The maximum energy density of a capacitor is comparatively small due to large leak currents that thermally degrade the system. We study a three-plate system with nanometer gaps between the plates. Two negatively charged plates (cathodes) sandwich a thin, positively charged inner plate (anode). The dynamics of the electrons, in gaps of such a capacitor, are quantized, even at room temperature, because the dimensions are so small. Under strong fields, eigenstates between the electrodes fill and reduce the leak current between the anode and cathode. We show that the self-discharge time for a three-plate nanocapacitor can be significantly longer than a comparable two-plate nanocapacitor, thus increasing maximum energy density of such a nanocapacitor.

## I. INTRODUCTION

Capacitors have long been proposed as an efficient means of energy storage. Electrochemical batteries are limited to a small number of charging cycles due to their volatile chemicals and low power densities due to the relatively slow nature of ion diffusion.^{1} Supercapacitors, a proposed alternative to electrochemical batteries, are stable across a large number of charging cycles and have much greater power density; however, the energy densities of capacitors are limited by “leakage currents.”^{2} Both electrochemical batteries and supercapacitors are also only operational under a narrow temperature range.

In this paper, we present theoretical calculations, which show that the presence of intermediate eigenstates in a three-plate capacitor can limit the leak current across the capacitor, thus increasing the self-discharge time and maximum energy density as compared to a traditional two-plate capacitor.^{3} In Fig. 1, the potential energy of the proposed three-plate nanocapacitor is plotted. The middle plate is a mono-layer or bi-layer of graphene or another almost 2D layer of a conducting material with a thickness of a few atomic layers. The cathodes are made of the same material, such as a 10-nm layer of tungsten or aluminum. The cathodes have the same voltage. In the gap is a vacuum or an insulating dielectric, such as a 10-nm layer of aluminum oxide.

In such nanocapacitors, eigenstates form between the cathodes at energies above the Fermi level of the anode. We assume that the anode is so thin that tunneling makes it possible for eigenstates to extend through the anode. When an electron enters the gap region, it occupies a state where its wave function is on both sides of the anode. Due to the overlap of the wave function of electrons in the gap with unoccupied states in the anode, the anode will eventually capture the electron. The electrons inside the gap are called space charges.

As the nanocapacitor is charged, the eigenstates in the gap become filled, and by the Pauli exclusion principle, the primary mechanism for current flow in the system eventually transitions between these filled states and the anode via spontaneous emission.^{4} When all eigenstates are filled, the current is almost completely blocked and the dielectric strength is much larger. This is called a Coulomb blockade. Three plate nanocapacitors with a Coulomb blockade are expected to be operational under a large range of temperatures and have high energy and power densities.

For gap sizes of 10 nm or less, the energy difference between ground and the first excited state *E*_{2,1} = (*E*_{2} − *E*_{1}) is about 20 times larger than *E _{t}* =

*k*at room temperature. In this case, a quantum model is more accurate than a thermodynamic model. If the gap size is increased or the temperature is increased, the thermodynamic approach is better. At lower temperatures all the way down to 0

_{b}T*K, the quantum approach is superior.*

## II. FOWLER-NORDHEIM TUNNELING

Flow of current across a metal barrier in a strong electric field is best described by Fowler-Nordheim tunneling.^{5} Fowler-Nordheim tunneling assumes free electrons in a metal are incident on a (roughly) triangular barrier. The current across the barrier is then proportional to the tunneling probability across the barrier. Forbes^{6} and others^{7–9} derived the Fowler-Nordheim current, and the general analytic form of the current is

where *ϕ* is the work function (or height) of the barrier, *F* is magnitude of the electric field, and *α*_{1} and *α*_{2} are the Fowler-Nordheim constants defined as

The dimensionless parameters *ν* and *γ* describe the deviation of the barrier from a perfectly triangular barrier and can be determined numerically for other potentials.^{6,10} They are typically close to 1.

## III. DENSITY OF STATES

In the three plate nanocapacitor, we assume that the conduction band of the anode is separated by a band gap of Δ*ϵ* from the eigenstates in between the cathode plates. Assuming the inner plate is thin compared to the separation of the cathode plates, the potential in between the cathode plates is approximately V-shaped

The energy levels for the eigenstates between the cathode plates can then be determined from the Wentzel–Kramers–Brillouin (WKB) approximation

where $ a \u2261 \u03f5 n z e \u2009 F $ are the classical turning points. Evaluating the integral and solving for the energy levels yields

If the plates of the nanocapacitor are macroscopic in the other two dimensions, each of the energy levels in Eq. (6) will introduce a subband^{8} with a density of states of $ \rho 2 D = m \pi \u2009 \u210f 2 $. Thus, the total density of states in the medium between the plates is

The summation over the $ \u03f5 n z $ energy levels can be approximated by multiplying by the average number of subbands, i.e., solving Eq. (6) for *n _{z}*

The average number of subbands at energy *ϵ* is then

and thus, the density of states in the void between the cathode plates is approximately

with the factor of two coming from the two spin states. Assuming all states are filled to the Fermi level of the system, *ϵ _{f}*, we integrate to find the concentration of states in the medium between the cathodes

The Fermi energy of the system is related to the voltage across the capacitor and the work function of the cathode

where *d* is the separation distance between the cathode plate and the anode plate.

## IV. RATE OF SPONTANEOUS EMISSION

The eigenstates in the anode of the nanocapacitor act similarly to the band of unoccupied states in the valence band of a semiconductor material. The primary mechanism for transitions from the band of states between the cathodes to the band in the anode is spontaneous emission, where a photon is released as an electron drops from the conduction band to the valence band.^{11} Detailed quantum mechanical discussions of spontaneous emission can be found in Bebb and Williams,^{12} Agrawal and Dutta,^{13} Dutta,^{14} Thompson,^{15} and others. At lowest order and at equilibrium, the rate of spontaneous emission is proportional to the concentration of filled states in the conduction band and the concentration of unoccupied states in the valence band. Thus, we assume the rate of spontaneous emission, Γ, is

where *n _{c}* and

*n*are the concentration of electrons and holes in the conduction band and valence band, respectively. Using Eq. (11) for

_{v}*n*and multiplying by

_{c}*e*, we find the spontaneous emission current

The constant *B* is commonly referred to as the bimolecular recombination coefficient, and in semiconductors has values on the order of ∼10^{−10} cm^{2} s^{−1} to ∼10^{−14} cm^{2} s^{−1}. Semiclassical models, such as the Van Roosbroeck-Shockley model,^{16} provide some theoretical and analytical framework for determining the value of the bimolecular recombination coefficient. Magnus and Schoenmaker^{4} and others^{17} have also calculated the leak current across multi-layer gates with full quantum mechanical models.

## V. SELF-DISCHARGE TIME

If a charged capacitor is disconnected from a voltage bias, the leakage current across the capacitor will act to discharge the capacitor. We define the self-discharge time to be the time it takes for the voltage across the capacitor to drop to one half its initial value. Assuming the fields are linear and the self-discharge process is quasi-static, classical electrodynamics yields

In a two-plate nanocapacitor, we assume the leak current is primarily due to Fowler-Nordheim tunneling.^{3} In the three-plate nanocapacitor, when the rate of spontaneous emission is slower than the rate of Fowler-Nordheim tunneling, the leak current will be limited. Figure 2 plots the Fowler-Nordheim current and the spontaneous emission current for the three-plate system. The measured leak current across the capacitor, *J*^{leak}(*F*), will be the smaller of the two current mechanisms.

For a two-plate capacitor starting from an initial field strength of *F*_{0}, and if we set *J*^{leak}(*F*) = *J*^{F-N}(*F*), Eq. (15) can be solved in closed form to find the self-discharge time, *τ*^{F-N}

For a three-plate capacitor, there are two regimes of leak current: Fowler-Nordheim tunneling at small field strengths when the eigenstates between the cathodes do not fill completely and spontaneous emission at large field strengths. In the Fowler-Nordheim tunneling regime, the self-discharge time is the same as Eq. (16) above. The self-discharge time in the spontaneous emission regime can be calculated from the spontaneous emission current in Eq. (14). Writing the spontaneous emission current as a function of the field strength *F*

where the parameter *β* is (with *d* in nanometers)

If the leak current *J*^{leak}(*F*) equals the spontaneous emission current *J*^{s-e} in Eq. (17), the differential equation Eq. (15) has a closed form solution. With this solution, we find

In Fig. 3, the self-discharge times are plotted as a function of initial field strength.

## VI. CONCLUSION

We show that a quantized charge limits the leak current across a three-plate nanocapacitor. At large voltages, for a two-plate nanocapacitor, *J*^{leak} ∼ *V*^{2}; however, for a three-plate nanocapacitor, *J*^{leak} ∼ *V*^{3∕2}. This reduction in leak current increases the self-discharge time and enables larger energy densities. Breakdown occurs when the leak current is roughly 10^{5} A/cm^{2}. Therefore, the dielectric strength of the capacitor is larger if the leak current is smaller. The maximum energy density is proportional to the square of the dielectric strength. Therefore, a reduction of the leak current usually results in an increase of the maximum energy density.

There is some experimental work on two-plate nanocapacitors. Recently, it was discovered that two-plate nanocapacitors have space charges^{18} and high energy densities.^{19} Earlier it was found that two-plate capacitors with a smaller gap size have larger energy densities.^{3} The experimental design of a three-plate nanocapacitor could be very similar to a two-plate nanocapacitor, except that there is a third plate and that the center plate should be very thin.

Currently, the maximum energy density of capacitors is still significantly below the energy density of Li-ion batteries, but a trend analysis by Magee^{20} predicts that capacitors will reach and exceed the energy density of Li-ion batteries. Three-plate capacitors are certainly a step in this direction.

## ACKNOWLEDGMENTS

This work was funded in part by the Office of Naval Research Grant No. N00014-15-1-2397, Air Force Research Laboratory Grant No. AF FA9453-14-1-0247, and a NASA subcontract of the contract NASA-STIR NNX 16 CM29P. We thank Dr. Charles P. Marsh of the Engineer Research and Development Center, Construction Engineering Research Laboratory (Champaign, IL), for fruitful discussions and supporting work under No. BPO W9132T-14-A-0001.