Research on thermionic emission dates back to 1853, when Becquerel (1853) first detected electrical current passing between two platinum wires, one hot and one cold, kept in a variety of gases. Becquerel was studying the electrical conductivity of gases but did not fully recognize the impact of thermal energy at the electrode on electrical current generation. Many others contributed to the understanding of thermionic emission in the late nineteenth century, not the least of which was Guthrie who in 1873 published his work on the relationship between heat and static electricity. Guthrie (1873) measured the discharge of a positively charged, red-hot sphere into air, helping establish understanding of the thermionic emission phenomenon. Later in the nineteenth century, Elster and Geitel (Angrist, 1976) would investigate sealed devices in which electric current was generated between two electrodes, one hot and one cold, measured utilizing an electrometer. Many of these early studies provided the foundations for cathode and x-ray tubes. Crookes and Hittorf (Crookes, 1896) were the foremost pioneers at developing and studying cathode rays, and their research was critical to the present understanding of thermionic emission, as well as the fourth state of matter – plasma.

While research on thermionic emission continued through the ensuing decades, many of the significant advances in thermionic emission, dedicated toward energy generation, occurred in the 1950s. In 1956, Murphy and Good (1956) published a rigorous study of thermionic and field-emission theory, and Hatsopoulos (Hatsopoulos and Kaye, 1958) presented new thermionic converter types while performing theoretical and experimental work on a thermoelectron engine. The interested reader should consult Angrist’s book (Angrist, 1976) on direct energy conversion that provides more details on the history of thermionic emission, along with a detailed thermodynamic analysis of the theory underlying thermionic emission and associated energy generation.

The foundations of photoemission largely derive from Albert Einstein’s Nobel Prize winning work on the quantum theory of the photoelectric effect. However, it was not until the late 1920s that modern photo-emission theory began to take shape. Much of the theory of photo-emission derives from field-emission studies, and no work was more critical to field-emission research than that performed by Fowler (1931) and Fowler and Nordheim (1928) on electron emission induced by high-intensity electric fields. Their studies led to the work performed by DuBridge (1933) in which the effect of temperature on photoelectron energy distributions was quantified. The pioneering work performed by these researchers generated a surge of interest in photoemission, but applying photo-emission theory to newly developed materials proved difficult, as the phenomenon is fundamentally difficult to interpret owing to dependencies on surface structure and material properties that are often challenging to measure.

Starting in the late 1950s and spanning over nearly four decades, Spicer, often in collaboration with Berglund (Berglund and Spicer, 1964a,b; Spicer and Herreragomez, 1993), advanced the understanding of experimental photoemission via a three-step model. The process is described by (1) photoexcitation of an electron, (2) electron motion through the material, and (3) escape of electrons over the surface barrier and into vacuum. Much of this work was dedicated to the theory and applications of photocathodes and provided deeper understanding of critical photo-emission processes such as electron scattering and quantum yield. Both Spicer and Berglund sought a more complete understanding of how absorption and scattering mechanisms influence photoemission in the hope that experimentalists could better interpret their data.

More recently, Jensen has reported many advances in the understanding of photoemission, both theoretically and experimentally. Examples of this work include the development of more rigorous photo-emission models (Jensen et al., 2006a), and experimentally derived wavelength-dependent quantum efficiencies (electrons emitted per photon of illumination) (Jensen et al., 2006b) for metals and coated materials during field and photoemission. This work incorporates the extreme complexity involved in field and photo-emission processes by including thermal effects, surface conditions, illumination source parameters, reflectivity, penetration depth, and scattering rates (amongst others), all in order to quantify accurately electron current distributions and quantum efficiencies. Possibly most useful to researchers in the electron-emission area is his recent work deriving a general formulation for thermal, field, and photo-induced electron emission, bridging the gaps between the three electron-emission processes (Jensen, 2007b).

Derivations of two and three-dimensional thermionic and photo-emission theory are briefly described here, but for more detailed derivations the interested reader should review work by Fisher (2013) and McCarthy (2013). General electron flux equation provides the best foundation for deriving thermionic and photo-emission equations. A general form for electron flux was developed by Gadzuk and Plummer (1973) and is described in Landauer form as Fisher (2013): (1)J=qe2∫0∞vgEDdDETEfFDT1−fFDT2dE where J is the electrical current density, E is electron energy, q_e is the magnitude of electron charge, v_g(E) is the electron group velocity, D_dD(E) is the electron density of states for a dimensionality dD, T(E) is the transmission function which accounts for the probability of electron transport through the device, and f_FD(T) is the Fermi–Dirac electron distribution function. For simplicity, the group velocity and density of states in Eq. 1 are replaced by the electron mode density to create a general expression for electron density (Fisher, 2013): (2)J=qeπℏ∫0∞MdDETEfFDT1−fFDT2dE where ℏ is the reduced Planck’s constant (h/2π), and M_dD(E) is the electron mode density for a dimensionality dD.

To derive thermionic current density from this equation, the temperature at the second state is assumed to be so low that no current flows from state two to state one, such that Eq. 2 is approximated as: (3)AM3DE=Agvme2πℏ2E−EcHE−Ec

The Fermi–Dirac electron distribution function is defined as: (4)fFDT1=1+expE−μkBT1−1 where k_B is Boltzmann’s constant (8.62 × 10⁻⁵ eV/K), and μ is the electrochemical potential of the emitter. For a parabolic conduction band with edge E_c, the number of modes is described by: (5)M1DE=HE−Ec (6)WxM2DE=Wxgv2meE−Ecπ hHE−Ec (7)AM3DE=Agvme2πℏ2E−EcHE−Ec where H is the Heaviside step function, g_v is the electronic band degeneracy, W_x is the width, and m_e is the electron mass. In this study, E_c is taken to be zero, g_v is taken to be unity, and m_e = 9.11 × 10⁻³¹ kg (Fisher, 2013).

To develop electron-emission theory fully, the transmission function must be derived. A general form for free-electron metals has been developed (Fisher, 2013) as: (8)TE=1−μ +ϕEd−12HE−μ+ϕ where ϕ is the work function. In this study, μ is considered equivalent to the E_F.

When evaluating Eq. 3, Fermi–Dirac statistics are typically replaced with Maxwell–Boltzmann statistics when ϕ ≫ k_BT, a condition that is almost always valid. For a three-dimensional emitter, the thermionic current density can be calculated by substituting Eqs 7 and 8 into Eq. 3, resulting in: (9)JT=∫0∞qeme2π2h3E−μ+ϕHE−μ+ϕexpE−μkBT1dE

Integral evaluation results in the well-known Richardson–Dushman equation (Murphy and Good, 1956). (10)JT=meqekB22π2ℏ3T12exp−ϕkBT1

The electron-emission intensity for a given energy is defined as: (25)IEED=dJdE where EED is the electron energy distribution of emission. By substituting Eq. 3 into Eq. 25 and evaluating, we define the thermionic EED (TEED) as: (26)ITEED=qπℏMdDETEfFDT1

Having a generalized equation for thermionic electron emission, dimension-specific equations can also be developed.

An equation for the three-dimensional TEED has been developed elsewhere (McMullen, 2010; Westover et al., 2010) but is re-derived here by substituting Eqs 7 and 8 with d = 3 into Eq. 26 such that: (27)ITEED−3D=qeme2π2ℏ3E−EF−ϕHE−EF−ϕ1+expE−EFkBT

A simple photo-emission model is developed here following work by Fowler and DuBridge (Fowler and Nordheim, 1928; DuBridge, 1933) and offers a generalized approach for predicting photoemission by ignoring specific material properties. As with the thermionic theory of Eq. 27, this model assumes a free-electron gas with a single parabolic energy band. The model also assumes that photon absorption is independent of photon energy and that all photon energy is converted into electron energy corresponding to momentum in the normal direction. Given these assumptions, the electron population energy distribution is simply increased by the photon energy, and the photo-emission EED (PEED) becomes (Vander Laan, 2011): (28)IPEED−3D=qeme2π2ℏ3E−EF−ϕHE−EF−ϕ1+expE−EF−ℏωkBT where ℏω is the photon energy. This model neglects photonic flux and assumes that all electrons are elevated to a higher energy equivalent to the photon energy. Eq. 28 is termed the “normal energy” model.

An alternative model was developed by Westover (2008) and Vander Laan (2011) as an extension of the theory developed by Fowler and DuBridge that modifies the normal model (Fowler, 1931) by weighting the number of electrons available for emission based on the angle of electron emission. The full derivation is provided here for clarification and to facilitate the two-dimensional derivation provided later.

The derivation closely follows work by Fowler (1931). To begin, a gas of electrons obeying Fermi–Dirac statistics has a distribution per unit volume with velocity components in the ranges u, u + du, v, v + dv, and w, w + dw, with u being the velocity component normal to the surface: (29)nu,v,w3Ddudvdw=14π3meℏ3dudvdwexp12meu2+v2+w2−EFkBT+1

where n(u,v,w) is the electron velocity distribution based on its three-dimensional components. A distribution of electrons based on velocity in the normal direction and in the range of u, u + du is described by: (30)ñu3Ddu=14π3meℏ3du∫−∞∞∫−∞∞dvdwexp12meu2+v2+w2−EFkBT+1

Figure 4 contains the top half of a polar coordinate system for three dimensions with angles (θ_p, ϕ_p) where θ_p is the polar angle from the surface normal and ϕ_p is the azimuthal angle in the surface of the plane. Converting to radial coordinates in the plane encompassing the velocity components of v and w, Eq. 30 can be rewritten as: (31)ñu3Ddu=14π3meℏ3du∫0∞∫02πρdρdϕpexp12meu2+ρ2−EFkBT+1 where ρ² = v² + w², v = ρcos(ϕ_p), and w = ρsin(ϕ_p). Eq. 31 is solved analytically to produce: (32)ñu3Ddu=kBT2π2meℏ3lnexpEF−12meu2kBT+1du

The energy associated with the velocity direction normal to the surface is defined as: (33)W=12meu2

Substitution of Eq. 33 in Eq. 32 gives a distribution of electrons based on their energy associated with velocity in the normal direction such that: (34)ñW3DdW=kBT2π22meWmeℏ3lnexpEF−WkBT+1dW

When developing the random energy model (REM), Eq. 34 must be expanded to account for the absorbed photon of energy hω. The resulting expression becomes: (35)ñW,θp,ϕp3DdW=kBT2π22meWmeℏ3dW∫ϕp=02π∫θp=0πlnexpEF−WkBT+1×Ψpθp,ϕpsinθpdθpdϕp

where Ψ_p(θ_p,ϕ_p) is the probability that an electron gains energy ℏω in the direction (θ_p,ϕ_p). As a probability, the sum of Ψ_p across the entire solid angle must equal unity, such that: (36)∫ϕp=02π∫θp=0πΨpθp,ϕpsinθpdθpdϕp=1

For clarity, Ψ_p is assumed to be independent of direction, and therefore, Ψ_p = 1/4π. Simplifying Eq. 35 gives: (37)ñW,θp3DdW=kBT8π32meWmeℏ3×lnexpEF−WkBT+1dW∫θp=0πsinθpdθp

If θ_p is then discretized into N small segments of angle interval Δθ_p, then the remaining integral simplifies to: (38)∫θp=0πsinθpdθp≈cosθp−Δθp2−cosθp+Δθp2

To obtain an expression for the number of electrons available for emission, Eq. 37 must be integrated over W to find all electrons with energy associated with velocity in the normal direction such that W + ℏωcos(θ_p) > E_F + ϕ. The resulting expression for the total number of electrons available for emission for a given angle θ_p is: (39)Navail,Δθp−3Dθp=∫EF+ϕ−ℏω cosθp∞kBT8π32meWmeℏ3×lnexpEF−WkBT+1dW×cosθp−Δθp2−cosθp+Δθp2

Eq. 39 is solved numerically, at which point the REM PEED for all dimensions is: (40)IPEEDEdE=∑n=1NNavail,ΔθpθpΔθpNavail,totalIPEED,ΔθpEdE where the total number of electrons available for emission and the θ-dependent PEED for three-dimensional emission are described by: (41)Navail,total=∑n=1NNavail,Δθpθp (42)IPEED,Δθp−3D=qeme2π2ℏ3E−EF−ϕHE−EF−ϕ1+expE−EF−ℏωcosθpkBT

For a broad-band illumination source, such as a solar simulator, Eq. 40 is integrated over all photon energies such that: (43)IPEEDEdE=∫ℏωminℏωmaxαIphotℏω×∑n=1NNavail,ΔθpθpNavail,totalIPEED,ΔθpEdEdω where α is the coefficient of photon absorption that is assumed to be constant for normalized PEEDs, and I_phot(ℏω) is the photon irradiance defined as the number of photons per unit time per unit area. For data fitting purposes, I_phot(ℏω) is normalized with respect to its peak intensity prior to inclusion in Eq. 43. Because the theory does not strictly account for photon flux, all data fitting is performed with normalized peak intensities, and total emission currents are not considered.

An expression for two-dimensional thermionic emission is produced by substituting Eq. 6 and 8 with d = 2 into Eq. 25 and evaluating such that: (44)ITEED−2D=qe2meπℏ2E−EF−ϕHE−EF−ϕ1+expE−EFkBT

When comparing to the three-dimensional TEED, a E^1/2 dependence exists in the numerator. A normal energy PEED for two-dimensional theory is once again derived by shifting the Fermi–Dirac function in the TEED such that: (45)IPEED−2D=qe2meπℏ2E−EF−ϕHE−EF−ϕ1+expE−EF−ℏωkBT

The two-dimensional REM is derived by the same method utilized for three-dimensional theory. We exclude most of the details for the two-dimensional derivation in the interest of brevity.

Ψ_p is once again assumed to be independent of direction, resulting in Ψ_p = 1/2π for two dimensions. The resulting expression for the total number of electrons available for emission is thus: (46)Navail,Δθp−2Dθp=14π3meℏ2×∫EF+ϕ−ℏωcosθp∞∫0∞1ΓWdΓdWexpW+Γ−EFkBT+1 where Γ is the energy associated with velocity in the transverse direction. θ_p is again discretized into N small segments of angle interval Δθ_p, and Eq. 46 is solved numerically. The PEED can again be defined by Eq. 40 and 41 where: (47)IPEED,Δθp−2D=qe2meπℏ2E−EF−ϕHE−EF−ϕ1+expE−EF−ℏωcosθpkBT

For a broad-band solar simulator, Eq. 43 still applies.

Eq. 43 predicts a PEED that is broader than that typically observed in experiments, particularly for three-dimensional theory. The broad theoretical distribution relative to experimental PEEDs may be a result of scattered electrons that are not accounted for in the theory. One means of accounting for scattering effects was developed by Sun et al. (2011) who modified photo-emission theory by accounting for electron scattering via optical phonons. Low-energy acoustic phonons do not significantly reduce electron-emission energy; therefore, their contributions are ignored.

The assumption underlying the model is that electrons are only scattered by optical phonons, all of which have the same energy. For each scattering event, a reduction of electron energy equivalent to the phonon energy takes place. Electrons can experience multiple scattering events, and the probability of an electron being scattered n times and still emitting follows a Poisson distribution. The modification to the original REM PEED can then be described by Sun et al. (2011): (48)ISCATEdE=∑n=0mCnexp−μsn!μsnIPEEDE+nEphdE

where E_ph is the photon energy. Eq. 48 gives the emission intensity contribution to energy E by electrons originally with higher energy E′ that have been scattered by optical phonons and as a result emit with energy E. In this calculation, electrons are assumed to scatter with equal probability. Here, n = (E′ − E)/E_ph, where contributions to an energy level E are only evaluated at an energy level E′such that n is a positive integer; m is taken to be the maximum number of scattering events an electron at energy E′ can encounter and still emit. While this term can be solved for m going to infinity, the higher m terms are ignored so that contributions of electrons with final energies outside of the original PEED are not included in the results (Sun et al., 2011). An integral of all scattering contributions added to the original REM PEED provides the final REM PEED. μ_s is the mean number of scattering events that an emitted electron experiences, and C(n) is a probability normalization coefficient used to conserve the number of electrons emitted and is defined by: (49)Cn=∑j=0nexp−μsμsjj!−1

Figure 5 contains a plot of a normalized PEED from nitrogen-doped diamond films excited by 340 nm light at room temperature (RT) originally presented by Sun et al. (2011). The normal photo-emission model does not capture the experimental distribution shape and peak location, but the addition of scattering by optical phonons leads to a close fit. When fitting to experimental data, uncertainties arising from electron energy detection must be included. Because some variation is allowed in the exact energy of the electrons during detection, a Gaussian instrument function centered on the desired electron energy range describes the extent to which an EED is “smeared” by the detector. The Gaussian instrument function is given by Reifenberger et al. (1979): (50)GE=1σ2πexp−12E−Eoσ2 where E_o is the center energy value of the recorded EED and σ is the analyzer resolution. When performing data fitting, the theory must account for the Gaussian instrument function of the detector. The final theoretical model should therefore be a convolution Eq. 50 and the thermionic or photo-emission theory of choice.

For further clarification of the impact that scattering has on the PEED, Figure 6 provides a plot of a three-dimensional REM PEED with ϕ = 1.5 eV, T = 500 K, E_F = 5.0 eV, and σ = 66 meV for μ_s = 0, 2, 4, 6, 8, and 10. No scattering events produces a broad PEED with peak energy near 2.2 eV. However, for μ_s = 2 the PEED is already substantially narrowed, and peak energy is shifted down to approximately 1.7 eV. As μ_s is further increased, the PEED narrows, and the peak energy decreases, but at a reduced rate owing to the exponential term in Eq. 48.

In this section, we compare basic EEDs acquired from graphene petals (McCarthy, 2013; McCarthy et al., 2013) to the models above, with particular attention to low-dimensional effects that are expected to appear for thin graphitic emitters. MATLAB computational software (MATLAB 2011b, 2011) was utilized for fitting electron-emission theory to experimental data derived from potassium-intercalated graphitic petals. Theoretical thermionic and PEEDs exhibit strong dimensional dependencies. The square-root dependence on electron energy in the numerator of the two-dimensional emission theory (see Eqs 44 and 45) produces a narrowing of the resulting energy distribution along with a decrease of the peak emission energy when compared to three-dimensional energy distributions.

This phenomenon is evident in Figure 7. The left plot of Figure 7 contains a comparison of two-dimensional and three-dimensional thermionic theory when sample parameters are kept the same: ϕ = 2.5 eV, T = 1280 K, and σ = 0.08 eV. The right plot in Figure 7 is of two-dimensional and three-dimensional thermionic fits to an experimental TEED for thermionic emission from graphitic petals at 1123 K (McCarthy, 2013). The estimated parameters from data fitting are provided in Table 1. A distinct narrowing of the distribution occurs for the same parameter values for two-dimensional emission compared to three-dimensional in the left plot in Figure 7. The narrowing is not apparent in the right plot as the fit allows for T to change, and an increased temperature relative to the three-dimensional fit causes a broadening of the distribution that offsets any narrowing caused by reduced dimensionality. A slight lowering of the peak energy is also apparent. The residual, R_x, in Table 1 is taken to be: (51)Rx=1−yexp−yfit2n where y_exp and y_fit are the normalized intensity values of the experimental data and theoretical fit respectively, and n is the number of data points for the EED.

The energy at peak intensity and FWHM of the TEEDs for two-dimensional and three-dimensional theory are derived in order to better understand the left-side plot of Figure 7. The energy at peak intensity is found by first solving: (52)dITEEDdEE=0 where the Fermi–Dirac statistics describing electron distribution is replaced by Maxwell–Boltzmann statistics.

Once the energy at peak intensity is known, the peak intensity is calculated along with the energies associated with half the peak intensity, allowing for direct calculation of the FWHM. Utilizing this method, the energies at peak intensity for three-dimensional and two-dimensional thermionic theory are E_Max3D = E_F + ϕ + k_BT and E_Max2D = E_F + ϕ + 1/2 k_BT. The corresponding FWHMs are FWHM_3D = 2.45 k_BT and FWHM_2D = 1.80 k_BT_. These effects are demonstrated in Figure 7. Two-dimensional theory provides less error than three-dimensional theory when data fitting, though neither fit matches exactly to the thermocouple temperature of 1123 K.

Recent advances in thermionic converters have derived from improvements on the classical metal electrode to exploit the electrode surfaces of thermionic generators. For example, El-Genk and Momozaki (2002) proved for low-temperature applications (<1800 K) that planar molybdenum electrodes with cesium vapor could produce as much as a 70% increase in power density relative to traditional tungsten electrodes with cesium vapor. Furthermore, El-Genk and Luke (1999) demonstrated increases in power output of a simple thermionic converter with macro-grooved molybdenum electrodes relative to smooth electrodes. The simple concept and design provided a directly proportional rise in power output with increased surface area, i.e., a 40% increase in surface area generated a 40% increase in power output.

Thermionic and photo-emission energy converters were often projected for use in extraterrestrial applications, as a high-intensity solar flux is available in space, and maintaining vacuum requires less infrastructure. In space, thermal waste can often be difficult to remove; consequently, utilizing the waste heat in thermionic emission applications is a logical approach. One example of a modified thermionic converter utilized in extraterrestrial applications is an alkali metal thermal to electric converter (AMTEC) (VanHagan et al., 1996), in which the temperature difference between hot and cold electrodes drives a liquid alkali metal “circuit.” Figure 9 provides a schematic of an AMTEC device with a liquid alkali metal circuit, originally published by Tournier (Tournier and El-Genk, 1999). While most AMTEC devices demonstrated conversion efficiencies less than 15%, Tournier and El-Genk (1999) predicted an efficiency of 22.5% for the schematic shown in Figure 9 using a stainless steel cell and molybdenum hot side structure and rhodium coatings. Similar investigations were undertaken in which concentrated light was utilized for energy-conversion applications, including work by Begg et al. (2002).

More recently, Jensen (Jensen et al., 2006a,b; Hawkes and Jensen, 2007; Jensen, 2007a,b) and Nemanich (Garguilo et al., 2005; Sun et al., 2011; Koeck and Nemanich, 2012) have produced studies on electron emission with applications in power generation. Combinations of multiple forms of electron emission may offer the most efficient form of energy-conversion moving forward. Jensen et al. (Jensen et al., 2006b; Jensen, 2007b) and Nemanich et al. (Garguilo et al., 2005; Sun et al., 2011) have developed conversion devices utilizing various combinations of thermionic, field, and photoemission.

Some of the most compelling work related to electron emission for energy generation comes from Schwede et al. (2010, 2013) and the PETE concept. PETE combines photovoltaic based photoemission and thermionic emission processes allowing for conversion of photon energy directly to emitted electron energy, while simultaneously utilizing thermalization and photon absorption losses to drive thermionic emission from electrons excited into the emitter band gap. PETE is designed with a structure similar to thermionic converters in which two parallel-plates are separated by a vacuum gap, except that the typically metallic emitter is replaced with a p-type semiconductor. A three-step energy-conversion process takes place. First, electrons are excited by solar irradiation into the conduction band. Second, the electrons rapidly thermalize in the conduction band based on the equilibrium thermal distribution at the emitter’s temperature. Lastly, electrons with energies greater than the electron affinity directly emit to vacuum and are harvested by the collector. Figure 10 provides an energy diagram and schematic for a potential PETE conversion device originally proposed by Schwede et al. (2010).

Given that PETE occurs by electrons directly emitted to vacuum after photon absorption, high efficiencies of photovoltaic cells are expected. Schwede demonstrated that increased power conversion efficiency is possible within solar concentrator systems for which large increases in temperature of the emitter are advantageous to electron emission. The PETE concept has led to other theoretical studies modeling the electron-emission process and energy generation, all of which have demonstrated impressive possibilities for PETE applications (Sahasrabuddhe et al., 2012; Varpula and Prunnila, 2012; Segev et al., 2013). For example, Segev et al. (2013) postulated a side-illuminated series of PETE devices with thermal cycle coupling that could generate conversion efficiencies greater than 50%. However, devices of this efficiency have yet to be proven experimentally.

Limitations in the conversion efficiency of emitter devices have motivated the fabrication of nanostructures that can increase surface area to volume ratios and confine electrons, forcing them to higher energy levels and thereby reducing work functions. For example, Huang et al. (1995) provided theoretical studies on atomically sharp silicon and demonstrated that the local density of states changes dramatically at the emitter tip. Tavkhelidze et al. (2006) demonstrated experimentally that quantum interferences occur in solids with nanoscale features. These quantum interfaces in turn led to reductions in emitter work functions, and with optimized materials, i.e., single crystalline, they anticipate dramatic increases in the quantum interference effects leading to further reductions in work function.