solutions appliesanalytic solutions applies For example, For example,

– If the surface concentration is near the solubility If the surface concentration is near the solubility limit, the continuous (erf) solution may be appliedlimit, the continuous (erf) solution may be applied

– If we have a low surface concentration, the If we have a low surface concentration, the limited source (Gaussian) solution may be appliedlimited source (Gaussian) solution may be applied

Irvin’s CurvesIrvin’s Curves If we describe the dopant profile by either the If we describe the dopant profile by either the

Gaussian or the erf modelGaussian or the erf model– The surface concentration becomes a parameter in The surface concentration becomes a parameter in

this integrationthis integration– By rearranging the variables, we find that the surface By rearranging the variables, we find that the surface

concentration and the product of sheet resistance and concentration and the product of sheet resistance and the junction depth are related by the definite integral the junction depth are related by the definite integral of the profileof the profile

There are four separate curves to be evaluatedThere are four separate curves to be evaluated– one pair using either the Gaussian or the erf function, one pair using either the Gaussian or the erf function,

and the other pair for n- or p-type materials because and the other pair for n- or p-type materials because the mobility is different for electrons and holesthe mobility is different for electrons and holes

Irvin’s CurvesIrvin’s Curves

Irvin’s CurvesIrvin’s Curves An alternative way of presenting the data may be An alternative way of presenting the data may be

found if we set found if we set effeff=1/=1/ssxxjj