1. Lecture #7 OUTLINE Carrier diffusion Diffusion current Einstein relationship Generation and recombination Read: Sections 3.2, 3.3

2. Spring 2007EE130 Lecture 7, Slide 2 Particles diffuse from regions of higher concentration to regions of lower concentration region, due to random thermal motion. Diffusion

3. Spring 2007EE130 Lecture 7, Slide 3 1-D Diffusion Example Thermal motion causes particles to move into an adjacent compartment every  seconds –Each particle has an equal probability of jumping to the left and to the right.

4. Spring 2007EE130 Lecture 7, Slide 4 D is the diffusion constant, or diffusivity. xx Diffusion Current

5. Spring 2007EE130 Lecture 7, Slide 5 J N = J N,drift + J N,diff = qn  n  + J P = J P,drift + J P,diff = qp  p  – J = J N + J P Total Current

6. Spring 2007EE130 Lecture 7, Slide 6 Non-Uniformly-Doped Semiconductor The position of E F relative to the band edges is determined by the carrier concentrations, which is determined by the dopant concentrations. In equilibrium, E F is constant; therefore, the band energies vary with position: E v (x) E c (x) EFEF

7. Spring 2007EE130 Lecture 7, Slide 7 In equilibrium, there is no net flow of electrons or holes  The drift and diffusion current components must balance each other exactly. (A built-in electric field exists, such that the drift current exactly cancels out the diffusion current due to the concentration gradient.) 0  dx dn qDqnJ NnN  J N = 0 and J P = 0 

8. Spring 2007EE130 Lecture 7, Slide 8 Consider a piece of a non-uniformly doped semiconductor: E v (x) E c (x) EFEF q kT n  

9. Spring 2007EE130 Lecture 7, Slide 9 0  dx dn qDqnJ NnN  Under equilibrium conditions, J N = 0 and J P = 0 kT qD qn N n   0 Similarly, Einstein Relationship between D and  Note: The Einstein relationship is valid for a non-degenerate semiconductor, even under non-equilibrium conditions  

10. Spring 2007EE130 Lecture 7, Slide 10 What is the hole diffusion constant in a sample of silicon with  p = 410 cm 2 / V s ? Solution : Remember: kT/q = 26 mV at room temperature. Example: Diffusion Constant

11. Spring 2007EE130 Lecture 7, Slide 11 Potential Difference due to n(x), p(x) The ratio of carrier densities (n, p) at two points depends exponentially on the potential difference between these points:

12. Spring 2007EE130 Lecture 7, Slide 12 Quasi-Neutrality Approximation If the dopant concentration profile varies gradually with position, then the majority-carrier concentration distribution does not differ much from the dopant concentration distribution. –n-type material: –p-type material:  in n-type material

13. Spring 2007EE130 Lecture 7, Slide 13 Summary Electron/hole concentration gradient  diffusion Current flowing in a semiconductor is comprised of drift & diffusion components for electrons & holes –In equilibrium, J N = J N,drift + J N,diff = 0 The characteristic constants of drift and diffusion are related: J = J N,drift + J N,diff + J P,drift + J P,diff