Skip to content Skip to navigation

OpenStax_CNX

You are here: Home » Content » Diffusion

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • OrangeGrove display tagshide tags

    This module is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange GroveAs a part of collection: "Introduction to Physical Electronics"

    Click the "OrangeGrove" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Rice Digital Scholarship display tagshide tags

    This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "Introduction to Physical Electronics"

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Bookshare

    This module is included inLens: Bookshare's Lens
    By: Bookshare - A Benetech InitiativeAs a part of collection: "Introduction to Physical Electronics"

    Comments:

    "Accessible versions of this collection are available at Bookshare. DAISY and BRF provided."

    Click the "Bookshare" link to see all content affiliated with them.

Also in these lenses

  • Lens for Engineering

    This module is included inLens: Lens for Engineering
    By: Sidney Burrus

    Click the "Lens for Engineering" link to see all content selected in this lens.

  • ElectroEngr display tagshide tags

    This module is included inLens: Electronic Engineering
    By: Richard LloydAs a part of collection: "Introduction to Physical Electronics"

    Click the "ElectroEngr" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

  • Tronics display tagshide tags

    This module is included inLens: Brijesh Reddy's Lens
    By: Brijesh ReddyAs a part of collection: "Introduction to Physical Electronics"

    Comments:

    "BJTs,FETs etc"

    Click the "Tronics" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Diffusion

Module by: Bill Wilson. E-mail the author

Summary: The module discusses the process of electrons moving across a p-n or n-p junction known as diffusion.

Introduction

Let us turn our attention to what happens to the electrons and holes, once they have been injected across a forward-biased junction. We will concentrate just on the electrons which are injected into the p-side of the junction, but keep in mind that similar things are also happening to the holes which enter the n-side.

As we saw a while back, when electrons are injected across a junction, they move away from the junction region by a diffusion process, while at the same time, some of them are disappearing because they are minority carriers (electrons in basically p-type material) and so there are lots of holes around for them to recombine with. This is all shown schematically in Figure 1.

Figure 1: Processes involved in electron transport across a p-n junction
Diffusion across a P-N Junction
Diffusion across a P-N Junction (f2_52.png)

Diffusion Process Quantified

It is actually fairly easy to quantify this, and come up with an expression for the electron distribution within the p-region. First we have to look a little bit at the diffusion process however. Imagine that we have a series of bins, each with a different number of electrons in them. In a given time, we could imagine that all of the electrons would flow out of their bins into the neighboring ones. Since there is no reason to expect the electrons to favor one side over the other, we will assume that exactly half leave by each side. This is all shown in Figure 2. We will keep things simple and only look at three bins. Imagine I have 4, 6, and 8 electrons respectively in each of the bins. After the required "emptying time," we will have a net flux of exactly one electron across each boundary as shown.

Figure 2
First example of a diffusion problem
First example of a diffusion problem (f2_53.png)
Figure 3
Diffusion from bins
Diffusion from bins (f2_54.png)

Now let's raise the number of electrons to 8, 12 and 16 respectively (the electrons may overlap some now in the picture.) We find that the net flux across each boundary is now 2 electrons per emptying time, rather than one. Note that the gradient (slope) of the concentration in the boxes has also doubled from one per box to two per box. This leads us to a rather obvious statement that the flux of carriers is proportional to the gradient of their density. This is stated formally in what is known as Fick's First Law of Diffusion:

Flux=( D e )dnxd x Flux D e x n x
(1)
Where De De is simply a proportionality constant called the diffusion coefficient. Since we are talking about the motion of electrons, this diffusion flux must give rise to a current density Je diff Je diff . Since an electron has a charge q q associated with it,
J e diff =q D e dnd x J e diff q D e x n
(2)

Now we have to invoke something called the continuity equation. Imagine we have a volume V V which is filled with some charge QQ. It is fairly obvious that if we add up all of the current density which is flowing out of the volume that it must be equal to the time rate of decrease of the charge within that volume. This ideas is expressed in the formula below which uses a closed-surface integral, along with the all the other integrals to follow:

SJd S =dQd t S S J t Q
(3)
We can write QQ as
Q=Vρvd V Q V V ρ v
(4)
where we are doing a volume integral of the charge density ρρ over the volume VV. Now we can use Gauss' theorem which says we can replace a surface integral of a quantity with a volume integral of its divergence:
SJd S =divJd V S S J V V J
(5)
So, combining Equation 3, Equation 4 and Equation 5, we have (note we are still dealing with surface and volume integrals):
divJd V =dρd t d V V V J V V t ρ
(6)
Finally, we let the volume VV shrink down to a point, which means the quantities inside the integral must be equal, and we have the differential form of the continuity equation (in one dimension)
divJ=J x =dρxd t J x J t ρ x
(7)

What about the Electrons?

Now let's go back to the electrons in the diode. The electrons which have been injected across the junction are called excess minority carriers, because they are electrons in a p-region (hence minority) but their concentration is greater than what they would be if they were in a sample of p-type material at equilibrium. We will designate them as n n , and since they could change with both time and position we shall write them as n xt n x t . Now there are two ways in which n xt n x t can change with time. One would be if we were to stop injecting electrons in from the n-side of the junction. A reasonable way to account for the decay which would occur if we were not supplying electrons would be to write:

dd t n xt= n xt τ r t n x t n x t τ r
(8)
Where τr τr called the minority carrier recombination lifetime. It is pretty easy to show that if we start out with an excess minority carrier concentration no no at t=0 t 0 , then n xt n x t will goes as
n xt= n 0 et τ r n x t n 0 t τ r
(9)
But, the electron concentration can also change because of electrons flowing into or out of the region xx. The electron concentration n xt n x t is just ρxtq ρ x t q . Thus, due to electron flow we have:
dd t n xt=1qdρxtd t =1qdivJxt t n x t 1 q t ρ x t 1 q J x t
(10)
But, we can get an expression for Jxt J x t from Equation 2. Reducing the divergence in Equation 10 to one dimension (we just have a J x x J ) we finally end up with
dd t n xt= D e d 2 n xtd x 2 t n x t D e x 2 n x t
(11)
Combining Equation 11 and Equation 8 (electrons will, after all, suffer from both recombination and diffusion) and we end up with:
dd t n xt= D e d 2 n xtd x 2 n xt τ r t n x t D e x 2 n x t n x t τ r
(12)
This is a somewhat specialized form of an equation called the ambipolar diffusion equation. It seems kind of complicated but we can get some nice results from it if we make some simply boundary condition assumptions. Let's see what we can do with this.

Using the Ambipolar Diffusion Equation

For anything we will be interested in, we will only look at steady state solutions. This means that the time derivative on the LHS of Equation 12 is zero, and so we have (letting n xt n x t become simply n x n x since we no longer have any time variation to worry about)

d 2 dt 2 n x1 D e τ r n x=0 t 2 n x 1 D e τ r n x 0
(13)
Let's pick the not unreasonable boundary conditions that n 0= n 0 n 0 n 0 (the concentration of excess electrons just at the start of the diffusion region) and n x0 n x 0 as x x (the excess carriers go to zero when we get far from the junction) then
nx= n 0 ex D e τ r n x n 0 x D e τ r
(14)
The expression in the radical D e τ r D e τ r is called the electron diffusion length, Le Le , and gives us some idea as to how far away from the junction the excess electrons will exist before they have more or less all recombined. This will be important for us when we move on to bipolar transistors.

Just so you can get a feel for some numbers, a typical value for the diffusion coefficient for electrons in silicon would be D e =25cm2sec D e 25 cm 2 sec and the minority carrier lifetime is usually around a microsecond. Thus

L e = D e τ r =25×10-6=5×10-3cm L e D e τ r 25-6 5-3 cm
(15)
which is not very far at all!

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks