Potential vs. Kinetic Energy

2. Chapter 3 Carrier Action Electron kinetic energy Ec Increasing electron energy Increasing hole energy Ev Hole kinetic energy Potential vs. Kinetic Energy • Ec represents the electron potential energy:

3. Chapter 3 Carrier Action Band Bending • Until now, Ec and Ev have always been drawn to be independent of the position. • When an electric field E exists inside a material, the band energies become a function of position. E Ec Ev x • Variation of Ec with position is called “band bending”

4. Chapter 3 Carrier Action Band Bending • The potential energy of a particle with charge –q is related to the electrostatic potential V(x): • Since Ec, Ev, and Ei differ only by an additive constant

5. Chapter 3 Carrier Action Diffusion • Particles diffuse from regions of higher concentration to regions of lower concentration region, due to random thermal motion (Brownian Motion).

6. Chapter 3 Carrier Action 1-D Diffusion Example • Thermal motion causes particles to move into an adjacent compartment every τ seconds.

7. Chapter 3 Carrier Action e- h+ Diffusion Currents n p x x Electron flow Hole flow • D is the diffusion coefficient [cm2/sec] Current flow Current flow

8. Chapter 3 Carrier Action Total Currents • Drift current flows when an electric field is applied. • Diffusion current flows when a gradient of carrier concentration exist.

9. Chapter 3 Carrier Action Current Flow Under Equilibrium Conditions • In equilibrium, there is no net flow of electrons or : • The drift and diffusion current components must balance each other exactly. • A built-in electric field of ionized atoms exists, such that the drift current exactly cancels out the diffusion current due to the concentration gradient.

10. Chapter 3 Carrier Action Ec(x) EF Ev(x) Current Flow Under Equilibrium Conditions • Consider a piece of non-uniformly doped semiconductor: n-type semiconductor Decreasing donor concentration • Under equilibrium, EF inside a material or a group of materials in intimate contact is not a function of position

11. Chapter 3 Carrier Action Einstein Relationship between D and m • But, under equilibrium conditions, JN= 0 and JP= 0 Similarly, • Einstein Relationship • Further proof can show thatthe Einstein Relationship is valid for a non-degenerate semiconductor, both in equilibrium and non-equilibrium conditions.

12. Chapter 3 Carrier Action Example: Diffusion Coefficient • What is the hole diffusion coefficient in a sample of silicon at 300 K with p = 410 cm2 / V.s ? • Remark:kT/q= 25.86 mVat room temperature

13. Chapter 3 Carrier Action Recombination–Generation • Recombination: a process by which conduction electrons and holes are annihilated in pairs. • Generation: a process by which conduction electrons and holes are created in pairs. • Generation and recombination processes act to change the carrier concentrations, and thereby indirectly affect current flow.

14. Chapter 3 Carrier Action Generation Processes Band-to-Band R–G Center Impact Ionization Release of energy ET: trap energy level • Due to lattice defects or unintentional impurities • Also called indirect generation EG • Only occurs in the presence of large E

15. Chapter 3 Carrier Action Recombination Processes Band-to-Band R–G Center Auger Collision • Rate is limited by minority carrier trapping • Primary recombination way for Si • Occurs in heavily doped material

16. Chapter 3 Carrier Action Phonon Photon Photon Direct and Indirect Semiconductors E-k Diagrams Ec Ec Ev Ev GaAs, GaN (direct semiconductors) Si, Ge (indirect semiconductors) • Large change in momentum is required for recombination • Momentum is conserved by mainly phonon (vibration) emission + photon emission • Little change in momentumis required for recombination • Momentum is conserved by photon (light) emission

17. Chapter 3 Carrier Action Values under arbitrary conditions Deviation from equilibrium values Equilibrium values Excess Carrier Concentrations • Positive deviation corresponds to a carrier excess, while negative deviations corresponds to a carrier deficit. • Charge neutrality condition:

18. Chapter 3 Carrier Action “Low-Level Injection” • Often, the disturbance from equilibrium is small, such that the majoritycarrier concentration is not affected significantly: • For an n-type material • For a p-type material • Low-level injection condition • However, the minority carrier concentration can be significantly affected.

19. Chapter 3 Carrier Action Indirect Recombination Rate • Suppose excess carriers are introduced into an n-typeSi sample by shining light onto it. At time t = 0, the light is turned off. How does p vary with time t > 0? • Consider the rate of hole recombination: NT : number of R–G centers/cm3 Cp : hole capture coefficient • In the midst of relaxing back to the equilibrium condition, the holegeneration rate is small and is taken to be approximately equal to its equilibrium value:

20. Chapter 3 Carrier Action Indirect Recombination Rate • The net rate of change in p is therefore: where • For holes in n-type material • Similarly, where • For electrons in p-type material

21. Chapter 3 Carrier Action Minority Carrier Lifetime • The minority carrier lifetimeτis the average time for excess minority carriers to “survive” in a sea of majority carriers. • The value of τ ranges from 1 ns to 1 ms in Si anddepends on the density ofmetallic impurities and the density of crystalline defects. • Thedeep trapsoriginated from impurity and defects capture electrons or holes to facilitate recombination and are calledrecombination-generation centers.

22. Chapter 3 Carrier Action Example: Photoconductor • Consider a sample of Si doped with 1016 cm–3 Boron, with recombination lifetime 1 μs. It is exposed continuously to light, such that electron-hole pairs are generated throughout the sample at the rate of 1020 per cm3 per second, i.e. the generation rate GL = 1020/cm3/s • a) What are p0 and n0? • b) What are Δn and Δp? • Hint: In steady-state, generation rate equals recombination rate

23. Chapter 3 Carrier Action Example: Photoconductor • Consider a sample of Si at 300 K doped with 1016 cm–3 Boron, with recombination lifetime 1 μs. It is exposed continuously to light, such that electron-hole pairs are generated throughout the sample at the rate of 1020 per cm3 per second, i.e. the generation rate GL = 1020/cm3/s. • c) What are p and n? • d) What are np product? • Note: The np product can be very different from ni2 in case of perturbed/agitated semiconductor

24. Chapter 3 Carrier Action Net Recombination Rate (General Case) • For arbitrary injection levels and both carrier types in a non-degenerate semiconductor, the net rate of carrier recombination is: where • ET : energy level of R–G center

25. Chapter 2 Carrier Action Homework 3 • 1. (4.27) • Problem 3.12, from (a) until (f), for Figure P3.12(a) and Figure P3.12(f),Pierret’s “Semiconductor Device Fundamentals”. • 2. (5.28) • The electron concentration in silicon at T = 300 K is given by • where x is measured in μm and is limited to 0 ≤ x ≤ 25 μm. The electron diffusion coefficient is DN = 25 cm2/s and the electron mobility is μn = 960 cm2/(Vs). The total electron current density through the semiconductor is constant and equal to JN = –40 A/cm2. The electron current has both diffusion and drift current components. • Determine the electric field as a function of x which must exist in the semiconductor. Sketch the function. • Deadline: 10 February 2011, at 07:30.