chaptedopant diffusion 7 dopant diffusion _ i

7/27/2019 ChaptedOPANT dIFFUSION 7 Dopant Diffusion _ I

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1. Introduction and application.

2. Dopant solid solubility and sheet resistance.

3. Microscopic view point: diffusion equations.

4. Physical basis for diffusion.

5. Non-ideal and extrinsic diffusion.

6. Dopant segregation and effect of oxidation.

7. Manufacturing and measurement methods.

Chapter 7 Dopant Diffusion

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NE 343: Microfabrication and thin film technologyInstructor: Bo Cui, ECE, University of Waterloo; http://ece.uwaterloo.ca/~bcui/Textbook: Silicon VLSI Technology by Plummer, Deal and Griffin


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Base Emitter Collector

p

p

n+n-p+ p+

n+ n+

BJT

p well

NMOS

Doping in MOS and bipolar junction transistors

Doping is realized by: Diffusion from a gas, liquid or solid source, on or above surface. (no longer popular) Ion implantation. (choice for todays IC) Nowadays diffusion often takes place unintentionally during damage annealing Thermal budget thus needs to be controlled to minimize this unwanted diffusion.

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In this chapter, diffusion means two very different concepts: one is to dope the substrate fromsource on or above surface the purpose is doping; one is diffusion inside the substrate thepurpose is re-distribute the dopant.


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Doping profile for a p-n junction

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Diffusion from gas, liquid or solid source

Pre-deposition (dose control) Drive-in (profile control)

Silicon dioxide is used as a mask against impurity diffusion in Silicon. The mixture of dopant species, oxygen and inert gas like nitrogen, is passed over the

wafers at order of 1000 oC (900 oC to 1100 oC) in the diffusion furnace. The dopant concentration in the gas stream is sufficient to reach the solid solubility

limit for the dopant species in silicon at that temperature. The impurities can be introduced into the carrier gas from solid (evaporate), liquid

(vapor) or gas source.

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Pre-deposition

Drive-in

Comparison of ion implantation with solid/gas phase diffusion

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NE 343 Microfabrication and thin film technologyInstructor: Bo Cui, ECE, University of WaterlooTextbook: Silicon VLSI Technology by Plummer, Deal and Griffin

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Dopant solid solubility

Solid solubility: at equilibrium, the maximum concentration for an impuritybefore precipitation to form a separate phase.

Figure 7-4 8


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Solid solubility of common impurities in Silicon

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Solubility vs. electrically active dopant concentration

Not all impurities are electrically active.

As has solid solubility of 2 10 21 cm -3.

But its maximum electrically active dopant concentration is only 2 10 20 cm -3 .

V: vacancyFigure 7-5

10

As insubstitutionalsite, active

Inactive


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Resistance in a MOS

For thin doping layers, it is convenient to find the resistance from sheet resistance .

Figure 7-1

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Al

R

A

l

jS x

R R

w x j

jS x R

Sheet resistance R S

: (bulk) resistivity

x j: junction depth, or film thickness

wl R

wl

xwxl

Al R S

j j

R=Rs when l=w (square)

Figure 7-2


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Ohms law: Mobility :

By definition:

Therefore:

Finally:

Where:

E J

pn pnq

J E E v

nh vnv pq J

xnx

x

hxnh

E vn

E v pq

E vn

E v pq

x

hx p E

v x

nxn E

v

Important formulas

: conductivity; : resistivity; J: current density; E: electrical fieldv: velocity; q: charge; n, p: carrier concentration.

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Sheet resistance

Qq Nxq x x R

j j jS

111

N is carrier density, Q is total carrier per unit area, x j is junction depth

j x

B j j

S

dx xn N xnq x x

R

0

11

For non-uniform doping:

This relation is calculated to generate the so- called Irvins curves.See near the end of this slide set.

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NE 343 Microfabrication and thin film technologyInstructor: Bo Cui, ECE, University of WaterlooTextbook: Silicon VLSI Technology by Plummer, Deal and Griffin

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Diffusion from a macroscopic viewpoint

Ficks first law of diffusion

F is net flux.

x

t xC Dt x F

,,

C is impurity concentration (number/cm 3), D is diffusivity (cm 2/sec).

D is related to atomic hops over an energy barrier (formation and migration of mobile

species) and is exponentially activated.

Negative sign indicates that the flow is down the concentration gradient.

This is similar to other lawswhere cause is proportional toeffect (Fouriers law of heat flow, Ohms law for current flow).

Figure 7-6

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A

x At xC t t xC ,,

t At x F t x x F t At x x F t x F ,,,,

t At x F t x x F x At xC t t xC ,,,,

Ficks second law

The change in concentration in a volume elementis determined by the change in fluxes in and outof the volume.Within time t, impurity number change by:

During the same period, impurity diffuses in andout of the volume by:

Therefore:

Or,

Since:

We have:

x

t x F

t

t xC ),(),(

x

t xC Dt x F

,,

x

t xC D

x xt x F

t t xC ,,,

If D is constant:

2

2 ,,

x

t xC D

t

t xC

Figure 7-7

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x

C I=0

C*C s

Cg

SiO 2 Si

022

xC D

t C bxaC

Solution to diffusion equation

2

2 ,,

x

t xC D

t

t xC

At equilibrium state, C doesnt change with time.

Diffusion of oxidant (O 2 or H2O)through SiO 2 during thermaloxidation.

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Gaussian solution in an infinite medium

C 0 as t 0 for x>0

C as t 0 for x=0C(x,t)dx=Q (limited source)

This corresponds to, e.g. implant a verynarrow peak of dopant at a particular depth,which approximates a delta function.

Dt x

t C Dt x

Dt Q

t xC 4exp,04exp2,

22

Important consequences: Dose Q remains constant Peak concentration (at x=0) decreases as 1/ t Diffusion distance from origin increases as 2 Dt

Figure 7-920

At t=0, deltafunction dopantdistribution.

At t>0


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21

Dt

x

Dt

Qt xC

4exp,

2

Gaussian solution near a surface

A surface Gaussian diffusion can betreated as a Gaussian diffusion withdose 2Q in an infinite bulk medium.

Note: Pre-deposition by diffusion can alsobe replaced by a shallow implantation step.

1. Pre-depositionfor dose control

2. Drive in forprofile controlFigure 7-10


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22

Dt C

Q Dt

C C

Dt x

B

T

B

s j

ln2

ln2

Dt

Qt C C T S

,0

t xC Dt x

xt xC ,

2,

B

s

j

B

x C C

xC

xt xC

j

ln2,

Gaussian solution near a surface

Surface concentration

decreases with time

Concentration gradient

Junction depth At p-n junction


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Error function solution in an infinite medium

An infinite source of material in the half-plane can be considered to be made up of a sum of Gaussians. The diffused solutionis also given by a sum of Gaussians,known as the error-function solution.

This corresponds to, e.g. putting athick heavily doped epitaxial layer on a

lightly doped wafer.At t=0C=0 for x>0C=C for x


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Evolution of erfc diffused profile

Important consequences of error function solution: Symmetry about mid-point allows solution for constant surface concentration to be derived. Error function solution is made up of a sum of Gaussian delta function solutions. Dose beyond x=0 continues to increase with annealing time.

Figure 7-12

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Properties of Error Function erf(z) and Complementary Error Function erfc(z)

x x

2)(erf For x > 1

00erf

1erf

x

duu x0

2-exp2

erf

2exp2erf xdx

xd

22

2

exp4erf

x xdx xd

x

du-u x x 2exp2erf 1erfc

0

1)(erfc

dx x

10erfc

0erfc

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Boundary condition: C(x,0)=0, x 0; C(0,t)=C s; C( ,t)=0

Error function solution near a surfaceConstant surface concentration at all times, corresponding to, e.g., the situation of diffusion from a gas ambient, where dopants saturate at the surface (solid solubility).

Constant 1/2

Dt x

u s s due

C

Dt

xC t xC

2

22

2erfc,

0

2

2erfc Dt

C dx

Dt x

C Q s s

Pre-deposition dose

Cs is surface concentration,limited by solid solubility,which doesnt change toofast with temperature.

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S i diff i


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Dt tot D ii

t i

Successive diffusions Successive diffusions using different times and temperatures Final result depends upon the total Dt product

neff t t t D Dt ...21 ..... .

1

221112211

D D

t Dt Dt Dt D Dt eff

When D is the same (same temperature)

When diffused at different temperatures

As D increases exponentially with temperature, total diffusion (thermalbudget) is mainly determined by the higher temperature processes.

For example, the profile is a Gaussianfunction at time t=t 0, then after furtherdiffusion for another 3t 0, the final profile isstill a Gaussian with t=4t 0=t0+3t 0.

(The Gaussian solution holds only if the Dt used tointroduce the dopant is small compared with thefinal Dt for the drive-in i.e. if an initial /deltafunction approximation is reasonable)

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j x

B

j jS

dx xn N xnq x x

R

0

11

Irvins curves Motivation to generate Irvins curves: both N B (background carrier concentration), R s (sheet resistance) and x j can be conveniently measured experimentally but not N 0 (surfaceconcentration). However, these four parameters are related by:

Irvins curves are plots of N 0 versus (R s, x j) for various N B, assuming erfc or half-Gaussianprofile. There are four sets of curves for (n-type and p-type) and (Gaussian and erfc).

j x

j

dx x x 0

1

1-


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Irvins curves

Four sets of curves: p-type erfc, n-type erfc, p-type half-Gaussian, n-type half-Gaussian

Explicit relationship between: N 0, x j, NB and R S.

Once any three parameters are know, the fourth one can be determined.

Figure 7-17

1-


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ExampleDesign a boron diffusion process (say for the well or tub of a CMOS process) such that

s=900 /square, x j=3 m, with C B=1 10 15 /cm 3.

From (half- Gaussian) Irvins curve, we find Cs


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Dt x

Dt Q

t xC 4exp,

2

29

15

17

242

cm107.3

10104

ln4

103

ln4

B

s

j

C C

x Dt

Dt

xC C

j s B 4exp

2

Example (cont.)

hours8.6seccm105.1

cm107.3 21329

indrivet

213917 cm103.4107.3104 Dt C Q s

Assume drive-in at 1100 oC, then D=1.5 10 -13cm 2/s.

Pre-deposition dose

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chaptedopant diffusion 7 dopant diffusion _ i

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