2.8 Carrier Transport
of Contents - 1
In this section:
Bulk scattering mechanisms
Electron and hole mobility in silicon - doping and
Surface and interface scattering
Summary: As one applies an electric field to a semiconductor, the
electrostatic force causes the carriers to first accelerate and then reach
a constant average velocity, v, as the carriers scatter due to impurities
and lattice vibrations. The ratio of the velocity to the applied field
is called the mobility. The velocity saturates at high electric fields
reaching the saturation velocity. Additional scattering occurs when carriers
flow at the surface of a semiconductor, resulting in a lower mobility due
to surface or interface scattering mechanisms.
Carriers within a semiconductor crystal move as if they were free particles
which are not affected by the presence of the atoms in the material except
for the fact that it effectively changes the mass of the particle. The
carriers can either be electrons or holes (missing electrons) wich carry
one negative (positive) unit of charge. These carriers move even when no
electric field is applied because of the thermal energy associated with
all particles. The thermal energy of non-relativistic electrons equals
kT/2 for each possible degree of freedom. At room temperature the
thermal velocity of electrons in bulk semiconductors is about 107
The carriers move through the semiconductor until a collision occurs.
The collisions, also called scattering events, are due to defects, impurities
or the emission/absorption of phonons. The dominant types of collision
mechanisms are charged impurity scattering and phonon emission/absorption.
These collisions cause an abrupt change in the carrier velocity and energy
at the time of the collision. The resulting carrier motion is semi-random
due to frequent changes in direction and velocity.
Carrier transport in a semiconductor in the presence of an applied field
can also be visualized as being semi-random except that in addition the
individual carriers also accelerate between collisions. And even though
the random velocity greatly exceeds the average velocity parallel to the
applied field, it can be ignored since the random motion does not result
in a net flow. The carrier acceleration follows Newtonís law, where the
force equals the product of the electric field with the charge of the particle.
The collisions cause an abrupt change in the carrier velocity and energy
at the time of the collision after which the acceleration resumes.
The net effect of the collisions is that the carriers on average do
not accelerate, but rather quickly reach a constant velocity. The collisions
contribute to a friction term in the equation of motion which is characterized
by a time constant, t, namely the time
during which the particle loses the momentum, mv, associated with
the average carrier velocity, v. For a particle which has a constant
acceleration between collisions, this time constant also equals the time
between two consequent collisions.
2.8.2 Drift Current
The motion of a carrier drifting in a semiconductor due to an applied electric
field is illustrated in the figure below. The field causes the carrier
to move with a velocity, v.
Fig.2.8.1 Drift of a carrier due to an applied electric field.
Assuming that all the carriers in the semiconductor move with the same
velocity, the current can be expressed as the total charge in the semiconductor
divided by the time needed to travel from one electrode to the other, or:
where tr is the transit time of a particle, traveling
with velocity, v, over the distance L. The current density
can then be rewritten as a function of either the charge density, r,
or the density of carriers, n in the semiconductor:
Carriers however do not follow a straight path along the electric field
lines, but instead bounce around in the semiconductor and constantly change
direction and velocity due to scattering. This behavior occurs even when
no electric field is applied and is due to the thermal energy of the electrons.
Electrons in a non-degenerate and non-relativistic electron gas have a
thermal energy which equals kT/2 per particle per degree of freedom.
A typical thermal velocity at room temperature is around 107
cm/s, which exceeds the typical drift velocity in semiconductors. The carrier
motion in the semiconductor in absence and in the presence of an electric
field can therefore be visualized as in the figure below:
Fig.2.8.2 Random motion of carriers in a semiconductor with
and without an applied electric field.
In the absence of an applied electric field, the carrier motion is random
and the carriers move quickly through the semiconductor and frequently
change direction. When an electric field is applied, the random motion
still occurs but in addition there is on average a net motion along the
direction of the field.
We now analyze the carrier motion considering only the net movement
without the random motion. Applying Newton's law we state that the acceleration
of the carriers is proportional to the applied force:
Where the force consists of the electrostatic force minus the force due
to the loss of momentum at the time of scattering divided by the average
time between scattering events:
Combining both relations yields an expression for the average particle
We now consider only the steady state situation in which the particle has
already accelerated and has reached a constant average velocity. Under
such conditions, the velocity is proportional to the applied electric field
and we define the mobility as the velocity to field ratio:
The mobility of a particle in a semiconductor is therefore expected to
be large if its mass is small and the time between scattering events is
The drift current can then be rewritten as a function of the mobility,
Throughout this derivation we simply considered the mass, m, of
the particle. However in order to incorporate the effect of the periodic
potential of the atoms in the semiconductor we must use the effective mass,
m*, rather than the free particle mass:
The linear relationship between the average carrier velocity and the
applied field breaks down when high fields are applied. As the electric
field is increased, the average carrier velocity and the average carrier
energy increases as well. As the carrier energy increases beyond the optical
phonon energy, the probability of emitting an optical phonon increases
abruptly. This mechanism causes the carrier velocity to saturate with increasing
electric field. For carriers in silicon and other materials which do not
contain accessible higher bands, the velocity versus field relation can
be described by:
The maximum obtainable velocity, vsat, is refered to
as the saturation velocity.
2.8.4 Bulk scattering mechanisms
Scattering by lattice waves: Scattering by lattice waves includes
the absorption or emission of either acoustical or optical phonons. Since
the density of phonons in a solid increases with temperature, the scattering
time due to this mechanism will decrease with temperature as will the mobility.
Theoretical calculations reveal that the mobility in non-polar semiconductors,
such as silicon and germanium, is dominated by acoustic phonon interaction,
and is expected to be proportional to T -3/2, while the
mobility due to optical phonon scattering only is expected to be
proportional to T -1/2. Experimental values of the temperature
dependence of the mobility in germanium, silicon and gallium arsenide is
provided in the table below:
m µ T -s
||µ T -1.7
||µ T -2.4
||µ T -1.0
||µ T -2.3
||µ T -2.2
||µ T -2.1
Scattering by impurities: By impurities we mean foreign atoms
in the solid which are efficient scattering centers especially when they
have a net charge. Ionized donors and acceptors in a semiconductor are
a common example of such impurities. The amount of scattering due to electrostatic
forces between the carrier and the ionized impurity depends on the interaction
time and the number of impurities. Larger impurity concentrations result
in a lower mobility. The dependence on the interaction time helps to explain
the temperature dependence. The interaction time is directly linked to
the relative velocity of the carrier and the impurity which is related
to the thermal velocity of the carriers. This thermal velocity increases
with the ambient temperature so that the interaction time increases, the
amount of scattering decreases, resulting in a mobility increase with temperature.
To first order the mobility due to impurity scattering is proportional
to T 3/2/NI, where NI
is the density of charged impurities.
2.8.5 Electron and hole mobility in silicon
The mobility of electrons and holes in silicon at room temperature is shown
in the figure below.
resistiv.xls - mobility.gif
Fig.2.8.3 Electron and hole mobility versus doping density
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:
These are empirical relations obtained by fitting experimental values.
Empirical relations for the temperature as well as doping dependence
of the carrier mobilities in silicon are available as well and are listed
These relations are valid between 250 and 500 K. A plot of the resulting
mobility as a function of temperature is shown in the figure below:
Fig.2.8.4 Electron and hole mobility versus temperature.
The doping density equals 1016 (top curve), 1017
and 1018 (bottom curve) cm-3.
2.8.6 Surface and Interface Scattering
The surface and interface mobility of carriers is affected by the nature
of the adjacent layer or surface. Even if the carrier does not transfer
into the adjacent region, its wavefunction does extend over 1 to 10 nanometer
so that there is a non-zero probability for the particle to be in the adjacent
region. The net mobility is then a combination of the mobility in both
layers. Carriers in the inversion layer of a MOSFET have an up to three
times lower mobility, since the mobility in the amorphous silicon dioxide
is much lower than that in the silicon. The presence of charged surface
states further reduces the mobility just like ionized impurities would.
© Bart J. Van Zeghbroeck, 1998