Solid-State Diffusion - University of Washington · exp exp 0 o aa Vo Bi B EEp DD D kT n kT R. B....

R. B. Darling / EE-527 / Winter 2013

EE-527: MicroFabrication

Solid-State Diffusion


The Process of Diffusion

• Diffusion is a process driven by temperature and described by statistical thermodynamics.

• It occurs in numerous circumstances and it is responsible for many important physical transport effects in all fields of science and engineering.

• In the field of microfabrication, solid-state diffusion is the process by which dopant impurity atoms are introduced into semiconductors to form localized n-type and p-type regions, and where these regions adjoin, pn-junctions.


Fick’s Laws

• Fick’s First Law: the diffusive flux is proportional to the concentration gradient:

• Particle Continuity Equation: conservation of matter:

• Fick’s Second Law: combine the above two:

F D C

C Ft

C D Ct


The Diffusion Coefficient

• The diffusion coefficient or “diffusivity” usually follows an Arrhenius relationship:

• Ea is the activation energy. • The activation energy is the

slope of the line on a plot of log(D) versus 1/kBT.

• Typical activation energies for solid-state diffusion are ~ 3.3 to 4.4 eV.

0 exp a

B

ED Dk T

1

Bk T

0

log DD

slopeaE


Atomic Mechanisms of Diffusion – 1• Impurity atoms may reside on a normal lattice site; this is a

substitutional impurity. • Or they may reside in an open space of the lattice; this is

an interstitial impurity. • For an impurity atom to behave as a dopant, it must be

substitutional so that it can ionize and either donate or accept electrons.

• Interstitial impurity atoms can diffuse fairly quickly because they do not need to break any bonds.

• Substitutional impurity atoms diffuse fairly slowly because they must break and make bonds in order to propagate.

• Moving an impurity atom from an interstitial into a substitutional position is termed dopant activation.


Atomic Mechanisms of Diffusion – 2

interstitial transport

interstitial transfer

direct exchange

Frank-Turnbull

vacancy exchange

kick-out


Atomic Mechanisms of Diffusion – 3• Diffusion of substitutional impurities can occur through:

– Direct exchange: The impurity atom an adjacent host lattice atom simply trade places. This requires breaking ~6 bonds.

– Vacancy exchange: The impurity atom moves into a lattice location which is vacant. The vacancy and the impurity effectively trade places. This requires breaking only ~3 bonds.

– Interstitial transfer: The impurity atom can get knocked out of a substitutional position and become an interstitial. As an interstitial, the impurity atom can move much more rapidly through the lattice. The impurity atom can then come to rest by becoming a substitutional atom again. This can occur through:

• Frank-Turnbull process: The interstitial impurity atom falls into and fills a vacancy.

• Kick-out process: The interstitital impurity displaces a host lattice atom and takes its place.


Dopant Atoms for Silicon• Boron (P-type)

– A relatively fast diffuser.

• Phosphorous (N-type)– A relatively fast diffuser, almost identical to B.

• Arsenic (N-type)– A relatively slow diffuser, about 10X slower than B or P.

• Aluminum (P-type)– An extremely fast diffuser, 10X faster than B or P. – Important for making p-type contacts.

• Gallium (P-type)– A relatively fast diffuser, about the same as B or P.

• Antimony (N-type)– A relatively slow diffuser, about the same as As.


Common Interstitial Elements in Silicon• Oxygen

– High concentrations in Czochralski (CZ) grown wafers. – Almost none in float-zone (FZ) grown wafers.

• Iron• Copper

– An extremely fast diffuser and precipitate in silicon. – Can be used to decorate point defects.

• Nickel• Zinc• Manganese• Gold

– A fast diffuser and a carrier lifetime killer.


Diffusion by Vacancy Exchange• This is the predominant mechanism for most dopants in Si. • Consider the example for boron (B): • As a substitutional impurity, boron is an acceptor and it

will be singly negative charged (−1) when ionized. • Vacancies have the ability to take on nearly any charge

state: {…, −2, −1, 0, +1, +2, …}. • Singly positive charged vacancies will interact strongly

with boron, because the resulting pair is neutral. • Diffusion of boron can be achieved through the movement

of (B−V0) pairs or (B−V+) pairs. • The neutral (B−V+) pairs have more than 10X higher

diffusivity.


Ionization of Vacancies• Vacancies can take on different charge states by providing

a point defect site that can trap a free electron or hole. • This can be viewed as a reaction: e.g. hole capture:

• Intrinsic case: n = p = ni,

• When p rises above ni, then NV+ will also rise above NVi+ in the same proportion.

• The combined diffusion coefficient becomes

0V h V 0 equilibrium rate constantVe

V

NKpN

0Vi

ei V

NKn N

V i

Vi i

N npN n n

0

0exp expo a aV o

B i B

E EpD D Dk T n k T


Diffusion by Vacancy Exchange

• Example: boron in silicon: – D00 = 0.037 cm2/V-s, Ea0 = 3.46 eV– D0+ = 0.41 cm2/V-s, Ea+ = 3.46 eV

• In general,

• Because n and p depend upon the local doping level, the diffusion coefficient becomes concentration dependent.

• For the majority of cases, the high temperatures of a diffusion furnace make ni quite large, and the sample is for the most part under intrinsic conditions, n = p = ni.

2 2*0 * * * * ...V

i i i i

p p n nD D D D D Dn n n n


Donor Impurity Diffusion Coefficients

• Phosphorous in silicon: – D00 = 3.85 cm2/V-s, Ea0 = 3.66 eV– D0− = 4.44 cm2/V-s, Ea− = 4.0 eV– D0−− = 44.2 cm2/V-s, Ea−− = 4.37 eV

• Arsenic in silicon: – D00 = 0.066 cm2/V-s, Ea0 = 3.44 eV– D0− = 12 cm2/V-s, Ea− = 4.05 eV

• Antimony in silicon: – D00 = 0.214 cm2/V-s, Ea0 = 3.65 eV– D0− = 15 cm2/V-s, Ea− = 4.08 eV

Data from R. B. Fair, S. K. Ghandhi; Runyan and Bean - SICPT


Acceptor Impurity Diffusion Coefficients

• Boron in silicon: – D00 = 0.037 cm2/V-s, Ea0 = 3.46 eV– D0+ = 0.41 cm2/V-s, Ea+ = 3.46 eV

• Aluminum in silicon: – D00 = 1.385 cm2/V-s, Ea0 = 3.41 eV– D0+ = 2480 cm2/V-s, Ea+ = 4.20 eV

• Gallium in silicon: – D00 = 0.374 cm2/V-s, Ea0 = 3.39 eV– D0+ = 28.5 cm2/V-s, Ea+ = 3.92 eV

Data from R. B. Fair, S. K. Ghandhi; Runyan and Bean - SICPT


Diffusion Coefficients for Dopants in Silicon

900 1000 1100 12001 10 17

1 10 16

1 10 15

1 10 14

1 10 13

1 10 12

1 10 11

DP Tdi

cm 2 s

DAs Tdi

cm 2 s

DSb Tdi

cm 2 s

DB Tdi

cm 2 s

DAl Tdi

cm 2 s

DGa Tdi

cm 2 s

TdiK 1 273.15

Al

GaPB

AsSb

ThreeSpeedGroups



‐22

‐20

‐18

‐16

‐14

‐12

‐10

0.6 0.7 0.8 0.9 1.0 1.1 1.2

Log10 of Diffusion Co

efficient, cm2/sec

1000/T, degrees K

Diffusion Coefficients in Silicon

Phosphorous

Arsenic

Antimony

Boron

Aluminum

Gallium



1.E‐22

1.E‐21

1.E‐20

1.E‐19

1.E‐18

1.E‐17

1.E‐16

1.E‐15

1.E‐14

1.E‐13

1.E‐12

1.E‐11

1.E‐10

600 700 800 900 1000 1100 1200

Diffusion Co

efficient, cm2/sec

Temperature, degrees C


Phosphorous

Arsenic

Antimony

Boron

Aluminum

Gallium



1.E‐17

1.E‐16

1.E‐15

1.E‐14

1.E‐13

1.E‐12

1.E‐11

900 1000 1100 1200

Diffusion Co

efficient, cm2/sec

Temperature, degrees C


Phosphorous

Arsenic

Antimony

Boron

Aluminum

Gallium


Electric Field Enhanced Diffusion – 1• Equilibrium electric field in a doped region:

• This electric field can produce ionic drift which can enhance or retard the diffusion of charged impurity atoms (ions).

• Drift and diffusion together give an ion flux of

• Here, μ is the mobility of the diffusing ions, Z is the charge state of the ions, and the Einstein relationship for this is

B Bk T k Tn pEq n q p

B

qZF D C CE D C D CEk T

Bk TDqZ


Electric Field Enhanced Diffusion – 2• If the impurity is a donor, the electron density will be

related to the impurity concentration as

• The built-in electric field is then

• Performing the calculus,

• Giving:

2 21 42 in C n C

1 1B Bk T k T nE n Cq n q n C

2 2

1 14 i

nn C C n

2 24B

i

k T CEq C n


Electric Field Enhanced Diffusion – 3• The ion diffusion flux can then be simplified to the form:

• The electric field enhancement factor is

• For a singly charged donor ion (Z = +1), this is maximized for C >> ni, for which gmax = 2.

• For acceptor ions (Z = −1), the analogous result is

• However, field retardation can occur then a charged impurity is diffusing through an electric field region caused by the opposite type impurity.

F gD C

2 21 donors

4 i

ZCgC n

2 21 acceptors

4 i

ZCgC n


Intrinsic Carrier Concentration at High Temperatures• High concentration and electric field enhanced diffusion

occur when the dopant and carrier density exceed the intrinsic carrier density:

0 200 400 600 800 1000 12001 1010

1 1012

1 1014

1 1016

1 1018

1 1020

ni Ti cm3

Ti K 1 273.15


Constant Surface Concentration – 1

• The concentration of the dopant is held constant at the surface of the wafer throughout the diffusion process.

• The solution is a complementary error function profile:

2

2

C CDt x

( ,0) 0 initial condition(0, ) boundary condition( , ) 0 boundary condition

S

C xC t CC t

( , ) erfc2SxC x t CDt

erfc(0) 1

erfc( ) 0erfc( ) 1 erf ( )x x

2

0

2( )x uerf x e du


Constant Surface Concentration – 2

• The dose is the diffused concentration integrated over depth:

• For the constant surface concentration case, the dose increases as the square root of time:

0( ) ( , )S t C x t dx

2( ) SS t C Dt


Constant Dose – 1

• The fixed dose S of the dopant is introduced and held constant throughout the diffusion process.

• The solution is a Gaussian profile:

2

2

C CDt x

0

( ,0) 0 initial condition

( , ) boundary condition

( , ) 0 boundary condition

C x

C x t dx S

C t

2

( , ) exp4

S xC x tDtDt


Constant Dose – 2

• The dose is the diffused concentration integrated over depth:

• For the constant dose case, the surface concentration decreases as the square root of time:

0( ) ( , )S t C x t dx

(0, ) ( )SSC t C tDt


Predeposit – Drive-In Processing – 1

• A common two-step diffusion process which is used to create deeper diffusions with a smaller surface concentration. – This allows another diffusion of opposite type to be more easily

performed. (Double-diffused and triple-diffused structures)

• The predeposit step introduces the dose. – Diffusion with a constant surface concentration. – Ion implantation to a prescribed dose and range.

• The drive-in step diffuses this dose deeper into the substrate while diluting the peak concentration. – Diffusion with a constant dose. – The drive-in step will also anneal (activate) the implanted ions.


Predeposit – Drive-In Processing – 2

Boron into Si: T = 1000°CD = 10−14 cm2/stpd = 1 hourtdi = 3 hours∆t = 10 minsCS = 1018 cm−3


Junction Depth

• The metallurgical junction depth is defined as where the doping polarity changes from n-type to p-type.

• Computationally, it is where the diffused dopant concentration becomes equal to the background doping of the substrate, CB.

• For the constant surface concentration case:

• For the constant dose case:

12 erfc Bj

S

Cx DtC

4 lnjB

Sx DtC Dt


Multiple Thermal Cycles

• The time-temperature profile of the overall thermal processing schedule can be accounted for by using an effective diffusivity-time product:

• Noting that,

• Note that increasing the temperature has a much more significant effect on the diffusion depths than does lengthening the time.

( )i ieffi

Dt D t D t dt

( ) ( ( ))D t D T t


Double Diffused BJT Structure

Net doping (Nd+ − Na

−) near junctions is reduced from diffused impurity profiles because of partial dopant compensation.

1020

1019

1018

1017

1016

1015

1014

1013

1012

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0x, m

N(x), cm3

Emitter

Base

Collector

N (x)E

N (x)B

NC

xjE xjC


Emitter Push Effect

• Emitter push is a two-dimensional diffusion effect where a second high concentration diffusion (an emitter) effectively increases the junction depth of a preceding diffusion (a base).

emitterbase

collector


Emitter Pull Effect

• Emitter pull is a two-dimensional diffusion effect where a second high concentration diffusion (an emitter) effectively decreases the junction depth of a preceding diffusion (a base).

emitterbase

collector


Diffusion Spiking or Piping• Some impurity atoms, in very high surface concentrations,

can diffuse along preferential crystal directions much faster than at lower concentrations.

• This can lead to dopant spikes which penetrate deep into the substrate and which can short out shallow diffusions.

• A good example is aluminum:

p-type

n-type

Al contact

shortedjunction


Dopant Sources for Silicon• Gaseous – metered out into the diffusion tube

– diborane, B2H6

– phospine, PH3

– arsine, AsH3

• Liquid – vaporized and carried into the diffusion tube– boron bromide, BBr3

– phosphorous oxychloride, POCl3 (“pockel”)– arsenic trichloride, AsCl3

• Solid – wafers that are loaded along beside the silicon wafers– boron nitride, BN – phosphorous pentoxide, P2O5

– arsenic trioxide, As2O3

– These are known as “planar solid sources.”


Boron Surface Chemistry• All dopant sources for boron first create a boron oxide on

the surface of the silicon wafer: – Gaseous: Diborane: B2H6 + 2O2 → B2O3 + 3H2O– Liquid: Boron bromide: 2BBr3 → 2B + 3Br2, 4B + 3O2 → 2B2O3

– Solid planar sources: Boron nitride: 4BN + 3O2 → 2B2O3 + 2N2

– The oxide is necessary to create a boron phase which will condense and adhere to the silicon wafer surface.

• Simultaneous oxidation of the Si will create a mixed oxide on the surface: SiO2 + B2O3.

• The boron oxide will decompose with high heat, yielding free boron at the surface of the wafer: – 2B2O3 + 3Si → 4B + 3SiO2.

• Excess boron will create tenacious silicon borides: – Excess B + Si → SiB4 or SiB6.


Phosphorous Surface Chemistry• Phosphorous sources must also create an oxide on the

surface of the silicon wafer: – Gaseous: Phosphine: 2PH3 + 4O2 → P2O5 + 3H2O– Liquid: “Pockel”: 2POCl3 + 2O2 → P2O5 + 3Cl2

– Solid planar source: Phosphorous pentoxide: P2O5

• Simultaneous oxidation of the Si will create a mixed oxide on the surface: SiO2 + P2O5.

• The phosphorous oxide will decompose with high heat, yielding free phosphorous at the surface of the wafer: – 2P2O5 + 5Si → 4P + 5SiO2.

• Excess phosphorous will create silicon phosphides: – Excess P + Si → SiP or SiP2. – Unlike SiB4 and SiB6, the SiP and SiP2 are easy to remove.


Spin-On Dopants for Silicon• Spin-on dopants are the least expensive and easiest to

handle dopant source. – They likewise have the least controllability and reproducibility. – They can be a good, economical solution when only a highly

doped contact region is required, such as in solar cells, and some rectifier applications.

• Typical composition: – Dopant oxide: e.g. B2O3, P2O5, As2O3, SnO, ZnO– Siloxane polymer: – O – Si – O –– Coating solvent: e.g. EtOH, MeOH, C6H13OH (hexanol)

• Typical processing: – Spin coat at 3000 rpm for a thickness of 150 to 250 nm. – Bake at 200°C for 15 min. to evaporate the solvent and cross-link

the siloxane into SiO2 which also incorporates the dopant oxide. – Furnace tube diffusion cycle and deglaze.


Diffusion Modeling and Simulation• Diffusion is perhaps the easiest physical process to

simulate because it is inherently stable. – Time evolution tends to smooth out profiles, reducing or

eliminating singularities. – Dependent variables remain everywhere differentiable. – Highly efficient solver routines can be applied.

• Commercial and academic diffusion process simulators: – SUPREM IV (the original from Stanford University)– TSUPREM IV (Avant!, Inc.)– ATHENA (Silvaco, Inc.) – Sentaurus (Synopsys, Inc.)– COMSOL – write it yourself, but obtain greater flexibility – good

for including the physics that other TCAD programs leave out.

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