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2.4. Scaling of classical electromagnetic systems

2.4.1. Basic assumptions

In considering the scaling of electromagnetic systems, it is convenient to assume that electrostatic field strengths (and hence electrostatic stresses) are independent of scale. With this assumption, the above constant-stress, constant-speed scaling laws for mechanical systems continue to hold for electromechanical systems, so long as magnetic forces are neglected. The onset of strong field-emission currents from conductors limits the electrostatic field strength permissible at the negative electrode of a nanoscale system; values of ~109 V/m can readily be tolerated (Section 11.6.2).

2.4.2. Major corrections

Chapter 11 describes several nanometer scale electromechanical systems, requiring consideration of the electrical conductivity of fine wires and of insulating layers thin enough to make tunneling a significant mechanism of electron transport. These phenomena are sometimes (within an expanding range of conditions) understood well enough to permit design calculations.

Corrections to classical continuum models are more important in electromagnetic systems than in mechanical systems: quantum effects, for example, become dominant and at small scales can render classical continuum models useless even as crude approximations. Electromagnetic systems on a nanometer scale commonly have extremely high frequencies, yielding large values of /kT300. Molecules undergoing electronic transitions typically absorb and emit light in the visible to ultraviolet range, rather than the infrared range characteristic of thermal excitation at room temperature. The mass of an electron is less than 10–3 that of the lightest atom, hence for comparable confining energy barriers, electron °wave functions are more diffuse and permit longer-range tunneling. At high frequencies, the inertial effects of electron mass become significant, but these are neglected in the usual macroscopic expressions for electrical circuits. Accordingly, many of the following classical continuum scaling relationships fail in nanoscale systems. The assumption of scale-independent electrostatic field strengths itself fails in the opposite direction, when scaling up from the nanoscale to the macroscale: the resulting large voltages introduce additional modes of electrical breakdown. In small structures, the discrete size of the electronic charge unit, ~ 1.6×10 –19C, disrupts the smooth scaling of classical electrostatic relationships (Section 11.7.2c).

2.4.3. Magnitudes and scaling: steady-state systems

Given a scale-invariant electrostatic field strength,

  voltage electrostatic field length L (2.19)

At a field strength of 109 V/m, a one nanometer distance yields a 1 V potential difference. A scale-invariant field strength implies a force proportional to area,

  electrostatic force area (electrostatic field)2 L2 (2.20)

and a 1 V/nm field between two charged surfaces yields an electrostatic force of ~ 0.0044 nN/nm2.

Assuming a constant resistivity,

  resistanse L-1 (2.21)

and a cubic nanometer block with the resistivity of copper would have a resistance of ~ 17 . This yields an expression for the scaling of currents,

  ohmic current L2 (2.22)

which leaves current density constant. In present microelectronics work, current densities in aluminum interconnections are limited to <1010 A /m2 or less by electromigration, which redistributes metal atoms and eventually interrupts -circuit continuity (Mead and Conway, 1980). This current density equals 10 nA/nm2 (as discussed in Section 11.6.1b, however, present electromigration limits are unlikely to apply to well-designed eutactic conductors).

For field emission into free space, current density depends on surface properties and the electrostatic field intensity, hence

  field emission current area L2 (2.23)

and field emission currents scale with ohmic currents. Where surfaces are close enough together for tunneling to occur from conductor to conductor, rather than from conductor to free space, this scaling relationship breaks down.

With constant field strength, electrostatic energy scales with volume:

  electrostatic energy volume (electrostatic field)2 L3 (2.24)

A field with a strength of 109 V/m has an energy density of ~ 4.4 maJ per cubic nanometer ( ≈kT300).

Scaling of capacitance follows from the above,

  capacitance L (2.25)

and is independent of assumptions regarding field strength. The calculated capacitance per square nanometer of a vacuum capacitor with parallel plates separated by 1 nm is ~ 9×10–21 F; note, however, that electron tunneling causes substantial conduction through an insulating layer this thin.

In electromechanical systems dominated by electrostatic forces,

  electrostatic power electrostatic force speed L2 (2.26)


  electrostatic power density L-1 (2.27)

These scaling laws are identical to those for mechanical power and power density. Like them, they suggest high power densities for small devices (see Section 11.7).

The power density caused by resistive losses scales differently, given the above current density:

  resistance power density (current density)2 constant (2.28)

The current density needed to power an electrostatic motor, however, scales differently from that derived from a constant-field scaling analysis. In an electrostatic motor, surfaces are charged and discharged with a certain frequency, hence

  motor current density frequency L-1 (2.29)

and the resistive power losses climb sharply with decreasing scale:

  motor resistive power density (motor current density)2 L-2 (2.30)


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