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# lec4 - 6.012 Electronic Devices and Circuits Lecture 4...

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6.012 - Electronic Devices and Circuits Lecture 4 - Non-uniform Injection (Flow) Problems - Outline Announcements Handouts - 1. Lecture Outline and Summary; 2. Thermoelectrics Review Thermoelectricity: temperature gradients as a driving force The 5 basic equations Non-uniform injection (flow) problems Five assumptions: 1. Uniform doping: r = e ∂E/∂x ≈ q(p' – n'); ∂n = ∂n', ∂p = ∂p' 2. Low level injection: [np - n i 2 ]r(T) ≈ n'/ t min 3. Quasi-neutrality: n' ≈ p', ∂n'/∂x ≈ ∂p'/∂x 4. Negligible minority carrier drift: J e ≈ qD e ∂n'/∂x (assumes p-type) 5. Quasi-static: ∂/∂t ≈ 0 The diffusion equation for minority carriers Getting J e (x), J h (x), E(x), and p'(x), knowing n'(x) Solving the diffusion equation for n'(x) (using p-type example) Homogeneous solutions; minority carrier diffusion length, L min Particular solution Boundary conditions: ohmic contact, reflecting boundary, internal boundary Total solution Clif Fonstad, 9/03 Lecture 4 - Slide 1

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Consider our electron current density expression with the drift term rewritten in terms of the potential and including the term that must be added if we have a temperature gradient*: This way of writing the current emphasizes the fact that the carriers respond to gradients in various quantities. Drift is the response to a gradient in the electrostatic potential, diffusion is the response to a gradient in the carrier concentration, and the Seebeck effect is the response to a gradient in temperature. The Seebeck effect is interesting because it can be used to generate electric power from a temperature difference. Conversely, a temp- erature difference can be generated by passing a current through a suitably designed semiconductor structure. This reverse effect is called the Peltier effect . Thermoelectric effects - the Seebeck and Peltier effects (current fluxes caused by temperature gradients, and visa versa) Clif Fonstad, 9/03 Lecture 4 - Slide 2 J e = - q -m e n ( ) - d f dx + - q ( ) D e - dn dx + - q ( ) S e n - dT dx * Don’t panic. We will only consider isothermal situations in 6.012 exams and problem sets.
Two examples: Right - The hot point probe , an apparatus for determining the carrier type of semiconductor samples. Below - A thermoelectric cooler array like that found in solid- state refrigerators. Thermoelectric effects - the Seebeck and Peltier effects (current fluxes caused by temperature gradients, and visa versa) Clif Fonstad, 9/03 Lecture 4 - Slide 3 Ref.: Appendix B in the course text.

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Non-uniform material with non-uniform excitations (laying the groundwork to model diodes and transistors) A. General description To model devices we must understand semiconductors in which we have (1) spatial variation of the doping and (2) spatial and temporal variation of the generation function: Because the doping and excitation are not uniform, we anticipate that the carrier population, currents, and electric field will also vary with space and time: To determine these five quantities, assuming we know the doping
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