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The p-n junction diode characteristics and operation is closely connected to the behaviour of the carriers in the junction formed at the boundary between the p-type and n-type materials. The p-n junction is also the basic building block of a variety of other devices, and the basic techniques used for the analysis of  the p-n junction can be extended to other problems. Here, we study the current-voltage characteristics of an ideal silicon p-n junction diode.




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Problem Definition

An ideal diode contains several simplifications to the properties and behavior of a more realistic device, but offers the opportunity to compare results with an analytic expression for the current-voltage relation. The ideal diode is modeled with the following approximations:

1D formulation,

ideal ohmic contacts,

an abrupt metallurgical junction,

and no recombination.


Based on these approximations, several key properties of the device can be formulated analytically. These values are used to validate the numerical model.


Built-in Voltage

The built-in voltage for a p-n junction, assuming perfect ohmic contacts, is independent of the junction profile. Assuming that the distance between the junction and contacts is sufficient to establish local thermal equilibrium in both the p-type and n-type regions, the built-in voltage Vbi can be determined from the following equation,



where k is the Boltzmann constant, NA is the density of acceptors in the p-type region, ND is the density of donors in the n-type region, and ni is the intrinsic carrier concentration.


Junction Width

When an abrupt junction is formed, the width (W) of the space-charge layer can be predicted using the full depletion approximation. Using the built-in voltage determined previously,



where ε is the dielectric permittivity of the material. Here, several assumptions about the dopants have been applied:

the dopants fully ionize

the junction is abrupt

the carrier density is neglected in the space - charge layer (NA >> p and ND >> n) which is known as the full depletion approximation.


When one side of the junction is doped more heavily than the other, e.g. NA >> ND, the expression simplifies: the majority of the space charge layer is distributed where the lowest dopant concentration exists.



Diode Current

Based on the carrier injection description of current through a p-n junction, and the assumption that recombination can be neglected in the space-charge layer, the diode equation is derived,



where A is the junction area, and V is the applied voltage, The reverse bias saturation current I0 is expressed in terms of the minority hole concentration pn in the n-type region and the minority electron concentration np in the p-type region. Dn,p is the diffusivity, which is related to the mobility of the electrons and holes according to the Einstein relation,



and Ln,p is the diffusion length,



related to the carrier lifetime (due to recombination) τn,p. When a reverse bias voltage (V < 0) is applied where |V| is more than a few multiples of kT/q, the diode will reach the reverse bias saturation current I0.


Noting that




far from the space-charge layer in the appropriate region (where the subscripts refer to that region – n-type or p-type) and using the equilibrium relation



we then have



A capacitance can be associated with the p-n junction due to the charge build up at the depletion layer,



It can be shown that this capacitance for an ideal p-n_junction is :



And When NA >> ND , this expression can be simplified:


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