Chapter 5

UEEP2613

Microelectronic Fabrication

Diffusion

Prepared by

Dr. Lim Soo King

24 Jun 2012

- i -

Chapter 5 Diffusion .........................................................................131

5.0 Introduction ............................................................................................ 131

5.1 Model of Diffusion in Solid .................................................................... 133

5.2 Fick’s Diffusion Equation ...................................................................... 134

5.2.1 Constant Diffusivity ........................................................................................ 135 5.2.1.1 Constant Surface Concentration ............................................................................... 135 5.2.1.2 Constant Total Dopant ............................................................................................... 136 5.2.1.3 Sheet Resistance of a Diffused Layer ........................................................................ 138 5.2.1.4 Effect of Successive Diffusion Steps ........................................................................... 139

5.2.2 Concentration Dependent Diffusivity ........................................................... 140 5.2.3 Temperature Dependent Diffusivity ............................................................. 142

Exercises ........................................................................................................ 144

Bibliography ................................................................................................. 146

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Figure 5.1: Schematic of a diffusion system using liquid source ..................................... 132 Figure 5.2: Mechanism of diffusion in solid .................................................................... 133 Figure 5.3: Plot of complementary error function ............................................................ 136 Figure 5.4: A surface Gaussian diffusion with total dopant QT at the center of silicon ... 138 Figure 5.5: A typical Irvin curve for p-type Gaussian profile in an n-type background

concentration .................................................................................................. 139 Figure 5.6: Diffusivity dependent on doping concentration ............................................. 141 Figure 5.7: Concentration dependent diffusivity of common dopant in single crystal

silicon ............................................................................................................. 142 Figure 5.8: Arrhenius plot of diffusivity of the common dopants in silicon .................... 143

Figure 5.9: Temperature dependence of the diffusivity coefficient of common dopant in

silicon ............................................................................................................. 143 Figure 5.10: Intrinsic diffusivity for silicon self diffusion of common dopants ................ 144

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Chapter 5

Diffusion

_____________________________________________

5.0 Introduction

Diffusion of impurity atom or dapant in silicon is an important process in

silicon integrated circuit. Using diffusion techniques, altering the conductivity

in silicon or germanium was disclosed in a patent by William Gardner Pfann in

1952. Since then, various ways of introducing dopant into silicon by diffusion

have been studied with the goal of controlling the distribution of dopant, the

concentration of total dopant, its uniformity, and reproducibility, and for

processing large number of device wafer in a batch to reduce the manufacturing

cost.

Diffusion is used to form emitter, base, and resistor for the bipolar device

technology. It is also used to form drain and source regions and to dope

polysilicon in MOS device technology. Dopant that spans a wide range of

concentration can be introduced by a number of ways. The most common way

of diffusion is from chemical source in vapor form at high temperature. The

other ways are diffusion from a doped oxide source and diffusion and annealing

from ion implanted layer. Ion implantation can provide 1011

cm-2

to greater than

1016

cm-2

. It is used to replace the chemical or doped oxide source wherever

possible and is extensively used in VLSI/ULSI device fabrication.

Diffusion of impurities is typically done by placing semiconductor wafers

in a carefully controlled, high temperature quartz-tube furnace and passing a gas

mixture that contains the desired dopant through it. Its purpose is to introduce

dopant into silicon crystal. Mixture of oxygen and dopants such as diborane and

phosphine are introduced in the furnace with the exposed wafer surface at

temperature ranges between 8000C and 1,200

0C for silicon and 600

0C and

1,0000C for gallium arsenide. The number of dopant atoms that diffused into the

semiconductor is related to the partial pressure of the dopant impurity in the gas

mixture.

The schematic of a diffusion system using liquid source is shown in Fig.

5.1.

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Figure 5.1: Schematic of a diffusion system using liquid source

Dopant can be introduced by solid source such as BN for boron, As2O3 for

arsenic, and P2O5 for phosphorus, gases source such as B2H6, AsH3, and PH3,

and liguid source such as BBr3, AsCl3, and POCl3. However, liquid source is the

commonly used method. The chemical reaction for phosphorus diffusion using

liquid source POCl3 is shown as follow.

4POCl3 + 3O2 → 2P2O5 + 6Cl2 ↑ (5.1)

The P2O5 forms a glass-on-silicon wafer and then reduces to phosphorus by

silicon following the equation.

2P2O5 + 5Si → 4P + 5SiO2 ↑ (5.2)

The phosphorus is released and diffused into silicon Si and chlorine Cl2 gas is

vented.

For diffusion in gallium arsenide, the high vapor pressure of arsenic

requires special method to prevent loss of arsenic by decomposition or

evaporation. These methods include diffusion in sealed ampules with over

pressure of arsenic and diffusion in an open-tube furnace with doped oxide

capping layer such as silicon nitride. Most of the studies on p-type diffusion

have been confined to the use of zinc in the forms of Zn-Ga-As alloys and

ZnAs2 for the sealed-ampule approach or ZnO-SiO2 for the open-tube approach.

The n-type dopants in gallium arsenide include selenium and tellurium. To

complete the process, 'drive in' or re-distribution of dopant is done in nitrogen or

wet oxygen where silicon dioxide SiO2 is grown at the same time.

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5.1 Model of Diffusion in Solid

At high temperature, point defects such as vacancy and self interstitial atom are

generated in a single crystalline solid. When concentration gradient of the host

or impurity atom exists, such point defect affects the atom movement namely as

diffusion. Diffusion in solid can be treated as the atomic movement of diffusant

either impurity atom or host atom in crystal lattice by vacancy and self

intersitial.

There are several types of diffusion mechanisms. They are vacancy

diffusion, intersitial diffusion, divacancy or diffusion assisted by a double

vacancies, and interstitialcy diffusion. Figure 5.2 illustrates the diffusion

mechanisms by vacancy, interstitial, and interstitialcy. At elevated temperature,

the atom in crystal lattice vibrates in its equilibrium site. Occasionally, the atom

acquires sufficient energy to leave its equilibrium site and becomes a self

interstitial atom. If there is an impurity atom (red color) around or a neigboring

host atom, it can occupy this vacant site and this type of diffusion is termed as

diffusion by a vacancy. This type of diffusion is illustrated by diffusion

mechanism 1 shown in Fig. 5.2(a). If the the migrating atom is a host atom, it is

called self diffusion. If it is a impurity atom, then it is called impurity diffusion.

(a) Vacancy diffusion and interstitial diffusion (b) Interstitialcy diffusion

Figure 5.2: Mechanism of diffusion in solid

If the movement of impurity atom is in between equilibrium site of the crystal

lattice that does not involve occupying lattice site, it is called interstitial

diffusion as illustrated by diffusion mechanism 2 shown in Fig. 5.2(a).

Diffusion assisted by a double vacancy or divacancy is a diffusion mechanism

involving impurity atom has to move to a second vacancy that is at the nearest

neighbor of the original vacancy site.

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Interstitialcy diffusion is shown in Fig. 5.2(b). The mechanisms are shown

by four steps. In step 1, a self interstitial host atom displaces an impurity atom

(read color) from the lattice site and makes this impurity atom as interstitial

atom (step 2). This interstitial impurity atom then displaces a host atom (step 3)

from its equilibrium site and makes this host atom to become interstitial atom

(step 4).

Vacancy and interstitialcy diffusions are commonly happened for

phosphorus P, boron B, arsenic As, antimony Sb impurity diffusion in silicon.

However, for phosphorus P and boron B diffusion, interstitialcy diffusion is

more dominant than vacancy diffusion. Vacancy diffusion is more dominant

than interstitialcy diffusion for arsenic As and antimony Sb diffusion. Group 1

and VIII elements have small ionic radii and are fast diffuser in silicon. The

diffusion is normally involved interstitial diffusion.

5.2 Fick’s Diffusion Equation

In 1855, Adolf Fick published the theory of diffusion. His theory was based on

the analogy between material transfer in a solution and heat transfer by

conduction. Fick’s assumed that in dilute liquid or gaseous solution, in the

absence of convection, the transfer solute atom per unit area in one direction

flow can be described by Fick’s first law of diffusion shown in equation (5.3).

x

)t,x(CDJ

(5.3)

where J is the local rate of transfer of solute per unit area or the diffusion flux, C

is the concentration of solute is a function of x and t, x is the coordinate axis in

the direction of solute flow, t is the diffusion time, and D is the diffusivity or at

time it is called diffusion coefficient or diffusion constant. The negative sign of

the equation denotes that the solute flows to the direction of lower concentration.

From the law of conservation of matter, the change of solute concentration

with time must be the same as the local decrease of the diffusion flux in the

absence of a source or sink. Thus,

x

)t,x(J

t

)t,x(C

(5.4)

Substitute equation (5.3) into equation (5.4) yields equation of Fick’s second

law in one dimensional form, which is

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x

)t,x(JD

xt

)t,x(C (5.5)

When the concentration of solute is low, the diffusivity at a given temperature

can be considered as a constant then equation (5.5) shall become

2

2

x

)t,x(JD

t

)t,x(C

(5.6)

Equation (5.6) is another form of Fick’s second law of diffusion. In equation

(5.6), D is given in unit of cm2/s or m

2/h and C(x, t) is in unit of atom/cm

3. The

solution for equation (5.6) for various initial condition and boundary condition

shall be dealt in next sub-section.

5.2.1 Constant Diffusivity

The solution of diffusion equation shown in equation (5.6) has constant

diffusivity or diffusion coefficient for constant surface concentration and

constant total dopant will be discussed in this sub-section. The sheet resistance

of a diffused layer of constant diffusivity will be discussed too.

5.2.1.1 Constant Surface Concentration

For the case of constant surface concentration, the initial condition at time t = 0

is C(x, 0) = 0 and the boundary conditions are C(0, t) = Cs and C(, t) = 0. The

solution of equation (5.6) is equal to

Dt2

xerfcC)t,x(C S (5.7)

where Cs is the surface concentration, D is the constant diffusivity, x is the

distance, t is the diffusion time, and erfc is the complementary error function.

The plot of complementary error function erfc of equation (5.7) is shown in

Fig. 5.3.

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Figure 5.3: Plot of complementary error function

Since erfc(x) = 1 – erf(x), equation (5.7) is also equal to

Dt2

xerf1C)t,x(C S (5.8)

From the result shown in Fig. 5.2, the error function solution is approximately a

triangular function, so that the total amount of dopant per unit area introduced

can be approximated by DtCQ ST . A more accurate answer for the total

amount of dopant introduced per unit area is

DtC2

dxDt2

xerf1CQ S

0

ST

(5.9)

5.2.1.2 Constant Total Dopant

If a thin layer of dopant is deposited onto the silicon surface with a fixed or

constant total amount of dopant QT per unit area. This dopant diffuses only into

the silicon and all the dopants remain in the silicon. The initial and boundary

conditions are initial condition C(x, 0) = 0 and boundary condition

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T

0

Qdx)t,x(C

and C(, t) = 0. The solution of the diffusion equation shown in

equation (5.6) shall be

Dt4

xexp

Dt

Q)t,x(C

2

T (5.10)

If x = 0, equation (5.10) is equal to surface concentration CS, which is

Dt

Q)t,0(CC T

S

(5.11)

Combining equation (5.10) and (5.11) yields equation (5.12).

Dt4

xexpC)t,x(C

2

S (5.12)

Equation (5.10) is often called the Gaussian distribution and the diffusion

concentration is referred to dopant concentration of the pre-deposited thin layer

source or drive-in diffusion from a fixed total dopant concentration. Impurity

atom distribution from ion implantation into amorphous material can be

approximated by Gaussian function.

For the case whereby a thin layer of dopant is deposited in the center the

silicon surface, the diffusion profile looks as what is shown in Fig. 5.4. The

diffusion flux will be equal to

Dt4

xexp

Dt2

Q)t,x(C

2

T (5.13)

whereby the assumption is that half of the QT will diffuse virtually.

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Figure 5.4: A surface Gaussian diffusion with total dopant QT at the center of silicon

5.2.1.3 Sheet Resistance of a Diffused Layer

For a diffused layer that form pn junction, an average sheet resistance RS is

defined and related to junction depth xj, the carrier mobility , and the impurity

distribution C(xi) by the equation (5.14).

jj x

0

eff

x

0

S

dx)x(Cq

1

dx)x(Cq

1R (5.14)

Empirical expression of mobility versus impurity distribution C has been

determined for concentration above 1016

cm-3

in silicon. The donor dopant

mobility n is

Vs/cm0.92)10x3.1/C(1

901360 2

91.017n

(5.15)

For acceptor concentration in silicon for acceptor dopant, the mobility p is

Vs/cm7.49)10x6.1/C(1

7.49468 2

7.017p

(5.16)

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The sheet resistance RS is also equal to

j

Sx

R

(5.17)

where is the resistivity. Thus, the effective conductivity is equal to

jSxR

1 (5.18)

Once the surface resistance, surface concentration, and junction depth are

known, one can design a diffused layer. There is a useful design curve called

Irvin curve that can be used to determine the surface concentration CS versus the

effective conductivity on background concentration CB, which shown in Fig. 5.5.

Figure 5.5: A typical Irvin curve for p-type Gaussian profile in an n-type background

concentration

5.2.1.4 Effect of Successive Diffusion Steps

Since there are often multiple diffusion steps in a fully integrated circuit

process, they must be added in some ways before the final profile can be

predicted. It is clear that if all the diffusion steps occurred at a constant

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temperature where the diffusivity is the same then the effective Dt product is

given by

.....tDtD...)tt(D)Dt( 2111211eff (5.19)

In other words doing a single step in a furnace for a total time of t1 + t2 is the

same as doing two separate steps, one for time t1 and one for time t2.

Mathematically, one could increase the time t2 by a numerical factor D2/D1

and re-write equation (5.19) as

22112

1

2111eff tDtDt

D

DDtD)Dt(

(5.20)

Thus, the derived formula for the total effective Dt for a dopant that is diffused

at a temperature T1 with diffusivity D1 for time t1 and then diffused at

temperature T2 with diffusivity D2 for time t2. The total effective Dt is given by

the sum of all the individual Dt products.

5.2.2 Concentration Dependent Diffusivity

At high concentration, when the diffusion conditions are closed to the constant

surface concentration case or the constant total dopant case, the measured

impurity profiles are not the same as the constant diffusivity cases. For high

concentration case, it can be represented by concentration dependent diffusivity.

Anderson and Lisak obtained the concentration dependent diffusivity equation

by changing the diffusivity of equation (5.5) with equation (5.19), which is

r

i

in

CD2D

(5.21)

where Di is the constant diffusivity at low concentration or intrinsic diffusivity;

C is doping concentration; and ni is the intrinsic concentration, and r is a

constant. We shall further discuss this equation.

Based on many experiment results, the diffusivity of common dopants in

silicon has been characterized and found to depend linearly or sometimes

quadratically on the carrier concentration as shown in Fig. 5.6.

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Figure 5.6: Diffusivity dependent on doping concentration

The effective diffusivity under extrinsic condition based on equation (5.6) can

be written as

2

i

2

i

0eff

A DDDD

n

n

n

nfor n-type dopant (5.22)

2

i

2

i

0eff

A DDDD

n

p

n

p for p-type dopant (5.23)

D0 and D

+ etc. are chosen because on the atomic level. These different terms are

thought to occur because of the interaction with neutral and charged point defect.

The diffusivity under intrinsic condition for an n-type dopant (n = p = ni) is

20*

A DDDD (5.24)

The concentration dependent diffusivity of common dopant in single crystal

silicon is shown in Fig. 5.7.

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By re-writing equation (5.22) and (5.24), the diffusivity measured under

extrinsic conditions can be elegantly described by equation (5.25).

1

n

n

n

n1

DD

2

ii*

A

eff

A (5.25)

where 0D/D and 02 D/D .

Si B In As Sb P Unit

Do, 0 560 0.05 0.6 0.011 0.214 3.85 cm

2s

-1

Do, E 4.76 3.5 3.5 3.5 3.65 3.66 eV

D+, 0 0.95 0.6 cm

2s

-1

D+, E 3.5 3.5 eV

D-, 0 31.0 15.0 4.44 cm

2s

-1

D-, E 4.15 4.08 4.0 eV

D2-

, 0 44.2 cm2s

-1

D2-

, E 43.7 eV

Figure 5.7: Concentration dependent diffusivity of common dopant in single crystal silicon

5.2.3 Temperature Dependent Diffusivity

The diffusivity determined experimentally over range of diffusion temperature

can be expressed in Arrhenius form as

kT

EexpDD a

o (5.26)

where k is the Boltzmann constant and T is the temperature. The activation

energy Ea has a typical value between 3.5 to 4.5eV for impurity dopant in

silicon. Plots of the diffusivity of common dopant in silicon are shown in Fig.

5.8 and 5.9 corresponding to the intrinsic diffusivity and activation energy

shown in Fig. 5.10. Figure 5.10 also represents an Arrhenius fit to the

diffusivity under intrinsic condition.

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Figure 5.8: Arrhenius plot of diffusivity of the common dopants in silicon

Figure 5.9: Temperature dependence of the diffusivity coefficient of common dopant in

silicon

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Si B In As Sb P Unit

Do 560 1.0 1.2 9.17 4.58 4.70 cm2s

-1

Ea 4.76 3.5 3.5 3.99 3.88 3.68 eV

Figure 5.10: Intrinsic diffusivity for silicon self diffusion of common dopants

Exercises

5.1. Given the solution of Fick’s diffusion equation that satisfies the initial

and boundary conditions is

Dt2

xerfcC)t,x(C S . Prove that the total

number of dopant atoms per unit area of the semiconductor is

DtC2

)t(Q sT

.

5.2. Find the diffusivity and total impurity from a known impurity profile.

Assume that boron is diffused into an n-type silicon single crystal

substrate with doping concentration of 1015

cm-3

and that the diffusion

profile can be described by a Gaussian function. Using diffusion time of

one hour, one obtains a measured junction depth of 2.0m and surface

concentration of 1.0x1018

cm-3

.

5.3. A boron diffusion process such that the surface concentration is

4.0x1017

cm-3

, thickness x = 3.0m, substrate concentration CB =

1.0x1015

cm-3

. Calculate the drive in time if the diffusion temperature is

1,100oC.

5.4. Boron pre-deposition is performed at 950oC for 30 minutes in a neutral

ambient. Given the activity energy of boron Ea = 3.46eV, D

0 = 0.76

cm2/sec and the boron surface concentration is Cs = 1.8x10

20cm

-3.

(i) . Calculate the diffusion length.

(ii). Determine the total amount of dopant introduced.

5.5. Assume the measured phosphorus profile can be represented by a

Gaussian function with a diffusivity D = 2.3x10-13

cm2s

-1. The measured

surface concentration is 1.0x1018

cm-3

, and the measured junction depth is

1.0m at a substrate concentration of 1.0x1015

atoms/cm3.

(i) . Calculate the diffusion time.

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(ii). Find the total dopant in the diffused layer.

5.6. For boron diffusion in silicon at 1,000oC, the surface concentration is

maintained at 1.0x1019

cm-3

and the diffusion time 1.0hr. If the diffusion

coefficient of boron at 1,000oC is 2.0x10

-14cm

2/s, calculate

(i). The total dopant QT diffused into silicon.

(ii). The location where the dopant concentration reaches 1.0x1018

cm-3

.

5.7. Calculate the effective diffusion coefficient at 1,000oC for two different

box shaped arsenic profile grown by silicon epitaxy, one doped at

1.0x1018

cm-3

and the other doped at 1.0x1020

cm-3

.

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Bibliography

1. JD Pummer, MD Del, and Peter Griffin, “Silicon VLSI Technology”

Fundamentals, Practices, and Modeling”, Prentice Hall, 2000.

2. Hong Xiao, “Introduction to Semiconductor Manufacturing Technology”,

Pearson Prentice Hall, 2001.

3. SM Sze, “VLSI Technology”, second edition, McGraw-Hill, 1988.

4. CY Chang and SM Sze, “ULSI Technology”, McGraw-Hill, 1996.

Index

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A

Activation energy ............................................... 142 Adolf Fick ........................................................... 134 Antimony ........................................................... 134 Arrhenius ........................................................... 142 Arsenic ....................................................... 132, 134 Arsenic trioxide .................................................. 132 As2O3 ......................................... See Arsenic trioxide

B

Boltzmann constant ........................................... 142 Boron ................................................................. 134

C

Chlorine ............................................................. 132 Complementary error function ......................... 135

D

Diborane ............................................................ 131 Diffusion ............................................................ 131 Diffusion coefficient .......................................... 134 Diffusion time .................................................... 135 Diffusivity ........................................................... 134 Divacancy diffusion ............................................ 133 Dopant ............................................................... 131

E

Effective conductivity ........................................ 139

F

Fick’s first law of diffusion ................................. 134 Fick’s second law of diffusion ............................ 135

G

Gaussian function .............................................. 137

I

Impurity diffusion .............................................. 133 Interstitial diffusion ........................................... 133 Interstitialcy diffusion ........................................ 134

Intrinsic diffusivity .............................................. 140 Irvin curve .......................................................... 139

M

Mobility .............................................................. 138

P

P2O5 ............................... See Phosphorus pentoxide Phosphine .......................................................... 131 Phosphorus ................................................ 132, 134 Phosporus pentoxide ......................................... 132 pn junction ......................................................... 138 Point defect ........................................................ 133

S

Selenium ............................................................ 132 Self diffusion ...................................................... 133 Semiconductor

Gallium arsenide .................................... 131, 132 Germanium .................................................... 131

Sheet resistance ................................. 135, 138, 139 Silicon ................................................................. 138 Silicon dioxide .................................................... 132

T

Tellurium ............................................................ 132

U

Ultra large scale integration............................... 131

V

VLSI .................................................................... 131

W

William Gardner Pfann....................................... 131

Z

Zinc arsenate ...................................................... 132 ZnAs2 ............................................ See Zinc arsenate Zn-Ga-As alloy .................................................... 132