On the brightness, transverse emittance, and transverse coherence of field emission beam

Field emission cathodes emit electrons upon application of a strong electric field at the solid surfaces via tunneling of electrons through the surface barrier into vacuum. Field emitters are characterized by their small sizes and high beam brightness.^1,2 Most notably, these properties make the field emitters advantageous to construct a high resolution electron microscope. Single field emitters in the shape of sharpened needles that were used in the past to observe atomic structures in the field emission microscopy and field ion microscopy^3,4 are typically integrated into the gun of electron microscopes.¹ Further processing of such cathodes could turn them into single-atom tips with single or a few number of atoms residing on top of the metal tip apexes.^5,6 By taking advantage of their small effective (or virtual) source sizes, these cathodes have been successfully utilized to conduct point projection electron holographic imaging of small objects.^7–13 Applications of field emitters as high current cathodes¹⁴ for x-ray sources^15–28 or for vacuum electric amplifiers^29,30 have been intensely studied with single emitters or field emitter arrays (FEAs) such as the Spindt cathode.^31,32 A Spindt cathode is an array of molybdenum emitters equipped with an on-chip electron extraction gate electrode and produces electron beamlets. Because of the close proximity of the gate electrode to the field emitter tips, the necessary gate voltage for electron emission is $∼100$ V, which is orders of magnitude smaller than typical field emitters with the tip apex radius of curvature of 10–100 nm. Micro-/nano-fabricated FEAs can have tip apex sizes down to a few nanometers by optimizing fabrication methods,^20,33 further reducing the necessary voltage biases via the increased field enhancement at the tip.

The etched-wire needle-shaped field emitters and individual emitter tips of FEAs emit electrons over the large angle range, typically reaching 30$°$ or larger.^4,17,34 The emission beam spreads similarly over the large angle. However, the field emission beam is locally much less divergent. The typical emission angle is the consequence of the spread of the emission area over the curved emitter tip apexes, which at the same time helps to attain the high field enhancement. Reducing the large emission angle of individual beamlets and in turn that of FEAs demand the focusing gate electrode for individual emitters. In such a double-gate FEA with the optimized gate geometry and operation conditions, the entire field emission beam comprises collimated beamlets^18,35,36; the beam divergence of the double-gate FEAs is minimal and dictated by the intrinsic transverse momentum spread of the field emission beam.²⁸ The advantage of double-gate FEAs^{22,25,28,37–47} as high current and high brightness cathodes has been pointed out theoretically,³⁵ and several groups have reported the fabrication and focusing characteristics of the double-gate FEAs.^38,40,41 In a recent work,²⁸ the measurement of the beam property of double-gate FEAs was reported, demonstrating the theoretical predictions. Each collimated beamlet of the electron beam of a double-gate FEA is produced from a nanometer-size field emitter, therefore expected to be transversely coherent. Although the coherence properties of typical wire-shaped field emitters have been studied in detail, reports on the coherence properties of the micro-/nano-fabricated field emitters are rare.^28,48,49 In a recent experiment, the electron diffraction from a suspended graphene layer was demonstrated using a double-gate FEA²⁸ and a double-gate single-nanotip field emitter.⁴⁹ These experiments highlight the importance of the average transverse energy of field emission beams³ on the transverse structure of the wave function of the field emission electron beam.^50–53 This raises the interesting issue of the impact of the phase coherence of electrons in the solid from which they are extracted on the transverse coherence of the field emission beam.

Accordingly, the purpose of the present work is to review the important characteristics of field emitters for beam applications and to consider the possible influence of the coherence properties of electrons in the solid on the field emission beam applications that demand high transverse coherence. In Sec. II, the brightness of the field emission beam and its importance in electron microscope applications are discussed. In Sec. III, the significance of the transverse emittance for beam applications and the relationship between the transverse emittance and the average transverse energy of the field emission beam are discussed. In Sec. IV, the transverse coherence of field emission beam in terms of the van Cittert–Zernike theorem as well as the theory of the partially coherent beam and their possible consequences in experiments are discussed. Finally, we conclude in Sec. V.⁵⁴ 

In 1931, the first electron microscope was built by Ruska and Knoll.⁵⁵ Although the field emission was known even before that time,⁵⁶ it was 1966 when the first integration of a field emitter into the electron microscope was reported by Crewe et al.⁵⁷ By taking advantage of its small size and the high beam brightness of the field emitter, the electron microscope with the field emission electron gun was instrumental to demonstrate single atom contrast.⁵⁸ Since an electron behaves as a wave as well as a particle as the electron diffraction experiments by Thomson⁵⁹ and Davisson and Germer⁶⁰ demonstrated, the resolution limit of the electron microscope is dictated by the diffraction limit given by $λ$/(2$θ$⁠), where $λ$ is the electron wavelength and $θ$ is the angular spread of the focusing lens. As a result, by using an electron beam with the wavelength in the range of sub-Å, the resolution of an electron microscope can exceed by orders of magnitude that of an optical microscope. Reaching atomic or sub-atomic resolution became possible with a reasonable magnitude of $θ$⁠. Such high resolution electron microscopes with routinely attainable atomic resolutions have become available recently with the advent of field-emission-electron-gun technology and with the widespread use of spherical aberration correctors or the Cs-correctors,⁶¹ which substantially remove the spherical aberration of electron lenses. The resolution of $∼100$ pm can be achieved with $λ$ of $∼2$ pm for electrons accelerated to 300 keV and $θ$ of 10 mrad.¹ 

In the electron microscopes with the Cs-correctors, the diffraction limit and the (effective) emitter size are the two main contributions for the beam spot size. Here, the emitter radius $rem$ is not the physical size. It is rather a quantity calculated from the probe current $Ip$⁠, the angular opening $θ$⁠, and a given beam brightness $B$⁠, as $Ip/(πBθ2)$⁠, where $θ$ is determined by an aperture. The beam brightness $B$ is given by $Iem/(AΩ)$ with the emission current $Iem$⁠, the physical source area $A$⁠, and the total emission angle $Ω$ of the emitter; using a field emission gun with a high $B$⁠, a high $Ip$ is available for users while reaching the atomic resolution. Referring to Ref. 62, $B$ of typical field emitters used in a Cs-corrected scanning transmission electron microscope (STEM) operating at 100 keV is equal to $2×1013$ A/(m$2$ sr) with the probe size of 66 pm at $Ip$ of 10 pA. When $Ip$ is increased to 1 nA, the probe size is increased to 145 pm.

In general, assuming a cylindrical symmetry, the beam brightness $B(ρ,α)$ of a cathode is a function of the radial distance $ρ$ from the optical axis and the angle $α$ of the beam direction both measured from the optical axis,

where $dS$ is the small area on the cathode surface and $dΩ$ is the differential stereo angle in the direction of $α$⁠, see Fig. 1. Considering an electron emitted from a position $(ρ0,z0)$ of a curved cathode surface which propagates to a position $(ρ,z)$⁠, its trajectory depends on the normal energy $En$ (the kinetic energy of the electron in the direction normal to the cathode surface), the transverse energy $Et$ (the kinetic energy of the electron parallel to the cathode surface), and the electric field distribution including the surface electric field as determined by the electron optics and the shape of the emitter.⁶³ Consequently, $B$ at $(ρ,z)$ also depends on all these characteristics. Therefore, general consideration of the beam brightness is a difficult task. $B$ also depends on the beam energy since the beam divergence given by the ratio of the transverse momentum to the longitudinal momentum can be reduced by acceleration with the increase in the longitudinal electron momentum (in the propagation direction $z$⁠). For comparing performances at different beam energies, the reduced beam brightness $β$ given by $B$ divided by the beam energy (corrected for the relativistic effect) is often used. $B$ or $β$ still depends on the gun optics, especially for electrons emitted from the off-axis area. Therefore, to facilitate the comparison of performances between cathodes, the axial brightness for electrons emitted from the small area at the emitter tip apex along the optic axis is used as a measure. Referring to the results by Shimoyama and Maruse,⁶³ the reduced axial rightness $β0$ of a field emitter is written as

In these equations, $J(F,T)$ is the field emission current density, $p$ = $kBT/dF$⁠, $kBTc=dF$⁠, $W$ is the work function, $e$ and $m$ are the elemental charge and electron mass, $ℏ$ is Planck’s constant, $kB$ is the Boltzmann constant, $b=(4/3)2m/(eℏ)≃6.830890eV−3/2$ V nm$−1$⁠, $tF∼1$⁠, and $vF$ is a function of $F$ representing the barrier lowering by the Schottky effect.⁶⁴ We further note that the average transverse energy $⟨Et⟩$ is equal to $dF$⁠.⁵² 

When trajectories of all the electrons emitted from a spherical-tip field emitter are calculated as a function of $Et$ and $ρ0$ and their average envelope at the anode plane are back-tracked to the cathode, a virtual cathode image is formed behind the emitter. From such a geometrical optics consideration, the virtual image size or the virtual source size defines the source size of the electron gun in microscopes. The virtual source size also determines the transverse coherence length of the beam along the column via the van Cittert–Zernike theorem (see Sec. IV). In a spherical emitter case, an analytical calculation shows that the virtual cathode size is smaller than the physical cathode size by the ratio $⟨Et⟩$/$V$⁠, where $V$ is the beam energy.^2,65,66 We note that since $kBTc$ in Eq. (2) is equal to $⟨Et⟩$⁠, $β$ (⁠$=β0×V$⁠) is inversely proportional to $⟨Et⟩/V$⁠. This is consistent to the fact that the ratio of the average transverse velocity to the longitudinal velocity is $⟨Et⟩/V$⁠. From more general considerations, it was found that the virtual source size is also a function of the electron gun optics.⁶⁷ 

In the case of a typical tungsten field emitter with $W$ of 4.5 eV and $F$ of 3 GeV/m, $β0$ is in the order of 10$14$ A/(m$2$ sr V). However, this value neglects the Coulomb repulsion between the emitted electrons. When the current density is increased and the space-charge becomes large enough to modify the electrons’ trajectories, it becomes difficult to further increase the beam brightness with the increase in current or $F$⁠, although the exact range of the current or electric field depends on the size of the emitters and the emitter geometry. In the analysis reported in Ref. 68, the authors concluded that the influence of the space-charge effect can amount to as high as six orders of magnitude: the maximum $β0$ is limited to $∼$10$8$ A/(m$2$ sr V); therefore, the maximum $B$ is limited to $∼1013$ A/(m$2$ sr) for the beam energy of ∼100 keV. This value is comparable to the above quoted $B$ for Cs-corrected STEMs. The same $β0$ or $B$ gives the value for the emitter size or the effective virtual source size equal to $∼3$ nm. We note that the Coulomb repulsion between electrons from sharp metal tips via optical excitation has been also reported,^69–71 which will have an influence on the ultrafast electron beam experiments.

For electron beam applications such as accelerator-based photon sources and vacuum electric amplifiers, the transverse emittance is often used to specify the requirement for the beam.^{17,23,28,34,72–74} Using the transverse emittances $εx$ and $εy$ in the $x$- and $y$-directions, respectively (the beam propagation direction is along the $z$-direction), the beam brightness is written as $Iem$/(⁠$εxεy$⁠). The transverse emittance is a measure of the phase space spread of the particle distribution. For a given particle distribution in the trace space $x$-$x′$⁠, the rms transverse emittance $εx$ is calculated by the following from the ensemble average of the particle position $x$ and the slope $x′=dx/dz≃Px$/$P$ (⁠$P$ and $Px$ are, respectively, the total momentum and the momentum in the $x$-direction),⁷² 

which gives the phase space spread of the particles. Similar to reduced beam brightness, since $εx$ in Eq. (6) is dependent on acceleration, the normalized rms transverse emittance $εn,x$ is defined as

with $v$ and $c$⁠, respectively, being the longitudinal electron velocity and the speed of light. On the surface of a flat cathode with no correlation between the position and the slope, $εn,x$ is given by the real space beam size times the average transverse velocity,⁷³ 

These definitions are consistent with the definitions of beam brightness in Sec. II as given by the current divided by the source area and beam divergence.

The conservation of the beam brightness and the transverse emittance along the beam column or the beamline are the consequence of Liouville’s theorem with the dominant particle momentum in the propagation direction. Liouville’s theorem states that the density of particles or the volume occupied by a given number of particles in phase space remains invariant when the particle dynamics follows Hamilton’s equation.⁷² When the longitudinal momentum (along the beam direction) is much larger than the transverse momentum and the correlation between the longitudinal emittance and the transverse emittance is negligible as in many situations, one can optimize the beamline and the cathode by considering these emittances separately.

As an example, let us consider an experiment depicted in Fig. 2, in which an electron beam emitted from a cathode with the diameter of 1 mm (0.25 mm-rms radius when the beam intensity is constant over the source ) is accelerated to 20 keV, is to be focused down to 50 $μ$m by using an ideal lens. When $εn,x$ is equal to 1 mm-mrad (or 1 $μ$m) with the corresponding average transverse energy of the beam of 11 eV, the beam can be focused down to 50 $μ$m by using a strong lens with a focal length of 2 mm. When $εn,x$ is equal to 0.1 $μ$m and the corresponding average transverse energy is equal to 0.1 eV, the required focusing needs a lens with the focal length of 10 mm. In this case, as a result of the long-focal length of the lens, the focal depth is increased to 2 mm: the beam with the 10 times smaller $εn,x$ can propagate 10 times the distance of the beam with $εn,x$ of 1 $μ$m with the minimal beam spot size. This shows the importance of the small transverse emittance of the beam for applications, e.g., to produce electromagnetic waves at high frequency^29,30 or to accelerate electrons by laser electric fields when the electron travels along submicrometer narrow channels.⁷⁵ 

For FEAs, the transverse emittance is large because of the geometrical spread of the field emission beamlets.^17,23,34 Nevertheless, since the electron emission is nearly perpendicular to the emitter surface locally, individual focusing of the field emission beam by an on-chip focusing gate should be able to reduce the transverse emittance of the whole FEA beam³⁵ (even though the transverse emittance of the individual beamlet is separately conserved irrespective to this focusing). The measurement of the transverse emittance and the demonstration of the reduction in the transverse emittance for a FEA beam produced from a double-gate FEA were reported recently in Ref. 28. In the experiment, FEA chips equipped with the double on-chip gate electrodes have been fabricated and integrated into a DC gun for measuring the normalized transverse emittance. Field emission beamlets were emitted from a 10$4$-molybdenum-nanotip array with the tip apex size of 5–10 nm, aligned with the pitch of 10 $μ$m over a circular area with the diameter of 1 mm. The field emission beam was accelerated to 20 keV in a diode configuration, transported through a 1 mm anode iris, and focused by a solenoid built into the anode block on a down-beam plane at a distance of $∼100$ mm from the anode. Subsequently, the variation of the focused beam profile upon propagation was measured by displacing a phosphor screen away from the focal plane. The normalized transverse emittance was evaluated by fitting the relationship between the beam size and the propagation distance. The experiment showed that by turning on the beam focusing for optimum collimation, the normalized transverse emittance decreased from 1.2 to 0.13 $μ$m, with the corresponding decrease in the average transverse energy from 11.8 to 0.13 eV. The observed average transverse energy of the maximally collimated beam was found to be in good agreement with the known value obtained from the standard field emission theory.³ 

Now we discuss the transverse coherence of the field emission electron beam. As a matter wave, an electron beam is diffracted by crystalline solids, forming Bragg spots on a screen in a direction determined by the lattice. To form sharp Bragg spots, the electron beam should be transversely coherent over the length spanning several lattice unit cells. Therefore, for studying samples such as protein crystals with large unit cell lengths at atomic resolution, the transverse coherence length of the electron beam on the sample should be tens of nanometers or larger. In imaging experiments based on electron holography^10,76,77 or coherent diffraction imaging,^78–80 the aim is to reconstruct the two-dimensional or three-dimensional structure of nanometer-scale objects from the holographic or diffraction patterns when the transverse coherence length is more than twice the size of the objects.⁸¹ The large transverse coherence is clearly a prerequisite for the holographic wavefront synthesis^82–87 and for the development of quantum electron microscopes.¹⁰⁹ For time-resolved electron diffraction experiments,⁸⁹ not only bright and coherent electron pulses are required, but also the temporal or longitudinal coherence length $Llc$ should be sufficiently larger than the object size or thickness. However, the condition on the longitudinal coherence is normally satisfied for field emitters. With the energy spectrum width $δE$ of $∼0.2$ eV, $Llc=2λδE/E$ is larger than a few hundred nanometers for the beam energy $E$ larger than a few hundred electron volts (⁠$λ$ is the wavelength of the electron at $E$⁠),⁸¹ and larger than the inelastic mean free path of electrons at the same energy.^90,91

The small source sizes and the high beam brightness make field emitters advantageous for such experiments demanding the transverse coherence. Assuming that the source is incoherent, that is, when two electrons emitted from any two points on the source $S$ have no phase relationship, the transvese coherence length of the beam on the sample can be calculated by the van Cittert–Zernike theorem.⁹² The theorem states that for a small incoherent source with the diameter of $ds$ located at a distance $z$ from a sample plane, the transverse coherence length $Lcoh$ is given by $λ/(2θs)$⁠, where the angle $θs$$=ds/z$ subtends the source from the sample and $λ$ is the wavelength of the beam.

This can be understood by considering a double-slit experiment by a monochromatic beam, see Fig. 3. Let us assume that the beam with the wavelength $λ$ emitted from the source illuminates the plane $Σ1$ with two slits $P1$ and $P2$⁠. The transmitted beams through $P1$ and $P2$⁠, in turn, illuminate the screen $Σ2$⁠, forming the interference fringes. Assuming that the beam is described by a scalar wave, we write the wave functions at $P1$ and $P2$ as $ψ1(P1)$ and $ψ2(P2)$⁠, respectively. In this setup, the interference fringe at point $Q$ on $Σ2$ is given by intensity $I(Q)$ written by the square modulus of the superposition of the two beams,

where $In(Q)=|ψn(Q)|2$ is the beam intensity when only $Pn$ (⁠$n=1$⁠, 2) is kept on $Σ1$⁠. The third term describes the interference of the two beams. Denoting the distance from $Pn$ to $Q$ as $Rn$⁠, the interference term is proportional to

The mutual coherence $J12$ given by the correlation of $ψ1(P1)$ and $ψ2(P2)$ dictates the strength of the interference.⁹² When the distance $z$ between the source and the plane $Σ1$ is in the Fraunhofer diffraction range⁹³ (⁠$z$$≫$$d2/λ$⁠), $J12$ is proportional to the spatial Fourier transform of the beam intensity of the source at the spatial frequency $f$ given by $(d/z)/λ$⁠. Assuming a circular symmetric source with the intensity $S(ρ)$ with the radial coordinate $ρ$⁠,⁹³ 

where $j=−1$⁠. Therefore, for a circular uniform source with the diameter of $ds$⁠, $J12$ is finite when $d$ is smaller than $λ$/(⁠$2θs$⁠). Accordingly, $Lcoh$ of an incoherent source is given by $λ$/(⁠$2θs$⁠), that is, $Lcoh$ is zero on the source plane but is finite at a plane separated from the source by $z$ in the farfield; the transverse coherence can be enhanced as much as one wants by decreasing $θs$ by way of using a smaller aperture or placing the source farther away with the reduction of the probe current,⁸⁸ hence the advantage of field emitters with small sizes and high beam brightness for experiments requiring the transverse coherence.

The assumption that a field emitter is fully incoherent appears to be mostly consistent in literature including the case of the single- or few-atom field emitters.^5,6 The analysis of finite beam spread and the finite virtual source size of cathodes are well-established in literature.² The latter is evaluated from the ray optics calculation of the trajectory spread of an electron beam with a finite transverse energy:⁹⁴ the trajectory of an electron emitted with an oblique angle $α$ from the optical axis with the velocity of $ve$ under the influence of an accelerating electric field is assumed and the tangential line of the electron trajectory at the anode plane is extrapolated back inside the cathode. The virtual image position is found from its intersection with the optical axis, which is located behind the physical cathode surface. By considering the average spread of the radial position of the lines at the virtual image position as a function of $α$ and/or $ve$ (or the electron initial energy $V0$ equal to $(1/2)mve2$⁠), the virtual source size is defined. From the optics point of view, the virtual source size is a consequence of the spherical aberration and the chromatic aberration of the source lens. Such a calculation is analytically tractable in a planar⁹⁵ or a spherical potential model.⁶⁵ 

In Ref. 94, Scheinfein, Qian, and Spence analyzed the virtual source size of field emitters with the emphasis on the single- or few-atom field emitters using the van Cittert–Zernike theorem (with the assumption of the incoherent cathode). They concluded that for a single-atom emitter with the physical source size (diameter) of 20 Å (as indicated by the field emission image, e.g., in Ref. 12), the virtual source size (diameter) is 4.6 Å. For a typical metal field emitter with the physical source size of 2000 Å, the virtual source size is equal to 84 Å. From these values, they found that $Lcoh$ of such single-atom emitter on the sample is larger than the beam width owing to the small sizes: $Lcoh$ is $∼1.7$ cm on the detector located at $∼10$ cm away from the source. The impact of the transverse coherence on the beam propagation of a field emitter has been studied theoretically by Latychevskaia.⁹⁶ They showed that the experimentally observed Gaussian-like propagation of the field emission beam indicates partial coherence and is not compatible with the fully incoherent source. Including this effect, the spatial resolution of a reconstructed object image from an electron in-line hologram was analyzed, which showed that, for the beam energy of 50 eV, atomic resolution (below 3.5 Å) demands the source size to be smaller than 2 Å.

In Ref. 97, Garcia and Rohrer discussed the transverse coherence of single-atom-tip emitters from a different point of view. They assumed that the field emitter is partially coherent and considered the transverse coherence of the beam on the sample by propagating the mutual coherence from the source to the sample assuming the Gaussian Schell-model beam. Here, following Mandel and Wolf,⁹⁸ we discuss the propagation of the partial coherence of a monochromatic beam with the frequency of $ν$ [$=E/(ℏ/2π$⁠), with beam energy $E$]. The cross-spectral density function $W(ρ1,z1;ρ2,z2,ν)$ is defined from the correlation of waves at two points $(ρ1,z1)$ and $(ρ2,z2)$⁠, wherein we assume the $z$-direction as the optical axis and $ρ$$=(x,y)$ is the transverse coordinate, see Fig. 4(a), and we consider the propagation of the beam from the source plane at $z=0$ to the sample plane at $z$⁠. For simplicity, we write $W$ on a given $z$-plane as $W(ρ1,ρ2,z)$ and omit writing $ν$⁠. When $W(ρ1,ρ2,z)$ is given, the beam intensities $S(ρ1,z)$ and $S(ρ2,z)$ and the spectral degree of coherence $μ(ρ2−ρ1,z)$⁠, which corresponds to the mutual coherence $J12$ in Eq. (19), are given by

In the Gaussian Schell-model beam (see the Sec. 5.6.4 in Ref. 98), $W(ρ1,ρ2,z=0)$ at the source is parameterized by the beam spot size $σs$ and the transverse coherence length $σg$ and written as

By applying the propagation law for each wave function from the source to the sample plane at $z$ (given by Fraunhofer diffraction for large $z$⁠), the beam intensities and the spectral degree of coherence at $z$ are given by

where $k=2π/λ$ and $j=−1$⁠.

Equation (18) shows that the $z$-dependence of beam divergence is given by $σsΔ(z)$⁠. The function $Δ(z)$ depends on $σs$ and $σg$ via $δ$ in Eq. (17). In Fig. 4, we show two examples of beam propagation. Figure 4(b) shows the case of an incoherent source with $σg<σs$⁠, and Fig. 4(c) shows the case of a partially coherent source with $σg=σs$⁠. $λ=0.1$ nm and $σs=3$ nm were assumed in both cases. From the comparison of these, it is apparent that the beam emitted from a partially coherent source spreads less than the beam emitted from the incoherent source.

Another impact of the partial coherence of the source manifests itself on the $z$ dependence of the ratio $ξ$ of the coherence width to the beam size. From Eqs. (18) and (19), we see that the coherence width and the beam size are, respectively, given by $σgΔ(z)$ and $σsΔ(z)$⁠, both of which vary with $z$ with the same function $Δ(z)$⁠. Therefore, $ξ$ is constant over $z$⁠,

Through this effect, Garcia and Rohrer ascribed the large transverse coherence length (in the order of 1 cm and in the order of $∼10$% of the beam size on the screen) of the electron beam produced from a single-atom-tip field emitter on the screen to the partial coherence of those emitters. Interestingly, the analysis in Ref. 94 is in agreement with this conclusion. The conservation of $ξ$ along the column of an electron microscope was shown by Pozzi within the assumption of the incoherent source but with the partial coherence of the electron beam introduced by limiting the source angle with an aperture.⁹⁹ 

Some theoretical works highlighted possible quantum mechanical effects at the small electron emission areas. In Ref. 100, Gadzuk analyzed the field emission through an adatom at the emitter tip apex via resonant tunneling. He showed the importance of the balance between the electron distribution in the solid and the momentum required for resonant tunneling. To produce a field emission beam with a finite current density with beam divergence at most 10$°$–15$°$⁠, the radius of the resonant tunneling channel by adatoms should be $∼5$ Å. This is because when the resonant tunneling channel is narrower, e.g., $∼1$ Å, the tunneling demands the higher transverse momentum. However, since electrons with higher transverse momentum have negligible longitudinal momentum, the tunneling current is reduced toward zero. Similar analysis was conducted by Lang et al. by the density functional calculation.¹⁰¹ In their work, by considering the extremely high field enhancement factor of the single-atom-tip, the authors focused on the effect of an extremely high electric field (⁠$>1$ V/Å) and predicted a beam narrowing as a result of the strong modification of the three-dimensional barrier profile. This is due to a horn-shaped channel opened up in the surface normal direction with the transverse radius in the order of 5 Å through which the electron tunnels into the vacuum at $∼1$ V/Å. However, when the electric field is a normal value (⁠$∼0.6$ V/Å), the electron leaving the atom tip encounters a potential barrier in all directions, which makes the beam divergence even larger than the typical field emitters with the tip apex size of 10–100 nm. We note that these works assumed coherent electrons at the field emitter tip; hence, the produced field emission beam is fully coherent. The decoherence process of electrons in solid upon field emission was not considered, of which rate would depend on the material and physical shape of emitters but nevertheless should be finite in experiments. How the finite phase coherence length of electrons in the solid would affect the emission profile predicted in their calculations is yet to be elucidated.

The calculations of the transverse coherence of an electron beam by Scheinfein et al. (Ref. 94) and Pozzi (Ref. 99) suggest that how the field emission beam becomes partially coherent, either via the physical process or by the insertion of an aperture, is not relevant. However, as the discussion by Garcia and Rohrer⁹⁷ indicates, such analysis may not describe all the physics relevant to the coherence of an electron beam. In this respect, an interesting example is provided by Cho et al. in Ref. 102. They studied the temperature dependence of the transverse coherence of a field emission beam in a projection imaging by using a nanotube as an electron bi-prism. They found that upon cooling the field emitter from 300 to 77 K, the transverse coherence length that was evaluated from the decay of the interference from the center of the beam on the screen increased 5.4 times (from 13 to 70 mm on the screen that was 165 mm away from the emitter). When the van Cittert–Zernike theorem is applied, this observation is interpreted as the 5.4-fold reduction in the source size (virtual and physical, since these two are approximately proportional) upon cooling. Instead, by applying the partial coherence beam theory, the observation is ascribed as the increase in the source coherence length (⁠$σg$⁠) at low temperature. Considering that the field emission is the quantum mechanical tunneling of electrons in the solid into vacuum, this is ascribed as the increase in the phase coherence length in the solid upon cooling. This is compatible with the observations that the phase-breaking, inelastic scattering probability decreases in the solid with the decrease in the temperature.¹⁰³ 

In the limit of fully coherent electron beam with $σs$ $≪$ $σg$⁠, the cross-spectral density function is a multiplication of the electron wave function at two points. This raises a question on the form of the wave function of field emission electron including the transverse direction. When we write the wave function as functions of the transverse coordinate $ρ$ and the normal coordinate $z$⁠, the variation of the wave function normal to the emitter surface at $z$ far from the surface is well known,¹⁰⁴ 

In Eq. (23), the sinusoidal function is the asymptotic form of the Airy function, wherein the wave number $kz$ in the propagating direction (the $z$-direction) is given by the initial electron energy $E0$ and the surface electric field $F$ as the following (when $F$ is constant):

However, the transverse part $f(ρ)$ has been less studied.^50–52,105 For a point electron source, when $f(ρ)$ takes a Gaussian form, its spatial spread $δρ$ at the source should be at least $ℏ/pt$ by the uncertainty principle and nominally larger than the size of a single atom. Even though the maximum momentum $ℏkF$ of electrons in a metal is given by the Fermi wave vector $kF$ with the atomic scale Fermi wavelength, electrons with the transverse momentum of $ℏkF$ do not contribute to the current since the momentum (and velocity) of such electrons in the tunneling direction is zero; $δρ$ is given by the inverse of the average transverse momentum over the electron distribution weighted by the field emission tunneling probability. Such a calculation led to $f(ρ)$ as follows⁵²:

where $k0=kz$ in Eq. (24) with $E0$ equal to the work function $W$⁠.

The form of the transverse function $f(ρ)$ shows that the field emission beam propagates as a Gaussian beam. The beam spread $σ(z)$ and the beam curvature $R(z)$ are functions of the material parameters as well as the field distribution at the emitter tip. For a beam emitted from a point emitter on a flat surface, $R(z)$ increases as 2 $z$ in far field. $σ(z)$ increases by the balance of $σ(0)=σ0$⁠,⁵² 

and the increase in the momentum in the propagation direction, wherein $σ0$ is inversely proportional to the square root of $⟨Et⟩=dF$ [$∝$$F/W1/2$⁠, see Eq. (5)] and gives the minimal wave function spread, determined by the electron distribution in the solid. The value of $σ0$ is equal to $∼0.5$ nm for a typical metal field emitter with $W=4.5$ eV, the Fermi energy of 10 eV, and $F=3$ GV/m. The beam divergence angle in far field is given by $λW/z$ and independent of $F$⁠, where $λW$ is equal to $2π/2mW/ℏ2$⁠. This agrees with the electron trajectory calculation in the planar cathode-lens case:^94,95 when electrons emitted from a point of a planar cathode is accelerated with a constant electric field $F$ (⁠$=V/z$ with $V$ the accelerating potential and $z=l$ is the gap size), the beam divergence calculated as a function of the average transverse energy is equal to $dF/(Fz)$⁠.⁹⁴ This is same as the result of the Gaussian wave function in Eq. (25).⁵² 

In such a geometrical optics calculation, the tangent of the electron trajectories at the anode plane is extrapolated back into the cathode, and the virtual cathode image is defined at their intercept with the optical axis. The position of the virtual cathode image is a function of the initial electron energy at the cathode and, therefore, has a distribution given by the energy spread of the source electrons. This gives the chromatic and spherical aberration of the cathode lens, resulting in the finite virtual source size. The virtual source size dictates the transverse coherence length of the produced beam in down stream with the assumption of the incoherent source as described in Sec. IV B. In contrast, within the monochromatic approximation and the coherence of the source, the electron beam with the Gaussian transverse function is fully coherent. However, it is yet to be elucidated if this leads to the aberration of the transverse structure of the wave function and limits the beam spot size on the sample beyond the diffraction limit.

When the source is of finite size with the radius of $a$ instead of a point, the coherent superposition of the wave function over the finite area leads to the narrowing of the beam.⁹⁸ In Fig. 5, we show the case when the Gaussian wave function is emitted from a source with $a$ (⁠$>σ0$⁠), over which the electrons are phase-coherent. We found that the beam divergence is reduced by a factor of $σ0/a$ from the point source case. There is no counter part of this effect in the classical case. In contrast to this fully coherent case, in the incoherent emitter case, the increase in the source size results in the increase in the virtual source size by convolution.

The transverse structure of the wave function of field emission electrons discussed in Sec. IV C showed that it has a minimal value of the transverse spread [=$σ0$⁠, see Eq. (26)]. This suggests that the coherence fraction $ξ$ ([see Eq. (22)] of any field emission beam is finite with the value approximately given by the ratio $ξmin≃σ0/a$ for an emitter with the rms radius of $a>σ0$⁠, even if the phase-coherence length of electrons in the solid is extremely short. As an example, in the case of molybdenum field emitters fabricated by molding,^25,49,52 $a$ is equal to 2–5 nm. In this case, $ξmin$ is equal to 0.1–0.3. When $ξmin$ conserves upon propagation as discussed in Sec. IV C, the transverse coherence length on the sample plane $Lcoh$ amounts to the significant fraction of the beam spot size (⁠$∼60$ $μ$m for the maximally collimated beam in Ref. 49, see below).

To have an insight into this effect and compare its significance with that of the van Cittert–Zernike theorem, we look into low-energy electron diffraction experiments of a suspended graphene reported in Refs. 28 and 49 using such molybdenum field emitters. We note that the Bragg reflection occurs when the reflection of electrons in a particular direction is reinforced by the finite transverse coherence of the beam. Therefore, when the beam spot size $D$ on the sample is small, and the sample size $S$ is larger than $D$⁠, the Bragg spot size $B$ on the screen will be as small as the direct beam spot size $Ds$ on the screen as long as $Lcoh$ is much larger than the unit cell length $d$⁠. However, when $Lcoh$ is only a few times of $d$⁠, $B$ will be larger than $Ds$ due to the spread of the diffraction angle. Conversely, when $S$ is smaller than $D$⁠, $B$ is determined by $S$ and small (⁠$B$ $≪$ $Ds$⁠) as long as $Lcoh$ is much larger than $d$⁠. This way, the analysis of $B$ with respect to $D$ would give an insight into $Lcoh$ and $S$⁠.

In the experiments reported in Refs. 28 and 49, the low-energy electron diffraction from suspended graphene was studied using a double-gate FEA with the array diameter of 1 mm and a double-gate single-tip field emitter, respectively. Both chips were fabricated by the same fabrication protocol. The emitters were molybdenum with the tip apex radius of 2.5–5 nm and equipped with an electron extraction gate $Gext$ and a beam collimation gate $Gcol$⁠. The nominal aperture diameters of $Gext$ and $Gcol$ were, respectively, $dext=1.2μ$m and $df=7μ$m. Applying a negative voltage -$Vext$ to the emitter with respect to $Gext$⁠, electrons are emitted by field emission over an angle of 30$°$–40$°$⁠. When a reverse voltage with the amount approximately same as $Vext$ is applied to $Gcol$ with respect to $Gext$⁠, the electrons are maximally collimated with the beam divergence limited by $⟨Et⟩$ [= $dF$ in Eq. (5)].²⁸ In the experiment, a graphene sheet suspended on a TEM grid was used as a sample and mounted on a sample holder biased at $Vacc=800$ V with $Gcol$ at the ground potential. Field emission beamlet was accelerated by $Vacc$ toward the sample holder separated by the gap $lgap$ of 2 mm from the source, propagated through the 2 mm-diameter hole over the distance of $lt=2$ mm, and irradiated the graphene. The beam spot size on the sample (evaluated by the shadow image of the TEM grid on the screen) was nominally same as the FEA size in the experiment using the FEA. In the single-tip experiment, the beam spot size on the sample was 59 $μ$m-rms. We note that assuming the planar cathode lens geometry^94,95 and neglecting the lens effect due to the non-uniform electric field near the entrance of the hole (see Fig. 6), the calculated beam radius on the sample is equal to (2 $lgap$+$lt$⁠) $⟨Et⟩/Vacc$ with the value of 76 $μ$m for $⟨Et⟩=0.13$ eV. This is in good agreement with the observation. This is also consistent with the previous experiment of the transverse emittance measurement using the double-gate FEA.

In the single-nanotip experiments, the transverse coherence length on the sample $Lcoh(vCZ)$ determined by the van Cittert–Zernike theorem is equal to 37.9 nm, as given by $λ$/(2 $θg)$ with $θg$= $dcol$/$(2lgap+lt)$⁠, wherein $λ$ given by $Vacc$ is equal to 0.043 nm, and the diameter of the collimated field emission beam at the cathode (where the nominal energy is zero) is assumed to be equal to the diameter of the collimation gate aperture $dcol$⁠. $Lcoh(vCZ)$ is orders of magnitude larger than the unit cell size of graphene. In comparison, when the theory of the partially coherent beam is applied, the transverse coherence length $Lcoh(pc)$ estimated from the beam spot size and $ξmin$ is equal to 6–18 $μ$m for the single-tip experiment. Because of the estimated large coherence fraction of the field emission beam and the macroscopic size of the beam spot size, $Lcoh(pc)$ is orders of magnitude larger than $Lcoh(vCZ)$⁠.

The field emission beam produced by the double-gate FEA with the array diameter of 1 mm showed clear hexagonal Bragg spots from the suspended graphene when the beam was maximally collimated. The Bragg spot size $B$ on the screen was as large as the direct beam size $Ds$ on the screen as expected. Now assuming $D>Lcoh$ on the sample plane, we consider three cases: (1) the sample is larger than the beam spot (⁠$S>D$⁠), (2) the sample is smaller than the beam spot and smaller than the transverse coherence length (⁠$D,Lcoh>S$⁠), and (3) the sample is smaller than the beam spot but larger than the transverse coherence length (⁠$D>S$ and $S≳Lcoh$⁠). In case (1), $B$ is equal to $Ds$⁠. This was the case in the FEA experiment with $D$ equal to $∼1$ mm (or $∼0.25$ mm-rms). Therefore, it is difficult to estimate the amount of $Lcoh$ except it should be several times larger than the unit cell size of the graphene.

In the double-gate single-nanotip field emitter experiment, $B$ also closely matched to $Ds$⁠. This itself again implies that the experiment was in case (1). However, close inspection of the diffraction image showed that, in addition to the main Bragg spots, satellite Bragg spots with the size of $Bsat$⁠, that is 10 times smaller than $B$ and $Ds$ and rotated from the main Bragg spots by 20$°$–30$°$⁠, were observed. This suggests the case (2) with $Lcoh$ in the order of the sample size $Ssat$ or larger, and $Bsat$ determined by $Ssat$⁠. From $Bsat/B$ equal to 1/10, $Ssat$ is estimated to be equal to $∼6μ$m (⁠$=D$/10). Hence, $Lcoh$ is in the order of $6μ$m. This is compatible to $Lcoh(pc)$ of 6–18 $μ$m. The small $Bsat$ is tentatively ascribed to the existence of small islands or defects with the satellite sample size $Ssat$ that are incommensurate to the orientation of the main large area $S$ of the graphene (⁠$S≫Ssat$⁠). We note that when we assume the experiment was case (3), the lower bound of $Lcoh$ equal to $∼20$ nm was evaluated from $Bsat$ by comparing it with a calculated relationship between the beam spot size and the transverse coherence length.^28,49 Incidentally, this value is in the same order of magnitude as $Lcoh(vCZ)$ of 37.9 nm and several orders of magnitude larger than the unit cell size of graphene. Still, the existence of small islands has to be assumed. In conclusion, although $Lcoh(vCZ)$ estimated from the optics consideration following the incoherence source approximation could explain the observation assuming the case (3), the partially coherent source approximation with $Lcoh(pc)$ is not contradictory with the experimental results either. Further experimental and theoretical studies are required to fully elucidate the intrinsic transverse coherence of these field emitters and our interpretation of the graphene island structure.

We review here the beam characteristics of field emitters as a basis for future developments and optimizations of established applications such as the high-resolution electron microscopy as well as for exploring novel microscopy methods and devices. The concepts of beam brightness and the transverse emittance that are closely related but utilized in different contexts have been described with the emphasis on the importance of the transverse energy spread of the field emission beam, which is governed by the electron distribution in the solids. The semi-classical beam characteristics of metal field emitters by geometrical optics approach have been intensely studied in the past, which are instrumental for the design and analysis of various experiments ranging from the interference experiments to ultrafast electron beam experiments by exciting sharp metal tips by femtosecond laser pulses.^106,107 In contrast, the impact of the quantum mechanical nature of electrons in the solid on the field emission beam has not been fully elucidated yet, despite the intense experimental and theoretical studies in relation to the point-source field emitters over the past decades including recent works.^12,108 This might be the key for the cathode development of quantum electron microscopes,¹⁰⁹ miniature on-chip accelerators,⁷⁵ and novel microscopes utilizing the coherent scattering and reconstructions.¹¹⁰ In addition, there are a number of questions that have to be addressed, for example, the influences of the acceleration of electrons from the emitter tip on the propagation of beam brightness and the partial coherence instead of geometrical optics approximation, and the electron wave function emitted from realistic surface structures instead of an ideal flat or atomically smooth surface, that are often assumed in literature, to name a few. Further theoretical and experimental studies are required to elucidate the nature of the transverse coherence of the field emission beam.

I acknowledge Stephen Purcell for helpful discussions, encouragements, and support; Mathieu Kociak and Elisabeth Müller for helpful discussions and providing materials on high-resolution TEMs; Tatiana Latysvkaia for helpful discussions on the coherence, holography, and diffraction imaging experiments; and the reviewers of this article for their valuable comments to improve the article. I also acknowledge following colleagues for discussions and collaborations on the research and development of FEAs : Hans-Heinrich Braun, Prat Das Kanungo, Micha Dehler, Thomas Feurer, Hans-Werner Fink, Jens Gobrecht, Vitaliy Guzenko, Patrick Helfenstein, Eugenie Kirk, Chiwon Lee, Simon Leemann, Dwayne R. J. Miller, Anna Mustonen, Mahta Monshipouri, Youngjin Oh, Martin Paraliev, and Albin F. Wrulich.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.