Incorporating photoemission into the theoretical unification of electron emission and space-charge limited current

Electron emission is a critical phenomenon in multiple applications, including high power microwaves, electronic thrusters, nanovacuum transistors, and time-resolved electron microscopy, and across multiple phases, such as vacuum, gas, liquid, and solid.^1–3 While many electron emission studies focus on vacuum,^1–3 recent experiments and theories have examined the implications of field emission (FE) on gas breakdown mechanisms for microscale and smaller gaps.^4–7 This has opened up a wider examination of the unification of electron emission mechanisms across various parameter spaces,⁴ including emitter temperature,⁸ charged particle mobility in the gap,⁹ external series resistance,¹⁰ and diode gap distance.¹¹ 

A seminal study unified field emission with space-charge limited current (SCLC) at vacuum by using the Fowler–Nordheim (FN) equation for field emission current density and solving the force equation to determine the transit time for a single electron to cross the diode.¹² Performing a matched asymptotic analysis, this study recovered the Child–Langmuir law (CL), given by^13,14

where $ε0$ is the permittivity of free space, e is the electron charge, m is the electron mass, D is the gap distance, and V is the bias voltage, for pure planar SCLC at high V or small D; at low V or high D, this study recovered the FN equation,¹² given by^15–17 

where

$Φ$ is the electrode work function, $Es$ is the electric field at the cathode, $y=Φ−14QeEs$⁠, where $Q=e2/(16πϵ0)$⁠, and $AFN=e3/(16π2ℏΦ)$ and $BFN=42mΦ3/(3ℏe)$ are FN constants, where $ℏ$ is the reduced Planck's constant. Table I summarizes the values of key physical parameters used in these equations and throughout this paper.

Motivated by a theoretical analysis of plasma sheath formation linking CL at vacuum with the Mott–Gurney law (MG)¹⁸ for SCLC when collisions with neutral particles (dissipation) are important, which is given by¹⁹ 

where $μ$ is electron mobility, which is generally inversely proportional to pressure, we modified the electron force law from the prior vacuum assessment¹² to incorporate collisions through $μ$⁠.⁹ Accounting for collisions causes electron emission to asymptotically approach MG for small D at low $μ$ and moderately high V. The SCLC solution always transitions from MG to CL with any combination of increasing $μ$⁠, increasing V, and decreasing D, all of which diminish collisional effects until the electron acts as if the gap were in vacuum.⁹ This approach yielded phase space plots based on equations (1), (2), and/or (5), representing the asymptotic solutions of the exact solution for electron transit time in the appropriate limits for FN, CL, and MG, respectively, to determine the transition point between dominant electron emission mechanisms.⁹ We subsequently defined matching such asymptotic solutions as “nexus theory.”⁴ We refer to matching two asymptotic solutions as a “second-order” nexus, matching three asymptotic solutions as a “third-order” nexus, and so forth. For the case of MG, CL, and FN, we obtained a third-order nexus where all three asymptotic solutions met and showed that defining any of V, $μ$⁠, or D uniquely defines the other two parameters at the third-order nexus, where it is necessary to simultaneously account for all three mechanisms.⁹ 

Practical devices are often more complicated than the simple planar diode described above. For instance, practical electron emitters use a series resistor to limit the current to the emitter to minimize damage.^20,21 Placing a resistor in series with the diode divides the applied voltage between the two elements, which may be determined by applying Kirchhoff's law to a circuit composed of the resistor in series with the impedance of the diode, which may be modeled as a parallel plate capacitor in the circuit.^10,21 This approach permitted the derivation of the transitions among FN, CL, and Ohm's law (OL);²¹ incorporating $μ$ in the electron force law introduced transitions involving MG and a fourth-order nexus among all these mechanisms.¹⁰ We emphasize that while OL is not an electron emission mechanism, it represents a limit on emission current (just as SCLC does) and is important for characterizing the contribution of the series resistor to overall device operation.

In addition to circuit elements, not all devices use “cold cathodes” (zero initial electron velocity) for emission; many devices use higher temperature thermionic emission (TE) and there is a vast literature considering cold and thermionic cathodes driven by field and thermionic emission, respectively.²² Thermionic emission is typically defined by the Richardson–Laue–Dushman (RLD) law, given by²³ 

where T is the cathode temperature, $kB$ is the Boltzmann constant, $ARLD=qm/(2π2ℏ3)$⁠, and q is the particle charge. The transition between thermionic and field emission is fully accounted for by the general thermo-field current density equation $JGTF$⁠.^2,24–26 Additionally, many studies have examined Miram curves, which assess the transition from thermionic emission to SCLC,^27,28 with recent theories and simulations extending this work to account for the influence of nonuniform emitters.^29–32 We considered the simple case of emission in a planar diode and applied the $JGTF$ equation to directly unify FN, RLD, and CL; we further used the asymptotic solutions for MG and OL to link all five mechanisms with nexus theory.⁸ Using an analogous approach, we directly unified quantum CL, nonquantum (standard) CL, MG, and FN and indirectly linked these mechanisms with field emission driven breakdown and Paschen's law.¹¹ We are currently performing nanoscale (gaps between ∼20 and 800 nm) experiments at atmospheric pressure to assess the transitions between these mechanisms.⁴ 

One important electron emission mechanism that we have not yet incorporated into nexus theory is photoemission (PE), where electron emission occurs due to the photoelectric effect, typically due to laser exposure.^2,33 This leads to the Fowler–DuBridge (FD) equation,^2,34 given by

where QE is the photoyield or quantum efficiency, $ℏω$ is the photon energy (product of $ℏ$ and photon frequency $ω$⁠), $JFD$ is the FD current density, and $Iω$ is the laser intensity. Due to the ability to generate high brightness, low emittance, and spatiotemporally coherent electron bunches, photoemission is critical for multiple applications, including time-resolved electron microscopy,^35,36 free-electron lasers,³⁷ carrier-envelope-phase detection,³⁸ and novel nanoelectronic devices.^39–42 

We rewrite (7) as^43–46 

where F is the cathode electric force (⁠$F=eE$⁠, where E is the electric field); $ω=2πc/λ$ is the angular frequency; $λ$ is the laser wavelength; c is the speed of light in vacuum;

is the emission probability, where $ζ(2)≡π2/6$ is the Riemann zeta function, $φ=Φ−4QF$ is the reduced work function, $Q=e2/(16πϵ0)$⁠; and

is the effective laser intensity, where $Ii$ is the initial laser intensity in units of $Wm−2$⁠, $Fλ$ quantifies the scattering, and $R(θ)$ is the reflectivity of a laser with incident angle $θ$⁠.

Photoemission devices may undergo transitions between photoemission and other mechanisms over a range of physical parameters. In practical photocathodes, local electric fields may be higher than the applied electric field due to surface imperfections increasing field enhancement, which causes field emission through the reduced photoelectric barrier.⁴⁷ This combined field and photoemission regime may also be applied practically by using laser intensities of ∼1 W cm⁻² with a bias voltage <10 V to activate a semiconductor-free device whose small dimensions would permit integration with semiconductor devices.⁴¹ Additionally, one may leverage thermionic emission of photoexcited electrons from a semiconductor cathode at high temperature to operate in a combined thermionic and photoemission regime to exceed the theoretical limits of single-junction photovoltaic cells.⁴⁸ 

As for thermionic and field emitters, extracting the maximum current from a photoemitter, or a device that operates with a combination of thermionic, field, and/or photoemission, ultimately requires including space-charge. As discussed in detail in multiple reviews and references therein,^1–4 fully accounting for the transitions between the multiple emission regimes becomes challenging, motivating the development of nexus theory. In the case of photoemission, while several theoretical studies have linked various combinations of thermionic, field, and photoemission,^{24,43–46,49–55} the connection between photoemission and space-charge limited emission (especially MG and CL simultaneously) has been less thoroughly studied.³ 

Thus, this paper seeks to unify CL, MG, FD, RLD, and FN in a single theoretical framework, as shown schematically in Fig. 1. In the process, we will demonstrate a consistent, step-by-step process for unifying mechanisms to develop general nexus curves for given conditions. This will provide a procedure for constructing nexus curves based on the specific conditions under consideration to determine which mechanism(s) and theory(s) must be considered. Note that we use the general thermal-field-photoemission (GTFP) current density^24,25 for our canonical current, adding photoemission to our prior nexus theory analysis,^4,8 so we do not explicitly study time-dependent phenomena here.³ Upon understanding the relevant regime of operation, one may either use the full theory outlined in Sec. II or, more realistically, an appropriate simplified theory from the literature that considers one, two, or three mechanisms depending on the proximity of desired parameters relative to relevant nexus curves.

Section II outlines the derivation of the exact theory unifying CL, MG, FD, RLD, and FE and the process for deriving nexus curves unifying two or more of these mechanisms. Section III details the process for constructing nexus curves as a function of different variables to demonstrate the conditions for various transitions. We provide concluding remarks in Sec. IV.

As in prior theoretical studies with planar diodes, we consider the motion of a single electron emitted from a grounded cathode with potential $ϕ=0$ at $x=0$ and an anode at $x=D$ biased to $ϕ=V$⁠. We can then write Poisson's equation, conservation of energy, and electron continuity as

and

respectively, where J is the current density, $ρ$ is the charge density, v is electron velocity, and $νi$ is the initial electron velocity.

Prior nexus theories have defined J as either $JFN$ or $JGTF$ when unifying space-charge limited emission with field emission^9,10,12,21 or thermo-field emission,⁸ respectively. Since this study seeks to unify space-charge limited emission, field emission, thermionic emission, and photoemission, we define J as the GTFP current density,^24,25 given by

where F is the cathode electric force (⁠$F=eE$⁠, where E is the electric field), $JFD(F,T)$ is the Fowler–DuBridge equation for generalized photoemission given by (8)–(10), and

where $ARLD=(emkb2)/(2π2ℏ3)$ and

where $n≡βT/βF$⁠, $s=βF(Em)(ℏω−φ)$⁠, $Q≈0.36eVnm$⁠,^2,37 and $Em=μF+φ$⁠, where $μF$ is the Fermi energy; this model is valid provided that the emission remains below the barrier energy.^26,52 Equation (16) accounts for the different regimes where field and thermionic emission occur.

The GTF equation becomes the GTFP equation when $N(n,s)$ goes to $N(n,−s)$⁠,^2,24,52–55 yielding

where $u=βF(Em)(μF+φ−ℏω)$ and $ns=βT(ℏω−φ)$ for PE.²⁶ Similar to $R(θ)$ in (10), the scattering factor $Fλ$ depends on various factors, including the incident angle; however, $Fλ$ is calculated using a different analytical approach.³⁶ Because of the complexity of $R(θ)$ and $Fλ$ due to their dependence on material-specific parameters, we will treat them as constants for this analysis. Performing a matched asymptotic analysis for nexus theory requires determining the appropriate limits for the transitions from photoemission. The emission mechanisms transition in $JGTFP$ based on the values of $βT$⁠, $βF$⁠, n, and s.

We note that this analysis is a first step in unifying these multiple mechanisms and, as such, we do not attempt to use the most complete theory of photoemission. This is analogous to our recent work incorporating quantum space-charge phenomena where we used an early model of quantum SCLC to demonstrate nexus theory.¹¹ The present study focuses on traditional photoemission due to multiphoton absorption at the large Keldysh parameter $γ≫1$⁠, which corresponds to low laser power where the FD and GTFP theories are acceptable. Higher laser powers can cause electron tunneling through the time varying barrier and lead to optical tunneling. At its core level, this paper focuses on unifying the GTFP with SCLC by using a simpler version of photoemission that neglects optical tunneling. As such, it does not specifically address conditions at the frontier of photoemission research where $1≲γ≲10$⁠, where both multiphoton and optical field emission are important.^56,57 Moreover, our approach does not directly account for laser-induced heating of the material, which is time-dependent and nonequilibrium during photoemission; this can also make the two-temperature assumption inaccurate.⁵⁸ As we have demonstrated through prior studies and in subsequent sections in this paper, the concept of nexus theory may be modified and expanded to include additional mechanisms or more details of a given mechanism. Specifically, this paper provides a primer for developing nexus curves by either incorporating additional mechanisms or adding details within given mechanisms.

To eliminate physical parameters and more intuitively assess asymptotic behavior, we define the following nondimensionalized parameters:

with physical scaling parameters (with units in square brackets)

where S is emission area, R is resistance, and the dimensions are denoted in square brackets. Table II summarizes the values of the scaling parameters defined in (19).

Using these values yields the same asymptotic limits derived previously for FN, RLD, CL, generalized CL (GCL) (which models SCLC with nonzero injection velocity), MG, and OL as^8–10 

and

respectively.

Next, we define the relevant asymptotic solution for FD to develop nexus phase space plots assessing transitions between these emission mechanisms and demonstrate the transitions between the exact solution from (8) to (17) above to the asymptotic solutions under appropriate limits. We obtain the asymptotic solution for FD when $F→0$ and $T→0$⁠. Nondimensionalizing (8) yields

with $χ1=em/(ℏAFNBFN2)=9.5×10−31$⁠, $χ2=ℏAFNE0/(ϵ0Φ)=0.3690$⁠, $χ3=e3BFN/(4πϵ0Φ2)=4.6368$⁠, and $χ4=μFΦ−1=1.5556$ for the parameters in Table I. When $T=0$⁠, (26) becomes

when $F=0$⁠, (26) becomes

and when $F=0$ and $T=0$⁠, (26) becomes

Note that (26)–(29) are not universal (i.e., independent of material) since $χ2,χ3$⁠, and $χ4$ all depend on the work function $Φ$⁠. Since (29) eliminates the temperature and field dependence and just retains the photon dependence, we use (29) as the asymptotic solution for the FD equation when developing nexus curves in subsequent sections. When assessing $J¯$ vs $V¯$ curves, we use the more complete version of FD from (26) to demonstrate the full, analytic transitions among FD, FN, and RLD since $J¯GTFP$ from (14) includes the effects of $E¯$ and $T¯$ upon photoemission.

The process to obtain the exact solution is identical to that reported for our prior unification of thermionic, field, and space-charge limited emission⁸ except that we replace $J¯GTF$ with $J¯GTFP$⁠, which gives

where $ξ=−1+[1+(2J¯/E¯2)((2V¯+v¯i2)1/2−v¯i)]1/2$ and

where we use (26) for $J¯FD(F¯,T¯)$ in the exact solutions (30) and (31).

As described briefly above and in detail elsewhere,⁴ setting two or more of the asymptotic solutions for current density equal yields nexuses representing the transition in dominance from one mechanism to another. A condition where two asymptotic solutions match is a second-order nexus; equating additional asymptotic solutions yields higher-order nexus points.

To begin, we write the conditions for the various second-order nexus curves by equating the various combinations of (20)–(25) and (29) as follows (the ones independent of FD have been derived previously):

where the subscript N denotes a nexus quantity, for MG to CL,

for MG to FN,

for MG to OL,

for MG to RLD,

for FN to CL,

for FN to RLD,

for FN to OL,

for CL to RLD,

for CL to OL,

for RLD to OL,

for FD to FN,

for FD to RLD,

for FD to CL,

for FD to MG, and

for FD to OL.

We may obtain higher-order nexuses by setting appropriate combinations of (32)–(46) equal. For instance, the third-order nexus between RLD, FN, and FD comes from equating the second-order nexuses between RLD and FN from (37) and FN and FD from (42) [or, alternatively, by matching their asymptotic solutions from (20), (21), and (29), or using the nexus between RLD and FD from (43) with (37) or (42)] to obtain

Note that (47) depends on $D¯N/V¯N$⁠, not on $V¯N$ and $D¯N$ individually (i.e., $V¯N$ and $D¯N$ are linked and defining one defines the other), and is not universal (true for any material) since $χ2$ is material dependent. Due to the numerous combinations available for the number of mechanisms studied here, we do not perform these calculations for all potential third-order and higher nexuses; however, we have demonstrated the straightforward use of this process and discussed the many options available to arrive at additional third-order and higher nexuses.

This leads to a critical point concerning nexus theory. Matching the various asymptotic solutions invariably violates the assumptions for the individual asymptotic solutions. For instance, the asymptotic solution for FN assumes negligible space-charge, meaning that $E=V/D$⁠; however, $E≠V/D$ for the CL equation due to the presence of space-charge, so the CL and FN asymptotic relations must both be invalid at the nexus. Accurately determining the behavior near the nexus requires a theory combining both mechanisms.¹² A device operating with parameters within one to two orders of magnitude of a nexus most likely requires using a theory combining multiple mechanisms, while parameters further from a nexus typically can be modeled asymptotically. In other words, nexuses can identify by proximity the mechanisms mostly likely to be relevant for a particular device. Adding phenomena makes deriving complete, exact solutions from first principles more tedious; however, the nexus process remains straightforward. For instance, adding MG, CL, and OL to (47) yields a sixth-order nexus between FD, RLD, FN, CL, MG, and OL, given by

The most critical parameters to determine when assessing the behavior for emission mechanism transitions across multiple regimes are the nexus points. Since we introduce photoemission to our prior nexus theory studies,^4,8–10 we focus here on the third-order nexus between the emission source current densities from FD, RLD, and FN from (47). Figure 2 shows the resulting third-order nexus plot as a function of $E¯=V¯/D¯,ω¯=λ¯−1$⁠, and $T¯$ where defining one of these three parameters uniquely defines the other two. Note that because the asymptotic solutions used to obtain (47) assume no space-charge, the third-order nexus point depends on $E¯$ and not separately on $V¯$ and $D¯$⁠, which means that defining either $V¯$ or $D¯$ uniquely determines the other parameter to achieve this order nexus point. Figure 2 provides important guidance for characterizing the mechanisms involved with changing parameters, much as the third-order nexus when coupling FN, MG, and CL determined when electron emission would either encounter or bypass the MG regime when transitioning from FN to CL with increasing $V¯$ or decreasing $D¯$⁠.⁹ As we shall see, depending on the relationship with the third-order nexus, electron emission may transition from FD to RLD before FN or from FD to FN directly depending on the parameters relative to the third-order nexus conditions.

As in our prior analyses,^8–11 we next examine the full solution of $J¯$ as a function of $V¯$ under various conditions to demonstrate the transition between electron emission mechanisms with increasing $V¯$ by comparing to the relevant asymptotic solutions. Figure 3 shows the initial effect of changing $T¯$ relative to $T¯N$⁠, the temperature representing the third-order nexus between FD, RLD, and FN [cf. (47) and Fig. 2], on $J¯$ as a function of $V¯$ with no series resistor with $D¯=0.5549(1nm)$⁠, $ω¯=0.7018(1.7703×1015s−1)$⁠, $λ¯=591(λ=1064×10−9m)$⁠, and effective intensity $I¯=3.4813×1025(8×1010Wm−2)$ for $T¯=0.03T¯N=0.0016(82K)$⁠, $0.15T¯N=0.0392(2047K)$⁠, $T¯N=0.0523(2729K)$⁠, and $1.5T¯N=0.0784(4094K)$⁠. For all $T¯$⁠, the exact solution follows FD at low $V¯$ since the input energy from the laser is sufficiently high for photoemission to dominate over either field or thermionic emission at these low electric fields. For $T¯<T¯N$⁠, the exact solution next follows FN rather than RLD since field emission dominates over thermionic emission; ultimately, emission becomes space-charge limited and follows GCL at higher voltages. For $T¯=T¯N$⁠, FD, RLD, and FN meet at the third-order nexus, and the exact solution undergoes a transition region after following FD and before following FN that follows neither asymptote. For $T¯=1.5T¯N$⁠, the exact solution now transitions from FD to RLD to FN to GCL with increasing $V¯$⁠. As observed previously when considering FN, RLD, and GCL,⁹ the full solution tracks with the highest $J¯$ from the emission laws until reaching GCL.

Figure 4 shows the initial effect of changing $λ¯$ relative to $λ¯N$⁠, representing the wavelength corresponding to the third-order nexus between FN, RLD, and FD from (47), on $J¯$ as a function of $V¯$ with no series resistor for $D¯=0.5549(1nm)$⁠, $T¯=0.0523$⁠, and effective intensity $I¯=3.4813×1025(8×1010Wm−2)$ for $λ¯=0.05λ¯N$⁠, $λ¯N$⁠, $50λ¯N$⁠, and $500λ¯N$⁠, where $λ¯N=416$ (750 nm). As for Fig. 3 with FN, the exact solution initially follows FD at low $V¯$ since $J¯FD$ is higher than both $J¯RLD$ and $J¯FN.$ With increasing $V¯$⁠, electron emission transitions to FN and GCL in all cases independent of $λ¯$⁠. For $λ¯>0.05λ¯N$ in Fig. 4(a), electron emission never follows RLD. Prior to reaching FN, the exact solution appears to undergo a transition from FD to RLD; however, the exact solution begins to approach the FN asymptote before reaching RLD, so the exact solution never follows RLD. Changing device operating conditions, most notably $T¯$⁠, would most likely make the transition to RLD more distinct. Figure 4(a) highlights the importance of characterizing the contribution of multiple emission mechanisms. Although emission never follows RLD in Fig. 4(a), thermionic emission does contribute to the emission between the FD and FN regimes, which would necessitate the use of the exact solution. For $λ¯≥λ¯N$ in Figs. 4(b)–4(d), thermionic emission is completely bypassed and emission transitions from FD directly to FN with transition occurring more rapidly (i.e., a smaller transition region in $D¯$ requiring the exact solution) with increasing $λ¯$ beyond $λ¯N$⁠. Again, the exact solution follows the emission law (FD, RLD, or FN) giving the highest $J¯$ until emission is space-charge limited (GCL).

We next construct nexus curves highlighting the transitions between the different emission mechanisms by matching the solutions for the different canonical currents, SCLC, and OL, which arise under appropriate asymptotic conditions⁴ as we have outlined previously for the following conditions: (a) FN, MG, and CL;⁹ (b) FN, MG, CL, and OL;¹⁰ (c) FN, MG, CL, RLD, and OL.⁸ While we have previously developed these nexus curves, we have not specifically reported the step-by-step process used to generate them. This is straightforward when only considering a second-order nexus (e.g., FN to CL),¹² which requires only matching two asymptotic solutions; however, this becomes increasingly challenging with additional mechanisms. Furthermore, as Refs. 8–10 and 12 demonstrate, adding mechanisms also requires considering more parameters for nexus development. For instance, FN to CL¹² requires only considering $D¯$ and $V¯$⁠. Incorporating MG (Ref. 9) requires also considering $μ¯$⁠, while including OL (Ref. 10) and RLD (Ref. 8) requires adding $R¯$ and $T¯$⁠, respectively. Incorporating photoemission requires including $λ¯$⁠, which we mathematically incorporate by studying $ω¯$ in dimensionless units (while we could more precisely consider $λ¯$ since most photoemission literature considers wavelength rather than angular frequency, the mathematical analysis is more convenient with $ω¯$⁠, and since $ω¯=λ¯−1$⁠, the transformation between the two is straightforward). Thus, Figs. 5–8 consider $V¯$ as a function of $D¯,μ¯,T¯,ω¯$ at low $I¯$ and $ω¯$ at high $I¯$⁠. For each figure, we provide the step-by-step procedure for developing the nexus curves by starting from a single second-order nexus and adding other second-order nexuses to form the full plot, which “slices” the phase space into regimes of primary influence for each mechanism. Each panel introduces a new nexus curve as a dashed curve, which is replaced by a solid curve in subsequent panels. In particular, we emphasize the slicing of phase space into the various emission regimes as we incorporate additional nexuses since adding a new mechanism will often introduce nexuses with other existing mechanisms and not just one. This slicing of multiple regimes is what introduces additional complexity in adding more mechanisms to the nexus plots and motivates us to detail the philosophy and procedure of plot formation in the following figures.