MIT6_012F09_lec03_five_eqn

# MIT6_012F09_lec03_five_eqn - 6.012 Electronic Devices and...

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6.012 - Electronic Devices and Circuits Solving the 5 basic equations - 9/17/09 Version The 5 basic equations of semiconductor device physics: We will in general be faced with finding 5 quantities: n (x,t) , p (x,t) , J e (x,t) , J h (x,t) , and E (x,t) , and we have five independent equations that relate them: (1) J e (x,t) = q µ e n (x,t) E (x,t) + q D e n ( x x,t) (2) J h (x,t) = q µ h p (x,t) E (x,t) - q D h p ( x x,t) (3) n ( t x,t) - q 1 J e ( x x,t) = g L (x,t) - [n (x,t) p (x,t) - n i 2 ] r (4) p ( t x,t) + q 1 J h ( x x,t) = g L (x,t) - [n (x,t) p (x,t) - n i 2 ] r (5) ε E ( x x,t) = q [p (x,t) - n (x,t) + N d (x) - N a (x) ] where the assumptions we made getting these equations are: N d + (x) N d (x) , N a - (x) N a (x), R (x,t) n (x,t) p (x,t) r r, ε , µ e , µ h , D e , and D h are assumed to be independent of position. Temperature is assumed to be constant (isothermal). (Note: n i , r, µ e , µ h , D e , and D h all depend on temperature). 1

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This is a set of 5 coupled, non-linear differential equations that are in general not solvable analytically. ... BUT we know the solution in three special cases already. ... 1) Uniform doping, thermal equilibrium: n o , p o (See page 4 below for a discussion of this case) 2) Drift: g = 0, uniform doping (See page 4 below for a discussion of this case) 3) Uniform Low-level Injection: n’, p’, τ min (See pages 5-6 below for a discussion of this case) AND we are able to find APPROXIMATE ANALYTICAL solutions for two very important new situations. ... 1) Doping Profile Problems: non-uniformly doped material in thermal equilibrium (an important subset of these problems are solved using the depletion approximation) (See pages 7-10 below for a discussion of these problems) 2) Flow Problems: non-uniform injection of excess carriers into uniformly doped material (See pages 11-14 below for a discussion of these problems) ******** With an understanding of these solutions to the five equations we will be able to model and understand all of the important semiconductor devices, including diodes, bipolar transistors, and MOSFETs. (See page 3 for an illustration of this point) 2
Why we care: Understanding Flow Problems, the Depletion Approximation, and Drift, we can under- stand all of the basic devices we see in 6.012: Diodes involve flow problems in two regions and the depletion approximation about one junction: Note: This is true not only for simple electronic diodes, but also for light emitting and laser diodes, and for photodiodes and solar cells. p-type n-type Flow problem Junctiion problem Flow problem Bipolar transistors involve flow problems in three regions and the depletion approximation about two junctions: n-type p n-type E B C Flow problems Junction problem MOSFETs involve two diodes, the depletion approxi- mation in the gate region, and drift in the channel: n+ n+ S G D Diodes Depletion approximation Drift 3

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Special Case: UNIFORM DOPING, THERMAL EQUIL.
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MIT6_012F09_lec03_five_eqn - 6.012 Electronic Devices and...

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