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Einstein Relation—the relation between and D
At equilibrium, if there is a diffusion current (e.g., due to doping gradient or band gap variation), it must be balanced by a built-in field to give zero net current flow, (4-27) Use Eq.(4-27) becomes: (4-28) Because And that EF is constant at equilibrium, Eq. (4-28) reduces to: Einstein relation (4-29) Valid for either carrier type!
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Example 4.5 An intrinsic Si sample is doped with donors from one side such that Nd = N0e-ax a. Find an expression for E(x) at equilibrium over the range for which Nd » ni b. Find an expression for E(x) when a = 1 m-1 c. Sketch a band diagram and indicate the direction of E(x)
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4.4.3 Diffusion and Recombination; The Continuity Equation
in our discussions of Diffusion, we have neglected Recombination. Recombination can cause significant variation in the carrier distribution Figure 4—16 Current entering and leaving a volume ∆xA.
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Consider both diffusion and recombination, we have:
Continuity Equations (4-31b) If the current is strictly by diffusion: (4-33a) Diffusion Equations (4-33b)
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A pulse of electrons in a semiconductor:
spreads out by diffusion, and disappears by recombination to find n(x,t) we would start with:
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4.4.4 Steady State Carrier Injection; Diffusion Length
In the steady state case the diffusion equations become (4-34a) Steady State Diffusion Equations (4-34b) Diffusion Length Let us assume that excess holes are injected into a semi-infinite semiconductor bar at x=0, and that at the injection point p(x=0) = p is constant. The solution to Eq. (4-34b) has the form: (4-35)
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We expect p decay to zero for large x due to recombination, therefore C1=0. Similarly, the
condition p = p at x=0 gives C2=p, and the solution is: (4-36) The physical significance of Lp (or Ln) p = (1/e) p at x = Lp (2) Lp is the average distance a hole diffuses before recombining. <x>=Lp (4-40a) (4-40b)
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Figure 4—17 Injection of holes at x = 0, giving a steady state hole distribution p(x) and a resulting diffusion current density Jp(x).
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4.4.6 Gradients in the Quasi-Fermi Levels
- At equilibrium, there is no gradient in the Fermi level EF In the steady state, there is a gradient in the quasi-Fermi level due to drift and diffusion The total electron current becomes: (4-51) (4-52a) Modified Ohm’s Law (4-52b)
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first performed in 1951at Bell Telephone Laboratories
4.4.5 The Haynes-Shockley Experiment (Please read Section 4.4.5) classic experiment demonstrating the drift and diffusion of minority carriers first performed in 1951at Bell Telephone Laboratories independent measurement of the minority carrier mobility and diffusion coefficient Figure 4—18 Drift and diffusion of a hole pulse in an n-type bar: (a) sample geometry; (b) position and shape of the pulse for several times during its drift down the bar.
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Figure 4—19 Calculation of Dp from the shape of the p distribution after time td. No drift or recombination is included
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Figure 4—20 The Haynes–Shockley experiment: (a) circuit schematic; (b) typical trace on the oscilloscope screen.
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Chapter 5 Junctions p-n junctions metal-semiconductor junctions heterojunctions Junctions Fabrication Equilibrium conditions Biased junctions; steady state conditions Reverse bias breakdown Transient and AC conditions Deviations from simple theory p-n junctions Strong qualitative understanding of the properties of p-n junctions Know how to use the mathematics of p-n junctions to make calculations Goals
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5.1 Fabrication of p-n junctions
Major process steps (Please read Section 5.1) Thermal Oxidation Diffusion Rapid Thermal Processing Ion Implantation Chemical Vapor Deposition (CVD) Photolithography Etching Metallization Crystal Growth and Wafer Preparation (Chap.1) Process Simulation Process Integration
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Figure 5—10 Simplified description of steps in the fabrication of p-n junctions. For simplicity, only four diodes per wafer are shown, and the relative thicknesses of the oxide, PR, and the Al layers are exaggerated.
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5.2 Equilibrium Conditions
What will happen if we bring a p-type semiconductor and a n-type semiconductor together to form a junction? Based on knowledge gained from previous chapters, we expect: Initially, current will flow due to diffusion No net current can flow across the junction at equilibrium An internal electric field E will build up as a result of uncompensated donor ions (Nd+) and acceptor ions (Na+) The electric field gives rise to a “contact potential” V0 across the junction E=-dV(x)/dx The Fermi levels will be aligned at equilibrium
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