Introduction to Junctions and Equilibrium Conditions in Semiconductor Devices

1. Chapter 5 Junctions

2. 5.1 Introduction (chapter 3) • 5.2 Equilibrium condition • 5.2.1 Contact potential • 5.2.2 Equilibrium Fermi level • 5.2.3 Space charge at a junction • 5.3 Forward bias • 5.3.1

3. Irradiation Mask/Shield/Pattern (negative) Photoresist Silicon Oxide Silicon Develop Metal Oxide Lift off

4. Fermi Gas and Density of State

5. Particle in a Infinite Well For three-dimensional box

6. Electron Energy Density

7. Density of State ρ(E)

8. Properties Dependent on Density of States Experiment provide information on density of state

13. N(E)f(E) N(E)[1-f(E)] Holes (a) Intrinsic (a) Intrinsic f(E): Probability of occupation (Fermi-Dirac distribution function) N(E): Density of state N=N(E)f(E)dE: Concentration of electrons in the conduction band N=N(E)dE: Total number of states per unit volume

14. Electrons Holes (a) Intrinsic (b) n-type (c) p-type

15. This density of state equation is derived from assumption of electron in the infinite well with vacuum medium, where the E is proportional to k2. We found that the free electron in the conduction band of semiconductor has local minimum of energy E versus wave number k. We can approximate the bottom portion of the curve as if E is still proportional to k2 and write down the similar energy-wave number equation as to describe the behavior of the free electrons, where mn* is the equivalent electron mass, which account for the electron accommodation to medium change. If we prefer to the energy at the bottom of the conduction band as a nun-zero value of Ec instead of Ec = 0, The density of state equation can be further modified as

17. Nc: Effective density of state at bottom of C.B. Nv: Effective density of state at top of V.B. no: Concentration of electrons in the conduction band po: Concentration of holes in the valence band Ec: Conduction band edge Ev: Valence band edge EF: Fermi level Ei: Fermi level for the undoped semiconductor (intrinsic)

18. Fermi Level and Carrier Concentration of Intrinsic Semiconductor

21. Example 3-5 A Si sample is doped with 1017 As atoms/cm3. What is the equilibrium hole concentration po at 300K? Where is EF relative to Ei?

22. 5.1 Introduction • 5.2 Equilibrium condition • 5.2.1 Contact potential • 5.2.2 Equilibrium Fermi level • 5.2.3 Space charge at a junction • 5.3 Forward bias • 5.3.1

23. Electric field

25. Einstein relationship (explained later) Electric field

27. Einstein Relationship drift diffusion • At equilibrium, no net current flows in a semiconductor. Jp(x) = 0 • Any fluctuation which would begin a diffusion current also sets up an electric field which redistributes carriers by drift. • An examination of the requirements for equilibrium indicates that the diffusion coefficient and mobility must be related.

28. Einstein Relationship Drift Diffusion

29. Einstein Relationship Drift and diffusion drift diffusion diffusion

40. The derivation of Poisson's equation in electrostatics follows. SI units are used and Euclidean space is assumed. Starting with Gauss' law for electricity (also part of Maxwell's equations) in a differential control volume, we have: is the divergence operator. is the electric displacement field. is the free charge density (describing charges brought from outside). Assuming the medium is linear, isotropic, and homogeneous (see polarization density), then: is the permittivity of the medium. is the electric field. By substitution and division, we have: Poisson's equation http://en.wikipedia.org/wiki/Poisson's_equation