CARRIER TRANSPORT PHENOMENA
1. Drift Current
2. Diffusion Current (Diff. Current)
Figure 4.1 Typical random behavior of a hole in a semiconductor (a) without an
electric field and (b) with an electric field
Drift Current:
 If a positive volume charge density ρ is moving at an
average drift velocity vd, the drift current density can be
expressed as
Jdrf = ρ . vd
[4.1]
Where J = current density (Amps/cm2 or coul/cm2-sec),
ρ = charge density and vd = average drift velocity.
 If the volume charge density is due to the positively
charged holes, then
Jpdrf = (ep) . vdp
[4.2]
 The charge density can be defined as
ρ = e .p
[4.3]
Where,
Jpdrf = holes current density and vdp = average drift
velocity of the holes
 The equation of motion of a positive charged hole in the
presence of an electric field is
F = m*p a = e E
[4.4]
Where e = electronic charge
a = acceleration
E = electric field
 For low electric field, the velocity of the carrier is
directly function of the electric field.
Vdp = µpE
[4.5]
Where µp = holes mobility
 By recombining Equations (4.2) and (4.4), the drift current
density due to holes may be expressed as
 The drift current density due to holes is in the same direction as
the electric field The drift current density due to the electrons:
Where
Jn/drf = drift current density of electrons and
vdn = average drift velocity
 vdn is directly function of the electric field and the net
transport of the electron is opposite of the electric field.
 So we can write
Where µn = electron mobility.
 The equation [4.6] may be expressed as
 The electron and hole mobility (µn and µp) are function
of temperature and doping concentrations.
 Since both electrons and holes contribute to the drift
current, the total drift current density is the sum of the
individual electron and hole drift current densities
expressed as:
 The equation [4.4] of motion (Lorentz Equation) can be
expressed by following equation by substituting the
holes acceleration:
Where v is the velocity of the hole influenced the electric field.
 The equation of motion is the Lorentz equation:
 If we assume that the effective mass and electric fields are
constant, the integration of equation [4.10] yields:
VELOCITY SATURATION:
 The total velocity of a particle is the sum of the random thermal
velocity and drift velocity.
Total Velocity = Random thermal velocity + Drift velocity
 At T = 300K, the average thermal energy can be
expressed as:
 Thermal velocity or thermal speed is a typical velocity of the
thermal motion of particles that make up a gas, liquid, etc.
 Thus, indirectly, thermal velocity is a measure of temperature.
 It is a measure of the width of the peak in the Maxwell–
Boltzmann particle velocity distribution
 If is defined as the root mean square of the velocity in any one
dimension (i.e. any single direction), then

 The energy translates into a mean thermal velocity of
approximately 1017 cm/sec for an electron in silicon.
 The two basic types of scattering mechanisms that influence electron and
hole mobility are lattice scattering and impurity scattering.
 In lattice scattering a carrier moving through the crystal is scattered by a
vibration of the lattice, resulting from the temperature.
 The frequency of such scattering events increases as the temperature
increases, since the thermal agitation of the lattice becomes greater.
 Therefore, it is expected the mobility to decrease as the sample is heated
(Fig. 3-22).
 On the other hand, the scattering from crystal defects such as ionized
impurities becomes the dominant mechanism at low temperatures.
 Since the atoms of the cooler lattice are less agitated, lattice scattering is
less important.
 However, the thermal motion of the carriers is also slower.
 Since a slowly moving carrier is likely to be scattered more strongly by
an interaction with a charged ion than is a carrier with greater
momentum, impurity scattering events cause a decrease in mobility with
decreasing temperature.
 As Fig. 3-22 indicates, the approximate temperature dependencies are T3/2
for lattice scattering and T3/2 for impurity scattering.
 Since the scattering probability of Eq. (3-32) is inversely proportional to
the mean free time and therefore to mobility, the mobilities due to two or
more scattering mechanisms add inversely:
 The mobility affects by (1) ionized impurity scattering effect (µI) and (2)
acoustic phonon scattering effect (µL).
 The mobility µL (acoustic phonon scattering) can be evaluated
by
qh 4 C11 8
L 
 ( m * ) 5 / 2 T 3 / 2
*5 / 2
3/ 2
3E ds m (kT )
Where C11 = average longitudinal elastic constant of the semiconductor, and
Eds = displacement of the edge of the band per unit dilation of the
lattice.
 The mobility µI (ionized impurity scattering) can be evaluated by
64(2kT ) 3 / 2  S2   
12 S kT 2  
I 
ln
1

(
) 


N I q 3 m *1 / 2
q 2 N I1 / 3  
 
1
 (m * ) 1 / 2 N I1T 3 / 2
Where NI is the total ionized impurity concentration in the semiconductor
NI = ND+ + NA
Phonon Scattering: Acoustic Phonon and Optical Phonon.
o The electron and hole mobilities have a similar doping dependence: For low doping
concentrations the mobility is almost constant and is primarily limited by phonon
scattering.
 At higher doping concentrations the mobility decreases due to ionized impurity
scattering with the ionized doping atoms. The actual mobility also depends on the type of
dopant. The above figure is for phosphorous and boron doped silicon and is calculated
using:
The drift current density evaluated in Equation [4.9] may be
expressed as:
Where σ is the conductivity of the semiconductor material.
The reciprocal of conductivity is resistivity (unit ohm-cm).
If we have a bar of semiconductor material as shown in Figure 4.5
applying a voltage to generate the current. So we can
write:
J = σ E, J = I/A
 Equation [4.22b] is ohm’s law for a semiconductor.
 The resistance is a function of resistivity or conductivity as well
as the geometry of the semiconductor.
 If p-type semiconductor with acceptor doping concentration
NA and ND = 0, so
NA >> ni
And assume electron mobility µn = hole mobility µp
The conductivity can be expressed as:

 The conductivity and resistivity of an extrinsic
semiconductor are function of majority carrier parameters.
RESISTIVITY
DIFFUSION CURRENT:
 It is the process whereby the particles flow from a region of
high concentration toward low concentration region.
 The diffusion of electrons from a region of high
concentration to a region of low concentration creates a flux
of electrons flowing in the negatives x directions.
 Since electrons have a negative charge, the conventional
current direction is in the positive x direction. The diffusion
current density may be expressed by following equation:
Where Dn = electron diffusion coefficient (cm2/sec)
 The holes diffusion current can be expressed
by
 Where Dp = holes diffusion coefficient
(cm2/sec)
-----------------------------------------End--------------------------------------------------------------
INDUCED ELECTRIC FIELD
 Consider the semiconductor is in thermal equilibrium.
 The Fermi energy level is constant through out the crystal.
 The doping concentration decreases with as x increases.
 Concentration gradient is formed
 Electrons are coming from donors atoms making the ionized
donors atoms.
 Separation of +ve and –ve charge induces an electric field
and that will oppose the migration of electrons.
 The electric potential Φ is correlated to the EPE (electron
potential energy) by the charge (-ve), so we can express:
 If we assume a QN (quasi neutral) condition, in which the
electron concentration is equal to the donor impurity
concentration, then we can write as: