Space charge
Space charge is an interpretation of a collection of electric charges 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 near "vacuum" has a pressure of 10 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: formula_1 where The reflection coefficient can be as low as 0.105 but is usually near 0.5. For tungsten, (1 − "ř")"A" = 0.6 to 1.0 × 10 A mK, and "φ  = 4.52 eV. At 2500 °C, the emission is 28207 A/m. 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: formula_4 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 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, formula_8, and the electric field, formula_9, is called the mobility, formula_10: formula_11 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: formula_14 where formula_3 is the voltage that  has been applied across the slab and formula_16 is the permittivity 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, 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, formula_10, 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: Derivation Consider a crystal of thickness formula_12 carrying a current formula_13. Let formula_22 be the electric  field at a distance formula_23 from the surface, and formula_24 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: formula_25 When formula_26 is the electrons mobility and formula_27 the diffusion coefficient. Laplace's equation gives for the field: formula_28 Hence, eliminating formula_29, we have: formula_30 After integrating, making use of the Einstein relation and neglecting the formula_31 term we obtain for the electric field: formula_32 where formula_33 is a constant. We may neglect the formula_31 term because we are supposing that formula_35 and formula_36 . Since, at formula_37, formula_38, we have: It follows that the potential drop across the crystal is: Making use of () and () we can write formula_13 in terms of formula_3. For small formula_3, formula_13 is small and formula_43, so that: Thus the current increases as the square of formula_3. For large formula_3, formula_46 and we obtain: formula_47 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/V-s, a dielectric constant of 11.9, an area of 10 cm and a thickness of 10 cm can be  calculated by an online calculator to 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, formula_48 the drift current density is given by the Mark-Helfrich equation, formula_49 where formula_50 is the elementary charge, formula_51 with formula_52 being the thermal energy, formula_53 is the effective density of states of the charge carrier type in the semiconductor, i.e., either formula_54  or formula_55, and formula_56 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: formula_57 Note that the equation describing the current in the low voltage regime follows the same thickness scaling as the Mott–Gurney law, formula_58, 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 formula_59 Note the different dependence of  formula_13 on formula_3 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, formula_62 where formula_53 is the effective density of states of the charge carrier type in the semiconductor. Shot noise. Space charge tends to reduce shot noise. 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.