Space charge

Space charge is a concept in which excess electric charge is treated as a continuum 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 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 media (including vacuum) 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 object when it is heated to incandescence in a vacuum. This effect was first observed by Thomas Edison in light bulb filaments, where it is sometimes called the Edison effect. Space charge is a significant phenomenon in many vacuum and solid-state electronic devices.
Cause.
Physical explanation.
When a metal object is placed in a vacuum and is heated to incandescence, the energy is sufficient to cause electrons to "boil" away from the surface atoms and surround the metal object in a cloud of free electrons. This is called thermionic emission. 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:
It has been suggested that in alternating current (AC) most carriers injected at electrodes 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 (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 "vacuum" has a pressure of 10−6 mmHg or less, the main vehicle of conduction is electrons. The emission current density "(J") from the cathode, as a function of its thermodynamic temperature "T", in the absence of space-charge, is given by Richardson's law:
where
The reflection coefficient can be as low as 0.105 but is usually near 0.5. For tungsten, (1 - ř)A0 = 0.6 to 1.0 × 106 A m−2 K−2, and "φ" = 4.52 eV. At 2500 °C, the emission is 28207 A/m2.
The emission current as given above is many times greater than that normally collected by the electrodes, except in some pulsed valves such as the cavity magnetron. Most of the electrons emitted by the cathode are driven back to it by the repulsion of the cloud 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 tubes. This has at times made life harder or easier for electrical engineers who used tubes in their designs. For example, space charge significantly limited the practical application of triode amplifiers which led to further innovations such as the vacuum tube tetrode.
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 dielectrics. For example, when gas near a high voltage electrode begins to undergo dielectric breakdown, 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 fields. 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 formula_3 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 formula_5 is the anode current and "S" the surface area of the anode receiving the current; formula_6 is the magnitude of the charge of the electron and formula_7 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 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:
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 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, formula_8, and the electric field, formula_9, is called the mobility, formula_10:
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 formula_12, sandwiched between two selective Ohmic contacts, the electric current density, formula_13, flowing through the slab is given by:
It follows that the potential drop across the crystal is:
Making use of formula_15 and formula_17 we can write formula_13 in terms of formula_3. For small formula_3, formula_13 is small and formula_24, so that:
the drift current density is given by the Mark-Helfrich equation,
where formula_27 is the elementary charge, formula_28 with formula_29 being the thermal energy, formula_30 is the effective density of states of the charge carrier type in the semiconductor, i.e., either formula_31 or formula_32, and formula_33 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: