Nanotechnology

Scaling of Electromagnetic Systems

K. Eric Drexler, “Nanosystems”, (John Wiley and Sons, NY, 1992), Chap. 2, has suggested a number of macroscopically-based scaling laws for electromagnetism some of which are summarized in the attached table. Again, these results assume particular models of behavior with length scaling that may or may not apply at the nm length scales.

Table 1: Electromagnetic Scaling Laws

All of these relationships imply that we can legitimately scale to the atomic level using continuum laws. This is unlikely to be true given that on this scale given the distribution of electrical charges (electrons, and their absence - known as “holes”) is discrete and that quantum mechanical, not classical, laws will probably apply.

Voltage is related to its electric field, E (which, in turn depends on the local distribution of electrical charges). Therefore you should be alerted to the fact that macroscopic descriptions of electromagnetics if extrapolated to the nanoscale may lead to erroneous local pictures of physical reality.

The electric field E is defined as volts/distance so that, for a fixed E (or more physically, to its underlying distribution of charges), V is proportional to L. A field strength of E = 10⁹ V/m (as encountered in an insulator) yields 1 volt over 1 nm distance.

Resistivity of a material relates to resistance as R = σ L/A where σ is the resistivity and L distance, and A is the cross-section through which the current is flowing. For a constant σ, resistance scales as L/L² or electrical resistance proportional to 1/L. If a 1 cm cubic resistor has a 1-ohm resistance, a 1 nm cube of it would have a calculated 10⁷ ohms resistance (of course, in reality a cube this small would not obey this scaling law due to the discrete nature of the material at this scale).

We can now scale electrical current by Ohm’s law, I = V/R so that electrical current is proportional to L². However if current density, j = current/area, is assumed invariant, the current density is proportional to L⁰. According to Drexler, as of 1980, the maximum possible current density through a conductor^[1] was about 10¹⁰ A/m². For a 1 nm² cross sectional conductor we can thus pass up to 10 nA. The voltage drop across our 1 nm cube with 10⁷ ohms is thus 0. 1 volts and the field strength is 10⁸ V/m.

Electrostatic terms maybe important because we might have to check that they do not conflict with chemical bonding forces. Electric energy density is equal to ˝εE² or ~5 × 10⁸ J/m³ at the maximum practical field strength in a non-conducting liquid or solid of ~ 10¹⁰ V/m (or 10 V/nm). ε is called the permittivity; and, for a vacuum (or essentially air), ε₀ = 8.85 × 10^-12 farads/m, and is typically 5 - 80 times that for many liquids or solids. Since E is invariant to length scale change in our model, the electric energy density is also invariant, i.e., it is proportional to L⁰; in the example with E = 10 V/nm, it scales to 5 × 10^-19 J/nm³.

Electrical capacity or capacitance is a vital quantity in electronic chip design. Capacitance introduces electrical delays – which can be beneficial but are quite often undesirable. Capacitance is measured farads, the number of coulombs/volt. Capacitance is the measure of how much charge can be accumulated compared to the voltage required to maintain it. In our current context it is easiest to work from the stored electric energy in a capacitor, which is easily derived as ˝ CV². Thus C scales as electric energy/voltage², i.e., L³/L² so that capacitance is proportional to L.

A parallel plate vacuum capacitor with 1 nm square plates separated by 1 nm has an apparent^[2] capacitance of about 10^-20 farads (common capacitors will range from μF (micro or 10^-6) to pF (pico or 10^-12). While 10^-20 farads is a very small capacitance, it may be important in modern electronics. There capacitors may be switched at a rate of a few gigahertz (10 GHz is a modest extrapolation for today’s machines) and thus at the typical applied 5 volts, a power density of ˝ × 10^-20 × 5² × 10¹⁰ [F][V]²[1/s] = 1.3 × 10^-9 W/nm³ is developed if switching rate is continuous. This is a troublingly large amount to the electronic chipmakers. (Most nuclear reactors have an internal heating rate of about 100 kW/liter or 10^-19 W/nm³ or ten billion times less than electronic chips at this worst-case condition!).

Magnetic fields scale as the current to produce them/distance or magnetic fields proportional to L. At 1 nm from a conductor carrying 10 nA, the field strength is ~ 2 × 10^-6 tesla. The magnetic force between two such conductors is proportional to area × magnetic field² or magnetic force proportional to L⁴, the most powerful dependence on scaling we have yet seen. Two conductors carrying 10 nA and separated by 1 nm interact with a force of 2 × 10^-23 N, i.e., truly infinitesimal, and is about 10^-14 of the strength of typical covalent chemical bonds and about 10^-11 of electrostatic force terms.

Magnetic energy is even more dramatic: magnetic energy µ volume × (magnetic field)² or magnetic energy proportional to L⁵. Magnetic energy density is conveniently expressed in terms of “magnetic induction”, symbol B and measured in “tesla”. The magnetic energy density for a non-ferro magnetic material is then B²/2μ₀ where μ₀ is the “permeability”. With these SI units, the result is in J/m³.

The Take Home: at the nanoscale you can nearly always forget about magnetic forces or potentials and usually also can forget about electrical forces or potentials.