4.3.4 Photoconductive Devices We have for a photoconductor the steady state excess carrier concentrations generated by an optical generation rate gop: n=ngOP and p=pgOP (n ≠p if trapping is present) (4-16) The change in photoconductivity is: ∆ = qgOP(nn + pp) (4-17) Considerations in choosing a photoconductor: (1) sensitive wavelength range (determined by Eg) (2) time response (, device geometry) (3) optical sensitivity ( and ) Examples of photoconductors: InSb (0.18eV); Ge (0.67eV); Si(1.11eV); CdS (2.42eV) Applications of photoconductive devices: light detectors that can be used in automatic night lights, burglar systems, moving object counters…

4.4 Diffusion of Carriers 4.4.1 Diffusion Processes Two basic processes of current conduction: Drift in an electric field Diffusion due to a concentration gradient Net flow of particles Diffusion is the result of the random motion the individual particles!

Figure 4—12 Spreading of a pulse of electrons by diffusion.

Figure 4—13 An arbitrary electron concentration gradient in one dimension: (a) division of n(x) into segments of length equal to a mean free path for the electrons; (b) expanded view of two of the segments centered at x0.

Diffusion Current Density The electron and hole flux density (rate of flow per unit area) due to diffusion: (4-22a) (4-22b) The electron and hole diffusion current density: (4-23a) (4-23b) Dn and Dp are electron and hole diffusion coefficients, respectively.

Total Current Density In the presence of both an electric field and a carrier gradient, the current density will each have a drift component and a diffusion component: (4-23a) (4-23b) The total current density is: J(x)=Jn(x) +Jp(x) (4-24) Minority carrier can contribute to current flow significantly by diffusion !

The Variation of Band Energy with E (x) —Assume an electric field E (x) in x direction, we expect: (1) Electrons will drift; (2) Electron potential energy U(x) will vary in x direction —If the electrostatic potential due to electric field E (x) is V(x), we have: V(x)=U(x)/(-q) Note: POTENTIAL=ENERGY/CHARGE —The definition of electric field, —Choose Ei as a convenient reference, (4-25) Fig. 4-15 (4-26)

Figure 4—14 Drift and diffusion directions for electrons and holes in a carrier gradient and an electric field. Particle flow directions are indicated by dashed arrows, and the resulting currents are indicated by solid arrows.

Figure 4—15 Energy band diagram of a semiconductor in an electric field %(x).

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!

4.4.3 Diffusion and Recombination; The Continuity Equation Consider both diffusion and recombination, we have: (4-31a) Continuity Equations (4-31b) If the current is strictly by diffusion: (4-33a) Diffusion Equations (4-33b)

Figure 4—16 Current entering and leaving a volume ∆xA.

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)

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)

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).

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)

4.4.5 The Haynes-Shockley Experiment 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.

Figure 4—19 Calculation of Dp from the shape of the dp distribution after time td. No drift or recombination is included

Figure 4—20 The Haynes–Shockley experiment: (a) circuit schematic; (b) typical trace on the oscilloscope screen.