dopant diffusion1

7/31/2019 Dopant Diffusion1

1/20

Diffusion 1

Dopant Diffusion (Jaeger Chapter 4 and Campbell Chapter 3)

As indicated previously the main front-end processing in building a device or

integrated circuit is to selectively introduction of dopant atoms into silicon wafer.

Dopant introduction by high temperature diffusion is one of the two major processes

for achieving this. Diffusion is carried out at high temperature ( 800 to 1000 C) in adopant-rich gaseous ambient.

Diffusion also covers the redistribution in the wafer of dopant atoms introduced into

the wafer by other methods such as ion implantation.

For simplicity the diffusion of dopant in silicon is modelled by the simple diffusion

theory described by Ficks law in which the flux of dopant atoms at any point isassumed to be proportional to the gradient of the dopant concentration at that point.

The proportionality constant is the diffusion coefficient, D . This is assumed to be a

constant for a given dopant species at a given temperature. Actually D is not constant

by is dependent on dopant concentration.

This section covers the diffusion techniques used and the simple 1-D analysis

describing the dopant distribution into the wafer.


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Diffusion 2

Diffusion System

The most common practice of introducing dopantinto wafers by diffusion is the horizontal quartztube furnaces similar to that used for thermal

oxidation of silicon wafers. A separate furnace,wafer holder etc are reserved for a particulardopant to avoid contamination by other dopants.

In general the dopant atoms are introduced in twoseparate diffusion steps, each in its own tubefurnace. The first step is referred to as pre-deposition diffusion, it is aimed to introduce acontrolled amount of dopant atoms into the silicon

surface (dopant atoms per unit surface area). Thesecond step known as the drive-in diffusion whichis aimed (at least theoretically) to redistribute thedopant atoms introduced during predepositionfurther into the silicon and reducing the dopantconcentration near the surface.

Predeposition Diffusion

In pre-deposition diffusion the wafer is exposed to

excess amount of dopant atoms in a gaseousambient to ensure that the wafer surface has themaximum concentration of dopant atomsdetermined by the diffusion temperature (referredto as the solid solubility limit). Not all introduceddopant atoms are electrically active.


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Diffusion 3

The introduction of dopant atoms into the silicon (particularly at high dopant

concentration) coupled with the high thermal gradients experienced by the wafers when

they are inserted or removed from the high temperature of the furnace can cause

significant damage to the crystal structure of the single crystal (dislocations and slip)which would affect the electrical properties of the material (lifetime reduction for example).

To alleviate this, wafers are pushed into and removed from the furnace in slow controlled

manner using mechanical puller which has a slow control pull rate (~a few cm per

minute). Mechanical puller is now used for all furnaces. In modern furnaces rather than

keeping the furnace at the diffusion temperature when the wafers are inserted, it is

possible to introduced the wafers into a cool furnace (~600 C) and then ramp up the

temperature to the diffusion temperature in a controlled fashion (similarly for a ramp downon wafers removal).

The dopant atoms can be introduced in solid, liquid or gaseous form. The following table

gives the sources used for Boron and Phosphorus:

Dopant Boron Phosphorus

Form

Solid B2O3,BN P2O5Liquid BBr3 POCl3

Gaseous BCl3 PH3

The BN solid source (in solid disc form) is actually converted to B2O3 in an oxidising

ambient in a separate furnace and then loaded in the predeposition furnace with each BN

disc facing the silicon wafers. The B2O3 is then transferred to the surface of the wafers.


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Diffusion 4

These diagrams show how these are done.


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Diffusion 5

Drive-in Diffusion

The drive-in diffusion is typically carried out in an oxidising ambient but with no dopant

atoms. This is carried out after in a furnace very similar to an oxidation furnace usually at

a temperature higher than the predeposition temperature. Prior to the drive-in diffusion, avery thin layer of dopant rich silicon and/or silicon dioxide formed during predeposition is

etched away (hydrofluoric acid dip).


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Diffusion 6

Prior to the drive-in diffusion, a very thin layer of dopant rich silicon and/or silicon dioxide

formed during predeposition is etched away (hydrofluoric acid dip). The drive-in oxidelayer formed on the wafer is to seal in the dopant atoms. In simple analysis. It isassumed that no dopant atoms are lost either to the ambient or to the oxide. Due to the

conversion of silicon into silicon dioxide and the redistribution of dopant atoms at Si-SiO2interface during oxidation, there is always loss of dopant atoms, more so for born due to

the suck-out effect.

As mentioned before, during drive-in the dopant atoms diffuse further into the silicon and it

also reduces the surface dopant concentration. The resistance (more correctly the sheet

resistance) of the diffused layer increases and the junction depth increases.


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Diffusion 7

Mathematical Modelling of Diffusion 1-dimensional

2

2

x

ND

t

N

Dopant diffusion is governed by Ficks Laws(diffusion laws).

Ficks 1st Law states that the diffusion flux F (i.e.dopant flow per unit cross-section area per unit

time ) is dependent on the dopant concentrationgradient:

The variation of dopant concentration with distance into the semiconductor is known

as the dopant profile. Consider the 1-dimension case in which we consider only

dopant concentration variation in x, the direction perpendicular to the semiconductor

surface:

where N(x) is the dopant concentration and D

is the diffusion coefficient (or diffusion

coefficient) at the diffusion temperature.

Ficks 2nd Law, conservation of matter, givesthe rate of change of N with time:

x

NDF


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Diffusion 8

Predeposition Diffusion

For this process, the dopant concentration at the surface N(x=0) is limited to the solid

solubility of the dopant at the predeposition temperature No for all t >0. Assuming thatthere are no dopant atoms of this element in the silicon prior to diffusion (N =0 for all x at

t 0.Then it can

be shown that the N(x) for t>0 is given by:

.d)exp()y(erf

asdefinedfunction,errorasknownfunctionalmathematicknownwelltheiserfwhere

)y(erfd)exp()y(erfc

asdefinedfunction,errorarycomplementtheiserfcwhere

Dt

xerfcNd)exp(N)t,x(N

y

y

o

)Dt(x

o

0

2

0

2

2

0

2

2

12

1

2

21

When x = 0, N(0,t) = No independent of t. This is expected because of the solid

solubility assumption at the surface. For any x >0, N(x,t) increases with t. This gives

dopant profile of the form:


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Diffusion 9

This profile is often referred to as the complementary error function profile or the

predeposition profile. The term 2(Dt) is known as the diffusion length (not to beconfused with the carrier diffusion length (Dt) when we consider carrier diffusion in theoperation of semiconductor devices). In some literature (Dt) is referred to as thecharacteristic length of the diffusion process. Increasing (Dt) extends the dopant atomsfurther into the bulk of the semiconductor wafer this simply means that by increasingthe predeposition diffusion time we have more dopant atoms moving further into the

semiconductor while keeping the surface concentration constant at No.


10/20

Diffusion 10

The diffusion coefficient, D, is a function of diffusion

temperature following the Arrhenius relationship:

Typical Do and activation energy EA as well as D-

temperature plots are given here. For first order

approximation, D is taken to be independent of the

actual dopant level.

kT

AE

oeDD


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Diffusion 11

For a given predeposition diffusion conditions (i.e. fixed No, D and t) the total amount of

dopant atoms introduced into the silicon wafer, Q, measured in atoms per unit surface

area of the diffusion window, is determined by predeposition diffusion. Q is given by:

Concept of Junction Depth, xj

When dopant of one type (p or n) is diffused into a wafer originally doped with dopant of

the opposite type, a junction will be formed. The distance of the junction from the

surface is known as the junction depth, xj. At xj the concentration of the diffused dopant

= the concentration of the substrate dopant (assumed to have a uniform concentration).

Consider a p-type substrate with uniform dopant concentration NA subjected to the n-

type predeposition diffusion with the profile:

.dy)y(erfc

:thatfacttheofusemadehavewewhere

DtN

dxDt

xerfcNdx)x(NQ

o

o

0

1

12

2

0

0

.NDt

xerfcN

:bygiventhenisxdepth,junctionthe

,DtxerfcN)x(N

Aj

Do

j

DoD

02

2


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Diffusion 12

For x xj it remains p-type. The net dopant concentration

on n-side is ND(x) NA and on p-side it is NA-ND(x). See Left plot below. Because thedopant concentration level varies several order of magnitude, the dopant concentration

axis is usually plotted on log scale. See npn bjt dopant concentration plot below right.


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Diffusion 13

Drive-in Diffusion

Dopant diffusion during drive-in is governed by the same diffusion equation i.e. Ficks 2ndLaw for the diffusion of dopant atoms inside the silicon, i.e.

The only difference is the boundary conditions for this equation. It is usually assumed that

a layer of SiO2 is formed immediately at the surface and this serves as a mask preventing

the inward (from ambient to silicon) or outward (silicon to ambient) diffusion of dopant

atoms. Mathematically, the boundary condition is that the dopant atom flux at x=0 is zero

for all t >0 i.e.

where D1 and t1 are the diffusion coefficient and diffusion time for predeposition diffusion.

In order to obtain analytical solution for the final profile, we have to make simplifying

assumptions based on our knowledge about the diffusion sequence.

s.x'allfor0tat,)tD

x

(erfcNN(x)

i.e.condition,initialtheasprofile

ionpredepositthehavewethenion,predepositafteroutcarriedisdiffusionthatAssuming

0tafor0xatx

N

o

112

0

2

2

x

ND

t

N


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Diffusion 14

In general, predeposition diffusion is carried out at a lower temperature than drive-in

diffusion and also for a shorter time. If D2 and t2 and the diffusion coefficient and

diffusion time for drive-in, then

D2t2 >> D1t1.Under this condition, it is usual to treat the predeposition diffusion profile as a delta

function at x=0 with the area of the delta function = Q, the dopant atoms per unitsurface area introduced by predeposition i.e.

N(x,t=0)= Q.d(x),

where,

In this case, the solution of the drive-in diffusion is:

The right hand side is the well known Gaussian function and the profile is referred to as

the Gaussian diffusion profile. The peak of the profile occurs at x =0, i.e. at the surfaceas we would expect it. However notice that the magnitude of this peak decreases as the

diffusion time t increases. This is the consequence that as diffusion time increases, the

fixed amount of dopant atoms, Q, is spread out further into the silicon wafer. The plot of

Gaussian dopant profile is shown below:

1

2 DtNQ o

.Dt

xexp

Dt

Qt)N(x,

4

2


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Diffusion 15

Erfc and Gaussian profiles are only

approximate description of the actual

profiles but they are used extensively for

estimation of junction depth, resistance

(known as sheet resistance) etc of diffusedlayers.

Some reasons for deviations from erfc and

Gaussian profiles:

1.Dopant concentration dependent

diffusion coefficient2.Lateral diffusion of dopant atoms

3.Boron suck-out and phosphorus pile-up

during drive-in

4.Shallow junctions where delta function

approximation of pre-deposition profile is

not accurate.


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Diffusion 16

Sheet Resistance and Irvin Curves

With varying dopant concentration

N(x), the conductivity and resistivity ofa diffused layer also change with

distance: (x)= q(mn.n+mp.p) =1/(x).However the layer can carry current

parallel to the wafer surface and will

obey Ohm Law. For the purpose of

characterising this layer and for the

use of the layer to form resistors, thelayer is characterised by a quantity

known as sheet resistance, Rs (ors).This is the resistance between the two

opposite sides of a unit square (side

length L) of the diffused layer as

shown below. It is assumed thatcurrent flows parallel to the surface.

Breaking this layer into elemental

layers each of thickness dx. They are

conductors in parallel.


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Diffusion 17

The elemental layer between x and x+dx has net dopant concentration [N(x) NB]. Ithas a conductance:

We have assumed here that only the majority carriers contribute to the current flow.We have also ignored the effect of the space charge region of the junction.

Since all elemental layers are in parallel, the total conductance of the diffused layer,GS, the sheet conductance, is:

Or, the sheet resistance, RS, is:

RS has the unit of and it is independent of the actual value of L i.e. so long it is asquare piece of the diffused layer. RS is more commonly referred to in per square.

Also notice that if we have the diffused layer of width L (direction perpendicular tocurrent flow) and length (direction parallel to current flow) 2L, its resistance is that oftwo square units connected in series i.e. R = 2RS.

dx]N)x(N[q

L

Ldx]N)x(N[q)x(g

B

B

m

m

mjx

Bs dx]N)x(N[qG

0

m

jx

Bs

s

dx]N)x(N[qG

R

0

11


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Diffusion 18

Diffused layers are used to produce resistors in bipolar technology, typically

the base diffusion is used for this purpose because it has higher sheet

resistance. The value of a diffused resistor is given by:

R = Rs. (L/W)Where Rs is the sheet resistance of the diffused layer and L and W are the

length and width of the diffused area as seen from the top.

L

W


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Diffusion 19

Sheet Resistance Measurement

Sheet resistance of a diffused layer ( or an

implanted layer) can be measured fairly

easily using what is referred to as the

(linear four-point probe measurement on atest wafer (of dopant type opposite to that

of the diffusion) on which the diffusion is

carried out on the whole wafer. The

measurement is setup is shown in

diagram. The four probes are in line equal

distance a apart (typically 1mm). Constantcurrent I is flowing into wafer at probe 1

and out of wafer at probe 4. Potential

difference between probes 2 and 3, V is

measured by a voltmeter.

Consider first the situation that there is

only current I into probe 1 (no current in or

out of probe 4). We expect this current toflow uniformly radially outward in the

diffused layer. At a radius r from probe 1,

the current density (sheet current density)

is:

r

IJs

2


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Diffusion 20

The sheet current density is related to the radial electric field E through Ohms Law:

E is related to the radial change in potential dV/dr by:

By integrating the above expression from probe 2 to probe 3, we gat the voltage drop

from probe 2 to probe 3 due to current I flowing into probe 1:

By similar consideration of a current I flowing out at probe 4 (with no current in or out of

probe 1), we get the voltage across probe 2 and 3 as:

The combination of the two above situations is equivalent to the case of the four point

probe measurement. By superposition theorem, the measured voltage between probe 2

and 3 for the four point probe measurement is:

If the current is set at 4.532 mA, then the voltage measured is the sheet resistance in

k per square.

r

IRJRE sss

2

r

IREdr

dVs

2

)ln(I

R)a

2aln(

IRV ss 2

22

123

)ln(I

RV s 22

423

IV.

IV

)ln(Ror)ln(IRV ss

2323

235324

22

dopant diffusion1

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