Space charge is an interpretation of a collection of electric charges in which excess
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
is treated as a
continuum
Continuum may refer to:
* Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes
Mathematics
* Continuum (set theory), the real line or the corresponding cardinal number ...
of charge distributed over a region of space (either a volume or an area) rather than distinct point-like charges. This model typically applies when
charge carriers
In physics, a charge carrier is a particle or quasiparticle that is free to move, carrying an electric charge, especially the particles that carry electric charges in electrical conductors. Examples are electrons, ions and holes. The term is used ...
have been emitted from some region of a solid—the cloud of emitted carriers can form a space charge region if they are sufficiently spread out, or the charged atoms or molecules left behind in the solid can form a space charge region.
Space charge only occurs in
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
media (including
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
) because in a conductive medium the charge tends to be rapidly neutralized or
screened. The sign of the space charge can be either negative or positive. This situation is perhaps most familiar in the area near a
metal
A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...
object when it is heated to
incandescence
Incandescence is the emission of electromagnetic radiation (including visible light) from a hot body as a result of its high temperature. The term derives from the Latin verb ''incandescere,'' to glow white. A common use of incandescence is ...
in a
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
. This effect was first observed by
Thomas Edison
Thomas Alva Edison (February 11, 1847October 18, 1931) was an American inventor and businessman. He developed many devices in fields such as electric power generation, mass communication, sound recording, and motion pictures. These inventio ...
in light bulb
filament
The word filament, which is descended from Latin ''filum'' meaning " thread", is used in English for a variety of thread-like structures, including:
Astronomy
* Galaxy filament, the largest known cosmic structures in the universe
* Solar filament ...
s, where it is sometimes called the
Edison effect
Thermionic emission is the liberation of electrons from an electrode by virtue of its temperature (releasing of energy supplied by heat). This occurs because the thermal energy given to the charge carrier overcomes the work function of the mate ...
. Space charge is a significant phenomenon in many vacuum and
solid-state electronic
Solid-state electronics means semiconductor electronics: electronic equipment using semiconductor devices such as transistors, diodes and integrated circuits (ICs). The term is also used as an adjective for devices in which semiconductor electr ...
devices.
Cause
Physical explanation
When a metal object is placed in a vacuum and is heated to incandescence, the energy is sufficient to cause
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
s to "boil" away from the surface
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
s and surround the metal object in a cloud of free electrons. This is called
thermionic emission
Thermionic emission is the liberation of electrons from an electrode by virtue of its temperature (releasing of energy supplied by heat). This occurs because the thermal energy given to the charge carrier overcomes the work function of the mate ...
. The resulting cloud is negatively charged, and can be attracted to any nearby positively charged object, thus producing an electric current which passes through the vacuum.
Space charge can result from a range of phenomena, but the most important are:
# Combination of the current density and spatially
inhomogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
resistance
#
Ionization
Ionization, or Ionisation is the process by which an atom or a molecule acquires a negative or positive charge by gaining or losing electrons, often in conjunction with other chemical changes. The resulting electrically charged atom or molecule i ...
of species within the dielectric to form heterocharge
# Charge injection from electrodes and from a stress enhancement
#
Polarization in structures such as
water trees. "Water tree" is a name given to a tree-like figure appearing in a water-impregnated polymer insulating cable.
It has been suggested that in
alternating current
Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...
(AC) most carriers injected at
electrode
An electrode is an electrical conductor used to make contact with a nonmetallic part of a circuit (e.g. a semiconductor, an electrolyte, a vacuum or air). Electrodes are essential parts of batteries that can consist of a variety of materials de ...
s during a half of cycle are ejected during the next half cycle, so the net balance of charge on a cycle is practically zero. However, a small fraction of the carriers can be trapped at levels deep enough to retain them when the field is inverted. The amount of charge in AC should increase slower than in
direct current
Direct current (DC) is one-directional flow of electric charge. An electrochemical cell is a prime example of DC power. Direct current may flow through a conductor such as a wire, but can also flow through semiconductors, insulators, or even ...
(DC) and become observable after longer periods of time.
Hetero and homo charge
Hetero charge means that the polarity of the space charge is opposite to that of neighboring electrode, and homo charge is the reverse situation. Under high voltage application, a hetero charge near the electrode is expected to reduce the breakdown voltage, whereas a homo charge will increase it. After polarity reversal under ac conditions, the homo charge is converted to hetero space charge.
Mathematical explanation
If the near "
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
" has a
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
of 10
−6 mmHg or less, the main
vehicle
A vehicle (from la, vehiculum) is a machine that transports people or cargo. Vehicles include wagons, bicycles, motor vehicles (motorcycles, cars, trucks, buses, mobility scooters for disabled people), railed vehicles (trains, trams), wa ...
of conduction is
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
s. The emission current density (''J'') from the
cathode
A cathode is the electrode from which a conventional current leaves a polarized electrical device. This definition can be recalled by using the mnemonic ''CCD'' for ''Cathode Current Departs''. A conventional current describes the direction in whi ...
, as a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
of its thermodynamic
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
''T'', in the absence of space-charge, is given by
Richardson's law
Thermionic emission is the liberation of electrons from an electrode by virtue of its temperature (releasing of energy supplied by heat). This occurs because the thermal energy given to the charge carrier overcomes the work function of the mater ...
:
where
*
* = elementary positive charge (i.e., magnitude of electron charge),
* = electron mass,
* =
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
= ,
* =
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
= ,
* =
work function
In solid-state physics, the work function (sometimes spelt workfunction) is the minimum thermodynamic work (i.e., energy) needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface. Here "immediately" m ...
of the cathode,
* = mean electron reflection coefficient.
The reflection coefficient can be as low as 0.105 but is usually near 0.5. For
tungsten
Tungsten, or wolfram, is a chemical element with the symbol W and atomic number 74. Tungsten is a rare metal found naturally on Earth almost exclusively as compounds with other elements. It was identified as a new element in 1781 and first isolat ...
, (1 − ''ř'')''A''
0 = , and . At 2500 °C, the emission is 28207 A/m
2.
The emission current as given above is many times greater than that normally collected by the electrodes, except in some
pulse
In medicine, a pulse represents the tactile arterial palpation of the cardiac cycle (heartbeat) by trained fingertips. The pulse may be palpated in any place that allows an artery to be compressed near the surface of the body, such as at the nec ...
d
valves
A valve is a device or natural object that regulates, directs or controls the flow of a fluid (gases, liquids, fluidized solids, or slurries) by opening, closing, or partially obstructing various passageways. Valves are technically fittings ...
such as the
cavity magnetron
The cavity magnetron is a high-power vacuum tube used in early radar systems and currently in microwave ovens and linear particle accelerators. It generates microwaves using the interaction of a stream of electrons with a magnetic field while ...
. Most of the electrons emitted by the cathode are driven back to it by the
repulsion of the
cloud
In meteorology, a cloud is an aerosol consisting of a visible mass of miniature liquid droplets, frozen crystals, or other particles suspended in the atmosphere of a planetary body or similar space. Water or various other chemicals may co ...
of electrons in its neighborhood. This is called the ''space charge effect''. In the limit of large current densities, ''J'' is given by the Child–Langmuir equation below, rather than by the thermionic emission equation above.
Occurrence
Space charge is an inherent property of all
vacuum tube
A vacuum tube, electron tube, valve (British usage), or tube (North America), is a device that controls electric current flow in a high vacuum between electrodes to which an electric voltage, potential difference has been applied.
The type kn ...
s. This has at times made life harder or easier for
electrical engineer
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
s who used tubes in their designs. For example, space charge significantly limited the practical application of
triode
A triode is an electronic amplifying vacuum tube (or ''valve'' in British English) consisting of three electrodes inside an evacuated glass envelope: a heated filament or cathode, a grid, and a plate (anode). Developed from Lee De Forest's 19 ...
amplifier
An amplifier, electronic amplifier or (informally) amp is an electronic device that can increase the magnitude of a signal (a time-varying voltage or current). It may increase the power significantly, or its main effect may be to boost the v ...
s which led to further innovations such as the vacuum tube
tetrode
A tetrode is a vacuum tube (called ''valve'' in British English) having four active electrodes. The four electrodes in order from the centre are: a thermionic cathode, first and second grids and a plate (called ''anode'' in British English). Th ...
.
On the other hand, space charge was useful in some tube applications because it generates a negative
EMF within the tube's envelope, which could be used to create a negative bias on the tube's grid. Grid bias could also be achieved by using an applied grid voltage in addition to the control voltage. This could improve the engineer's control and fidelity of amplification. It allowed to construct ''space charge tubes'' for
car radios that required only 6 or 12 volts anode voltage (typical examples were the 6DR8/EBF83, 6GM8/ECC86, 6DS8/ECH83, 6ES6/EF97 and 6ET6/EF98).
Space charges can also occur within
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
s. For example, when gas near a high voltage electrode begins to undergo
dielectric breakdown
Electrical breakdown or dielectric breakdown is a process that occurs when an electrical insulating material, subjected to a high enough voltage, suddenly becomes an electrical conductor and electric current flows through it. All insulating mate ...
, electrical charges are injected into the region near the electrode, forming space charge regions in the surrounding gas. Space charges can also occur within solid or liquid dielectrics that are stressed by high
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
s. Trapped space charges within solid dielectrics are often a contributing factor leading to dielectric failure within high voltage power cables and capacitors.
Space-charge-limited current
In vacuum (Child's law)
First proposed by
Clement D. Child in 1911, Child's law states that the space-charge-limited current (SCLC) in a plane-parallel vacuum diode varies directly as the three-halves power of the anode voltage
and inversely as the square of the distance ''d'' separating the cathode and the anode.
For electrons, the current density ''J'' (amperes per meter squared) is written:
where
is the anode current and ''S'' the surface area of the anode receiving the current;
is the magnitude of the charge of the electron and
is its mass. The equation is also known as the "three-halves-power law" or the Child–Langmuir law. Child originally derived this equation for the case of atomic ions, which have much smaller ratios of their charge to their mass.
Irving Langmuir
Irving Langmuir (; January 31, 1881 – August 16, 1957) was an American chemist, physicist, and engineer. He was awarded the Nobel Prize in Chemistry in 1932 for his work in surface chemistry.
Langmuir's most famous publication is the 1919 art ...
published the application to electron currents in 1913, and extended it to the case of cylindrical cathodes and anodes.
The equation's validity is subject to the following assumptions:
# Electrons travel ballistically between electrodes (i.e., no scattering).
# In the interelectrode region, the space charge of any ions is negligible.
# The electrons have zero velocity at the cathode surface.
The assumption of no scattering (ballistic transport) is what makes the predictions of Child–Langmuir law different from those of Mott–Gurney law. The latter assumes steady-state drift transport and therefore strong scattering.
In recent years, various models of SCLC current have been revised as reported in two review papers.
In semiconductors
In semiconductors and insulating materials, an electric field causes charged particles, electrons, to reach a specific drift velocity that is parallel to the direction of the field. This is different from the behavior of the free charged particles in a vacuum, in which a field accelerates the particle. The proportionality factor between the magnitudes of the drift velocity,
, and the electric field,
, is called the
mobility
Mobility may refer to:
Social sciences and humanities
* Economic mobility, ability of individuals or families to improve their economic status
* Geographic mobility, the measure of how populations and goods move over time
* Mobilities, a contemp ...
,
:
Drift regime (Mott–Gurney law)
The Child's law behavior of a space-charge-limited current that applies in a vacuum diode doesn't generally apply to a semiconductor/insulator in a single-carrier device, and is replaced by the Mott–Gurney law. For a thin slab of material of thickness
, sandwiched between two selective Ohmic contacts, the electric current density,
, flowing through the slab is given by:
where
is the voltage that has been applied across the slab and
is the
permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
of the solid. The Mott–Gurney law offers some crucial insight into charge-transport across an intrinsic semiconductor,
namely that one should not expect the drift current to increase linearly with the applied voltage, i.e., from
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
, as one would expect from charge-transport across a metal or highly doped semiconductor. Since the only unknown quantity in the Mott–Gurney law is the charge-carrier mobility,
, the equation is commonly used to characterize charge transport in intrinsic semiconductors. Using the Mott–Gurney law for characterizing amorphous semiconductors, along with semiconductors containing defects and/or non-Ohmic contacts, should however be approached with caution as significant deviations both in the magnitude of the current and the power law dependence with respect to the voltage will be observed. In those cases the Mott–Gurney law can not be readily used for characterization, and other equations which can account for defects and/or non-ideal injection should be used instead.
During the derivation of the Mott–Gurney law, one has to make the following assumptions:
# There is only one type of charge carrier present, i.e., only electrons or holes.
# The material has no intrinsic conductivity, but charges are injected into it from one electrode and captured by the other.
# The carrier mobility,
, and the permittivity,
, are constant throughout the sample.
# The current flow is not limited by traps or energetic disorder.
# The current is not predominantly due to doping.
# The electric field at the charge-injecting electrode is zero, meaning that the current is governed by drift only.
Derivation
Consider a crystal of thickness
carrying a current
. Let
be the electric field at a distance
from the surface, and
the number of electrons per unit volume.
Then the current is given has two contributions, one due to drift and the other due to diffusion:
When
is the electrons mobility and
the diffusion coefficient. Laplace's equation gives for the field:
Hence, eliminating
, we have:
After integrating, making use of the
Einstein relation and neglecting the
term we obtain for the electric field:
where
is a constant. We may neglect the
term because we are supposing that
and
.
Since, at
,
, we have:
It follows that the potential drop across the crystal is:
Making use of () and () we can write
in terms of
. For small
,
is small and
, so that:
Thus the current increases as the square of
. For large
,
and we obtain:
As an application example, the steady-state space-charge-limited current across a piece of intrinsic silicon with a charge-carrier mobility of 1500 cm
2/V-s, a relative dielectric constant of 11.9, an area of 10
−8 cm
2 and a thickness of 10
−4 cm can be calculated by a
online calculatorto be 126.4 μA at 3 V. Note that in order for this calculation to be accurate, one must assume all the points listed above.
In the case where the electron/hole transport is limited by trap states in the form of exponential tails extending from the conduction/valence band edges,
the drift current density is given by the Mark-Helfrich equation,
where
is the
elementary charge
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
,
with
being the thermal energy,
is the effective
density of states
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
of the charge carrier type in the semiconductor, i.e., either
or
, and
is the trap density.
Low voltage regime
In the case where a very small applied bias is applied across the single-carrier device, the current is given by:
Note that the equation describing the current in the low voltage regime follows the same thickness scaling as the Mott–Gurney law,
, but increases linearly with the applied voltage.
Saturation regimes
When a very large voltage is applied across the semiconductor, the current can transition into a saturation regime.
In the velocity-saturation regime, this equation takes the following form
Note the different dependence of
on
between the Mott–Gurney law and the equation describing the current in the velocity-saturation regime. In the ballistic case (assuming no collisions), the Mott–Gurney equation takes the form of the more familiar Child–Langmuir law.
In the charge-carrier saturation regime, the current through the sample is given by,
where
is the effective
density of states
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
of the charge carrier type in the semiconductor.
Shot noise
Space charge tends to reduce
shot noise
Shot noise or Poisson noise is a type of noise which can be modeled by a Poisson process.
In electronics shot noise originates from the discrete nature of electric charge. Shot noise also occurs in photon counting in optical devices, where shot ...
. Shot noise results from the random arrivals of discrete charge; the statistical variation in the arrivals produces shot noise.
A space charge develops a potential that slows the carriers down. For example, an electron approaching a cloud of other electrons will slow down due to the repulsive force. The slowing carriers also increases the space charge density and resulting potential. In addition, the potential developed by the space charge can reduce the number of carriers emitted.
When the space charge limits the current, the random arrivals of the carriers are smoothed out; the reduced variation results in less shot noise.
See also
*
Thermionic emission
Thermionic emission is the liberation of electrons from an electrode by virtue of its temperature (releasing of energy supplied by heat). This occurs because the thermal energy given to the charge carrier overcomes the work function of the mate ...
*
Vacuum tube
A vacuum tube, electron tube, valve (British usage), or tube (North America), is a device that controls electric current flow in a high vacuum between electrodes to which an electric voltage, potential difference has been applied.
The type kn ...
*
Grid leak
A grid leak detector is an electronic circuit that demodulates an amplitude modulated alternating current and amplifies the recovered modulating voltage. The circuit utilizes the non-linear cathode to control grid conduction characteristic and ...
References
*
*
{{Thermionic valves
Electricity
Theories
Microwave technology
Vacuum tubes
Mass spectrometry
Semiconductors