In this section:
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 10^{7} cm/s.
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 Newton’s 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.
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.
fmob2.gif
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:
(mob15)
where t_{r} 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:
(mob16)
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 10^{7} 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:
fmob1.gif
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:
(mob11)
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:
(mob12)
Combining both relations yields an expression for the average particle velocity:
(mob13)
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:
(mob14)
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 large.
The drift current can then be rewritten as a function of the mobility, yielding:
(mob17)
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:
(mob3)
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:
(mo2)
The maximum obtainable velocity, v_{sat}, is refered to as the saturation velocity.
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} |
Germanium |
Silicon |
Gallium Arsenide |
Electron mobility |
µ T^{ -1.7} |
µ T^{ -2.4} |
µ T^{ -1.0} |
Hole mobility |
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}/N_{I}, where N_{I} is the density of charged impurities.
The mobility of electrons and holes in silicon at room temperature is shown in the figure below.
Fig.2.8.3 Electron and hole mobility versus doping density for silicon
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:
and
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 below:
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:
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