# 1. Open-Circuit Properties

- Page ID
- 5966

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The *p*-*n* junction has two important open-circuit properties -- that is, properties that arise when there are no external connections to the device. These are the **barrier potential** and the **space-charge region**. These properties emerge from the interactions among free electrons and holes in the *n*- and *p*-type sections.

A *p*-*n* junction consists of one piece of *p*-type semiconductor adjacent to a piece of *n*-type semiconductor.

## Barrier Potential

Recall that the *p*-type material has an excess of holes, while the *n*-type material has an excess of electrons; when moving from the *p*-type material to the *n*-type material, the concentration of holes (or free electrons) will change. If you were to physically move a positive charge (a hole) from a point within the region with a high concentration of holes (point *x*_{1}, below) to a point within the region with a high concentration of electrons (point *x*_{2}), energy would be released, since the positively-charged hole is attracted to the negatively-charged electrons. Recall that **voltage** is defined as the work required to move a positive charge across a distance; therefore, we see that a voltage will be established across any distance where there is a concentration gradient of charges.

Moving a charge from *x*_{1} to *x*_{2} involves a release of energy - therefore, there is a voltage difference.

The same voltage exists between *x*_{1}' and *x*_{2}' (explained below); therefore the voltage difference is across the intersection point.

According to Bobrow^{1}, the formula for the voltage difference across a hole gradient is the following:

v_{0} = *V _{T }*ln(

*p*

_{1}/

*p*

_{2}),

where *V _{T}* =

*T*/ 11,586 ( = 0.0259 at room temperature of

*T*= 300 K),

*p*

_{1}is the hole concentration at

*x*

_{1}, and

*p*

_{2}is the hole concentration at point

*x*

_{2}. Note that this formula depends only on the difference in hole concentration between the two points and not the distance separating them; therefore, if we assume the hole and electron concentrations are constant throughout the p-type and n-type sections, respectively, we can see that the exact same voltage difference would be found between points

*x*

_{1}' and

*x*

_{2}' as between

*x*

_{1}and

*x*

_{2}. Furthermore, we could move

*x*

_{1}' and

*x*

_{2}' arbitrarily close to the junction line and still find the same voltage. Therefore, the voltages

*within*the

*p*- and

*n*-type materials must be constant, and the difference must occur only at the junction line.

Now let's simplify the equation. The concentration of holes in the *p*-type section (*p*_{1}) is approximately equal to the concentration of acceptor atoms, *N _{A}*. The

**mass-action law**says that, at all times, the product of the hole concentration (

*p*) and free electron concentration (

*n*) is equal to the square of the intrinsic concentration (

*n*) of the semiconductor:

_{i}np = *n _{i}*

^{2}.

Therefore, the hole concentration in the *n*-type section is *p* = *n _{i}*

^{2}/

*n*. Since the free electron concentration is approximately equal to the donor atom concentration

*N*, we have

_{D}*p*=

*n*

_{i}^{2}/

*N*, an expression which contains all easily measureable quantities. Plugging these expressions for

_{D}*p*

_{1}and

*p*

_{2}into Bobrow's equation, we find:

*v*_{0} = *V _{T}* ln (

*N*/

_{A}N_{D}*n*

_{i}^{2})

This is the formula for the barrier potential of a p-n junction. Note that this is the voltage difference in the direction from n-type towards p-type. In reality, the hole and electron concentrations will not be constant throughout the material and will decrease as you move towards the junction line. However, the approximations made here still produce a formula for the barrier voltage that is reasonably accurate with empirical results.

## Space-Charge Region

Consider again the locations and concentrations of charge carriers within the p- and n-type material. The p-type material will have a high concentration of holes and low concentration of free electrons, and vice versa for the n-type material. But any time particles are contained in a crowded environment, they will naturally diffuse out of that environment. Therefore, this steep concentration gradient will result in the diffusion of majority carriers across the junction line -- holes will diffuse into the n-type side and free electrons will diffuse into the p-type side. The free electrons will recombine with the holes, leaving this area depleted of charge carriers. This area is called the space-charge region, or the depletion region.

Recall that the p-type material is doped with Group 3 atoms, which have an ionic core charge of +3 and are surrounded by three valence electrons (charge -3). When placed in the semiconductor lattice, they create an excess hole. If this hole is filled by an electron that diffuses across the junction line, then the acceptor will still have an ionic core charge of +3 but will now be surrounded by 4 valence electrons. Therefore, the acceptor atoms in the space-charge region will have a charge of -1. The opposite is true for the Group 5 donor atoms in the n-type material, which normally have an ionic core charge of +5 and are surrounded by five valence electrons. When put into the semiconductor lattice, however, the fifth electron cannot covalent bond and is therefore a free electron. If the free electron diffuses across the junction-line, then the +5 ionic core will only be surrounded by 4 valence electrons; it will then carry a charge of +1. Therefore, the space-charge region consists of a region of negative ions in the p-type material, and a region of positive ions in the n-type material (see figure below). This juxtaposition of positive and negative ions creates an electric field within the material, pointed from n-type to p-type.

The diffusion of electrons and holes across the junction line creates uncovered ions in the space-charge region. These ions

create an electric field within the material, which further affect the carriers (explained below). Image courtesy of Wikimedia Commons.

The electric field resulting from the uncovered ions in the space-charge region will tend to push free electrons back into the n-type material and holes back into the p-type material. This force, which opposes diffusion, is called "drift". Drift gets stronger as diffusion increases, so it will prevent all the majority carriers from diffusing across the junction line. The charges will eventually orient themselves in such a manner that the diffusive and drift forces are exactly opposite.

## References

- "Chapter 6: Diodes."
*Fundamentals of Electrical Engineering*. 2nd ed. New York, New York: Oxford UP, 1996. 352-54. Print.