1 Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University Nov. 5, 2012 DEE4521 Semiconductor Device Physics Lecture 5 G-R Process G-R Process

2 Optical injection: A powerful means to address g-r process.

3 The electromagnetic spectrum. (The frequencies of acoustic waves are given for comparison although they are not electromagnetic waves.) Figure 11.1 Photon Energy (eV) = 1.24/Wavelength (um)

4 Some Concepts on Photons Golden Rule: E photon (eV)  photon (  m) = 1.24 Photons, being massless entities, carry very little momentum, and a photon-assisted transition is essentially vertical on an E-K plot. (K photon = 2  / photon. If E photon = 1.42 eV, then photon = 0.87  m. Clearly, photon >> a and K photon << 2  /a) (see p.228 of the textbook)

5 Overlap of Electron E-K Locus and Photon E-K Locus can determine some of interesting properties and even can be related to Einstein’s photoelectric effect.

6 (a) A photon of energy 2.06 eV is incident on a material of energy gap 2.5 eV. The photon cannot be absorbed. (b) The band gap is small enough that allowed states separated by 2.06 eV exist, thus the photon can be absorbed. The photon’s energy is given to the electron. (c) In emission, the electron goes to a lower energy state, releasing the extra energy in the form of a photon. Figure

(a) At equilibrium, electrons and holes are generated and destroyed at equal rates, thus maintaining some constant equilibrium n 0 and p 0. (b) When light shines on the sample, the photons can be absorbed, producing extra electron-hole pairs. Figure 3.13

For absorption to occur, K must be conserved as well as E. (a) A direct gap semiconductor; on the left is the E-K diagram, and on the right the conventional energy band diagram. (b) An indirect gap material (so called because conduction band minimum and the valence band maximum do not occur at the same value of K). Figure 3.14

9 Optical emission. The electron loses energy, giving off the excess as a photon of E H hv. Figure 3.15

Emission on the E-K diagram. Both K and E must be conserved. (a) A direct gap material; (b) an indirect semiconductor. Figure 3.16

11 Phonon E-K(Phonon = Lattice Vibrations)

For Direct-gap semiconductors, the upward transition probability for electron-hole pair separation = f 1 f 2 f 3. f 1 : the probability of finding a filled state in valence band f 2 : the probability of finding a photon f 3 : the probability of finding a final, unfilled state in conductance band For Indirect-gap semiconductors, upward transition probability for electron-hole pair separation = f 1 f 2 f 3 f 4. f 4 : the probability of finding a phonon Electron-Hole Pair Separation

13 Illustration of optical absorption in an indirect semiconductor involving an electron, a phonon, and a photon. The wave vector needed to make the transitions comes almost entirely from the phonon; the phonon contributes a small amount of energy, h , as well, but most of the energy is supplied by the photon. Figure S1B.12 S1B-12 _

Energy band diagram of the semiconductor of Figure 3.18, under electrical bias and optical illumination. The combination rate R, thermal generation rate G th, and the optical generation rate G op are illustrated. Figure 3.19 Rate: per unit volume per unit time

15 Absorption coefficients of some semiconductor materials. The indirect-gap materials are shown with a broken line. Based on data from References 1 and 2. Figure 11.4 Only case of E photon > E G is shown Indirect edge Direct edge

16 Quasi-Fermi Levels

Illustration of quasi Fermi levels for electrons and holes for the steady-state nonequilibrium case of Figure 3.22, with external field = 0. Figure 3.24 Level Split due to Carrier (Optical) Injection Equilibrium n o = N C exp(-(E C -E F )/k B T) = n i exp((E F -E i )/k B T) p o = N V exp(-(E F -E V )/k B T) = n i exp((E i -E F )/k B T) Quasi-Equilibrium (or non-equilibrium with a small field) n = N C exp(-(E C -E Fn )/k B T) = n i exp((E Fn -E i )/k B T) p = N V exp(-(E Fp -E V )/k B T) = n i exp((E i -E Fp )/k B T)

18 R – G = Equilibrium Quasi-equilibrium Net recombination rate Principle of detailed balance Hole lifetime Electron lifetime n-type bulk p-type bulk For optical injection (photon absorption) case Thermal generation rate Assume trap level E t at midgap Thermal recombination rate

Minority carrier diffusion lengths L n and L p as functions of impurity concentration N A or N D in uncompensated high quality Si. Figure 3.23 L n = (D n  n ) 1/2 L p = (D p  p ) 1/2 The critical distance a carrier can move without being annihilated. Diffusion Length:

The geometry for determining the continuity equation. The rate at which carriers accumulate in the incremental volume depends on the incoming and outgoing currents as well as the recombination and generation within the region dx. Figure 3.17 Then we can write the Continuity Equation according to the Conservation of Flux in two channels (one of conduction band and one of valence band): For p-type semiconductor: dn/dt =  dF n /dx + (G n –R n ) = (1/q)(dJ n /dx) + (G – R) = d  n/dt = (1/q)(dJ n /dx) + (G op –(  n/  n )) dp/dt =  dF p /dx + (G p –R p ) =  (1/q)(dJ p /dx) + (G – R) = d  p/dt =  (1/q)(dJ p /dx) + (G op –(  n/  n ))

The geometry for determining the continuity equation. The rate at which carriers accumulate in the incremental volume depends on the incoming and outgoing currents as well as the recombination and generation within the region dx. Figure 3.17 Then we can write the Continuity Equation according to Conservation of Flux in two channels (one of conduction band and one of valance band): For n-type semiconductor: dn/dt =  dF n /dx + (G n –R n ) = (1/q)(dJ n /dx) + (G – R) = d  n/dt = (1/q)(dJ n /dx) + (G op –(  p/  p )) dp/dt =  dF p /dx + (G p –R p ) =  (1/q)(dJ p /dx) + (G – R) = d  p/dt =  (1/q)(dJ p /dx) + (G op –(  p/  p ))

22 These expressions may be misleading from the aspect of the the conservation of flux.

23 Six Working Examples

24 A generic photodiode. Figure 11.2 Example 1

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28 A solar cell illuminated from the left. Figure 11.9 Steady State Example 2

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(a) Illustration of minority carrier diffusion in a surface-illuminated p-type semiconductor. The absorption is assumed to occur at the surface (how to make it real?). (b) Plots of the excess minority carrier concentration as a function of distance into the bar with increasing time. As the excess carriers are generated at the surface, they diffuse to regions of lower concentration, where they recombine. Figure 3.22 Transient Example 3

32 The Haynes-Shockley Experiment: Carrier Transient Behaviors Light Flash Example 4

33 Consider a very intense light source which produces an extremely large number G L (or G OP ) of excess carrier pairs per unit volume per unit time, over an extremely small length , and an extremely short time T (<< carrier lifetime). The net result is a flash light, producing instantaneously at t = 0 a plane of excess carrier pairs at x = 0 having  P o excess carriers per unit area:  P o = lim(G L T  ), G L  ,  0, T  0 Then by solving the continuity equation for holes in a uniform n-type semiconductor rod in the presence of an electric field E o : One can derive one solution for the excess hole concentration:

Schematic of a circuit used to measure minority carrier lifetime in semiconductors. Figure 3.18 Steady State Example 5

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36 Valid only for a sample thickness of no more than the reciprocal of the absorption coefficient .

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39 Variation of excess carriers in a semiconductor under pulsed illumination. (a) When the light is turned on, the excess carrier concentration increases exponentially. For the complete pulse, (b) the rise and fall time constants are equal to the minority carrier lifetimes. Figure 3.20

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41 Plot of minority carrier lifetime in uncompensated high quality Si as a function of doping concentration N A or N D. Figure 3.21

42 Example 6

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