Fick

1
Ficks Laws

• Combining the continuity equation with the first
  law, we obtain Ficks second law

2

• Solutions to Ficks Laws depend on the boundary
  conditions.
• Assumptions
• D is independent of concentration
• Semiconductor is a semi-infinite slab with either
• Continuous supply of impurities that can move
  into wafer
• Fixed supply of impurities that can be depleted

3
Solutions To Ficks Second Law

• The simplest solution is at steady state and
  there is no variation of the concentration with
  time
• Concentration of diffusing impurities is linear
  over distance
• This was the solution for the flow of oxygen from
  the surface to the Si/SiO2 interface in the last
  chapter

4
Solutions To Ficks Second Law

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

5
Solutions To Ficks Second Law

• Complimentary error function (erfc) is defined as
  erfc(x) 1 - erf(x)
• The error function is defined as
• This is a tabulated function. There are several
  approximations. It can be found as a built-in
  function in MatLab, MathCad, and Mathematica

6
Solutions To Ficks Second Law

• This solution models short diffusions from a
  gas-phase or liquid phase source
• Typical solutions have the following shape

7
Solutions To Ficks Second Law

• Constant source diffusion has a solution of the
  form
• Here, Q is the does or the total number of dopant
  atoms diffused into the Si
• The surface concentration is given by

8
Solutions To Ficks Second Law

• Limited source diffusion looks like

9
Comparison of limited source and constant source
models
1
10-1
exp(- )
2
10-2
erfc( )
10-3
Value of functions
10-4
10-5
10-6
0 0.5 1
1.5 2 2.5
3 3.5
10
Predep and Drive

• Predeposition
• Usually a short diffusion using a constant source
• Drive
• A limited source diffusion
• The diffusion dose is generally the dopants
  introduced into the semiconductor during the
  predep
• A Dteff is not used in this case.

11
Diffusion Coefficient

• Probability of a jump is
• Diffusion coefficient is proportional to jump
  probability

12
Diffusion Coefficient

• Typical diffusion coefficients in silicon

Element Do (cm2/s) ED (eV)
B 10.5 3.69
Al 8.00 3.47
Ga 3.60 3.51
In 16.5 3.90
P 10.5 3.69
As 0.32 3.56
Sb 5.60 3.95
13
Diffusion Of Impurities In Silicon

• Arrhenius plots of diffusion in silicon

14
Diffusion Of Impurities In Silicon

• The intrinsic carrier concentration in Si is
  about 7 x 1018/cm3 at 1000 oC
• If NA and ND are ltni, the material will behave as
  if it were intrinsic there are many practical
  situations where this is a good assumption

15
Diffusion Of Impurities In Silicon

• Dopants cluster into fast diffusers (P, B, In)
  and slow diffusers (As, Sb)
• As we develop shallow junction devices, slow
  diffusers are becoming very important
• B is the only p-type dopant that has a high
  solubility therefore, it is very hard to make
  shallow p-type junctions with this fast diffuser

16
Limitations of Theory

• Theories given here break down at high
  concentrations of dopants
• ND or NA gtgt ni at diffusion temperature
• If there are different species of the same atom
  diffuse into the semiconductor
• Multiple diffusion fronts
• Example P in Si
• Diffusion mechanism are different
• Example Zn in GaAs
• Surface pile-up vs. segregation
• B and P in Si

17
Successive Diffusions

• To create devices, successive diffusions of n-
  and p-type dopants
• Impurities will move as succeeding dopant or
  oxidation steps are performed
• The effective Dt product is
• No difference between diffusion in one step or in
  several steps at the same temperature
• If diffusions are done at different temperatures

18
Successive Diffusions

• The effective Dt product is given by
• Di and ti are the diffusion coefficient and time
  for ith step
• Assuming that the diffusion constant is only a
  function of temperature.
• The same type of diffusion is conducted (constant
  or limited source)

19
Junction Formation

• When diffuse n- and p-type materials, we create a
  pn junction
• When ND NA , the semiconductor material is
  compensated and we create a metallurgical
  junction
• At metallurgical junction the material behaves
  intrinsic
• Calculate the position of the metallurgical
  junction for those systems for which our
  analytical model is a good fit

20
Junction Formation

• Formation of a pn junction by diffusion

21
Junction Formation

• The position of the junction for a limited source
  diffused impurity in a constant background is
  given by
• The position of the junction for a continuous
  source diffused impurity is given by

22
Junction Formation
Junction Depth
Lateral Diffusion
23
Design and Evaluation

• There are three parameters that define a diffused
  region
• The surface concentration
• The junction depth
• The sheet resistance
• These parameters are not independent
• Irvin developed a relationship that describes
  these parameters

24
Irvins Curves

• In designing processes, we need to use all
  available data
• We need to determine if one of the analytic
  solutions applies
• For example,
• If the surface concentration is near the
  solubility limit, the continuous (erf) solution
  may be applied
• If we have a low surface concentration, the
  limited source (Gaussian) solution may be applied

25
Irvins Curves

• If we describe the dopant profile by either the
  Gaussian or the erf model
• The surface concentration becomes a parameter in
  this integration
• By rearranging the variables, we find that the
  surface concentration and the product of sheet
  resistance and the junction depth are related by
  the definite integral of the profile
• There are four separate curves to be evaluated
• one pair using either the Gaussian or the erf
  function, and the other pair for n- or p-type
  materials because the mobility is different for
  electrons and holes

26
Irvins Curves
27
Irvins Curves

• An alternative way of presenting the data may be
  found if we set ?eff1/?sxj

28
Example

• Design a B diffusion for a CMOS tub such that
  ?s900?/sq, xj3?m, and CB1?1015/cc
• First, we calculate the average conductivity
• We cannot calculate n or ? because both are
  functions of depth
• We assume that because the tubs are of moderate
  concentration and thus assume (for now) that the
  distribution will be Gaussian
• Therefore, we can use the P-type Gaussian Irvin
  curve to deduce that

29
Example

• Reading from the p-type Gaussian Irvins curve,
  CS?4x1017/cc
• This is well below the solid solubility limit for
  B in Si so we may conclude that it will be driven
  in from a fixed source provided either by ion
  implantation or possibly by solid state
  predeposition followed by an etch
• In order for the junction to be at the required
  depth, we can compute the Dt value from the
  Gaussian junction equation

30
Example

• This value of Dt is the thermal budget for the
  process
• If this is done in one step at (for example) 1100
  C where D for B in Si is 1.5 x 10-13cm2/s, the
  drive-in time will be
• Given Dt and the final surface concentration, we
  can estimate the dose
• This is easy to deposit by ion implantation

31
Example

• Let us also look at doing it by predep from the
  solid state (as is done in the VT lab course)
• The text uses a predep temperature of 950 C
• In this case, we will make a glass-like oxide on
  the surface that will introduce the B at the
  solid solubility limit
• At 950 C, the solubility limit is 2.5x1020cm-3
  and D4.2x10-15 cm2/s
• Solving for t

32
Example

• This is a very short time and hard to control in
  a furnace thus, we should do the pre-dep at
  lower temperatures
• In the VT lab, we use 830 860 C
• Does the predep affect the drive in?
• There is no affect on the thermal budget because
  it is done at such a low temperature

33
DIFFUSION SYSTEMS

• Use open tube furnaces of the 3-Zone design
• Wafers are mounted in quartz boat in center zone
• Use solid, liquid or gaseous impurities for good
  reproducibility
• Use N2 or O2 as carrier gas to move impurity
  downstream to crystals
• Common gases are extremely toxic (AsH3 , PH3)

34
SOLID-SOURCE DIFFUSION SYSTEMS
35
LIQUID-SOURCE DIFFUSION SYSTEMS

36
GAS-SOURCE DIFFUSION SYSTEMS
37
DIFFUSION SYSTEMS

• Al and Ga diffuse very rapidly in Si B is the
  only p-dopant routinely used
• Sb, P, As are all used as n-dopants

38
DIFFUSION SYSTEMS

• Typical reactions for solid impurities are

39
PRODUCTION DIFFUSION FURNACES

• Commercial diffusion furnace showing the furnace
  with wafers (left) and gas control system
  (right). (Photo courtesy of Tystar Corp.)

40
PRODUCTION DIFFUSION FURNACES

• Close-up of diffusion furnace with wafers.

41
Rapid Thermal Annealing

• An alternative to the diffusion furnaces is the
  RTA or RTP furnace

42
Rapid Thermal Anneling

• In this system, we try to heat the wafer quickly
  (but not so as to introduce fracture stresses)
• RTAs usually use infrared lamps and heat by
  radiation
• It is possible to ramp the wafer at 100 C /sec
• Such devices are used to diffuse shallow
  junctions and to anneal radiation damage
• In such a system, for the thermal conductivity of
  Si, a 12 in wafer can be heated to a uniform
  temperature in milliseconds
• Therefore, 1 100 s annealing times are very
  reasonable

43
Rapid Thermal Annealing
44
Concentration-Dependent Diffusion

• If the concentration of the doping exceeds the
  intrinsic carrier concentration at the diffusion
  temperature, another effect occurs
• We have assumed that the diffusion coefficient,
  D, is independent of concentration
• This is not valid if the concentration of the
  diffusing species is greater than the intrinsic
  carrier concentration
• In this case, we see that diffusion is faster in
  the higher concentration regions

45
Concentration-Dependent Diffusion

• The concentration profiles for P in Si look more
  like the solid lines than the dashed line for
  high concentrations (see French et al)

46
Concentration-Dependent Diffusion

• If we define the diffusivity to be a function of
  composition, then we can still use Ficks law to
  describe the dopant diffusion
• Usually, we cannot directly integrate/solve the
  differential equations when D is a function of C
• We thus must solve the equationnumerically

47
Concentration-Dependent Diffusion

• It has been observed that the diffusion
  coefficient usually depends on concentration by
  either of the following relations
• Look, for example, at the diffusion of P in Si
  observed by French et al
• How do we obtain information about the
  concentration dependence of diffusivity?
• There is a lovely experiment done with B

48
Concentration-Dependent Diffusion

• B has two isotopes B10 and B11
• We create a wafer with a high concentration of
  one isotope (say B10) and then we diffuse the
  second isotope into this material
• We use SIMS to determine the concentration of B11
  as a function of distance
• This gives us the diffusion of B as a function of
  the concentration of B
• These experiments have been done for a great many
  of the dopants in Si

49
Concentration-Dependent Diffusion

• We find that the diffusivity can usually be
  written in the formfor n-type dopants
  andfor p-type dopants

50
Concentration-Dependent Diffusion

• The superscripts are chosen because we believe
  the interaction is between charged vacancies and
  the charged diffusing species
• For an n-type dopant in an intrinsic material,
  the diffusivity is
• All of the various diffusivities are of the
  Arrhenius form

51
Concentration-Dependent Diffusion

• The values for all the pre-exponential factors
  and activation energies are known (see next
  Table)
• If we substitute into the expression for the
  effective diffusion coefficient, we
  findhere, ?D-/D0 and ?D/D0

52
Concentration-Dependent Diffusion
53
Concentration-Dependent Diffusion

• Expressed this way, ? is the linear variation
  with composition and ? is the quadratic variation
• Simulators like SUPREM include these effects and
  are capable of modeling very complex structures