Doping Profiles and 1D Approximations

in the Diode Structure

ELEC 3908, Physical Electronics, Lecture 6

ELEC 3908, Physical Electronics: Doping Profiles and 1D Approximations Page 6-2

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 6-3

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 6-4

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 6-5

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 6-6

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 6-7

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 6-8

1D Doping Profile

•

Plotting |NA-ND | (log scale) vs. depth intothe substrate (x) in the internal regionleads 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 6-9

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 -∞

NsubstrateNimplant

N - Nimplant substrate |N Nimplant - substrate|

x

x x

x

ELEC 3908, Physical Electronics: Doping Profiles and 1D Approximations Page 6-10

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 6-11

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 6-12

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 6-13

Current Density•

The diode area AD can be used to define a current density JD given by

•

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.

JIAD

D

D=

ELEC 3908, Physical Electronics: Doping Profiles and 1D Approximations Page 6-14

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

•

The term JS is the saturation current density, normally in A/cm2, given by

( )J J eD SqV kTD= −/ 1

JIAS

S

D≡

ELEC 3908, Physical Electronics: Doping Profiles and 1D Approximations Page 6-15

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 6-16

Example 6.1: Solution

•

The absolute currents are calculated as (note the conversion of μm to cm)

•

The currents are therefore identical, 300 μA

I J AI J A

D D D

D D D

= = ⋅ × ⋅ × = ×

= = ⋅ × ⋅ × = ×

− − −

− − −

100 30 10 10 10 3 1075 20 10 20 10 3 10

4 4 4

4 4 4

AA

ELEC 3908, Physical Electronics: Doping Profiles and 1D Approximations Page 6-17

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 6-18

Example 6.2: Solution

•

The saturation current density is found as

•

Using the simple ideal diode equation, at 0.7 V of bias

•

The required area is therefore

•

This would correspond to a square area approx. 34 μm on a side

( ) 202586.0/7.011 A/cm55.81105.1 =−×= − eJ D

( )211

24

14

A/cm105.1105001075.3 −

−

−

×=×

×==

D

SS A

IJ

AIJD

D

D= =

×= × =

−−100 10

8 55117 10 1170

65

.. cm m2 2μ

ELEC 3908, Physical Electronics: Doping Profiles and 1D Approximations Page 6-19

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 6-20

Example 6.3: Solution

•

To find the necessary area from the current spec., write

•

Substituting values gives

•

The corresponding current density is then

•

The process is therefore suitable for this requirement

( ) ( )JIA

J e AI

J eDD

DS

qV kTD

D

SqV kT

D

D= = − → =

−/

/11

( ) ( )AeD =

× −= × = ≈−

−115 10 1

51 10 5120 71511 0 9 0 025865 2

.. .. / . cm m m2 2μ μ

JIAD

D

D= =

×≈ ×−

1512 10

2 1054

.A / cm2

ELEC 3908, Physical Electronics: Doping Profiles and 1D Approximations Page 6-21

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