Page 176 - Sami Franssila Introduction to Microfabrication

Diffusion 155

(a) (b) (c)
Figure 14.5 Diffusion mechanisms: (a) interstitial; (b) substitutional/vacancy and (c) interstitialcy

diffusion necessitates that empty lattice site is available Table 14.1 D o and E a values for boron
next to the diffusing atom. At high temperatures and phosphorus
substitutional sites are thermally created. Antimony
Boron Phosphorus
and arsenic demonstrate substitutional mechanisms. The
interstitialcy mechanism is related to the substitutional D o (cm /s) 0.76 3.85
2
mechanism: the self-interstitial atoms move to the lattice E a (eV) 3.46 3.66
sites, and kick the dopants to the interstitial sites, and
from there they move to the lattice sites. Boron and
phosphorus are expected to diffuse via interstitialcy
mechanism, but there are still some open questions even k is the Boltzman’s constant, k = 1.38 × 10 −23 J/K or
in diffusion of the best-known dopants. 8.62 × 10 −5 eV/K
The substitutional and interstitialcy mechanism with T is the temperature in Kelvin.
activation energies of ca. 3.5 to 4 eV are the most ◦
The boron diffusion coefficient at 950 C is 4 ×
important for doping in silicon technology. Boron, −15 2 ◦ −14 2
10 cm /s and at 1050 C it is 4.7 × 10 cm /s
phosphorus, arsenic as well as antimony, indium and
(see Table 14.1). The characteristic diffusion length is
gallium all have activation energies in this range.
given by
Therefore, doping by diffusion must take place at √
x ≈ 4Dt (14.4)
a high temperature. Many metallic impurities diffuse
with the interstitial mechanism with activation energies ◦
so that at 1050 C boron diffusion for one hour
round 1 to 1.5 eV, and they are mobile at much lower
corresponds to roughly 0.26 µm diffusion depth. This
temperatures than substitutional dopants.
distance is a characteristic length scale only: diffusion
profiles are gently sloping and there is no clear cut-
off depth.
14.2 DOPING PROFILES IN DIFFUSION The sheet resistance of doped layers is given by
Equation 14.5a and it is approximated for a box profile
Concentration dependent diffusion flux is described by by Equation 14.5b.
Fick’s first law:

x j
j = −D(∂N/∂x) (14.2) 1/R s = qµ(N(x) − N b )dx (14.5a)
o
2
where D is the diffusion coefficient (cm /s), N is 1/R s = qµx j N(x) (14.5b)
−3
2
concentration (in cm ). The unit of flux is atoms/s*cm . where q is the elementary charge, µ is the mobility,
Diffusion coefficients can be presented by
N(x) is the dopant concentration, N b is the background
D = D o e (−E a /kT ) (14.3) concentration and x j is the junction depth. The mobilities
2
of n-type and p-type silicon are ca. 1400 cm /Vs
2
and 500 cm /Vs respectively, at low concentrations
where 15 3 2
(<10 /cm ) and ca. 50 cm /Vs at high concentrations
19
3
D o is the frequency factor (related to lattice vibrations, (>10 /cm ), irrespective of dopant. In 1 µm CMOS
14
10 13 to 10 Hz) technology source/drain diffusions are made by 5 ×
2
15
E a is the activation energy (related to energy barrier 10 /cm ion implant doses, and the depth is ca. 200 nm,
that the dopant must overcome) which translates to ca. 25 ohm/sq. For more advanced