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1 Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University October 2, 2014 DEE4521 Semiconductor Device Physics Lecture 3a: Transport: Drift and Diffusion Transport: Drift and Diffusion

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2 This lecture accompanies pp. 111–131 on drift and diffusion, as well as pp on non-uniform doping, of textbook. Textbook pages involved

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3 Hole drift current density J p,x,drift = qp = qp p x vxvx f(v x ) Maxwellian velocity distribution (equilibrium) for holes ( = 0) Shifted Maxwellian velocity distribution for holes (electric field x : must be small) xx : drift (NOT thermal) velocity p : hole mobility Similarly, for electrons J n,x,drift = qn n x Note: Polarity Holes Drift

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4 Electron diffusion current density J n,x,diffusion = qD n dn/dx p or n x Gradient of carrier density D p : Hole diffusion coefficient D n : Electron diffusion coefficient very hot very cold Diffusion Hot to Cold: Diffusion

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5 Why D and its unit? -3d -2d -1d 0 1d 2d 3d x N = 0 N = 1 N = 2 This is a random walk problem. Variance [x 2 ] =Nd 2 1 1/2 1/4 f(x)

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6 For electrons in a band, J n = J n, drift + J n,diffusion For holes in another band, J p = J p,drift + J p,diffusion Total J = J n + J p = ( n + p ) + diffusion components = + diffusion components Electron conductivity n = qn n Hole conductivity p = qp p Total conductivity = n + p = V/L I/A = J Applied V must be small. I-V in a biased semiconductor

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7 4-2 Non-uniformly doped semiconductor is a Good Vehicle, 1.to derive Einstein’s relationship. 2.to prove that in equilibrium case, Fermi level remains constant, through any direction in all spaces (real space, energy space).

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8 4-8

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9 Built-in Field in Non-uniform Semiconductors 4-9 You must be able to distinguish between built-in electric field and applied electric field. (hint: Superposition principle)

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The experimentally measured dependence of the drift velocity on the applied field. Figure 3.9 drift = Focus on low field region (<10 3 V/cm)

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Mobility as a function of temperature. At low temperatures, impurity scattering dominates, but at high temperatures, lattice vibrations dominate. Figure 3.8

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12 (a) An electron approaching an ionized donor is deflected toward it, but a hole is deflected away from the donor. (b) Electrons deflect away from the negatively charge ionized acceptors but holes deflect toward them. Figure Coulomb (or Impurity) Scattering

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13 Displacement of atomic planes under the influence of a pressure wave. For a longitudinal wave (a), the displacement is in the direction of motion. For a transverse wave (b), the displacement is transverse to the direction of motion. For a three-dimensional crystal, for each longitudinal wave there are two transverse waves. The dashed lines represent the equilibrium positions, and the solid lines indicate the deflected positions at a given time. Figure S1B.6 S1B-6 Lattice Vibrations

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14 Room temperature majority and minority carrier mobility as functions of doping in p-type and n-type silicon. Solid lines: minority carriers; dashed lines: majority carriers. Figure

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15 n = q fe /m* ce Electron Mobility Electron Mean Free Time or Electron Average Scattering Time Electron Conductivity Effective Mass Time constants are relevant in device physics. So, the ability to experimentally extract those is essential.

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16 How to derive n = q fe /m* ce ? This is a collision (scattering) event. This event is a Poisson event. Given applied in a conductor with a length L l l l l l : time to scatter, free time l: free path, scattering length Two random variables: and l n: total number of collision events v Slope = a = q /m aa t v = (a + a +…….+ a )/n v drift = = na /n =a … L = a n/2 follows exponential distribution n = L/v drift

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