1. Semiconductor Device Physics Lecture 4 Dr. GauravTrivedi, EEE Department, IIT Guwahati

2. 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. 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

4. 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”

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

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

7. Diffusion Currents n p x x Electron flow Hole flow • D is the diffusion coefficient [cm2/sec] Current flow

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

9. 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. 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. 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. 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. 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. 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. 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. Phonon Photon Photon Direct and Indirect Semiconductors 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. 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. “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. 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. 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. 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. 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

23. 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