How would you maximize the energy density of a capacitor?
This is the formula for capacitance:
C = ε A / d
That A/d is “area over distance”. So permittivity ε = C d / A, which means it has units equivalent to farads per meter. Specifically, ε₀ ≈ 8.85 pF/m. A farad is a joule per square volt.
The energy content of a capacitor at a given voltage V is E = CV²/2; its maximum energy capacity is CV²/2 where V is its maximum voltage.
Permittivity, then, is one characteristic of a dielectric that determines the energy capacity of the capacitor. Energy stored scales linearly with permittivity; double the permittivity means double the joules per square volt. The other relevant characteristic is its dielectric strength, which is measured in volts per meter; when the field strength reaches its dielectric strength, you get avalanche breakdown and your capacitor is, usually, destroyed. Energy capacity is quadratic in dielectric strength: twice the dielectric strength means that an otherwise-unchanged capacitor can withstand twice the voltage, and thus contain four times the energy.
The maximum voltage is V = d S, where S (a variable name I just made up) is the dielectric strength (or “breakdown field” or “breakdown voltage”) of the dielectric.
So the energy capacity of a capacitor is proportional to the square of the dielectric strength and of the permittivity of its dielectric. What else does it depend on?
Well, clearly it depends on the area of the plates and the separation between them, since those are also factors in the capacitance. It would seem that you can make your capacitor arbitrarily large in capacitance by making its plates bigger and closer together, and indeed real-world capacitors range over 12 orders of magnitude of capacitance, of which almost all is due to variations in these factors. You eventually run into a limit in capacitance once your dielectric layer is only a few atoms thick, as in double layer capacitors.
But, holding the dielectric volume constant, the area of the plates and the separation between them doesn’t affect the energy capacity at all!
Consider how the energy capacity varies with plate area and plate separation.
As I said before, the maximum energy capacity is E = CV²/2, where V = d s. So this works out to
E = d² S² ε A / 2d = d A S² ε / 2
Now d A is just the volume of the dielectric! So for a given volume of dielectric, it doesn’t matter whether you have a high-voltage, low-capacitance capacitor or a low-voltage, high-capacitance one; the energy stored is the same. And S² ε / 2 gives you the energy per unit volume!
That means that, for example, air has a dielectric energy capacity of about 40 joules per cubic meter (½(3 MV/m)² · ε₀/2), while glass, with a relative permittivity of 4.7 and dielectric strength of about 10 MV/m, can hold ½(10 MV/m)² · 4.7 · εₒ ≈ 2000 joules per cubic meter. Tantalum pentoxide has not only a remarkable relative permittivity of about 25, but also a dielectric strength of 400 MV/m, giving about 18 MJ/m³. (And that’s half of why tantalum capacitors explode when you overload them. The other half is that most tantalum capacitors are made of an explosive mixture of manganese dioxide and tantalum.)
Diamond’s dielectric strength is supposedly 20 MV/cm, or 2000 MV/m, and so even at its lower relative permittivity of 5.7, diamond-dielectric capacitors should have a higher energy density of 100 MJ/m³. Diamond dielectrics are not in current use, perhaps because as a semiconductor, even slight impurities give it unacceptably high conductance. (Or perhaps even without them? I’m not sure.)
More typical dielectrics include PZT (relative permittivity 300–5000, dielectric strength 10–25 MV/m), mica (relative permittivity 3-6, dielectric strength 118 MV/m), and fused quartz (relative permittivity 3.75, dielectric strength 30 MV/m).
Summarizing:
| material | relative | dielectric | energy |
| | permittivity | strength | density |
| | | (MV/m) | (J/ℓ) |
| air | 1 | 3 | 0.04 |
| glass | 4.7 | 10 | 2 |
| fused quartz | 3.75 | 30 | 15 |
| PZT (low) | 300 | 10 | 130 |
| mica | 3–6 | 118 | 250 |
| PZT (high) | 5000 | 25 | 14000 |
| tantalum oxide | 25 | 400 | 18000 |
| diamond | 5.7 | 2000 | 100 000 |
PZT is a little weird because its breakdown behavior is very time-dependent, and in a temperature-dependent way. Water, which I didn’t include in the table, is even more so; it has a very promising relative permittivity of around 80, but its breakdown voltage goes to zero as time goes to infinity — eventually, apparently, a streamer will form through the water and discharge your capacitor. There are nevertheless water-dielectric capacitors in use for special purposes such as particle accelerators.
It’s instructive to compare the capacitive energy densities above with more typical bulk energy storage systems: gasoline is 36 MJ/ℓ, lithium batteries are 4.3 MJ/ℓ, and lead-acid batteries are 340 kJ/ℓ. The above table suggests that diamond-dielectric capacitors might be in the ballpark of lead-acid batteries for energy storage.
The bottom line is that, among the solid capacitors commercially available today, tantalum capacitors offer an order of magnitude higher energy density for long-term energy storage.