Electron mobility
Encyclopedia
In solid-state physics
, the electron mobility characterizes how quickly an electron
can move through a metal
or semiconductor
, when pulled by an electric field
. In semiconductors, there is an analogous quantity for holes
, called hole mobility. The term carrier mobility refers in general to both electron and hole mobility in semiconductors.
Electron and hole mobility are special cases of electrical mobility
of charged particles in a fluid under an applied electric field.
When an electric field E is applied across a piece of material, the electrons respond by moving with an average velocity called the drift velocity
,
. Then the electron mobility μ is defined as
.
Electron mobility is almost always specified in units of cm2/(V
·s
). This is different from the SI
unit of mobility, m2
/(V
·s
). They are related by 1m2/(V·s) = 104cm2/(V·s).
Conductivity is proportional to the product of mobility and carrier concentration. For example, the same conductivity could come from a small number of electrons with high mobility for each, or a large number of electrons with a small mobility for each. For metals, it would not typically matter which of these is the case, most metal electrical behavior depends on conductivity alone. Therefore mobility is relatively unimportant in metal physics. On the other hand, for semiconductors, the behavior of transistor
s and other devices can be very different depending on whether there are many electrons with low mobility or few electrons with high mobility. Therefore mobility is a very important parameter for semiconductor materials. Almost always, higher mobility leads to better device performance, with other things equal.
Semiconductor mobility depends on the impurity concentrations (including donor and acceptor concentrations), defect concentration, temperature, and electron and hole concentrations. It also depends on the electric field, particularly at high fields when velocity saturation
occurs. It can be determined by the Hall effect
, or inferred from transistor behavior.
) move around randomly
. Therefore, on average there will be no overall motion of charge carriers in any particular direction over time.
However, when an electric field is applied, each electron is accelerated by the electric field. If the electron were in a vacuum, it would be accelerated to faster and faster velocities (called ballistic transport
). However, in a solid, the electron repeatedly scatters off crystal defects
, phonons, impurities, etc. Therefore, it does not accelerate faster and faster; instead it moves with a finite average velocity, called the drift velocity
. This net electron motion is usually much slower than the normally occurring random motion.
In a semiconductor the two charge carriers, electrons and holes
, will typically have different drift velocities for the same electric field.
Quasi-ballistic transport
is possible in solids if the electrons are accelerated across a very small distance (as small as the mean free path
), or for a very short time (as short as the mean free time
). In these cases, drift velocity and mobility are not meaningful.
.
where:
The hole mobility is defined by the same equation. Both electron and hole mobilities are positive by definition.
Usually, the electron drift velocity in a material is directly proportional to the electric field, which means that the electron mobility is a constant (independent of electric field). When this is not true (for example, in very large electric fields), the mobility depends on the electric field.
The SI unit of velocity is m/s
, and the SI unit of electric field is V
/m
. Therefore the SI unit of mobility is (m/s)/(V/m) = m2
/(V
·s
). However, mobility is much more commonly expressed in cm2/(V·s) = 10−4 m2/(V·s).
Mobility is usually a strong function of material impurities and temperature, and is determined empirically, mobility values are typically presented in table or chart form. Mobility is also different for electrons and holes in a given semiconductor
.
of electrons, and let μe be their mobility. In the electric field E, each of these electrons will move with the velocity vector
, for a total current density of 
(where e is the elementary charge
). Therefore, the electrical conductivity σ satisfies:
.
This formula is valid when the conductivity is due entirely to electrons. In a p-type semiconductor
, the conductivity is due to holes instead, but the formula is essentially the same: If p is the density of holes and μh is the hole mobility, then the conductivity is
.
If a semiconductor has both electrons and holes
, the total conductivity is
at room temperature (300 K) is 1400 cm2/ (V·s) and the hole mobility is around 450 cm2/ (V·s).
Very high mobility has been found in several low-dimensional systems, such as two-dimensional electron gases (2DEG
) (3,000,000 cm2/(V·s) at low temperature), carbon nanotubes (100,000 cm2/(V·s) at room temperature) and more recently, graphene
(200,000 cm2/ V·s at low temperature).
Organic semiconductors (polymer
, oligomer
) developed thus far have carrier mobilities below 10 cm2/(V·s), and usually much lower.
As the electric field is increased, however, the carrier velocity increases sub-linearly and asymptotically towards a maximum possible value, called the saturation velocity vsat. For example, the value of vsat is on the order of 1×107 cm/s for both electrons and holes in Si. It is on the order of 6×106 cm/s for Ge. This velocity is a characteristic of the material and a strong function of doping
or impurity levels and temperature. It is one of the key material and semiconductor device properties that determine a device such as a transistor's ultimate limit of speed of response and frequency.
This velocity saturation phenomenon results from a process called optical phonon scattering. At high fields, carriers are accelerated enough to gain sufficient kinetic energy
between collisions to emit an optical phonon, and they do so very quickly, before being accelerated once again. The velocity that the electron reaches before emitting a phonon is:
where ωphonon(opt.) is the optical phonon angular frequency and m*is the carrier effective mass in the direction of the electric field. The value of Ephonon (opt.) is 0.063 eV for Si and 0.034 eV for GaAs and Ge. The saturation velocity is only one-half of vemit, because the electron starts at zero velocity and accelerates up to vemit in each cycle. (This is a somewhat oversimplified description.)
Velocity saturation is not the only possible high-field behavior. Another is the Gunn effect, where a sufficiently high electric field can causes intervalley electron transfer, which reduces drift velocity. (This is unusual; increasing the electric field almost always increases the drift velocity, or else leaves it unchanged.) The result is negative differential resistance.
In the regime of velocity saturation (or other high-field effects), mobility is a strong function of electric field. This means that mobility is a somewhat less useful concept, compared to simply discussing drift velocity directly.
time, i.e., how long the carrier is ballistically accelerated
by the electric field until it scatters (collides) with something that changes its direction and/or energy. The most important sources of scattering in typical semiconductor materials, discussed below, are ionized impurity scattering and acoustic phonon scattering (also called lattice scattering). In some cases other sources of scattering may be important, such as neutral impurity scattering, optical phonon scattering, surface scattering, and defect
scattering.
Elastic scattering means that energy is (almost) conserved during the scattering event. Some elastic scattering processes are scattering from acoustic phonons, impurity scattering, piezoelectric scattering, etc. In acoustic phonon scattering, electrons scatter from state k to k', while emitting or absorbing a phonon of wave vector q. This phenomenon is usually modeled by assuming that lattice vibrations cause small shifts in energy bands. The additional potential causing the scattering process is generated by the deviations of bands due to these small transitions from frozen lattice positions.
. The amount of deflection depends on the speed of the carrier and its proximity to the ion. The more heavily a material is doped, the higher the probability that a carrier will collide with an ion in a given time, and the smaller the mean free time
between collisions, and the smaller the mobility. When determining the strength of these interactions due to the long-range nature of the Coulomb potential, other impurities and free carriers cause the range of interaction with the carriers to reduce significantly compared to bare Coulomb interaction.
If these scatterers are near the interface, the complexity of the problem increases due to the existence of crystal defects and disorders. Charge trapping centers that scatter free carriers form in many cases due to defects associated with dangling bonds. Scattering happens because after trapping a charge, the defect becomes charged and therefore starts interacting with free carriers. If scattered carriers are in the inversion layer at the interface, the reduced dimensionality of the carriers makes the case differ from the case of bulk impurity scattering as carriers move only in two dimensions. Interfacial roughness also causes short-range scattering limiting the mobility of quasi two-dimensional electrons at the interface.
, the vibrating atoms create pressure (acoustic) waves in the crystal, which are termed phonon
s. Like electrons, phonons can be considered to be particles. A phonon can interact (collide) with an electron (or hole) and scatter it. At higher temperature, there are more phonons, therefore increased phonon scattering which tends to reduce mobility.
where q is the elementary charge
, m* is the carrier effective mass, and is the average scattering time.
If the effective mass is anisotropic (direction-dependent), m* is the effective mass in the direction of the electric field.
in 1864):
.
where µ is the actual mobility,
is the mobility that the material would have if there was impurity scattering but no other source of scattering, and 
is the mobility that the material would have if there was lattice phonon scattering but no other source of scattering. Other terms may be added for other scattering sources, for example
.
Matthiessen's rule can also be stated in terms of the scattering time:
.
where τ is the true average scattering time and τimpurities is the scattering time if there was impurity scattering but no other source of scattering, etc.
Matthiessen's rule is an approximation and is not universally valid. This rule is not valid if the factors affecting the mobility depend on with each other, because individual scattering probabilities cannot be summed unless they are independent of each other. The average free time of flight of a carrier and therefore the relaxation time is inversely proportional to the scattering probability. For example, lattice scattering alters the average electron velocity (in the electric-field direction), which in turn alters the tendency to scatter off impurities. There are more complicated formulas that attempt to take these effects into account.
With increasing temperature, phonon concentration increases and causes increased scattering. Thus lattice scattering lowers the carrier mobility more and more at higher temperature. Theoretical calculations reveal that the mobility in non-polar
semiconductors, such as silicon and germanium, is dominated by acoustic phonon
interaction. The resulting mobility 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. Experimentally, values of the temperature dependence of the mobility in Si, Ge and GaAs are listed in table.
As
, where Σ is the scattering cross section for electrons and holes at a scattering center and 
is a thermal average (Boltzmann statistics) over all electron or hole velocities in the lower conduction band or upper valence band, temperature dependence of the mobility can be determined. In here, the following definition for the scattering cross section is used: number of particles scattered into solid angle dΩ per unit time divided by number of particles per area per time (incident intensity), which comes from classical mechanics. As Boltzmann statistics are valid for semiconductors
.
For scattering from acoustic phonons for temperatures
(Θ=Debye temperature), the estimated cross section Σph is determined from the square of the average vibrational amplitude of a phonon to be proportional to T. The scattering from charged defects (ionized donors or acceptors) leads to the cross section 
. This formula is the scattering cross section for “Rutherford scattering”, where a point charge (carrier) moves past another point charge (defect) experiencing Coulomb interaction.
The temperature dependencies of these two scattering mechanism in semiconductors can be determined by combining formulas for τ, Σ and
, to be for scattering from acoustic phonons
and from charged defects 
.
The effect of ionized impurity scattering, however, decreases with increasing temperature because the average thermal speeds of the carriers are increase. Thus, the carriers spend less time near an ionized impurity as they pass and the scattering effect of the ions is thus reduced.
These two effects operate simultaneously on the carriers through Matthiessen's rule. At lower temperatures, ionized impurity scattering dominates, while at higher temperatures, phonon scattering dominates, and the actual mobility reaches a maximum at an intermediate temperature.
Carrier mobility is most commonly measured using the Hall effect
. The result of the measurement is called the "Hall mobility" (meaning "mobility inferred from a Hall-effect measurement").
Consider a semiconductor sample with a rectangular cross section as shown in the figures, a current is flowing in the x-direction and a magnetic field
is applied in the z-direction. The resulting Lorentz force will accelerate the electrons (n-type materials) or holes (p-type materials) in the (−y) direction, according to the right hand rule and set up an electric field ξy. As a result there is a voltage across the sample, which can be measured with a high-impedance
voltmeter. This voltage, VH, is called the Hall voltage
. VH is negative for n-type material and positive for p-type material.
Mathematically, the Lorentz force
acting on a charge Q is given by
For electrons:
For holes:
In steady state this force is balanced by the force set up by the Hall voltage, so that there is no net force
on the carriers in the y direction. For electron,
For electrons, the field points in the -y direction, and for holes, it points in the +y direction.
The electron current
I is given by
. Sub vx into the expression for ξy,
where RHn is the Hall coefficient for electron, and is defined as
Since
Similarly, for holes
From Hall coefficient, we can obtain the carrier mobility as follows:
Similarly,
Here the value of VHp (Hall voltage), t (sample thickness), I (current) and B (magnetic field) can be measured directly, and the conductivities σn or σp are either known or can be obtained from measuring the resistivity.
(FET). The result of the measurement is called the "field-effect mobility" (meaning "mobility inferred from a field-effect measurement").
The measurement can work in two ways: From saturation-mode measurements, or linear-region measurements. (See MOSFET
for a description of the different modes or regions of operation.)
where L and W are the length and width of the channel and Ci is the gate insulator capacitance per unit area. This equation comes from the approximate equation for a MOSFET in saturation mode:
where Vth is the threshold voltage. This approximation ignores the Early effect
(channel length modulation), among other things. In practice, this technique may underestimate the true mobility.
with slope mlin. Then the mobility is:
.
This equation comes from the approximate equation for a MOSFET in the linear region:
In practice, this technique may overestimate the true mobility, because if VDS is not small enough and VG is not large enough, the MOSFET may not stay in the linear region.
s in semiconductors are electrons and holes. Their numbers are controlled by the concentrations of impurity elements, i.e. doping concentration. Thus doping concentration has great influence on carrier mobility.
While there is considerable scatter in the experimental data
, for noncompensated material (no counter doping) for heavily doped substrates (i.e.
and up), the mobility in silicon is often characterized by the empirical relationship
:
where N is the doping concentration (either ND or NA), and Nref and α are fitting parameters. At room temperature
, the above equation becomes:
Majority carriers:
Minority carriers:
These equations apply only to silicon, and only under low field.
Solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from...
, the electron mobility characterizes how quickly an electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
can move through a metal
Metal
A metal , is an element, compound, or alloy that is a good conductor of both electricity and heat. Metals are usually malleable and shiny, that is they reflect most of incident light...
or semiconductor
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
, when pulled by an electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
. In semiconductors, there is an analogous quantity for holes
Electron hole
An electron hole is the conceptual and mathematical opposite of an electron, useful in the study of physics, chemistry, and electrical engineering. The concept describes the lack of an electron at a position where one could exist in an atom or atomic lattice...
, called hole mobility. The term carrier mobility refers in general to both electron and hole mobility in semiconductors.
Electron and hole mobility are special cases of electrical mobility
Electrical mobility
Electrical mobility is the ability of charged particles to move through a medium in response to an electric field that is pulling them...
of charged particles in a fluid under an applied electric field.
When an electric field E is applied across a piece of material, the electrons respond by moving with an average velocity called the drift velocity
Drift velocity
The drift velocity is the average velocity that a particle, such as an electron, attains due to an electric field. It can also be referred to as Axial Drift Velocity since particles defined are assumed to be moving along a plane. In general, an electron will 'rattle around' in a conductor at the...
,
. Then the electron mobility μ is defined as.Electron mobility is almost always specified in units of cm2/(V
Volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...
·s
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
). This is different from the SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
unit of mobility, m2
Square metre
The square metre or square meter is the SI derived unit of area, with symbol m2 . It is defined as the area of a square whose sides measure exactly one metre...
/(V
Volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...
·s
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
). They are related by 1m2/(V·s) = 104cm2/(V·s).
Conductivity is proportional to the product of mobility and carrier concentration. For example, the same conductivity could come from a small number of electrons with high mobility for each, or a large number of electrons with a small mobility for each. For metals, it would not typically matter which of these is the case, most metal electrical behavior depends on conductivity alone. Therefore mobility is relatively unimportant in metal physics. On the other hand, for semiconductors, the behavior of transistor
Transistor
A transistor is a semiconductor device used to amplify and switch electronic signals and power. It is composed of a semiconductor material with at least three terminals for connection to an external circuit. A voltage or current applied to one pair of the transistor's terminals changes the current...
s and other devices can be very different depending on whether there are many electrons with low mobility or few electrons with high mobility. Therefore mobility is a very important parameter for semiconductor materials. Almost always, higher mobility leads to better device performance, with other things equal.
Semiconductor mobility depends on the impurity concentrations (including donor and acceptor concentrations), defect concentration, temperature, and electron and hole concentrations. It also depends on the electric field, particularly at high fields when velocity saturation
Velocity saturation
In semiconductors, when a strong enough electric field is applied, the carrier velocity in the semiconductor reaches a maximum value, saturation velocity. When this happens, the semiconductor is said to be in a state of velocity saturation...
occurs. It can be determined by the Hall effect
Hall effect
The Hall effect is the production of a voltage difference across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current...
, or inferred from transistor behavior.
Drift velocity in an electric field
Without any applied electric field, in a solid, electrons (or, in the case of semiconductors, both electrons and holesElectron hole
An electron hole is the conceptual and mathematical opposite of an electron, useful in the study of physics, chemistry, and electrical engineering. The concept describes the lack of an electron at a position where one could exist in an atom or atomic lattice...
) move around randomly
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
. Therefore, on average there will be no overall motion of charge carriers in any particular direction over time.
However, when an electric field is applied, each electron is accelerated by the electric field. If the electron were in a vacuum, it would be accelerated to faster and faster velocities (called ballistic transport
Ballistic transport
Ballistic transport is the transport of electrons in a medium with negligible electrical resistivity due to scattering. Without scattering, electrons simply obey Newton's second law of motion at non-relativistic speeds....
). However, in a solid, the electron repeatedly scatters off crystal defects
Crystallographic defect
Crystalline solids exhibit a periodic crystal structure. The positions of atoms or molecules occur on repeating fixed distances, determined by the unit cell parameters. However, the arrangement of atom or molecules in most crystalline materials is not perfect...
, phonons, impurities, etc. Therefore, it does not accelerate faster and faster; instead it moves with a finite average velocity, called the drift velocity
Drift velocity
The drift velocity is the average velocity that a particle, such as an electron, attains due to an electric field. It can also be referred to as Axial Drift Velocity since particles defined are assumed to be moving along a plane. In general, an electron will 'rattle around' in a conductor at the...
. This net electron motion is usually much slower than the normally occurring random motion.
In a semiconductor the two charge carriers, electrons and holes
Electron hole
An electron hole is the conceptual and mathematical opposite of an electron, useful in the study of physics, chemistry, and electrical engineering. The concept describes the lack of an electron at a position where one could exist in an atom or atomic lattice...
, will typically have different drift velocities for the same electric field.
Quasi-ballistic transport
Ballistic transport
Ballistic transport is the transport of electrons in a medium with negligible electrical resistivity due to scattering. Without scattering, electrons simply obey Newton's second law of motion at non-relativistic speeds....
is possible in solids if the electrons are accelerated across a very small distance (as small as the mean free path
Mean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...
), or for a very short time (as short as the mean free time
Mean free time
Molecules in a fluid constantly collide off each other. The mean free time of a molecule in a fluid is the average time between collisions. The mean free path of the molecule is the product of the average speed and the mean free time...
). In these cases, drift velocity and mobility are not meaningful.
Definition and units
The electron mobility is defined by the equation:.where:
- E is the magnitude of the electric fieldElectric fieldIn physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
applied to a material, - vd is the magnitude of the electron drift velocity (in other words, the electron drift speedSpeedIn kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...
) caused by the electric field, and - µ is the electron mobility.
The hole mobility is defined by the same equation. Both electron and hole mobilities are positive by definition.
Usually, the electron drift velocity in a material is directly proportional to the electric field, which means that the electron mobility is a constant (independent of electric field). When this is not true (for example, in very large electric fields), the mobility depends on the electric field.
The SI unit of velocity is m/s
Metre per second
Metre per second is an SI derived unit of both speed and velocity , defined by distance in metres divided by time in seconds....
, and the SI unit of electric field is V
Volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...
/m
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...
. Therefore the SI unit of mobility is (m/s)/(V/m) = m2
Square metre
The square metre or square meter is the SI derived unit of area, with symbol m2 . It is defined as the area of a square whose sides measure exactly one metre...
/(V
Volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...
·s
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
). However, mobility is much more commonly expressed in cm2/(V·s) = 10−4 m2/(V·s).
Mobility is usually a strong function of material impurities and temperature, and is determined empirically, mobility values are typically presented in table or chart form. Mobility is also different for electrons and holes in a given semiconductor
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
.
Relation to conductivity
There is a simple relation between mobility and electrical conductivity. Let n be the number densityNumber density
In physics, astronomy, and chemistry, number density is an intensive quantity used to describe the degree of concentration of countable objects in the three-dimensional physical space...
of electrons, and let μe be their mobility. In the electric field E, each of these electrons will move with the velocity vector
, for a total current density of (where e is the elementary chargeElementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
). Therefore, the electrical conductivity σ satisfies:
.This formula is valid when the conductivity is due entirely to electrons. In a p-type semiconductor
P-type semiconductor
A P-type semiconductor is obtained by carrying out a process of doping: that is, adding a certain type of atoms to the semiconductor in order to increase the number of free charge carriers ....
, the conductivity is due to holes instead, but the formula is essentially the same: If p is the density of holes and μh is the hole mobility, then the conductivity is
.If a semiconductor has both electrons and holes
Electron hole
An electron hole is the conceptual and mathematical opposite of an electron, useful in the study of physics, chemistry, and electrical engineering. The concept describes the lack of an electron at a position where one could exist in an atom or atomic lattice...
, the total conductivity is
Examples
Typical electron mobility for SiSilicon
Silicon is a chemical element with the symbol Si and atomic number 14. A tetravalent metalloid, it is less reactive than its chemical analog carbon, the nonmetal directly above it in the periodic table, but more reactive than germanium, the metalloid directly below it in the table...
at room temperature (300 K) is 1400 cm2/ (V·s) and the hole mobility is around 450 cm2/ (V·s).
Very high mobility has been found in several low-dimensional systems, such as two-dimensional electron gases (2DEG
2DEG
A two-dimensional electron gas is a gas of electrons free to move in two dimensions, but tightly confined in the third. This tight confinement leads to quantized energy levels for motion in that direction, which can then be ignored for most problems. Thus the electrons appear to be a 2D sheet...
) (3,000,000 cm2/(V·s) at low temperature), carbon nanotubes (100,000 cm2/(V·s) at room temperature) and more recently, graphene
Graphene
Graphene is an allotrope of carbon, whose structure is one-atom-thick planar sheets of sp2-bonded carbon atoms that are densely packed in a honeycomb crystal lattice. The term graphene was coined as a combination of graphite and the suffix -ene by Hanns-Peter Boehm, who described single-layer...
(200,000 cm2/ V·s at low temperature).
Organic semiconductors (polymer
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
, oligomer
Oligomer
In chemistry, an oligomer is a molecule that consists of a few monomer units , in contrast to a polymer that, at least in principle, consists of an unlimited number of monomers. Dimers, trimers, and tetramers are oligomers. Many oils are oligomeric, such as liquid paraffin...
) developed thus far have carrier mobilities below 10 cm2/(V·s), and usually much lower.
Electric field dependence and velocity saturation
At low fields, the drift velocity vd is proportional to the electric field E, so mobility μ is constant. This value of μ is called the low-field mobility.As the electric field is increased, however, the carrier velocity increases sub-linearly and asymptotically towards a maximum possible value, called the saturation velocity vsat. For example, the value of vsat is on the order of 1×107 cm/s for both electrons and holes in Si. It is on the order of 6×106 cm/s for Ge. This velocity is a characteristic of the material and a strong function of doping
Doping (semiconductor)
In semiconductor production, doping intentionally introduces impurities into an extremely pure semiconductor for the purpose of modulating its electrical properties. The impurities are dependent upon the type of semiconductor. Lightly and moderately doped semiconductors are referred to as extrinsic...
or impurity levels and temperature. It is one of the key material and semiconductor device properties that determine a device such as a transistor's ultimate limit of speed of response and frequency.
This velocity saturation phenomenon results from a process called optical phonon scattering. At high fields, carriers are accelerated enough to gain sufficient kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
between collisions to emit an optical phonon, and they do so very quickly, before being accelerated once again. The velocity that the electron reaches before emitting a phonon is:
where ωphonon(opt.) is the optical phonon angular frequency and m*is the carrier effective mass in the direction of the electric field. The value of Ephonon (opt.) is 0.063 eV for Si and 0.034 eV for GaAs and Ge. The saturation velocity is only one-half of vemit, because the electron starts at zero velocity and accelerates up to vemit in each cycle. (This is a somewhat oversimplified description.)
Velocity saturation is not the only possible high-field behavior. Another is the Gunn effect, where a sufficiently high electric field can causes intervalley electron transfer, which reduces drift velocity. (This is unusual; increasing the electric field almost always increases the drift velocity, or else leaves it unchanged.) The result is negative differential resistance.
In the regime of velocity saturation (or other high-field effects), mobility is a strong function of electric field. This means that mobility is a somewhat less useful concept, compared to simply discussing drift velocity directly.
Relation between scattering and mobility
Recall that by definition, mobility is dependent on the drift velocity. The main factor determining drift velocity (other than effective mass) is scatteringScattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
time, i.e., how long the carrier is ballistically accelerated
Ballistic transport
Ballistic transport is the transport of electrons in a medium with negligible electrical resistivity due to scattering. Without scattering, electrons simply obey Newton's second law of motion at non-relativistic speeds....
by the electric field until it scatters (collides) with something that changes its direction and/or energy. The most important sources of scattering in typical semiconductor materials, discussed below, are ionized impurity scattering and acoustic phonon scattering (also called lattice scattering). In some cases other sources of scattering may be important, such as neutral impurity scattering, optical phonon scattering, surface scattering, and defect
Crystallographic defect
Crystalline solids exhibit a periodic crystal structure. The positions of atoms or molecules occur on repeating fixed distances, determined by the unit cell parameters. However, the arrangement of atom or molecules in most crystalline materials is not perfect...
scattering.
Elastic scattering means that energy is (almost) conserved during the scattering event. Some elastic scattering processes are scattering from acoustic phonons, impurity scattering, piezoelectric scattering, etc. In acoustic phonon scattering, electrons scatter from state k to k', while emitting or absorbing a phonon of wave vector q. This phenomenon is usually modeled by assuming that lattice vibrations cause small shifts in energy bands. The additional potential causing the scattering process is generated by the deviations of bands due to these small transitions from frozen lattice positions.
Ionized impurity scattering
Semiconductors are doped with donors and/or acceptors, which are typically ionized, and are thus charged. The Coulombic forces will deflect an electron or hole approaching the ionized impurity. This is known as ionized impurity scatteringIonized impurity scattering
Ionized impurity scattering is the scattering of charge carriers by ionization in the lattice. The most primitive models can be conceptually understood as a particle responding to unbalanced local charge that arises near a crystal impurity; not dissimilar to an electron encountering an electric field...
. The amount of deflection depends on the speed of the carrier and its proximity to the ion. The more heavily a material is doped, the higher the probability that a carrier will collide with an ion in a given time, and the smaller the mean free time
Mean free time
Molecules in a fluid constantly collide off each other. The mean free time of a molecule in a fluid is the average time between collisions. The mean free path of the molecule is the product of the average speed and the mean free time...
between collisions, and the smaller the mobility. When determining the strength of these interactions due to the long-range nature of the Coulomb potential, other impurities and free carriers cause the range of interaction with the carriers to reduce significantly compared to bare Coulomb interaction.
If these scatterers are near the interface, the complexity of the problem increases due to the existence of crystal defects and disorders. Charge trapping centers that scatter free carriers form in many cases due to defects associated with dangling bonds. Scattering happens because after trapping a charge, the defect becomes charged and therefore starts interacting with free carriers. If scattered carriers are in the inversion layer at the interface, the reduced dimensionality of the carriers makes the case differ from the case of bulk impurity scattering as carriers move only in two dimensions. Interfacial roughness also causes short-range scattering limiting the mobility of quasi two-dimensional electrons at the interface.
Lattice (phonon) scattering
At any temperature above absolute zeroAbsolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....
, the vibrating atoms create pressure (acoustic) waves in the crystal, which are termed phonon
Phonon
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...
s. Like electrons, phonons can be considered to be particles. A phonon can interact (collide) with an electron (or hole) and scatter it. At higher temperature, there are more phonons, therefore increased phonon scattering which tends to reduce mobility.
Piezoelectric scattering
Piezoelectric effect can occur only in compound semiconductor due to their polar nature. It is small in most semiconductors but may lead to local electric fields that cause scattering of carriers by deflecting them, this effect is important mainly at low temperatures where other scattering mechanisms are weak. These electric fields arise from the distortion of the basic unit cell as strain is applied in certain directions in the lattice.Surface roughness scattering
Surface roughness scattering caused by interfacial disorder is short range scattering limiting the mobility of quasi two-dimensional electrons at the interface. From high-resolution transmission electron micrographs, it has been determined that the interface is not abrupt on the atomic level, but actual position of the interfacial plane varies one or two atomic layers along the surface. These variations are random and cause fluctuations of the energy levels at the interface, which then causes scattering.Alloy scattering
In compound (alloy) semiconductors, which many thermoelectric materials are, scattering caused by the perturbation of crystal potential due to the random positioning of substituting atom species in a relevant sublattice is known as alloy scattering. This can only happen in ternary or higher alloys as their crystal structure forms by randomly replacing some atoms in one of the cubic lattices (sublattice) that form zinc-blende structure. Generally, this phenomenon is quite weak but in certain materials or circumstances, it can become dominant effect limiting conductivity. In bulk materials, interface scattering is usually ignored.Inelastic scattering
During inelastic scattering processes, significant energy exchange happens. As with elastic phonon scattering also in the inelastic case, the potential arouses from energy band deformations caused by atomic vibrations. Optical phonons causing inelastic scattering usually have the energy in the range 30-50 meV, for comparison energies of acoustic phonon are typically less than 1 meV but some might have energy in order of 10 meV. There is significant change in carrier energy during the scattering process. Optical or high-energy acoustic phonons can also cause intervalley or interband scattering, which means that scattering is not limited within single valley.Electron-electron scattering
Due to the Pauli exclusion principle, electrons can be considered as non-interacting if their density does not exceed the value 1017 cm-3 or electric field value 103 V/cm. However, significantly above these limits electron-electron scattering starts to dominate. Long range and nonlinearity of the Coulomb potential governing interactions between electrons make these interactions difficult to deal with.Relation between mobility and scattering time
A simple model gives the approximate relation between scattering time (average time between scattering events) and mobility. It is assumed that after each scattering event, the carrier's motion is randomized, so it has zero average velocity. After that, it accelerates uniformly in the electric field, until it scatters again. The resulting average drift velocity is:where q is the elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
, m* is the carrier effective mass, and is the average scattering time.
If the effective mass is anisotropic (direction-dependent), m* is the effective mass in the direction of the electric field.
Matthiessen's rule
Normally, more than one source of scattering is present, for example both impurities and lattice phonons. It is normally a very good approximation to combine their influences using "Matthiessen's Rule" (developed from work by Augustus MatthiessenAugustus Matthiessen
Augustus Matthiessen, FRS , the son of a merchant, was a British chemist and physicist who obtained his PhD in Germany at the University of Gießen in 1852 with Johann Heinrich Buff. He then worked with Robert Bunsen at the University of Heidelberg from 1853 to 1856...
in 1864):
.where µ is the actual mobility,
is the mobility that the material would have if there was impurity scattering but no other source of scattering, and is the mobility that the material would have if there was lattice phonon scattering but no other source of scattering. Other terms may be added for other scattering sources, for example.Matthiessen's rule can also be stated in terms of the scattering time:
.where τ is the true average scattering time and τimpurities is the scattering time if there was impurity scattering but no other source of scattering, etc.
Matthiessen's rule is an approximation and is not universally valid. This rule is not valid if the factors affecting the mobility depend on with each other, because individual scattering probabilities cannot be summed unless they are independent of each other. The average free time of flight of a carrier and therefore the relaxation time is inversely proportional to the scattering probability. For example, lattice scattering alters the average electron velocity (in the electric-field direction), which in turn alters the tendency to scatter off impurities. There are more complicated formulas that attempt to take these effects into account.
Temperature dependence of mobility
| Si | Ge | GaAs | |
|---|---|---|---|
| Electrons | ∝T −2.4 | ∝T −1.7 | ∝T −1.0 |
| Holes | ∝T −2.2 | ∝T −2.3 | ∝T −2.1 |
With increasing temperature, phonon concentration increases and causes increased scattering. Thus lattice scattering lowers the carrier mobility more and more at higher temperature. Theoretical calculations reveal that the mobility in non-polar
Chemical polarity
In chemistry, polarity refers to a separation of electric charge leading to a molecule or its chemical groups having an electric dipole or multipole moment. Polar molecules interact through dipole–dipole intermolecular forces and hydrogen bonds. Molecular polarity is dependent on the difference in...
semiconductors, such as silicon and germanium, is dominated by acoustic phonon
Phonon
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...
interaction. The resulting mobility 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. Experimentally, values of the temperature dependence of the mobility in Si, Ge and GaAs are listed in table.
As
, where Σ is the scattering cross section for electrons and holes at a scattering center and is a thermal average (Boltzmann statistics) over all electron or hole velocities in the lower conduction band or upper valence band, temperature dependence of the mobility can be determined. In here, the following definition for the scattering cross section is used: number of particles scattered into solid angle dΩ per unit time divided by number of particles per area per time (incident intensity), which comes from classical mechanics. As Boltzmann statistics are valid for semiconductors.For scattering from acoustic phonons for temperatures
(Θ=Debye temperature), the estimated cross section Σph is determined from the square of the average vibrational amplitude of a phonon to be proportional to T. The scattering from charged defects (ionized donors or acceptors) leads to the cross section . This formula is the scattering cross section for “Rutherford scattering”, where a point charge (carrier) moves past another point charge (defect) experiencing Coulomb interaction.The temperature dependencies of these two scattering mechanism in semiconductors can be determined by combining formulas for τ, Σ and
, to be for scattering from acoustic phonons and from charged defects . The effect of ionized impurity scattering, however, decreases with increasing temperature because the average thermal speeds of the carriers are increase. Thus, the carriers spend less time near an ionized impurity as they pass and the scattering effect of the ions is thus reduced.
These two effects operate simultaneously on the carriers through Matthiessen's rule. At lower temperatures, ionized impurity scattering dominates, while at higher temperatures, phonon scattering dominates, and the actual mobility reaches a maximum at an intermediate temperature.
Hall mobility
Hall effect
The Hall effect is the production of a voltage difference across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current...
. The result of the measurement is called the "Hall mobility" (meaning "mobility inferred from a Hall-effect measurement").
Consider a semiconductor sample with a rectangular cross section as shown in the figures, a current is flowing in the x-direction and a magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
is applied in the z-direction. The resulting Lorentz force will accelerate the electrons (n-type materials) or holes (p-type materials) in the (−y) direction, according to the right hand rule and set up an electric field ξy. As a result there is a voltage across the sample, which can be measured with a high-impedance
High impedance
In electronics, high impedance means that a point in a circuit has a relatively high impedance to other points in the circuit.-Digital electronics:...
voltmeter. This voltage, VH, is called the Hall voltage
Hall effect
The Hall effect is the production of a voltage difference across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current...
. VH is negative for n-type material and positive for p-type material.
Mathematically, the Lorentz force
Lorentz force
In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...
acting on a charge Q is given by
For electrons:
For holes:
In steady state this force is balanced by the force set up by the Hall voltage, so that there is no net force
Net force
In physics, net force is the total force acting on an object. It is calculated by vector addition of all forces that are actually acting on that object. Net force has the same effect on the translational motion of the object as all actual forces taken together...
on the carriers in the y direction. For electron,
For electrons, the field points in the -y direction, and for holes, it points in the +y direction.
The electron current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
I is given by
. Sub vx into the expression for ξy,
where RHn is the Hall coefficient for electron, and is defined as
Since
Similarly, for holes
From Hall coefficient, we can obtain the carrier mobility as follows:
Similarly,
Here the value of VHp (Hall voltage), t (sample thickness), I (current) and B (magnetic field) can be measured directly, and the conductivities σn or σp are either known or can be obtained from measuring the resistivity.
Field-effect mobility
The mobility can also be measured using a field-effect transistorField-effect transistor
The field-effect transistor is a transistor that relies on an electric field to control the shape and hence the conductivity of a channel of one type of charge carrier in a semiconductor material. FETs are sometimes called unipolar transistors to contrast their single-carrier-type operation with...
(FET). The result of the measurement is called the "field-effect mobility" (meaning "mobility inferred from a field-effect measurement").
The measurement can work in two ways: From saturation-mode measurements, or linear-region measurements. (See MOSFET
MOSFET
The metal–oxide–semiconductor field-effect transistor is a transistor used for amplifying or switching electronic signals. The basic principle of this kind of transistor was first patented by Julius Edgar Lilienfeld in 1925...
for a description of the different modes or regions of operation.)
Using saturation mode
In this technique, for each fixed gate voltage VGS, the drain-source voltage VDS is increased until the current ID saturates. Next, the square root of this saturated current is plotted against the gate voltage, and the slope msat is measured. Then the mobility is:where L and W are the length and width of the channel and Ci is the gate insulator capacitance per unit area. This equation comes from the approximate equation for a MOSFET in saturation mode:
where Vth is the threshold voltage. This approximation ignores the Early effect
Early Effect
The Early effect is the variation in the width of the base in a bipolar junction transistor due to a variation in the applied base-to-collector voltage, named after its discoverer James M. Early...
(channel length modulation), among other things. In practice, this technique may underestimate the true mobility.
Using the linear region
In this technique, the transistor is operated in the linear region (or "ohmic mode"), where VDS is small and with slope mlin. Then the mobility is:.This equation comes from the approximate equation for a MOSFET in the linear region:
In practice, this technique may overestimate the true mobility, because if VDS is not small enough and VG is not large enough, the MOSFET may not stay in the linear region.
Doping concentration dependence in heavily-doped silicon
The charge carrierCharge carrier
In physics, a charge carrier is a free particle carrying an electric charge, especially the particles that carry electric currents in electrical conductors. Examples are electrons and ions...
s in semiconductors are electrons and holes. Their numbers are controlled by the concentrations of impurity elements, i.e. doping concentration. Thus doping concentration has great influence on carrier mobility.
While there is considerable scatter in the experimental data
Experimental data
Experimental data in science is data produced by a measurement, test method, experimental design or quasi-experimental design. In clinical research any data produced as a result of clinical trial...
, for noncompensated material (no counter doping) for heavily doped substrates (i.e.
and up), the mobility in silicon is often characterized by the empirical relationshipEmpirical relationship
In science, an empirical relationship is one based solely on observation rather than theory. An empirical relationship requires only confirmatory data irrespective of theoretical basis. Sometimes theoretical explanations for what were initially empirical relationships are found, in which case the...
:
where N is the doping concentration (either ND or NA), and Nref and α are fitting parameters. At room temperature
Room temperature
-Comfort levels:The American Society of Heating, Refrigerating and Air-Conditioning Engineers has listings for suggested temperatures and air flow rates in different types of buildings and different environmental circumstances. For example, a single office in a building has an occupancy ratio per...
, the above equation becomes:
Majority carriers:
Minority carriers:
These equations apply only to silicon, and only under low field.
External links
- The minority Carrier Lifetime in silicon wafer
- semiconductor glossary entry for electron mobility
- Resistivity and Mobility Calculator from the BYU Cleanroom
- Online lecture- Mobility from an atomistic point of view