Excess Carriers in Semiconductors

Chapter 4

• Until now, we were only focusing on semiconductors in equilibrium conditions

• When voltage is applied, or a light is shined, the semiconductor is operating under nonequilibrium conditions

• If an external excitation (i.e voltage, or light) is applied, concentration of excess carriers are generated in addition to the equilibrium concentration

• When excess holes or electrons are created increase in current

Excess Carriers in Semiconductors

• Direct recombination: excess electrons fall from conduction band to holes in the valence band

• Net rate of change of conduction band electrons:

• Initial excess electron concentration = Δn

• Initial excess hole concentration = Δp

• Net rate of change of instantaneous excess carrier concentration:

• Case of low level injection:

Direct Recombination of Excess Electrons and Holes

Direct Recombination of Excess Electrons and Holes

• In certain semiconductors, recombination happens indirectly

• Recombination occurs via recombination centers within the bandgap

• An impurity or defect can serve as a recombination center

• Each EHP recombines in two steps: hole capture and electron capture.

• Example: Er is a recombination center

• If a carrier (hole or electron) is captured at the recombination center and then reexcited without recombining Trapping

Indirect Recombination and Trapping

• Fermi level is meaningful only when no excess carriers are present

• Fn : quasi-fermi level for electrons

• Fp : quasi-fermi level for holes

• Carrier concentration equations:

• no =

• po =

• Deviation of Fn and Fp from EF indicates how far the carrier concentrations are from the equilibrium values of no and po .

Indirect Recombination and Trapping

• When excess carriers are created nonuniformly in a semiconductor, the n and p concentration varies with position

• Diffusion: net motion of carriers from region of high concentration to region of low concentration.

• There are two basic sources of current in semiconductors: diffusion due to carrier gradient and drift in an electric field

• Example of diffusion: perfume bottle opened in a closed room

Diffusion

• Carriers in semiconductors diffuse in a carrier gradient by random thermal motion and by scattering

• Example: excess electrons injected at t = 0 and x = 0. As time passes, electrons will diffuse to regions of low electron concentration until n(x) is constant

• Calculation of the rate at which carriers diffuse:

• Dn : electron diffusion coefficient

• Dp : hole diffusion coefficent

Diffusion of Carriers

• Diffusion current (Jdiff) :

• If an electric field is present in addition to diffusion, the current will have two components: a diffusion component and a drift component:

• Relation between carrier flow and current:

• Minority carriers can contribute significantly to the current through diffusion

Diffusion and Drift of Carriers

• Example: Assume that in n-type GaAs semiconductor at T = 300K, the electron concentration varies linearly from 1 ×1018

to 7 ×1017 cm-3 over a distance of 0.10cm. Calculate the diffusion current density if the electron diffusion coefficient is Dn = 225 cm2/s

Diffusion and Drift of Carriers

• Electrons drift in a direction opposite to the electric field their potential energy increases in the direction of the field

• Relation between electric field and electrostatic potential V(x):

• In band diagram to the right, since electrons drift downhill, the E-field points uphill

• At equilibrium, no net current flows. If any fluctuation causes diffusion current a drift current will counteract the diffusion current

• There must be a relation between diffusion coefficient and mobility such that net current is zero

Electric Field

• Set current density = 0

• Einstein relation:

• Einstein relation allows us to calculate D or µ from a measurement of the other.

• D/µ at room T = 0.026

• Built-in E-field results from balance of drift and diffusion at equilibrium due to the gradient in Ei . This is due to doping gradients

Einstein Relation

Dn (cm2/s) Dp (cm2/s) µn (cm2/s) µp (cm2/s)

Ge 100 50 3900 1900

Si 35 12.5 1350 480

GaAs 220 10 8500 400

• We need to consider effects of recombination on diffusion of carriers

• Continuity equation :

• If current is carried mainly by diffusion:

• Diffusion equation:

Continuity Equation

Steady State Carrier Injection and Diffusion Length

• Expect distribution of excess holes to decay to zero for large values of x due to recombination

• Solution form:

• Evaluate C1 and C2 from boundary conditions:

• Solution:

• Diffusion length(Ln or Lp) represents average distance an electron or hole diffuses before recombining

• Diffusion current:

Steady State Carrier Injection and Diffusion Length

Example II