Semiconductor Device Physics Lecture 6 Dr. Gaurav Trivedi, EEE Department, IIT Guwahati

Doping profile Metallurgical Junction

Poisson’s Equation Poisson’s equation is a well-known relationship in electricity and magnetism. It is now used because it often contains the starting point in obtaining quantitative solutions for the electrostatic variables. In one-dimensional problems, Poisson’s equation simplifies to :

Equilibrium Energy Band Diagram pn-Junction diode

Qualitative Electrostatics Band diagram Equilibrium condition Electrostatic potential

Qualitative Electrostatics Electric field Equilibrium condition Charge density

Formation of pn Junction and Charge Distribution qN A – qN D +

Formation of pn Junction and Charge Distribution

Built-In Potential V bi For non-degenerately doped material, V bi for several materials: Ge ≤ 0.66 V Si ≤ 1.12 V GeAs ≤ 1.42 V

The Depletion Approximation with boundary E (–x p )  0 with boundary E (x n )  0 The Depletion Approximation On the p-side, ρ = –qN A On the n-side, ρ = qN D

Step Junction with V A   0 Solution for ρ Solution for E Solution for V

Step Junction with V A   0 At x = 0, expressions for p-side and n-side for the solutions of E and V must be equal:

Relation between ρ(x), E (x), and V(x) 1.Find the profile of the built-in potential V bi 2.Use the depletion approximation  ρ(x) With depletion-layer widths x p, x n unknown 3.Integrate ρ(x) to find E (x) Boundary conditions E (–x p )  0, E (x n )  0 4.Integrate E (x) to obtain V(x) Boundary conditions V(–x p )  0, V(x n )  V bi 5.For E (x) to be continuous at x  0, N A x p  N D x n Solve for x p, x n

Depletion Layer Width Eliminating x p, Eliminating x n, Summing Exact solution, try to derive

One-Sided Junctions If N A >> N D as in a p + n junction, If N D >> N A as in a n + p junction, Simplifying, where N denotes the lighter dopant density

Step Junction with V A  0 To ensure low-level injection conditions, reasonable current levels must be maintained  V A should be small

Step Junction with V A  0 In the quasineutral, regions extending from the contacts to the edges of the depletion region, minority carrier diffusion equations can be applied since E ≈ 0. In the depletion region, the continuity equations are applied.

Step Junction with V A  0 Built-in potential V bi (non-degenerate doping): Depletion width W :

Effect of Bias on Electrostatics If voltage drop , then depletion width  If voltage drop , then depletion width 

Quasi-Fermi Levels Whenever Δn = Δp ≠ 0 then np ≠ n i 2 and we are at non- equilibrium conditions. In this situation, now we would like to preserve and use the relations: On the other hand, both equations imply np = n i 2, which does not apply anymore. The solution is to introduce to quasi-Fermi levels F N and F P such that: The quasi-Fermi levels is useful to describe the carrier concentrations under non-equilibrium conditions

Example: Quasi-Fermi Levels a)What are p and n? b)What is the np product?

Example: Quasi-Fermi Levels Consider a Si sample at 300 K with N D = cm –3 and Δn = Δp = cm –3. c)Find F N and F P ? EcEc EvEv EiEi FPFP eV FNFN eV

Linearly-Graded Junction

Linearly-Graded Junction Homework Assignment Derive the relations of electrostatic variables, charge density and builtin voltage V bi for the linearly graded junction.