Lecture 7 OUTLINE Carrier diffusion Diffusion current Einstein

Lecture #7 OUTLINE • Carrier diffusion • Diffusion current • Einstein relationship • Generation and recombination Read: Sections 3. 2, 3. 3

Diffusion Particles diffuse from regions of higher concentration to regions of lower concentration region, due to random thermal motion. 2 Spring 2007 EE 130 Lecture 7, Slide 2

1 -D Diffusion Example • Thermal motion causes particles to move into an adjacent compartment every t seconds – Each particle has an equal probability of jumping to the left and to the right. 3 Spring 2007 EE 130 Lecture 7, Slide 3

Diffusion Current x x D is the diffusion constant, or diffusivity. 4 Spring 2007 EE 130 Lecture 7, Slide 4

Total Current J = JN + JP e+ JN = JN, drift + JN, diff = qn n e– JP = JP, drift + JP, diff = qp p 5 Spring 2007 EE 130 Lecture 7, Slide 5

Non-Uniformly-Doped Semiconductor • The position of EF relative to the band edges is determined by the carrier concentrations, which is determined by the dopant concentrations. • In equilibrium, EF is constant; therefore, the band energies vary with position: Ec(x) EF Ev(x) 6 Spring 2007 EE 130 Lecture 7, Slide 6

• In equilibrium, there is no net flow of electrons or holes JN = 0 and JP = 0 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. ) dn = + =0 e J N qn n q. DN dx 7 Spring 2007 EE 130 Lecture 7, Slide 7

Consider a piece of a non-uniformly doped semiconductor: Ec(x) EF =Ev(x) n qe k. T 8 Spring 2007 EE 130 Lecture 7, Slide 8

Einstein Relationship between D and m Under equilibrium conditions, JN = 0 and JP = 0 J N = qn ne + q. DN 0 = qn n e - qn dn =0 dx q. DN e k. T Similarly, Note: The Einstein relationship is valid for a non-degenerate semiconductor, even under non-equilibrium conditions 9 Spring 2007 EE 130 Lecture 7, Slide 9

Example: Diffusion Constant What is the hole diffusion constant in a sample of silicon with p = 410 cm 2 / V s ? Solution: Remember: k. T/q = 26 m. V at room temperature. 10 Spring 2007 EE 130 Lecture 7, Slide 10

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: 11 Spring 2007 EE 130 Lecture 7, Slide 11

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 12 Spring 2007 EE 130 Lecture 7, Slide 12

Summary • Electron/hole concentration gradient diffusion • Current flowing in a semiconductor is comprised of drift & diffusion components for electrons & holes – In equilibrium, JN = JN, drift + JN, diff = 0 J = JN, drift + JN, diff + JP, drift + JP, diff • The characteristic constants of drift and diffusion are related: 13 Spring 2007 EE 130 Lecture 7, Slide 13