ELEC 3908, Physical Electronics, Lecture 6
Doping Profiles and 1D
Approximations
in the Diode Structure
Lecture Outline
• Previous lecture examined the fabrication of three planar
diode structures
• Now look in some detail at two aspects of the basic
structure
– Doping profile: the spatial (with distance) variation of doping
concentration within the structure, and one-dimensional or 1D
approximations
– The flow and spreading of current, and the relevant area to use in
scaling currents in varying sized devices
– The concept of current density, an area-independent quantity
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-2
Dopant Diffusion in Three Dimensions (3D)
•
•
•
When dopant is introduced into
the substrate, atoms can move
in all 3 directions (3D)
The doped region therefore
extends down into the substrate
and outside the masking
window
Following terminology used:
– Internal region: the area inside
the window opening, away
from the area of lateral
diffusion
– Peripheral region: the area
outside the window, i.e. at the
edges
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-3
Constant Doping Contours in 3D
•
•
•
In the internal area, constant
doping contours are flat sheets,
since concentration is uniform
across a given depth
In the peripheral area, contours
bend upwards to reflect lateral
diffusion outwards from
window
A doping vs. dimension
characteristic is called a doping
profile
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-4
2D Approximation to 3D Doping Profile
•
If a slice in the x-y plane is
taken through the device at a
value of z away from the end
regions, a two-dimensional
(2D) approximation is obtained
(i.e. the “front” of the previous
structure)
• Constant doping contours
represent the effect of lateral
diffusion towards the sides of
the device
• Internal region is again
characterised by flat contours no lateral component
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-5
Surface Plot of 2D Doping Profile
•
•
•
•
The spatial variation of doping
can also be visualized using a
surface plot
For generality, plot ln of the
absolute value of the difference
NA-ND vs. position in the x-y
plane
When NA=ND, ln(|NA-ND|) →
−∞, so a change in doping
between n and p-type is
indicated by a sharp drop
towards -∞
Lateral diffusion is evident
from the curved side regions
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-6
Surface Plot of 2D Doping Profile - Half Plot
•
•
•
Plot to the right is the same as
the previous, but only half the
surface is shown - from the
middle of the internal region
outwards
The front edge of this plot
illustrates the variation of
doping density with depth in the
internal region
This 1D doping profile holds
for any point in the internal
region
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-7
1D Doping Profile
• Plotting |NA-ND| (log scale) vs. depth into
the substrate (x) in the internal region
leads to the plot shown to the right
• The presence of the
metallurgical junction is
indicated by the drop of
the curve at x=1.0 μm
• When the doping of the
implanted region is much
higher than that of the
substrate, the junction is
termed one-sided
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-8
1D Doping Plot – Implant and Substrate Dopings
•
•
•
The figures on the right show
how the doping profile is
constructed, on linear axes (so
the negative portion can be
shown)
Nimplant - Nsubstrate is positive
where, Nimplant > Nsubstrate and
negative where Nimplant <
Nsubstrate
The absolute value of the
difference |Nimplant – Nsubstrate|
passes through zero where the
two are equal – on a log
ordinate, the curve dips to -∞
Nimplant
Nsubstrate
x
x
Nimplant - Nsubstrate
|Nimplant - Nsubstrate|
x
x
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-9
Uniform Doping Approximation
•
Analytic models difficult to
derive if accurate doping
variation taken into account
• Use a uniform approximation to
the characteristic - assume
implanted region has constant
doping at maximum value
• Not the only possible choice for
the approximation - could use
average value, etc.
• Note that although this profile
is shown for the substrate diode,
it would be found in the
diffusion regions of the other
structures
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-10
2D Current Flow in pn-Junction
•
•
•
Flow lines tend to spread as
current passes through substrate
Current flow is therefore
inherently two dimensional
This is another effect which is
difficult to model accurately
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-11
1D Internal Current Flow Approximation
•
•
•
To obtain first order analytic
solutions, assume that all
current flow is 1D (through
substrate) and confined to the
internal region
Current flow area is then the
internal area of the junction,
labelled AD
Accuracy of approximation
depends on diode area:
– large area structure is
dominated by internal
– small area structure has
significant peripheral effect
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-12
Current Density
•
The diode area AD can be used to define a current density JD given by
JD =
•
•
ID
AD
Current density is useful since it allows comparison between different
area devices
In the diagram below, all devices have the same current density: 1
mA/μm2 = 105 A/cm2.
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-13
Simple Ideal Diode Equation with Current Densities
• By dividing both sides of the original simple ideal diode
equation by the diode area, the relationship can be
expressed in terms of current densities
J D = J S (e
qV D / kT
− 1)
• The term JS is the saturation current density, normally in
A/cm2, given by
IS
JS ≡
AD
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-14
Example 6.1: Basic Current Density
Which is carrying more current, a device with a current
density of 100 A/cm2 and an area of 30 μm by 10 μm,
or a device with a current density of 75 A/cm2 and a
square area 20 μm on a side?
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-15
Example 6.1: Solution
• The absolute currents are calculated as (note the
conversion of μm to cm)
I D = J D AD = 100 ⋅ 30 × 10 −4 ⋅ 10 × 10 −4 = 3 × 10 −4 A
I D = J D AD = 75 ⋅ 20 × 10 − 4 ⋅ 20 × 10 − 4 = 3 × 10 − 4 A
• The currents are therefore identical, 300 μA
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-16
Example 6.2: Area Calculation
What area is required if an integrated diode is to conduct 100
μA of current at a junction potential of 0.7V? The
saturation current of a 500 μm by 500 μm device is
3.75x10-14 A.
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-17
Example 6.2: Solution
• The saturation current density is found as
3.75 ×10 −14
IS
−11
2
=
=
1
.
5
×
10
A/cm
JS =
2
AD
500 ×10 − 4
(
)
• Using the simple ideal diode equation, at 0.7 V of bias
(
)
J D = 1.5 ×10 −11 e 0.7 / 0.02586 − 1 = 8.55A/cm 2
• The required area is therefore
I D 100 × 10−6
AD =
=
= 117
. × 10−5 cm 2 = 1170 μm 2
8.55
JD
• This would correspond to a square area approx. 34 μm on
a side
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-18
Example 6.3: Maximum Current Density Spec.
An IC process for power applications specifies that a diode’s
current density cannot exceed 106 A/cm2. The saturation
current density is 1.5x10-11 A/cm2. If an application
requires a diode to conduct 1A of current at a terminal
voltage of 0.9V, can a diode from this process be used?
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-19
Example 6.3: Solution
• To find the necessary area from the current spec., write
ID
ID
qV D / kT
JD =
= J S (e
− 1) → AD =
AD
J S (e qVD / kT − 1)
• Substituting values gives
1
2
−5
2
2
AD =
= 51
. × 10 cm = 5120 μm ≈ ( 715
. μm )
−11
0.9 / 0.02586
15
− 1)
. × 10 (e
• The corresponding current density is then
ID
1
4
2
2
10
A
/
cm
JD =
=
≈
×
AD 512
. × 10 −5
• The process is therefore suitable for this requirement
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-20
Lecture Summary
• A doping profile is a plot of the spatial variation of dopant
concentration in a device, usually |NA-ND| on a log scale
• A uniform doping approximation ignores the spatial
variation and assumes a constant value, in our case the
peak value
• For devices with larger internal areas, a 1D approximation
ignores the peripheral region in favor of the internal
region, and considers current to be determined by the
internal area
• Current density is the current per unit internal area, and is a
useful means of comparing devices with different areas
and currents
ELEC 3908, Physical Electronics:
Doping Profiles and 1D Approximations
Page 6-21