PN Junction Electrostatics Chapter 5. PN Junction Electrostatics Sung June Kim kimsj@snu.ac.kr http://nanobio.snu.ac.kr

Subject Analysis of Field and potential distribution in the junction. - Quantitative Electrostatic Relationships

Junction Terminology/Idealized Profiles Preliminaries Junction Terminology/Idealized Profiles Net doping profile

Step (abrupt) junction Linearly graded junction The step junction is an acceptable approximation to an ion-implantation or shallow diffusion into a lightly doped starting wafer

Poisson’s Equation 1-Dimension Ks is the semiconductor dielectric constant and 0 is the permittivity of free space.  is the charge density (charge/cm3)

Let us assume an equilibrium conditions Qualitative Solution Let us assume an equilibrium conditions It is reasonable to expect regions far removed from the metallurgical junction to be identical to an isolated semiconductor.

Under equilibrium conditions, the Fermi level is a constant

P-N Diode Junction Energy Band Ec Ef Ev Ec Ef Ev

Equilibrium P-N Junction V=0 P N Ec Ef Ev Ec Ef Ev

Forward Biased P-N Junction V>0 P N Ec Ef Ev Ec Ef Ev

Reverse Biased P-N Junction Ec Ef Ev Ec Ef Ev

V versus x relationship must have the same functional form as the “ upside-down” of Ec Eref

The voltage drop across the junction under equilibrium conditions and the appearance of charge near the metallurgical boundary Where does this charge come from?

Charge neutrality is assumed to prevail in the isolated, uniformly doped semiconductors hole electron Charge redistribution Space charge region or depletion region Charge density

The build-up of charge and the associated electric field continues until the diffusion is precisely balanced by the carrier drift The individual carrier diffusion and drift components must of course cancel to make JN and JP separately zero

The Built-in Potential (Vbi) Consider a nondegenerately-doped junction Integrating Solving for and making use of the Einstein relationship, we obtain

The Depletion Approximation It is very hard to solve The carrier concentrations are negligible in The charge density outside the depletion region=0

Exact Depletion Approximation

Assumptions/definitions Quantitative Electrostatic Relationships Assumptions/definitions

Step Junction with VA=0 Solution for  Solution for

For the p-side of the depletion region Similarly on the n-side

Solution for V ( ) With the arbitrary reference potential set equal to zero at x=-xp and Vbi across the depletion region equilibrium conditions

For the p-side of the depletion region Similarly on the n-side of the junction @ x=0

Solution for xn and xp Depletion width

Step Junction with VA0 When VA > 0, the externally imposed voltage drop lowers the potential on the n-side relative to the p-side

The voltage drop across the depletion region, and hence the boundary condition at x=xn, becomes Vbi-VA

The voltage drop across the depletion region, and hence the boundary condition at x=xn, becomes Vbi-VA

Examination/Extrapolation of Results Depletion widths decrease under forward biasing and increase under reverse biasing A decreased depletion width when VA > 0 means less charge around the junction and a correspondingly smaller -field. Similarly, the potential decreases at all points when VA >0 The Fermi level is omitted from the depletion region

pn junction energy band diagrams.

Linearly Graded Junctions The linearly graded profile is modeled by The profile is continuous through x = 0 The profile is symmetrical about x = 0, all of the electrostatic variables exhibit a symmetry about x = 0 Although V is taken to be a constant outside of the depletion region, a modification of the equation for Vbi is still necessary

The charge density is symmetrical about x = 0, … … The charge density is symmetrical about x = 0, … …

Where ,