Chapter 1
Diffusion current
1.1
Problem
(a) The phosphorous (donor) concentration in a region of a silicon crystal varies linearly
from a concentration of n0 = 1014 cm−3 at x = 0mm to a concentration of n1 = 1017 cm−3
at x = 1mm. The diffusion constant for electrons is Dn = 22.5cm2 /s, the diffusion constant
for holes is Dp = 5.2cm2 /s, and the temperature is 300 K. What is the diffusion current
density in the positive x-direction?
(b) Plot (approximately) the current density versus electric field for silicon. How can the
mobility be determined from this plot? The electron drift velocity saturates at fields above
about 1000 V/cm. Why?
1.2
Solution
The diffusion current density can be calculated by equation 1.1. Where e− = −1.602·10−19 C
is the elemental charge. Dp,n are the diffusion constants for hole and electrons, respectivly.
p and n are the hole and electron concentrations.
JDif f = |e− | · Dn ·
dn
dp
− |e− | · Dp ·
dx
dx
(1.1)
The hole concentration can be determined by equation 1.2 and the intrinsic charge density
ni . For silicon ni,Si = 1.5 · 1010 cm−3 .
n2i,Si
=
p
=
p0
=
p1
=
n·p
n2i,Si
n
n2i,Si
1.5 · 1010 cm−3 )2
=
= 2.25 · 106 cm−3
n0
1014 cm−3
n2i,Si
1.5 · 1010 cm−3 )2
=
= 2.25 · 103 cm−3
n1
1017 cm−3
1
(1.2)
Because the doping varies linearly the derivation of p and n can be evaluated by equation 1.3
and 1.4.
dp
dx
dn
dx
=
=
∆p
p1 − p0
2.25 · 103 cm−3 − 2.25 · 106 cm−3
=
=
= −2.248 · 107 cm−4 (1.3)
∆x
∆x
0.1cm
∆n
n1 − n0
1017 cm−3 − 1014 cm−3
=
=
= 9.99 · 1017 cm−4
(1.4)
∆x
∆x
0.1cm
The diffusion current density equals
JDif f
dn
dp
− |e− | · Dp ·
dx
dx
1.602 · 10−19 C · 22.5cm2 /s · 9.99 · 1017 cm−4
= |e− | · Dn ·
=
−1.602 · 10−19 C · 5.2cm2 /s · (−2.248) · 107 cm−4
= 3.6A/cm2 + 1.873 · 10−11 A/cm2 = 3.6A/cm2
Figure 1.1 shows the plot for the current density JDrif t and the absolute value of the drift
velocity, respectively over the electric field E. The mobility of holes and electrons µp,n can
be evaluted using the tangential of the drift velocity in the origin.
Figure 1.1: Drift current density and carrier drift velocity over electric field.
For higher electric fields the velocity starts to saturate because of increased scattering of
electrons with atoms.
2