Figure 1 is a scatter diagram illustrating the capabilities of several diodes as a function of signal energy versus frequency [6].

Given the suitability of MIM-class diodes for high frequency, high sensitivity rectification, this kind of diode has been selected as a vehicle for the following analysis with the goal of optimally rectifying the Johnson-Nyquist noise.

The emergence of time-reversal asymmetry is due to a dynamic change in configuration associated with a state variable (thickness of depletion zone, disk position) modulated by the polarity and amplitude of the input signal. This state variable affects the resistance and bandwidth of the device. A change in this state variable requires a reconfiguration energy provided by the signal and is limited by the bandwidth of the device.

Figure 1. Performance map for several diode architectures. The diagonal line maps the peak spectral energy of the Johnson-Nyquist noise: E_Peak = hf_Peak = n_Peakk_BT, where the peak power frequencies and signal levels are located as a function of temperature. MIM-class diodes with one or more insulating layers, overlap this high frequency, low amplitude region. They include PN, Pin, Schottky, Tunnel, and MIM-class diodes (which include MIMs, MIIMs, MIIIMs and other multilayers variants). The line at the bottom right corner represents the frequency range and signal level of the Johnson-Nyquist noise as described by equation E_Peak = hf_Peak, where E_Peak is the energy of a plasmon at the peak spectrum frequency f_Peak (This topic is discussed in Section 8). As the diagram shows, the MIM-class diodes are the only ones capable of operating in that region. They stand out for their performance at terahertz and even optical frequencies.

MIM diode models by Tsu-Esaki [7], Simmons [8][9], Grover [10] and others [11]-[13] do include a state variable in the form of the changing Fermi distribution across the gap, but they do not account for the configuration energy and bandwidth required to change this distribution. This omission renders them instantaneous and inappropriate for modeling the rectification of Johnson-Nyquist fluctuations. As far as this author knows, no diode model exists which is suitable for thermal noise rectification. It should include:

1.A reconfiguration energy φ_Configsupplied by the signal, to modify the state variable.

2.Atime constant linked to the change in the state variable.

For the sake of simplicity this paper discusses the simplest MIM diode which includes a single insulating layer. In such a diode, the state variable is associated with the movement of charges that affect the occupancy and vacancy distribution on either side of the gap and change its effective width (in analogy to the change in thickness of the depletion zone in a PN diode). Figure 2 shows a MIM diode fabricated with:

1. On the left: a high density of occupied states and low density of vacant states, and

2. On the right: a low density of occupied states and high density of vacant states.

The effective width of the gap varies depending on the polarity of the applied voltage V:

(a)

(b)

Figure 2. The sides of the gap can be designed to have high occupancy and low vacancy on the left, and low occupancy and high vacancy on the right. a) In forward mode, the charge layers on the surfaces are thin, causing the gap to have a thin effective width. b) In reverse mode they are thick, causing the gap to have a thick effective width.

1. A positive voltage +V shifts the Fermi levels causing the diode to operate in forward mode with a high occupancy on the left and high vacancy on the right. Therefore, a carrier transiting across the gap, from left to right, originates near the surface of the metal on the left and ends up near the surface on the right. As a result, the gap has a narrow effective width, and its probability of transmission is high.

2. Conversely, a negative voltage −V causes the diode to operate in reverse mode with a low occupancy on the right and low vacancy on the left. Therefore, a carrier transiting from right to left originates far from the surface on the right, and it ends up far from the surface on the left. The gap has a large effective width, and its probability of transmission is low.

Accordingly, tunneling requires traversing three regions: a left one of width d_L, the central gap of width d_G and a right one of width d_R. Therefore, the effective expected gap size is d_eff= d_L + d_G + d_R. Since d_L + d_R is smaller in forward bias, d_eff is also smaller in forward bias. A sub-nanometer change in a one-nanometer gap results in a large difference in its conductivity. Full integration of this concept into existing models is the subject of a future paper. It is briefly discussed in Appendix C.

The change in the configuration of the electronic states isT-Oddsymmetric because it is synchronized with the input signal. It always favors a forward current over a reverse current. Unlike the Seebeck effect which requires a temperature difference, or Ohmic conduction which requires a voltage, this driving mechanism is purely entropic, caused by the synchronicity between the signal and the dynamic configuration of the states. At issue is not whether a diode can rectify a signal, it is how small this signal can be. This underlines the importance of the configuration energy φ_Config: it is commensurate with the signal to be rectified and must be included in the following analysis.

It is assumed that carriers tunnel through the nanometer gap much faster than the average period of a Johnson-Nyquist fluctuation and that, despite the fluctuations,they remain in local equilibrium. When a fluctuation produces a forward voltage bias, carriers maintain their equilibrium by tunneling (reversibly) up the energy barrier across the gap. This process is both isothermal and adiabatic:

1.Carriers remain isothermal in compliance with the second law (a Fermi gas in equilibrium in a field is isothermal) and with the proof of the isothermality of a Fermi gas in AppendixB.

2.Carriers remain adiabatic because direct tunneling implies no collision that changes the carrier’s energy.

Conventional thermodynamics admits one process which is both isothermal and adiabatic. It is called a free expansion. A gas freely expanding in a chamber retains its temperature and internal energy but increases its volumetric entropy.

In contrast to a free expansion, carriers crossing the gap lower their positional entropy, while remaining isothermal and adiabatic. Therefore, their change in positionisthe exact reverse of the free expansion. Let us call it “free compression.” The process is not driven by a temperature difference as in the Seebeck effect, yet it does lower the temperature equally on both sides of the junction:

1. The emitting side of the gap drops in temperature because it loses the most energetic carriers which are more prone to tunneling.

2. The receiving side also drops in temperature because the incoming carriers lose kinetic energy as they tunnel adiabatically against the barrier potential.

Overall, both sides of the gap remain in isothermal equilibrium with respect to each other because tunneling affects them equally. However, the junction, as a whole, experiences a drop in temperature with respect to its surroundings. This temperature difference causes an inflow of heat that replenishes the thermal energy converted to electricity. This temperature difference and heat flow is a Fourier heat conduction effect and not a Carnot thermodynamic requirement. There is no need for a heat sink, but the device must still be colder than the surroundings.

In summary, the thermodynamic cycle includes a free (isothermal) compression of the carriers in the diode, followed by an isothermal expansion in the load. The work function difference Φ defines the T-Odd asymmetry of the diode and provides it with the free energy used in rectifying heat to electrical energy. The potential across the gap acts as a catalyst in the sense that it is not consumed in the heat to work conversion process.

To determine the best area of a diode, one may begin with the observation that the Johnson Nyquist noise is uncorrelated at different points on a surface if these points are spaced further apart than the coherence length of the noise or the wavelength λ_cof a plasmon. To calculateλ_c in the material, one can first observe that in free space, at T = 300K, the wavelength at peak power is,

where the dielectric constant κ is assumed to be unity. However, in a MIM structure with a nanometer-scale insulator (1 - 5 nm), the surface plasmon polaritons (SPPs) are strongly compressed by the effective mode index which is about n_eff ≈ 30 for copper at that frequency [20]-[26]. Therefore, the plasmon wavelength for the copper example along the surface is about:

The area of a coherent pixel is therefore:

One could imagine a large flat MIM diode subdivided into little pixels of area A = λ c 2 , and thickness λ_p, with each pixel connected in parallel between two commonly shared conductive plates, each operating as a small diode, each operating uncorrelated and independently of each other, each rectifying its own local correlated noise, and each feeding its DC outputs in parallel to the plates. Figure 5 shows a stack of MIM diodes.

Figure 5. Large area MIM-class diodes can be stacked in layers forming a multiplicity of independently operating diodes, the volume of each diode defined by the correlation area of a surface plasmon and the penetration depth of the peak power frequency.

This visualization of a diode as an assembly of distinct pixels is useful as an explanatory tool but in practice there is no need for explicitlydividing the diode because separate correlation areas are formed spontaneously by the signal itself. The lithographic complexity of fabricating individual diodes is thereby eliminated. In principle, there is no limit to the size of the area of the diode.

Conductance through the gap can be promoted by favoring the transmission of electrons. Since the dielectric constant of the gap is lower than that of the metal, this can be achieved by setting the thickness of the gap to one half the de Broglie wavelength of an electron to cancel destructive interference due to reflections as in a Fabry-Perot resonator. Setting the width of the gap d_G to half the de Broglie wavelength, yields the thickness for maximum transmission:

Using m_eff= m₀ at T = 300K, provides an upper bound for the optimal thickness of the gap³:

This thickness is an upper bound because:

1. The effective mass of the electrons is assumed to be m₀ and

2. The change in effective gap thickness due to variations in occupancy and vacancy states as discussed in Section 4.2 and shown in Figure 2 is not included in d_G.

Fabricating a device with this dimension is challenging but can be achievable with tailored self- assembling monolayers (SAM), atomic layer deposition (ALD), and other semiconductor technology.

The diode’s metal layers need to be as thick as the skin penetration depth λ_p of the Johnson-Nyquist noise at the peak power frequency f_Peak:

where µ_mag is the magnetic permeability, and ρ is resistivity. For a non-magnetic metal, one can assume µ = 4π × 10⁻⁷ H/m, reducing the equation to:

For example, using copper with a resistivity of about ρ= 0.167 × 10⁻⁷ Ω∙m, the penetration depth at 17.64 THz is:

The classical skin-depth formula in Equation (34) is marginal at 17.64 THz, where copper’s Drude relaxation time makes conductivity complex. The true penetration depth there is set partly by the plasma frequency. Accordingly, the result of Equation (36) shall be treated as an approximation. Diode layers arranged in a stack, much thinner than this roughly estimated penetration depth would have to share thermal plasmons, and, therefore, become less efficient.

Unlike a battery with a fixed chemical potential, the free energy available from a diode is not fixed: it is a function of the load. An unloaded diode has an open-circuit potential difference Φ across the gap due to the different work functions of its metal plates. However, if a load is connected to the diode, the rectification of the Johnson-Nyquist noise produces a current. This current generates potentials eV_Load across the load and φ_Config across the diode and φ_Resist due to internal diode resistance. These three potentials oppose and reduce the potential across the gap from Φ to Φ – eV_Load – φ_Config − φ_Resist. Hence, the gap potential is variable:

Therefore, a diode optimized such that Z_asym = 1 and according to equations (3) and (11) has an actual gap potential:

The power spectrum of the Johnson-Nyquist noise per unit normalized frequency Δn_jn is given by Equation (21):

The total power over all frequencies is obtained by integrating from 0 to ∞ as shown in Equation (22).

Using copper plates with κ = n e f f 2 = 900, [20][26] and the total power at 300K is:

The diode is a threshold device, conductive only when the energy of a fluctuation n_jnk_BT exceeds the threshold potentialn_Threshk_BT determined by the energy barrieracross the gap as per Equation (38):

Furthermore, each successful rectification event delivers a fixed energy n_Threshk_BT = eV_Load + φ_Config + φ_Resist to be shared by the load, the configuration process, and the internal resistive loss, regardless of how much the plasmon energy exceeds the threshold; the excess is thermalized. The work function difference Φ acts as a catalyst and remains unchanged.

Accordingly, the power rectified at the threshold is the product of the rate of above-threshold events and the fixed energy delivered per event.

The rate of all candidate events (regardless of the threshold) within a normalized bandwidth Δn_jn is obtained by dividing the plasmon energy spectrum of Equation (22) by the energy per plasmon n_jnk_BT:

where d²N/dAdt is the number of plasmon crossing a unit area. A single plasmon with energy n_jnk_BT > n_Threshk_BT traverses the diode and generates an energy fixed by the threshold potential n_Threshk_BT. Taking the product of n_Threshk_BT and the plasmon distribution in Equation (43) and integrating from n_Thresh to infinity yields the total power carried by rectified plasmons through the diode:

To maximize power, we maximize the definite integral with respect to n_Thresh. A numerical analysis yields the optimum normalized threshold frequency:

which corresponds to a maximum integral of 2.8494. Note that n_Thresh = 2.1657 is not equal to n_Peak = 2.8214 as per Equation (23) and shown in Figure 4. Hence

Comparing Equation (46) with Equation (40) yields the maximum power efficiency through the threshold:

For the sake of simplicity,we shall assume in the rest of this paper that the diode resistive lossφ_Resist= 0. Therefore, as established in Equation (42), the actual power that reaches the load is only one half of the power through the diode: (eV_Load = ½ n_Threshk_BT). Dividing Equation (46) by two yields:

At T = 300K, with copper layers and κ = n e f f 2 ≈ 900 , the power to the load is:

This is approximately 42.8 times the power per unit area defined by the Shockley-Queisser efficiency limit [27][28] of 0.337 mW/mm² of solar cells.

Since the thickness of MIM diodes is typically in the submicron range, stacking such devices as shown in Figure 5 could achieve substantial power output per unit volume. The power per unit volume is obtained by dividing the power per unit area in Equation (48) by the optimal thickness of the metal layers λ_p. At 300K, and for λ_pCopper + d_G = 15.5 nm + 2.27 nm as per Equation (36) and (33):

Of course, the rapid conversion of thermal energy to DC power quickly lowers the temperature of the device. Heat exchangers interleaved within the stack then become the real limitation to the power generated per unit volume.

When a plasmon’s energy is equal to n_Threshk_BT, it is exactly and fully allocated: some of it, φ_Resist, is dissipated in the diode—The rest is equally shared by the load, and by the configuration, and nothing is left over at threshold.

This equation is illustrated by Figure 6 for an ideal diode with φ_Resist = 0 and operating under optimal design conditions:

Figure 6. Potential energy profile around circuit for an ideal diode with φ_Resist = 0. The load potential eV_Load and configuration potential φ_Config decrease the potential across the gap from the work function difference, Φ, to Φ −eV_Load − φ_Config. The left work function Φ_L is arbitrarily selected at zero relative potential.

From Equation (45), n_Thresh = 2.1657, and with Equation (51) assuming an ideal optimized diode with φ_Resist = 0:

At 300K,

Thus, a stack of about 36 diodes, each less than a micron thick, connected in series could produce up to 1V.

Two proofs that a Maxwellian gas column is isothermal are presented herein.

A1.Isothermality of Maxwellian Gas Due to Exponential Nature of Maxwellian Distribution.

At ground level Maxwellian gas molecules have a given Maxwellian distribution:

where A is the normalization factor, and E is the average kinetic energy of the particles. At a given elevation h, potential energy mgh must be added to E, yielding:

Factoring out mgh produces:

and renormalizing yields:

where:

Comparing equations (63) and (66) indicates that the exponential Boltzmann factor is left unchanged, but the normalization constant is altered. Hence, the gas column is isothermal, but its density changes with elevation.

A2.Isothermality of Maxwellian Gas Due to Relative Nature of Potential Energy

Consider gas particles observed at elevation h with a distribution defined by the Boltzmann factor:

Since potential energy mghis relative to the observer, a more accurate version of Equation (68) should include the observer’s reference “ground” energy mgh_Ref:

A change in potential energy mgΔh of gas particles is equivalent to a change in elevation of the frame of reference of the observer.To remain invariant across multiple observers this change must be applied equally to the potential mgh and to the reference potential mgh_Reffor any observer. Hence:

Applying these changes to Equation (69) yields:

leaving the distribution invariant in any frame:

Hence a Maxwellian gas column at equilibrium is isothermal. The key idea in this proof is that potential energy is relative to the frame of the observer, leaving the Boltzmann factor invariant.

This appendix presents a proof that fermions remain isothermal in a potential energy gradient. To avoid circular reasoning, this proof does not depend on the second law. It relies on the relative nature of potential energy which requires observers in frames withdifferent elevationsto perceive changes in potentialenergy equally.

For the purpose of providing an educational introduction to this topic, Appendix A provides two proofs that a column of Maxwellian gas remains isothermal in the absence of any forcing agent.

B1.A Fermi Gas Column is Isothermal

Consider a change in potential energy Δμ in a Fermi gas characterized by the distribution:

Does the change Δμ cause a change in the temperature of the gas? Since the distribution is not strictly exponential, one cannot prove isothermality by factoring a change in potential energy term Δμ out of the distribution as done in Appendix A for a Maxwellian gas.

The proof presented below relies on the relative nature of potential energy with respect to an observer’s frame.This property leads to the invariance of the Boltzmann factor for any observer.The dependency of the Fermi-Dirac distribution on the Boltzmann factor renders the distribution isothermal with elevation. A detailed explanation follows.

In a Fermi gas, quantum state can only be in one of two conditions: occupied by an electron or empty (occupied by a hole), and these two distributions must add up to unity:

Furthermore, the ratio of occupied to empty states follows the Boltzmann factor. Physically, this means the probability of finding an occupied state relative to the probability of finding an unoccupied state is governed by the energy difference represented by chemical potential μ:

Since potential energy μis relative to the observer, a more accurate version of Equation (75) should include the observer’s reference“ground”energyμ_Ref:

The Fermi-Dirac distribution can now be derived using Equation (76) which includes the ground energy. Rearranging terms yields:

A change in potential energyΔμ of the fermions is equivalent to a change in elevation of an observer.To remain invariant across multiple observers,this change must be applied equally to the potential μ and to the reference potential μ_Reffor any observer. Hence a change Δμ produces a change in μ and in μ_Ref: μ→ μ +Δμ and μ_Ref → μ_Ref +Δμ. Applying these changes to Equation (77) yields:

The Δμ’s cancel each other, thereby leaving the distribution invariant:

Therefore, a Fermi gas column in equilibrium has been proven to be isothermal, independently of the second law. The key concept in this proof is that potential energy is relative to the observer, allowing the Boltzmann factor to be invariant across observers, and independent of the potential energy measured in any given frame of reference. The proof also applies to any gas such as bosons, whose statistics can be derived from the Boltzmann factor as shown in Equation (75).

This appendix provides a bird’s eye view of the model required for rectifying very small amplitude and high frequency signals such as the Johnson-Nyquist noise. The formulation of an actual model is left for a future paper.

C1.Basic Requirements for Johnson-Nyquist Noise Rectification

Requirements for the model include:

1. It must be T-Odd symmetric: the configuration of the diode must change in response to the polarity of the signal. In a PN diode, the thickness of the depletion zone changes depending on forward or reverse bias. In a fluidic valve, the disk or piston moves in response to the direction of fluid flow.

2. The configuration should be described by a state variable which changes between forward and reverse bias and affects the transmittivity of the diode. This change requires energy φ_Config provided by the signal and it occurs over a time period τ_Config.

Without such a state variable, current models cannot properly assess how a diode behaves when the signal—the Johnson-Nyquist noise—has an energy E_Thermal and characteristic frequency ω_Thermal respectively matching the diode’s φ_Config and 1/τ_Config. For example:

1. If φ_Config is much greater than E_Thermal, the noise cannot pay the reconfiguration cost; the configuration remains frozen, the device is merely spatially asymmetric, and it does not rectify.

2. If φ_Config is much smaller than E_Thermal, the noise flips the configuration freely in both directions; with no energetic cost distinguishing SET from RESET, the modulation is itself an equilibrium fluctuation, and the device does not rectify.

3. If 1/τ_Config is much smaller than ω_Thermal, the diode cannot follow the signal; it remains static and does not rectify.

4. If 1/τ_Config is much greater than ω_Thermal, the diode follows the signal quasi-statically; it reduces to the instantaneous static model of Section C.2 and cannot rectify.

These four failure modes are the two endpoints of m-Scale approached along two different axes. Cases 1 and 3, in which the configuration is frozen, and no reconfiguration energy is dissipated, correspond to μ-Scale where Z_asym → ∞. Cases 2 and 4, in which reconfiguration is free or instantaneous, and all free energy is dissipated, correspond to M-Scale where Z_asym → 0. Rectification is only possible between these endpoints. As shown in this paper, the optimum between the configuration energy and the thermal noise is φ_Config = ½E_Thermal and the optimal point occurs at m-Scale where Z_asym = 1.

C2.Why Existing Models Are Inapplicable

No diode model currently satisfies these requirements. The models of Tsu-Esaki [7], Simmons [8][9], Grover [10] and Brinkman-Dynes-Rowell [11] compute the current from the instantaneous voltage across a barrier of fixed geometry. Writing the Tsu-Esaki current explicitly in terms of the transmission function T_Tunnel and the gap width d_G:

where E_x is the carrier energy normal to the gap and the supply function S_Supply counts the net carriers available for tunneling at that energy:

The transmission function is computed in the WKB approximation through a barrier profile φ_B(x) of fixed width d_G:

where φ ¯ B is the mean barrier height and the last form holds for carriers near the Fermi level. Three observations show why this formulation cannot describe thermal-noise rectification:

1. Equation (80) is “Voltage Asymmetric” and not T-Odd, because all parameters except the voltage are fixed. The width d_G appears only as a fixed parameter in the exponent of Equation (82): the barrier geometry cannot respond to the signal. When the device is in thermal equilibrium and in the absence of an input voltage V, the equation is perfectly time symmetric (T-Even). Therefore, by the argument of Section 3, it cannot rectify the equilibrium Johnson-Nyquist noise.

2. The response is instantaneous. The equations do not include any device time constant τ_Config, or noise frequency ω_Thermal.

3. No energy is assigned to the signal, and no cost in energy φ_Config is extracted from the signal to achieve a change in configuration. Furthermore, the density of states enters only through the free-electron supply function of Equation (81) and is completely decoupled from the transmission function—yet it is precisely the asymmetry in the DOS that generates the space-charge layers whose modulation renders the device T-Odd.

These are precisely the omissions identified in Section 4.2. It is worth noting how sensitively the transmission depends on the gap width: for φ ¯ B ≈ 1eV the decay constant in Equation (82) is 2sqrt( 2 m e φ ¯ B )/ħ ≈ 10.2nm⁻¹, so a modulation of d_eff by only 0.2nm changes the transmission by a factor of about 8. A sub-nanometer configurational change is sufficient to produce a strong rectification ratio, as anticipated in Section 4.2.

C3.The Brillouin Diode Paradox

Furthermore, Equation (80) shows its inadequacy by giving the appearance of a second law violation. If a voltage-asymmetric static I-V curve is naively driven by a zero-mean Johnson-Nyquist voltage, time-averaging produces a spurious DC output—the well-known Brillouin diode paradox [33]. The conventional resolution is that a self-consistent kinetic treatment—one that includes the diode’s own thermal fluctuations and its back-action on the signal by van Kampen—restores detailed balance and yields zero output [34].

But this argument negates the very essence of a diode: its configuration is not static; it is modulated by the input signal. The present framework agrees with the conventional resolution of the paradox at the endpoints of m-Scale (μ-Scale and M-Scale) but shows that a dynamic device—one that borrows reconfiguration energy from the signal—retains a net output within m-Scale, bounded by equations (42) and (46).

C4.Proposed Model Outline

It is proposed that the configuration that changes dynamically, as required by T-Odd symmetry, is the thickness of the space charges on either side of the gap. These thicknesses depend on the spatial asymmetry in the density of states (DOS) between the two metal plates across the gap. In combination they become T-Odd—thin in forward mode and thick in reverse mode, as illustrated in Figure 2—because of the input signal. This change in configuration changes the effective width d_eff of the gap, causing preferential tunneling in the forward mode. Hence, for a proper description, d_eff must be expressed in terms of the DOS, and the change in d_eff must require energy φ_Config and occur over a time period τ_Config.

Perhaps, the most suitable starting point is the transfer-matrix formulation used by Grover and Moddel [10] for multi-insulator diodes, typically solved by numerical methods. The potential profile across the junction is decomposed into N thin slices, each with constant potential Uj and thickness δj. Within slice j the wave function is a superposition of forward and backward waves:

where k_j is imaginary inside classically forbidden slices. Continuity of ψ and dψ/dx at each interface relates adjacent amplitude pairs through a 2 × 2 matrix M_j, and the product over all slices,

yields the transmission through the entire structure.

The strength of this formulation for the present purpose is that the potential is already discretized: the space-charge layers on either side of the gap are naturally represented as additional slices whose potentials U_j depend on the local density of states and occupancy, and whose extents d_L and d_R depend on the configuration state. It is therefore suggested to modify this model so that:

1. The slice potentials adjacent to the gap are functions of a configuration state variable determined by the DOS asymmetry of Figure 2, giving the effective width d_eff = d_L + d_G + d_R its bias dependence.

2. Each change of configuration draws energy φ_Config from the signal.

3. The configuration relaxes over a time τ_Config set by the charge-relaxation dynamics of the space-charge layers—a time constant distinct from, and much longer than, the femtosecond transit time of a tunneling carrier. The brevity of the transit time is often invoked to justify applying static models at terahertz frequencies; that argument addresses the wrong time constant, since it is the reconfiguration of the space-charge layers, not the transit of a single carrier, which governs the dynamic response.

C5.Relation to Photon-Assisted Tunneling

Dynamic tunneling models do exist, and the best known is photon-assisted tunneling (PAT), introduced by Tien and Gordon [29], generalized by Tucker and Feldman [30], and applied to rectenna solar cells by Grover, Joshi, and Moddel [31]. However, PAT is dynamic in a fundamentally different sense. In PAT, the time-varying potential modulates the energy of the carriers, producing sidebands at E ± nℏω weighted by Bessel functions, while the barrier geometry remains fixed and no energy cost is charged for any reconfiguration. PAT correctly describes the rectification of coherent, non-equilibrium illumination. Driven by equilibrium noise, it respects detailed balance and yields zero output. In the language of this paper, PAT is an endpoint theory: it governs the μ-Scale and M-Scale limits where the conventional second law holds, while the interior of m-Scale is opened only by a configurational state variable carrying an energy cost.

A structural analog of such a state variable appears in the theory of resonant-tunneling diodes, where the frequency response is modeled through the potential change of the quantum well caused by its own electron charge, with an associated relaxation dynamic [32]. That approach, however, is directed at oscillator bandwidth and includes no thermodynamic accounting of the reconfiguration energy. To the author’s knowledge, no published model combines a configurational state variable tied to the DOS asymmetry, an explicit reconfiguration energy φ_Config drawn from the signal, and an explicit relaxation time τ_Config, applied to the rectification of equilibrium noise. The formulation of such a model is the subject of a future paper.

The thermodynamic cycle of the diode shall assume an idea diode with zero internal resistance dissipation, φ_Resist = 0.

1. Optimal index of asymmetry Z_asym = 1 in the center of m-Scale, and source load matching as in Equation (3):

2. Source load matching:

3. Johnson-Nyquist noise matching threshold level at gap:

which implies:

Furthermore, since electrons cross the gap of a MIM diode much quicker than the period of the Johnson-Nyquist noise, electrons shall be assumed to be in quasi thermal equilibrium and isothermal across the gap as proven in Appendix B.

As illustrated in Figure 7, the cycle comprises two stages, BA, and AB.

Initial Conditions.

To monitor the changes in entropy of the electrons, and the environment around the cycle, we begin by setting them to zero:

D1.Stage BA

This stage begins when a fluctuation with energy n_Threshk_BT = 1/2Φ has barely enough energy to clear the gap potential step 1/2Φ = Φ − eV_Load−φ_Config. This stage comprises 1) a configuration process and 2) a compression of the electron gas.

Figure 7. Thermodynamic cycle of Temperature versus Entropy. Electrons cycling through the diode and load follow two stages: 1) A free compression in blue due to the tunneling through the gap where they go from low potential and high entropy, to high potential and low entropy, and 2) an isothermal expansion in green through the rest of the circuit from high potential and low entropy, to low potential and high entropy.

D.1.1 Configuration BA

Half of the fluctuation energy φ_Config = 1/4Φ= 1/2n_Threshk_BT is used to configure the electronic distribution. It SETs the diode into a FORWARD mode that favors forward tunneling of electrons through the gap. Since the configuration energy is potential, the only change in entropy is due to the transfer of heat from the surroundings:

D.1.2. Compression BA

This upward tunneling step is the exact reverse of the well-known “free expansion” which is isothermal, adiabatic, and entropic. This upward step can be called a “free compression” because it is isothermal, adiabatic and negentropic:

1.Isothermal because electrons undergo unforced equilibrium tunneling as explained in Appendix B. (Isothermality of a gas column in equilibrium is also mandated by the second law.)

2.Adiabatic. Electrons are not in thermal contact with their surroundings, as they undergo direct tunneling through the gap, therefore:

3.Negentropic. Electrons lower their positional entropy by −1/4Φ/T as they narrow down their position, and step into a higher potential across the gap. Only the eV_Load = 1/4Φ share is stored in the electron gas; the φ_Config share is stored in the diode’s configuration and returned to the surroundings on RESET —The cumulative changes in entropies are:

D2.Stage AB

This stage comprises a reconfiguration of the diode and an expansion of the electron gas:

D.2.1 Reconfiguration AB

This stage is initiated when the fluctuation changes sign and RESETS the diode to REVERSE mode. The potential configuration energy φ_Config = 1/4Φ= 1/2n_Threshk_BT is dissipated in the environment. Therefore, the cumulative changes in entropies are:

D.2.2 Expansion AB

As electrons expand isothermally against the load voltage, they receive Q = eV_Load = 1/4Φ from the surroundings and perform work W = eV_Load = 1/4Φ. Therefore, the cumulative change in entropy is:

The diode is back to its starting point, but work W = 1/4Φ has been performed, heat Q = 1/4Φ has been extracted from the surroundings, and the entropy of the surrounding has decreased by 1/4Φ/T.

µ: Potential energy

µ_mag: Magnetic permeability

A: Constant

A_Helm: Helmholtz free energy

B: Magnetic field vector

c: speed of light

d_eff: Effective thickness of gap

d_G: Nominal thickness of gap

d_L: Thickness of left addition to gap

d_R: Thickness of right addition to gap

e: Charge of electron

E: Electric field vector

E_dissip: Energy earmarked for dissipation

E_Peak: Energy of Johnson-Nyquist plasmon at the spectrum peak

E_Thermal: Energy of a thermal plasmon

eV_Load: Potential energy across the load

f: Frequency

f_FD: Fermi-Dirac distribution

f_Peak: frequency at the spectrum peak

g: acceleration of gravity

h: Planck’s constant

J: Current through MIM diode

k_B:Boltzmann’s constant

m: mass

m_eff: Effective mass of electron

M-Scale: Macroscale

m-Scale: Mesoscale

n_eff: = square root of dielectric constant κ

n_jn: normalized frequency expressed as the number of “k_BT” units in one Johnson-Nyquist plasmon

n_Thresh: normalized frequency for threshold potential in k_BT units

n_Peak: value of n_jn at the peak power in the spectrum

N: number of plasmon quanta

P: Power

P_Load:Power to the load

R:Resistance

S: Entropy

T: Temperature

T-Even: time-reversal symmetric

T-Odd: time-reversal asymmetric

V_RMS:RMS voltage of the Johnson-Nyquist noise

Z_asym: Index of asymmetry

Δf: bandwidth

θ_asym: asymmetry angle = atan(Z_asym)

κ: dielectric constant

λ: Wavelength

λ_c: Plasmon’s wavelength for Johnson-Nyquist noise

λ_p: Penetration depth

λ_Peak: Wavelength at the spectrum peak

μ-Scale: Microscale

ρ: resistivity

τ_Config: time period of the configuration

Φ: Work function difference between metal plates

φ ¯ B : Mean barrier height

φ_Config: Configuration energy of the diode

φ_Resist: Potential energy drop due to internal diode resistance

χ_ins: electron affinity of the insulator

ω_Thermal: Characteristic frequency of thermal noise