Equilibrium vs. Excess carriers in semiconductors

(1) Equilibrium vs. Excess carriers in semiconductors
At thermal equilibrium, a piece of Si doped with 10¹⁷/cm³ As atoms  (As atoms are donor impurities in Si) has the equilibrium concentrations at 300K, of electrons and holes, respectively, of

•  n₀ = 10¹⁷/cm³ = 100,000,000,000,000,000/cm³ for electrons  [majority carriers]
•  p₀ = n_i²/n₀ = (10¹⁰/cm³)²/10¹⁷cm³ = 1,000/cm³ for holes.  [minority carriers]

Application of an external stimulus to a semiconductor, such as a beam of laser light with an above-bandgap photon energy or a bias voltage across a pn junction, creates excess carriers.  The generated access carriers get to 'live' for a finite length of time, which on the average is called the excess carrier lifetime, until they recombine with an opposite-type carrier to their mutual annihilation.  For example, suppose that the pice of Si doped with 10¹⁷/cm³ As donor atoms is uniformly excited optically at room temperature such that 10¹⁹/cm³ electron-hole pairs are generated per second with the excess carrier lifetime of t_n = t_p = 10^-4 s.  Then there will be an excess carrier concentration of dn = 10¹⁷/cm³ * 10^-4 s = 10¹⁵/cm³ = dp.  The total concentration will then be

•  n = n₀ + dn = 10¹⁷ + 10¹⁵ » 10¹⁷ = n₀  for electrons.     [majority carriers]
•  p = p₀  + dp = 10³ + 10¹⁵ » 10¹⁵ = dp for holes.            [minority carriers]

This shows that the concentration of  minority carriers is dominated by the excess carriers; while the concentration of majority carriers  is little affected by the excess carriers.  [This condition is called a low-level injection condition.]
In this applet, we shall focus on the processes of excess minority carriers  (i.e., electrons in p-type Ge; and holes in the n-type Ge).
Note that the transport of  excess minority carriers under the influence of the externally applied bias voltage is responsible for the physical operation of such minority-carrier devices as the pn junction diode and  the bipolar junction transistor.  [In contrast, the metal-semiconductor Schottky diodes and the Field Effect Transistors are majority-carrier devices.]
 
(2) Excess Carrier Processes
The excess carriers go through recombination, diffusion and drift.
i) Recombination
Semiconductor with excess carriers is in non-equilibrium.  Unless the external stimulus, responsible for creating the excess carriers, is present, the semiconductor will 'try' to return to thermal equilibrium state by removing the excess carriers.   The removal process of excess carriers is carrier Recombination, in which a carrier recombines with an opposite-type carrier to their mutual annihilation.  Each excess carrier gets to survive an average of t seconds, the excess carrier lifetime, before the electron-hole recombination takes place.
If the external stimulus, which generated the uniform excess electron concentration Dn  in the semiconductor was shut off at t=0, then the total concentration of electrons at time t will be

 n = n₀ + Dn  exp(-t/t_n)

This recombination process exists in semiconductor whether the external stimulus exists or not.  In steady state, the rate of excess carrier generation by the external stimulus is exactly matched by the rate of the excess carrier recombination.  In thermal equilibrium, the rate of thermal generation of carriers is exactly matched by the rate of carrier recombination.
In this applet, we shall visually observe that the number of excess minority carriers decay in time after the laser light is turned off.
ii) Drift
Charge carriers (or charged particles) is transported in space by an electric field produced by an externally applied voltage bias.  The net displacement per unit time is proportional to the strength of externally applied electric field: Dx/Dt = mE,  where m is a proportional constant (called mobility) and E is the electric field strength.  Therefore, the position of the charge carrier at time t is

 x = x₀ + mEt.

iii) Diffusion
Any  spatial nonuniformity in the carrier concentration will cause displacement of the charge carreirs over time.  A net displacement of charge carriers occurs from a region of higher concentration to a region of lower concentration via a process called 'diffusion'.  The rate of such a spatial displacement of charge carriers is proportional to how steep the concentration nonuniformity is, i.e., the gradient of concentration profile.  The diffusion flux f(x), defined as the number of carriers crossing a unit cross-sectional area per unit time, is given by

f_n(x) = - D_n dn(x)/dx
f_p(x) = - D_p dp(x)/dx

(3) Some Useful Semiconductor Equations
i) Diffusion and Recombination => Continuity Equation
In one dimension, the continuity equation for excess electrons, dn, and excess holes, dp, are

 ddn/dt = D_n d²dn/dx² – dn/t_n
 ddp/dt = D_p d²dp/dx² – dp/t_n

These equations says that the number of carriers increased at x per unit time, ddn/dt, is equal to the number of carriers that 'emergies' at x per unit time, D_n d²dn/dx², less the number that recombines per unit time, dn/t_n.
ii) Drift and Diffusion => Current Density Formula
The drift and diffusion processes actually displace charge carriers in space, and thus are responsible for the current.  The current density, i.e., the current per unit cross-sectional area, is given by

 J_n(x) = q m_nnE + qD_n dn(x)/dx
 J_p(x) = q m_ppE – qD_p dp(x)/dx

The first term on the right hand side of erquation is the drift current density; and the second term is the diffusion current density. For electrons, for example, J_drift = q m_nnE and J_diffusion = qD_ndn(x)/dx.
 
Aderlis S. Marquez G
Electronica del Estado Solido
http://jas.eng.buffalo.edu/education/semicon/diffusion/intro.html