1. President UniversityErwin SitompulSDP 4/1 Lecture 4 Semiconductor Device Physics Dr.-Ing. Erwin Sitompul President University http://zitompul.wordpress.com 2013

2. President UniversityErwin SitompulSDP 4/2 EvEv EcEc EcEc EvEv GaAs, GaN (direct semiconductors) Si, Ge (indirect semiconductors) Photon Phonon Direct and Indirect Semiconductors Chapter 3Carrier Action E-k Diagrams Little change in momentum is required for recombination Momentum is conserved by photon (light) emission Large change in momentum is required for recombination Momentum is conserved by mainly phonon (vibration) emission + photon emission

3. President UniversityErwin SitompulSDP 4/3 Equilibrium values Deviation from equilibrium values Excess Carrier Concentrations Chapter 3Carrier Action Positive deviation corresponds to a carrier excess, while negative deviation corresponds to a carrier deficit. Values under arbitrary conditions Charge neutrality condition:

4. President UniversityErwin SitompulSDP 4/4 Often, the disturbance from equilibrium is small, such that the majority carrier concentration is not affected significantly. However, the minority carrier concentration can be significantly affected. For an n-type material For a p-type material “Low-Level Injection” Chapter 3Carrier Action This condition is called “low-level injection condition”. The workhorse of the diffusion in low-level injection condition is the minority carrier (which number increases significantly) while the majority carrier is practically undisturbed.

5. President UniversityErwin SitompulSDP 4/5 N T : number of R–G centers/cm 3 C p : hole capture coefficient Indirect Recombination Rate Chapter 3Carrier Action Suppose excess carriers are introduced into an n-type Si 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: In the midst of relaxing back to the equilibrium condition, the hole generation rate is small and is taken to be approximately equal to its equilibrium value:

6. President UniversityErwin SitompulSDP 4/6 where Indirect Recombination Rate Chapter 3Carrier Action The net rate of change in p is therefore: For holes in n-type material For electrons in p-type material Similarly,

7. President UniversityErwin SitompulSDP 4/7 Minority Carrier Lifetime Chapter 3Carrier Action The minority carrier lifetime τ is 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 and depends on the density of metallic impurities and the density of crystalline defects. The deep traps originated from impurity and defects capture electrons or holes to facilitate recombination and are called recombination-generation centers.

8. President UniversityErwin SitompulSDP 4/8 Photoconductor Chapter 3Carrier Action Photoconductivity is an optical and electrical phenomenon in which a material becomes more electrically conductive due to the absorption of electro-magnetic radiation such as visible light, ultraviolet light, infrared light, or gamma radiation. When light is absorbed by a material like semiconductor, the number of free electrons and holes changes and raises the electrical conductivity of the semiconductor. To cause excitation, the light that strikes the semiconductor must have enough energy to raise electrons across the band gap.

9. President UniversityErwin SitompulSDP 4/9 Example: Photoconductor Chapter 3Carrier Action Consider a sample of Si at 300 K doped with 10 16 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 10 20 per cm 3 per second, i.e. the generation rate G L = 10 20 /cm 3 /s. a)What are p 0 and n 0 ? b) What are Δn and Δp? Hint: In steady-state (equilibrium), generation rate equals recombination rate

10. President UniversityErwin SitompulSDP 4/10 Example: Photoconductor Chapter 3Carrier Action Consider a sample of Si at 300 K doped with 10 16 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 10 20 per cm 3 per second, i.e. the generation rate G L = 10 20 /cm 3 /s. c)What are p and n? d) What are np product? Note: The np product can be very different from n i 2 in case of perturbed/agitated semiconductor

11. President UniversityErwin SitompulSDP 4/11 E T : energy level of R–G center Net Recombination Rate (General Case) Chapter 3Carrier Action For arbitrary injection levels and both carrier types in a non- degenerate semiconductor, the net rate of carrier recombination is: where

12. President UniversityErwin SitompulSDP 4/12 JN(x)JN(x)J N (x+dx) dx Area A, volume A.dx Continuity Equation Chapter 3Carrier Action Consider carrier-flux into / out of an infinitesimal volume: Flow of current Flow of electron

13. President UniversityErwin SitompulSDP 4/13 Continuity Equation Chapter 3Carrier Action Taylor’s Series Expansion The Continuity Equations

14. President UniversityErwin SitompulSDP 4/14 Minority Carrier Diffusion Equation Chapter 3Carrier Action The minority carrier diffusion equations are derived from the general continuity equations, and are applicable only for minority carriers. Simplifying assumptions: The electric field is small, such that: For p-type material For n-type material Equilibrium minority carrier concentration n 0 and p 0 are independent of x (uniform doping). Low-level injection conditions prevail.

15. President UniversityErwin SitompulSDP 4/15 Minority Carrier Diffusion Equation Chapter 3Carrier Action Starting with the continuity equation for electrons: Therefore Similarly

16. President UniversityErwin SitompulSDP 4/16 Carrier Concentration Notation Chapter 3Carrier Action The subscript “n” or “p” is now used to explicitly denote n-type or p-type material. p n is the hole concentration in n-type material n p is the electron concentration in p-type material Thus, the minority carrier diffusion equations are: Partial Differential Equation (PDE)! The so called “Heat Conduction Equation”

17. President UniversityErwin SitompulSDP 4/17 Simplifications (Special Cases) Chapter 3Carrier Action Steady state: No diffusion current: No thermal R–G: No other processes: Solutions for these common special-case diffusion equation are provided in the textbook

18. President UniversityErwin SitompulSDP 4/18 Similarly, Minority Carrier Diffusion Length Chapter 3Carrier Action Consider the special case: Constant minority-carrier (hole) injection at x = 0 Steady state, no light absorption for x > 0 The hole diffusion length L P is defined to be:

19. President UniversityErwin SitompulSDP 4/19 Minority Carrier Diffusion Length Chapter 3Carrier Action The general solution to the equation is: A and B are constants determined by boundary conditions: Therefore, the solution is: Physically, L P and L N represent the average distance that a minority carrier can diffuse before it recombines with a majority carrier.

20. President UniversityErwin SitompulSDP 4/20 Example: Minority Carrier Diffusion Length Chapter 3Carrier Action Given N D =10 16 cm –3, τ p = 10 –6 s. Calculate L P. From the plot,

21. President UniversityErwin SitompulSDP 4/21 Quasi-Fermi Levels Chapter 3Carrier Action 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

22. President UniversityErwin SitompulSDP 4/22 Example: Quasi-Fermi Levels Chapter 3Carrier Action Consider a Si sample at 300 K with N D = 10 17 cm –3 and Δn = Δp = 10 14 cm –3. The sample is an n-type a)What are p and n? b)What is the np product?

23. President UniversityErwin SitompulSDP 4/23 Example: Quasi-Fermi Levels Chapter 3Carrier Action Consider a Si sample at 300 K with N D = 10 17 cm –3 and Δn = Δp = 10 14 cm –3. c)Find F N and F P ? EcEc EvEv EiEi FPFP 0.238 eV FNFN 0.417 eV

24. President UniversityErwin SitompulSDP 4/24 1. (6.17) A certain semiconductor sample has the following properties: D N = 25 cm 2 /s τ n0 = 10 –6 s D P = 10 cm 2 /sτ p0 = 10 –7 s It is a homogeneous, p-type (N A = 10 17 cm –3 ) material in thermal equilibrium for t ≤ 0. At t = 0, an external light source is turned on which produces excess carriers uniformly at the rate G L = 10 20 cm –3 s –1. At t = 2×10 –6 s, the external light source is turned off. (a) Derive the expression for the excess-electron concentration as a function of time for 0 ≤ t ≤ ∞. (b) Determine the value of the excess-electron concentration at (i) t = 0, (ii) t = 2×10 –6 s, and (iii) t = ∞. (c) Plot the excess electron concentration as a function of time. Chapter 3Carrier Action Homework 3 2. (4.38) Problem 3.24 Pierret’s “Semiconductor Device Fundamentals”. Due date: 14.10.2013.