(Diffusion, Drift, and Recombination Applet)
(1) Equilibrium vs. Excess carriers in semiconductors
At thermal equilibrium, a piece of Si doped with 10^{17}/cm^{3} As atoms (As atoms are donor impurities in Si) has the equilibrium concentrations at 300K, of electrons and holes, respectively, of
- n_{0} = 10^{17}/cm^{3 }= 100,000,000,000,000,000/cm^{3 }for electrons [majority carriers]
- p_{0} = n_{i}^{2}/n_{0} = (10^{10}/cm^{3})^{2}/10^{17}cm^{3} = 1,000/cm^{3} 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 10^{17}/cm^{3} As
donor atoms is uniformly excited optically at room temperature such that
10^{19}/cm^{3} 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
= 10^{17}/cm^{3} * 10^{-4} s = 10^{15}/cm^{3}
= dp. The total concentration will then
be
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 = n_{0} + 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 = x_{0} + 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(3) Some Useful Semiconductor Equations
f_{p}(x) = - D_{p} dp(x)/dx
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} d^{2}dn/dx^{2} – dn/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} d^{2}dn/dx^{2}, less the number that recombines per unit time, dn/t_{n}.
ddp/dt = D_{p }d^{2}dp/dx^{2} – dp/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_{n}nE + qD_{n} dn(x)/dxThe 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_{n}nE and J_{diffusion} = qD_{n}dn(x)/dx.
J_{p}(x) = q m_{p}pE – qD_{p} dp(x)/dx