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 usually 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, but 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:

1. Combination of the current density and spatially inhomogeneous resistance
2. Ionization of species within the dielectric to form heterocharge
3. Charge injection from electrodes and from a stress enhancement
4. 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.^[1][2]

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:

$J = (1-\tilde{r})A_0T^2\exp\left(\frac{-\phi}{kT}\right)$

where

$A_0 = \frac{4\pi e m_e k^2}{h^3} \approx 1.2 \times 10^6$ A m⁻² K⁻²

e = elementary positive charge (i.e., magnitude of electron charge),

m_e = electron mass,

k = Boltzmann's constant = 1.38 x 10⁻²³J/K,

h = Planck's constant = 6.62 x 10⁻³⁴ J s,

φ = work function of the cathode,

ř = mean electron reflection coefficient.

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 3000 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.

Child's Law

Graph showing Child-Langmuir Law. S and d are constant and equal to 1.

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 V_a and inversely as the square of the distance d separating the cathode and the anode.^[3]

For electrons, the current density J (amperes per meter squared) is written:

$J = \frac{ I_a }{ S } =\frac{4 \epsilon_0}{9}\sqrt{2 e / m_e} \frac{V_a^{3/2}}{d^2}$.

where I_a is the anode current and S the anode surface inner area; $e$ is the magnitude of the charge of the electron and $m_e$ is its mass. The equation is also known as the "Three-halves Power" Law and as 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:

1. Electrons travel ballistically between electrodes (i.e., no scattering).
2. In the interelectrode region, the space charge of any ions is negligible.
3. 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.

Mott–Gurney law

In semiconductors and insulating materials, an electric field causes charged particles like electrons to reach a specific "drift velocity" that is parallel to the direction of the field. It 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 $v$ and the electric field $\mathcal E$ is called the mobility $\mu$:

$v = \mu \mathcal{E}$

The Child's Law behavior of a space charge limited current that applies in a vacuum diode doesn't generally apply in a solid, and is replaced by the Mott-Gurney form. For a thin slab of material of thickness $L$, the electric current density $J$ flowing through the slab is:

$J=\frac{9{\epsilon}{\mu}{V_a}^{2}}{8{L}^3}$ ,

where $V_a$ is the voltage that has been applied across the slab and $\epsilon$ is the dielectric constant of the solid.^[4][5]

In the velocity-saturation regime, this equation takes the following form

$J=\frac{2{\epsilon}{v}{V_a}}{{L}^2}$

Note the different dependence of $J$ on $V_a$ in each of the two cases. Interestingly, in the ballistic case (assuming no collisions), the Mott-Gurney equation takes the form of the more familiar Child-Langmuir law.

It should be noted that the above derivations make the following assumptions:

1. There is only one type of charge carrier present.
2. The material has no intrinsic conductivity, but charges are injected into it from one electrode and captured by the other.
3. The carrier mobility $\mu$ and the dielectric permittivity $\epsilon$ are constant throughout the sample.
4. The electric field at the charge-injecting cathode is zero.

As an application example, the steady-state space-charge-limited current across a piece of silicon with a charge carrier mobility of 1500 cm²/V-s, a dielectric constant of 11.9, an area of 10⁻⁸cm² and a thickness of 10⁻⁴cm can be calculated by an on line calculator as 126.4mA at voltage 3V.

Shot noise

Space charge tends to reduce shot noise.^[6] Shot noise results from the random arrivals of discrete charge; the statistical variation in the arrivals produces shot noise.^[7] 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.^[8] When the space charge limits the current, the random arrivals of the carriers are smoothed out; the reduced variation results in less shot noise.^[7]

See also

• Thermionic emission
• Vacuum tube
• Grid leak

References

1. Moreau, E.; Mayoux, C.; Laurent, C.; Boudet, A. (February 1993), "The Structural Characteristics of Water Trees in Power Cables and Laboratory Specimens", IEEE Transactions on Electrical Insulation (IEEE) 28 (1): 54–64, doi:10.1109/14.192240, ISSN 0018-9367
2. Hennuy, Blandine; Marginet, Joachim; François, Alain; Platbrood, Gérard; Tits, Yvan; De Clerck, Quentin (June 2009), Water Trees in Medium Voltage XLPE Cables: Very Short Time Accelerated Ageing Tests (PDF), Prague, Paper 1060
3. Child, C. D. (1 May 1911). "Discharge From Hot CaO". Physical Review. Series I 32 (5): 492–511. Bibcode:1911PhRvI..32..492C. doi:10.1103/PhysRevSeriesI.32.492.
4. Mott, Nevill F.; Gurney, R. W. (1940). Electronic Processes in Ionic Crystals, 1st ed. Oxford University Press.
5. Murgatroyd, P. N. J. (1970). "Theory of space-charge-limited current enhanced by Frenkel effect". Phys. D: Appl. Phys. 3: 151. Bibcode:1970JPhD....3..151M. doi:10.1088/0022-3727/3/2/308.
6. Terman, Frederick Emmons (1943), Radio Engineers' Handbook (first ed.), New York: McGraw-Hill, pp. 286–294
7. Terman 1943, pp. 292–293
8. Terman 1943, pp. 286–287

• Starr, A. T. (1958), Telecommunications (second ed.), London: Sir Isaac Pitman & Sons, Ltd
• Coelho, R. (1979), Physics of Dielectrics for the Engineer, Amsterdam: Elsevier Scientific Pub. Co.