Development of a 3-D energy-momentum analyzer for meV-scale energy electrons

Over the past several decades, the study of photoelectron energy and momentum distributions has revealed to us a great deal regarding the chemistry and physics of materials.¹ Today, commercially available angle resolved electron energy analyzers are capable of measuring the energy and angular distributions (or equivalently, the 3-D momentum distributions) with sub-millielectronvolt energy resolutions and are routinely used for X-ray Photoelectron Spectroscopy (XPS) or Angle Resolved Photoelectron Spectroscopy (ARPES) to deduce the chemical composition and electronic structure of materials. Such analyzers either use concentric hemispheres as energy filters and image the emission angles of the electrons on a screen² or utilize a series of electron lenses with a delay-line-detector to measure the time-of-flight³ to deduce the electron energies and momenta. Typically, these analyzers are designed to study the energy and momentum distributions of electrons with (kinetic) energies ranging from ∼1 eV to a few kiloelectron volts.⁴ 

Usually, in XPS or ARPES applications, the photon energies used to emit electrons are much larger than the work-function and hence, the energies of the photoelectrons emitted are of the same order of magnitude as the photon energies (∼few 10s of eV to few keV).¹ More recently, with the advent of laser-based-ARPES, energy-momentum distributions of photoelectrons emitted with photon energies only a few electron volts larger than the work function have been investigated.⁵ The photoelectrons emitted and detected in such laser-based-ARPES techniques have energies ranging from several 100 meV to a few electronvolts. Thus, commercially available analyzers are sufficient to study the range of electron energies of interest to all well-developed XPS and ARPES applications.

Photoemission is also widely used as a source of electrons for ultra-fast electron diffraction and microscopy⁶ and linear particle accelerator applications like energy recovery linacs⁷ and free electron lasers.⁸ The brightness of electron beams is a key figure of merit that determines the performance of such applications. With significant advancements in beam shaping and space-charge management techniques in the past several decades, today the beam brightness is often limited by the mean transverse energy (MTE) (or equivalently the rms transverse momentum) of the electrons photoemitted from the cathode.⁹ Here, the transverse direction is the direction perpendicular to the direction of electron beam propagation (or the direction parallel to the cathode surface). A smaller mean transverse energy (MTE) implies brighter electron beams. Hence, in the past decade, a lot of effort has gone into studying the photoemission process and minimizing the MTE obtained from photocathodes. Often, reducing the MTE is a trade-off between other crucial performance metrics of photocathodes like the quantum efficiency (QE) and response time.^10,11

The smallest MTEs are obtained when the photon energy is very close to the photoemission threshold, and the energies of the emitted electron are near zero (up to a couple of 100 meV). Measuring and studying the energy and momentum distributions of such low energy electrons is crucial to understanding the photoemission process at photon energies very close to the photoemission threshold and develop cathode materials that can minimize the MTE without significantly impacting other crucial photocathode metrics. Such low energy electrons are very slow and have long (up to 15 nm) de Broglie wavelengths. Hence, the physics of photoemission of these electrons can be very different from the higher energy electrons typically used for photoelectron spectroscopy. Therefore, developing an analyzer capable of measuring energy and 3-D momentum distributions of very low energy (sub-100 meV) electrons is essential and has fundamental physics relevance. Commercially available energy-momentum analyzers used for photoelectron spectroscopy are not designed to measure such low energy electrons. Furthermore, such analyzers only accept electrons emitted in a fairly narrow cone about the surface normal and cannot measure the electrons emitted at large angles. It is crucial to measure these electrons due to their significant contribution to the MTE.

One challenge in measuring such low energy electrons is the sensitivity of their trajectories to stray magnetic and electric fields. In the past, several techniques that minimize the sensitivity to stray fields have been developed to measure the transverse and longitudinal energy distributions of such low energy electrons. So far, these techniques have been successfully implemented to either measure the 2-D transverse momentum distributions,^12,13 2-D longitudinal and transverse energy distributions,^14,15 or 1-D longitudinal energy distributions.¹⁶ 

In this paper, we report of the development of a technique to measure the complete 3-D energy and momentum distributions of very low energy electrons using the time-of-flight approach. Using this approach, we have developed an electron energy analyzer that can measure these distributions with a sub-millielectronvolt expected resolution. In Sec. II, we discuss the principle of the time-of-flight based measurement along with the design of the analyzer and the various contributions to its resolution and possible sources of error. In Sec. III, we show the measurements of the surface state on the (111) surface of silver as an example and discuss the calibration and operational details of the analyzer. A techniques similar to the one presented below was developed by Kirchmann et al. However, their implementation did not allow accurate measurement of very low energy electrons and was capable of detecting electrons emitted only in a 22° cone about the normal.¹⁷ 

Figure 1(a) shows the cross section of the analyzer. The analyzer has a parallel plate geometry, with one plate being the sample (photoemission surface) along with an electrostatic shield electrode and the other being the time-of-flight detector. The electrostatic shield is a cylindrical disk with a central hole into which the sample is inserted. The electrostatic shield and the sample are isolated electrically. Figures 1(b) and 1(c) show the cathode and electrical shield assembly from different views to show the detailed mechanical configuration. As seen in Fig. 1, the sample (blue) is mounted onto a plug (green) which is inserted into a copper holder. The dimensions of the assembly are designed such that the sample surface is flush with respect to the electrostatic shield surface. The sample diameter is 1 mm smaller than the diameter of the hole in the electrostatic shield so that the sample does not touch the shield and remains electrically isolated from it. This gap is small enough that the equipotential lines are not distorted significantly.

Figure 2 shows the connection of the copper holder and the electrostatic shield to the liquid Helium (LHe) cryostat via flexible copper straps. The flexible copper straps are essential to allow for the length contraction upon cooling. The cryostat is mounted onto a 4-axis x-y-z-ϕ manipulator. ϕ is the angle about the axis of the cryostat. The x-y-z directions have a precision of 100 μm, where as the ϕ motion has a precision of 0.1°.

The electrostatic shield is uniformly coated with a graphite film to obtain a uniform work function. The voltage applied to the electrostatic shield roughly is equal to the voltage applied to the sample plus the work function difference between the sample and the shield. This causes the electric field in the region between the sample and the detector to be uniform. The detailed procedure for obtaining the uniform electric field is discussed in Sec. III. A small accelerating voltage of a few volts (∼0–10 V) is applied between the sample and the detector. The whole setup is placed in an ultra high vacuum chamber with walls made of mu-metal to minimize the magnetic fields within the chamber. Combined with nonmagnetic construction, the fields at the sample are only a few milli-Gauss. The sample is thermally connected to the LHe cryostat to enable cooling to cryogenic temperatures. Figure 2 shows the 3-D model of the analyzer.

A femtosecond to picosecond scale laser pulse is focused into a spot of size less than 100 μm FWHM on the sample center. The intensity of the laser pulses is kept small enough that no more than one electron is emitted per pulse. Due to the uniform electric field, the emitted electrons are accelerated only in the longitudinal direction toward the detector and do not experience any significant transverse (x or y, along the direction parallel to the sample surface) acceleration (more details discussed in Sec. II D). The detector detects the x and y position of the electrons and the time required by the electrons to travel to the detector (t). The laser spot size on the sample is much smaller than the spot formed by the electrons on the detector; hence, we can assume that the electrons are emitted from a single point on the sample. From these measurements, momenta of the emitted electrons (ℏk_x, ℏk_y, and ℏk_z) can be calculated as

and the total energy can be calculated as

where m_e is the electron rest mass, e is the absolute value of the electron charge, d is the effective distance between the sample and the detector, V is the voltage applied to the sample (the detector is grounded), and V_off is the work function difference between the sample and the detector (work function of the sample minus the work function of the detector). $E⊥=ℏ22mekx2+ky2=ℏ2kr22me$ is the transverse energy, and $E∥=ℏ2kz22me$ is the longitudinal energy. Here, $r=x2+y2$ is the radial coordinate at which the electron hits the detector.

The detector records x, y, and t over a period of time for many electrons in order to generate a distribution $Nx,y,tdxdydt$⁠. This distribution can be converted into a distribution over the total energy E and the two transverse momenta k_x and k_y as $Nkx,ky,EJkx,ky,EdkxdkydE$⁠, where $Jkx,ky,E$ is the determinant of the appropriate Jacobian matrix and is given by

From Eqs. (1) and (2), we can see that the transverse momenta depend only on the measured values of x, y, and t and do not require the knowledge of the distance d and the work function difference V_off. However, calculating the longitudinal momentum ℏk_z and the total energy E requires the accurate knowledge of d and V_off. Obtaining these values accurately requires the use of some known feature in the energy or longitudinal momentum distribution like the energy of a surface state or the Fermi level. An example method of estimating the values of d and V_off using the surface state of Ag(111) has been shown in Sec. III.

A time resolved imaging MCP detector based on delay line detection (DLD4444 manufactured by Surface Concepts¹⁸) is used to detect the electrons and measure x, y, and t. The detector has a spatial resolution better than 40 μm and an intrinsic time resolution better than 30 ps. In practice, due to binning, the spatial resolution used for the measurements presented in this paper is 0.11 mm in the x direction and 0.12 mm in the y direction. The spatial resolution is also limited by the laser spot size on the sample to 100 μm FWHM. The delay line detector is triggered by an electrical pulse generated by a small fraction of the laser pulse incident onto a photodetector. The practically achieved temporal resolution of 170 ps FWHM is limited by the timing jitter between the laser and the trigger to the delay line detector.

The detector has an active area of 40 mm and comprises of a fine grid followed by a MCP. Behind the MCP, two meandering delay lines (one for the x direction and the other for the y direction) detect the amplified electron pulse to give the x and y positions and the time t of the electron hitting the MCP. For the measurement of the time of emission (t = 0), the sample (along with the holder assembly) is rotated about the axis of the cryostat to make the laser pulse reflected off the sample incident onto the delay line detector. The laser pulse reflected off the sample surface is detected by the delay-line-detector and is used to set the time of emission as t = 0. The temporal width of the reflected femtosecond laser pulse gives the δ function response of the detector and thus the effective temporal resolution of the detector after including the effect of the jitter. For an isotropic distribution of the emitted electrons, x = 0 and y = 0 can be set as the location of the centroid of the x and y positions of all the electrons detected.

The detector is capable of reliably detecting at most one count per laser pulse. The laser power incident on the sample must be reduced to ensure this condition.

The total energy resolution δE can be expressed as the sum of the longitudinal and transverse energy resolutions as δE = δE_∥ + δE_⊥.

From Eqs. (1)–(3), we can calculate

and

where $κd=2E∥med1t−Veet2med2$⁠, $κt1=2E∥med−dt2+Vee2med$⁠, $κVe=2E∥medet2med$⁠, $κr=mert2$⁠, $κt2=mer2t3$⁠, and V_e = V − V_off. The time t can be given as a function of the longitudinal energy E_∥ to obtain the terms κ_dδd, κ_t1δt, and $κVeδVe$ as functions of E_∥, d, and V_e. The value of d is constrained to approximately 40 mm due to the geometry of the setup. While the value of temporal uncertainty δt can be measured easily as the temporal resolution of the detector (see Sec. II B), the possible values of the uncertainties in d(δd) and V_e(δV_e) are more difficult to determine. Here, we make educated guesses to the values of these uncertainties.

The variation in d arises from a combination of the angle offset between the sample and the detector, vibrations in the system, roughness of the sample and the shield electrode, and the surface nonuniformity of the MCP in the detector. Figure 3(a) shows the contribution of the uncertainty in d to the energy resolution (κ_dδd) for several values of V_e. The contribution of the uncertainty in d reduces as the voltage between the sample and the detector V_e is reduced. When δd is assumed to be equal to 100 μm, the value of κ_dδd is less than 1 meV even for longitudinal energies as large as 100 meV at V_e = −1 V. Even for very large values of uncertainty in d (as large as 1 mm), the contribution to the energy resolution can be made small by reducing the effective voltage between the sample and the detector.

The variation in V_e can arise from the noise in the voltages supplied to bias the sample and the shield electrode or from the work function variations on the sample, shield electrode, and detector surfaces. The noise in the bias voltages has been measured to be less than 1 mV. Although care has been taken to maintain the uniformity of the work functions on the surfaces of the shield and the detectors (by coating them with a graphite film), one might still expect work function variations over mm scales due to adsorption of residual gas atoms and impurities limiting the value of δV_e to higher than 1 mV. As seen in Fig. 3(b), even for very large variations in the voltage (δV_e = 100 meV) due to possible work function variations, we can get a sub-10 meV energy resolution contribution of voltage uncertainty at 100 meV longitudinal energy at V_e = −1 V. For values of δV_e limited to 1 meV only by noise in the applied voltages, the contribution of voltage uncertainty is less than 1 meV, as shown in Fig. 3(b). The voltage uncertainty contribution to the energy resolution reduces with the effective voltage between the detector and the sample.

The effect of the temporal resolution on the longitudinal energy resolution is shown in Fig. 3(c). We see that for a temporal resolution as large as 200 ps (slightly larger than the measured resolution of 170 ps), at V_e = −1 V, the contribution to the energy resolution is about 1 meV.

As seen from Fig. 3, the uncertainty contributions of d and t to the energy resolution reduce with V_e. However, the uncertainty contribution of V_e to energy resolution increases with V_e. Hence, there is an optimum voltage to get the maximum energy resolution. The longitudinal energy resolution (assuming δd = 0.1 mm, δV_e = 1 mV, and δt = 170 ps) as a function of the longitudinal energy is shown in Fig. 3(d) for various values of V_e. For the values of the uncertainties assumed here, the longitudinal energy resolution is 3 meV for a longitudinal energy of 100 meV at V_e = −1 V and lower at lower energies and voltages. Although this is a realistic uncertainty estimate for the current setup, it can be significantly improved by an appropriate mechanical design to minimize vibrations, align the sample, shield and detector; by improving the electrical design to minimize the voltage fluctuations; and by improving the temporal jitter between the laser and the detector. The ultimate resolution will be determined by the temporal resolution of the detector and is below 0.1 meV even for 100 meV longitudinal energy electrons.

If the maximum longitudinal energy is much smaller than −eV_e, for the purpose of calculating the resolution, we can assume that all electrons start with near zero energies and $t≈2me−Veed$ for all electrons. Using this value of t, we get

Under the above assumption, we see that the longitudinal energy resolution is proportional to the square root of the longitudinal energy. This assumption also allows us to obtain a straight forward, quantitative estimate of the transverse energy and momentum resolution.

Figure 4 gives the transverse energy and transverse momentum resolution. The uncertainty in the radial coordinate is the sum of the effective detector pixel size after binning (∼120 μm) and the laser spot size on the sample (∼100 μm). The temporal resolution for the calculations in Fig. 4 is assumed to be 200 ps. We can see that the transverse energy resolution is ∼2 meV for 100 meV transverse energy at V_e = −1 V. This translates to a transverse momentum resolution of ∼1.5 × 10⁻³ 1/Å at V_e = −1 V, as seen in Fig. 4(b). This is comparable to the transverse momentum resolution obtained in most ARPES systems and a tiny fraction of a typical Brillouin zone width. Nearly a factor of 3 better resolution can easily be achieved by reducing δr. This can be done by reducing the laser spot size and by reducing the binning to increase the detector’s position resolution.

It is critical to establish a constant electric field along with a near zero magnetic field in the space between the sample/electrostatic shield and the detector. The mu-metal vacuum chamber acts as a magnetic shield reducing the magnetic field below 10 mG. The sample/electrostatic shield and the detector form a parallel plate capacitor like geometry. The outer diameter of the detector is roughly 2 times larger than the detector sample distance, ensuring sufficiently constant electric field in the region in which electrons travel toward the detector. In this section, we obtain a quantitative estimate of the errors produced in the measurement due to the nonuniform electric fields and the stray magnetic fields.

The nonuniformity in the electric field is due to the fringe effects from the finite size of the electrostatic shield and the detector and inaccurate determination of the work function difference between the sample and the electrostatic shield. For assessing the contribution of each of these, the electric fields between the sample and the detector were calculated for various values of the effective voltage between the sample and the detector (V_e) using the 2-D electrostatic solver POISSON-Superfish.¹⁹ Figure 5 shows the cross section of the cylindrically symmetric geometry used for the calculation and the equipotential lines calculated for one example voltage of V_e = −4 V.

The sample and the electrostatic shield, in general, have different work functions producing a significant nonuniformity in the electric fields. If the work function difference is known, an equivalent offset can be added to the voltages applied to the sample and the electrostatic shield. Determining the work function difference accurately is nontrivial, and a technique to do so to an accuracy of ∼5 mV has been presented in Sec. III. To account for this inaccuracy in the determination of the work function difference, a difference of 5 mV between the effective voltages of the sample and the electrostatic shield has been assumed in the calculations of the electric fields.

Trajectories of electrons launched with various longitudinal and transverse energies from the center of the sample were traced in the calculated electric fields to obtain the final transverse position and the time-of-flight to the detector. The longitudinal and the transverse energies were then calculated from the final transverse position and the time of flight using Eqs. (1)–(3) and compared to the initial energies that the electrons were launched with.

Figure 6 shows the difference between the longitudinal energy calculated from the simulated time-of-flight and the initial launch longitudinal energy (ΔE_∥) as a function of the initial longitudinal energy for various values of V_e. Figure 6(a) corresponds to electrons tracked in the electric field generated by assuming a 5 mV difference between the effective voltages of the sample and the electrostatic shield, while Fig. 6(b) corresponds to electrons tracked in the electric field generated by assuming the effective voltage difference to be 20 mV.

For higher values of V_e, the effects of the fringe distortions due to finite size of the detector and the shield electrode dominate causing ΔE_∥ to become larger than 10 meV even at small (∼10 meV) longitudinal energies, as seen in Fig. 6. For smaller values of V_e < −2 V, if the effective voltage difference between the sample and the shield is 5 mV or less, the effects of the fringe distortions are minimal and ΔE_∥ is limited to sub-millielectronvolt values for longitudinal energies as small as 10 meV, as seen in Fig. 6(a). However, for larger values of the effective voltage difference [20 mV as seen in Fig. 6(b)], ΔE_∥ is several millielectronvolt for longitudinal energy of 10 meV even at small values of V_e. It was noted that the values of ΔE_∥ did not depend strongly on the transverse energy of the electrons.

Similar calculations were performed for the inaccuracy in the transverse energy, and the electric field nonuniformities were found to have a negligible effect on the transverse energy inaccuracy at all values of V_e, even for 20 mV difference in the effective voltages of the sample and the shield electrode. It was found that adding a magnetic field of 10 mG in all three directions did not affect the trajectories significantly and produced errors of less than 1% in the calculated energies.

The above results suggest that the operation at low values of V_e results in a higher accuracy in the longitudinal energy and reducing the effective voltage difference between the sample and the electrostatic shield is critical. We also see that the field variations do not significantly affect the transverse energy measurement making this a very robust technique to measure the mean transverse energy of photoemitted electrons.

In this section, using the Ag(111) surface as an example, we demonstrate the procedure developed to obtain the uniform electric field in the region between the detector and the sample and also the procedure to obtain the exact distance d and voltage offset V_off between the sample and the detector.

A commercially bought Ag(111) single crystal was exposed to several cycles of ion bombardment with 1 keV Ar⁺ ions and annealing to 450 °C. This produces an atomically ordered surface with a Shockley surface state. The cleanliness and atomic ordering of the surface was confirmed using low energy electron diffraction and Auger electron spectroscopy. The clean, ordered Ag(111) surface was also later confirmed by the existence of the surface state. The base pressure in the chamber was in the low 10⁻¹⁰ Torr range. The work function of this surface is known to be 4.45 eV.¹¹ This surface was then used to perform the calibrations presented in this section.

Electrons were photoemitted using a frequency tripled Ti-sapphire laser with a tunable photon energy and a pulse length of ∼150 fs. The photon energy could be tuned from 4.2 to 4.9 eV. The laser had a repetition rate of 76 MHz. However, this was reduced by a factor of 20–3.8 MHz using an acousto-optic Bragg cell pulse picker to make the repetation rate less that 7 MHz (the maximum trigger rate for the DLD4444 detector). This laser was also used to generate an electrical pules to trigger the DLD4444 detector. During measurement, the laser power was adjusted to keep the count rate of electron hitting the detector to less than 50 000 counts/s to avoid damage to the MCP in the detector and ensure that not more than one electron is emitted per laser pulse.

As shown in Sec. II D, ensuring that the electric field is uniform between the sample and the detector is crucial to obtain the longitudinal energies and, to a certain extent, essential to calculate the transverse energy/momentum of the emitted electrons from the x, y, and t measurements. This has been achieved through the use of an electrostatic shield electrode. Ideally, the sample surface and the electrostatic shield surface facing the detector should lie in the same plane. However, due to manufacturing imperfections, it was found that the plane of the sample surface was slightly recessed from the plane of the electrostatic shield surface.

In order to ensure a uniform electric field, the voltages applied to the sample and the electrostatic shield electrode need to be offset. This offset is due to the work function difference between the sample and the electrostatic shield and also due to the fact that the plane of the sample is slightly recessed from the plane of the electrostatic shield. Figure 7(a) shows a schematic of the geometry with the slightly recessed sample. Let the distance between the plane of the sample surface and the plane of the electrostatic shield surface be d₁. The electric field between the sample and the detector will be uniform when the following condition is satisfied:

Here, V_sam and V_sh are the voltages applied to the sample and the shield, respectively, and W_sam and W_sh are the work functions of the sample and the shield, respectively. This relationship can be rewritten as

where K₁ = −K₂W_sam + W_sh and $K2=d−d1d$⁠. Since K₂ ≈ 1, K₁ = W_sh − W_sam.

When the condition in Eq. (13) [and hence also in Eq. (14)] is satisfied, the spot formed by all the electrons hitting the detector does not change position for small changes in the position of the sample + shield assembly in the direction parallel to the plane of the sample surface, as shown in Fig. 7(b). Note that the position of the laser spot is held fixed with respect to the detector as the sample + shield assembly is moved and electrons are emitted from an off-center location on the sample.

When the condition in Eq. (13) is not satisfied, the electrostatic shield essentially acts like an electron lens. In such a scenario, the electrons emitted from an off-center location on the sample get deflected by the lensing effect of the shield. Thus, when the position of the sample + shield assembly changes with respect to the detector, the electron spot on the detector changes position, as shown in Figs. 7(c) and 6(d).

For a given sample voltage, the shield voltage can be adjusted until the centroid of the electron spot on the detector stops moving with small changes in the position of the sample + shield assembly. This is the shield voltage for which the condition in Eq. (13) is satisfied. The values of K₁ and K₂ can be obtained by fitting a line through a plot of various values of V_sam and the corresponding values of V_sh. For the case of the Ag(111) surface, the values of K₁ = −375 mV and K₂ = 0.99 were obtained.

Once a uniform electric field is established between the sample and the detector, the transverse momentum and energy can be obtained directly from the x, y, and t measurements as seen from Eqs. (1) and (2). However, obtaining the longitudinal energy/momentum requires the knowledge of the distance between the sample and the detector d and the voltage offset V_off as evident from Eq. (3).

From the design, the value of d is expected to be around 40 mm. However, owing to the mechanical imperfections in manufacturing and assembly and the unknown exact point of detection of the electrons within the DLD4444 detector, it is essential to calibrate the exact value of d from the energy distribution measurements.

V_off is essentially the work function of the sample relative to that of the detector. A rough value of this can be obtained increasing the voltage on the sample [while adjusting the voltage on the shield according to Eq. (14)] until the electrons stop reaching the detector. This voltage provides a rough estimate of the work function difference between the sample and the detector as the front face of the detector is grounded. However, the uncertainty in V_off obtained from this technique is a couple of 100 meV due to the energy spread of the electrons and due to the fact that the electron spot on the detector becomes larger than the detector size at very small values of the effective accelerating voltage V_e, making detection of all emitted electrons impossible. A value of V_off ≈ 400 meV was obtained using this technique.

Below, we demonstrate another technique that allows us to determine both d and V_off to a much better accuracy using a known feature in the longitudinal or total energy distributions. The surface state on the Ag(111) surface is one such feature which manifests itself as a peak in the longitudinal (and total) energy distribution.²⁰ 

In our case, the longitudinal energy distributions obtained from the time-of-flight measurements using Eq. (3) need to satisfy the following two conditions of physicality:

• the location of the peak corresponding to the surface state in the longitudinal energy distribution (and more generally the entire longitudinal energy distribution) must remain invariant with the voltage V applied to the sample during the measurement [assuming an appropriate corresponding voltage given by Eq. (14) is applied to the shield to ensure uniform electric field].

• the surface state peak in the longitudinal energy should change by an equivalent amount with a change in photon energy.

When calculating the longitudinal energy distributions from the time-of-flight (t) measurements, both the above conditions are satisfied for a unique combination of the values of d and V_off.

Figure 8(a) shows the temporal distributions measured for two voltages V = −1 V and −2 V at two photon energies of 4.660 eV and 4.605 eV. Longitudinal energy distributions are calculated from these temporal distributions for various values of d around d = 40 mm and V_off around V_off = 400 mV. The difference of the energy corresponding to the peaks in the longitudinal energy distributions calculated from the temporal distributions measured at the two voltages at the photon energy of 4.660 eV is shown in Fig. 8(b) for various values of d and V_off. We see that the difference becomes nearly zero along a line in the d-V_off plane. All values of d and V_off on this line satisfy the first condition of physicality.

Figure 8(c) shows the difference of the energies corresponding to the peaks of the longitudinal energy distributions for the two photon energies minus the difference between the two photon energies (55 meV) for various values of d and V_off. We see that the difference also becomes nearly zero along a line in the d-V_off plane. All values of d and V_off on this line satisfy the second condition of physicality.

Both the conditions are satisfied at the intersection of these two lines which happens at d = 40.5 ± 0.1 mm and V_off = 405 ± 20 mV. Figure 8(d) shows the longitudinal energy distributions calculated from the temporal distributions in Fig. 8(a) using the values of d = 40.5 mm and V_off = 405 mV. Note that the accuracy of d and V_off produced by such a procedure depends strongly on the sharpness and intensity of the feature.

Here, we used a feature [the surface state on the Ag(111) surface] in the longitudinal energy distribution to obtain the values of d and V_off accurately. In the absence of such a feature, one may similarly use the Fermi energy in the total energy distribution for obtaining the exact values of d and V_off.

Upon obtaining the accurate values of d and V_off, we can calculate the 3-D energy and transverse momentum distribution (or equivalently the energy resolved transverse momentum distribution) using Eqs. (1)–(4) from the measured 3D x, y, and t distributions. Various cross sections of the k_x-k_y-E distributions are shown in Fig. 9.

Figure 9(a) shows the k_y = 0 cross section for the E vs k_x distributions for photon energy of 4.660 eV. This cross section shows the parabolic 2-D Shockley surface state. A parabola can be fitted to the maximum intensity points for each value of k_x. The effective mass computed from this parabola is found to be 0.53m_e and is in good agreement with previous results at room temperature.²¹ Figure 9(b) shows the k_x-k_y cross section for an energy of E = 200 meV for a photon energy of 4.660 eV. This shows a ring in the momentum distribution due to the 2-D surface state. Figures 9(c) and 9(d) show the k_y = 0 cross section for the E vs k_x distributions for photon energies of 4.605 eV and 4.550 eV along with the same parabolic fit used for the 4.660 eV case. Within the resolution of the system (this includes the band broadening due to thermal effects at room temperature and photon energy spread of the laser along with the resolution of the instrument), the surface state is identical except for a shift in energy that corresponds to the changing photon energy, demonstrating the ability of this instrument to measure millielectronvolt energy scale photoelectrons accurately. All measurements shown in Fig. 9 were performed at V = −1 V.

The MTE is a crucial figure of merit for photoemission based electron sources. As shown in Sec. II, the time-of-flight setup can give very accurate transverse energy distributions that do not require the knowledge of d and V_off and are relatively robust toward stray fields. This makes the time-of-flight an excellent technique to measure the MTE. Figure 10 shows the MTE obtained as a function of photon energy. These measurements agree well with previous measurements performed using other techniques.¹¹ We see that the MTE first reduces with photon energy, reaches a minimum, and then increases again. At threshold photon energies, we are exciting electrons from the Fermi level, where the surface state has a significant transverse momentum. As the photon energy is increased, electrons closer to the bottom of the surface state band with lower transverse momentum are emitted. At the binding energy of the surface state, electrons are emitted with zero transverse momentum. Thus, as we increase the photon energy from threshold by a value of the surface state energy, MTE reduces to a minimum value and then increases again as high transverse momentum bulk states at the Fermi level start to emit.

In this paper, we have reported the design and development of a 3-D energy-momentum analyzer based on the time-of-flight measurement. This analyzer can obtain sub-millielectronvolt resolution while being capable of measuring millielectronvolt-scale energy electrons. Such an analyzer will be an ideal tool for investigating photoemission processes close to the photoemission threshold and aid in the development of electron sources with very small energy and transverse momentum spreads. We have presented a detailed analysis of all possible sources of error and concluded that one should be able to achieve a sub-millielectronvolt resolution in both the longitudinal and transverse energy. Using the example of the Ag(111) surface state, we have also demonstrated the operation of this analyzer and discussed various techniques to calibrate to unknown parameters required for accurate measurements of the longitudinal and transverse energies and momenta.

In practice, demonstrating a sub-millielectronvolt energy resolution requires a very sharp feature in the energy distribution and a very narrow band-width laser. Sharp features in the energy distribution can be obtained at cryogenic (LHe) temperatures and the narrow band-width can be obtained by using picosecond laser pulses. In this case, this was done by dispersing light from the 150 fs Ti Sapphire laser oscillator and energy selecting with a slit. Such a reduction in bandwidth results in pulse length broadening. With the implementation of these two, it should be possible to demonstrate the sub-millielectronvolt resolution of this analyzer at sub-100 meV electron energies.

This work was supported by the U.S. National Science Foundation under Award No. PHY-1549132, the Center for Bright Beams, and the Director, Office of Science, Office of Basic Energy Sciences of the U.S. Department of Energy, under Contract Nos. KC0407-ALSJNT-I0013 and DE-AC02-05CH11231. We would also like to acknowledge the useful discussions with Dr. Andreas Olsner regarding the operation of the delay line detector.