Expansion of Momentum Space and Full 2$π$ Solid Angle Photoelectron Collection in Laser-Based Angle-Resolved Photoemission Spectroscopy by Applying Sample Bias

TL;DR

Bias ARPES leverages a small sample bias to bend low-energy photoelectrons into the analyzer, expanding the accessible momentum space from a laser ARPES setup and enabling nearly full $2\pi$ solid-angle collection with a $6.994\,$eV photon source. The authors establish a robust detector-to-emission-to-momentum conversion framework, grounded in a parallel-plate-capacitor field model, and correct spectral-weight distributions via Jacobians, with the sample work function $W_S$ determined from bias-enhanced photoemission. They demonstrate the method on Bi2212 and CsV$_3$Sb$_5$, achieving near-complete momentum coverage and access to antinodal, Dirac, and van Hove features, while carefully characterizing how bias degrades energy and angular resolutions and how beam size and off-normal geometry can mitigate these effects. The approach preserves laser-ARPES advantages—high energy/momentum resolution and bulk sensitivity—while dramatically expanding momentum-space reach, and is applicable across photon energies and material systems, pending careful determination of the effective bias $\eta U^*$ under experimental conditions.

Abstract

Angle-resolved photoemission spectroscopy (ARPES) directly probes the energy and momentum of electrons in quantum materials, but conventional setups capture only a small fraction of the full 2$π$ solid angle. This limitation is acute in laser-based ARPES, where the low photon energy restricts momentum space despite ultrahigh resolution. Here we present systematic studies of bias ARPES, where applying a sample bias expands the accessible momentum range and enables full 2$π$ solid angle collection in two dimension using our 6.994 eV laser source. An analytical conversion relation is established and validated to accurately map the detector angle to the emission angle and the electron momentum in two dimensions. A precise approach is developed to determine the sample work function which is critical in the angle-momentum conversion of the bias ARPES experiments. Energy and angular resolutions are preserved under biases up to 100 V, and minimizing beam size is shown to be crucial. The technique is effective both near normal and off-normal geometries, allowing flexible Brillouin zone access with lower biases. Bias ARPES thus elevates laser ARPES to a new level, extending momentum coverage while retaining high resolution, and is applicable across a broad photon-energy range.

Figures (16)

• Figure 1: Illustration of the Momentum Coverage in the Laser ARPES Measurements with a Photon Energy $h\nu$=6.994 eV and the Analyzer Detection Angle of $\pm$15$^\circ$. Here we take the cuprate superconductors as an example. The gray square represents the first Brillouin zone of the cuprate superconductors. The blue solid lines correspond to typical Fermi surface of cuprate superconductors. The red circle indicates the maximum accessible momentum range when the full 2$\pi$ solid angle of photoelectrons are collected in the laser ARPES measurements with a photon energy of 6.994 eV. Here the sample work function of 4.3 eV is assumed. The green line marked with Cut1 represents the momentum space that can be covered by a single momentum cut with the detector angle of $\pm15^\circ$ near the Brillouin zone center. The other green line marked with Cut2 represents the momentum space that can be covered near the Brillouin zone boundary by the same $\pm15^\circ$ detector angle. It is evident that, to cover the entire Brillouin zone, a lot of momentum cuts need to be carried out and combined together.
• Figure 2: Schematic Diagram of the Instrument Setup for Bias ARPES Measurement.a, Schematic layout of the cryostat (left side) and the electron energy analyzer (right side). The sample is mounted on top of the sample post which is screwed into a copper holder. The copper holder is connected with another copper block by a flexible copper braid. The copper block is mounted onto the cold head of the cryostat with a sapphire piece in between. The sapphire piece serves to electrically isolate the sample and is thermally conductive. A cable is connected to the sample holder which can either ground the sample or apply bias voltage on the sample. The electron energy analyzer measures the energy and angle of photoelectrons which are emitted from the sample surface along different directions. The detector angle $\varphi^D$ is defined along the angle direction of the detector. b, Schematic view of the bias ARPES in the X-Z plane. The emission angle $\phi^S$, the acceptance angle $\psi^A_H$ and effective emission angle $\beta_H$ are defined. In the Y-Z plane the corresponding angles are $\theta^S$, $\psi^A_V$, and $\beta_V$. c, Three-dimensional schematic view of the bias ARPES. When a negative bias voltage is applied to the sample, the photoelectrons are bent towards the lens axis and enter the cone of the electron energy analyzer. When the bias voltage is high enough, all the photoelectrons can be collected into the analyzer. In the analyzer entrance plane, the position of a photoelectron is defined by (X$_A$,Y$_A$) in the X-Y coordinate system. d, Definitions of the sample orientation and the velocity of a photoelectron. The sample orientation is determined by three angles $\phi^S$, $\theta^S$ and $\omega$. The velocity of the photoelectron right at the sample surface is represented by $\vec{V_{0}}$ which has three components (V$_{0x}$, V$_{0y}$, V$_{0z}$) in the x-y-z coordinate system.
• Figure 3: The Maximum Emission Angle of Photoemitted Electrons that can be Collected at Different Bias Voltages.a, Calculated color plot showing the maximum emission angle that can be collected by applying different bias voltages (horizontal axis) on the sample with different work functions (vertical axis). The plot is calculated based on the condition that the photon energy is 6.994 eV, the analyzer work function is 4.3 eV and the analyzer detection angle is $\pm$15$^\circ$. The maximum collected emission angle is represented by both the color and lines of constant emission angle. b, The maximum collected emission angle as a function of the sample bias voltage for three typical sample work functions of 2.5 eV (red line), 4.3 eV (green line) and 5.5 eV (blue line). The inset shows the maximum momentum coverage for the three work functions when the entire 2$\pi$ solid angle of photoelectrons is collected. The dashed square represents the first Brillouin zone of cuprate superconductors with a lattice constant $a=3.8$ Å.
• Figure 4: The Angular Magnification Factor ($M_A$) between the Detector Angle and the Emission Angle under different Bias Voltages. A small region ($dA_D$) at the detector angle $A_D$ is mapped into a region ($dA_E$) at the emission angle $A_E$ as shown in the inset in a. The angular magnification factor ($M_A$) is defined as $d A_E /d A_D$ for the detector angle $A_D$ and the emission angle $A_E$. a, Calculated angular magnification factor ($M_A$) as a function of the detector angle under different bias voltages which is varied every 10 Volts. b, Calculated angular magnification factor ($M_A$) as a function of the emission angle. Here, the photon energy is $h\nu=6.994\,eV$, the analyzer work function is $W_A=4.3621\,eV$ and the sample work function is taken as $W_S=4.3\,eV$.
• Figure 5: Sample work function measurements by using the normal emission photoemission spectrum. The photon energy h$\nu$ is 6.994 eV.(a-e) Constant energy contours of polycrystalline gold measured at a bias voltage |U|=90 V, at the binding energy (E$_B$) of 0 eV (a), 0.5 eV (b), 1.0 eV (c) , 1.5 eV (d) and 1.7 eV (e). The spectral intensity progressively concentrates to the normal emission region with increasing binding energy. f-j, Photoemission images taken along five cuts at different angle $\vartheta^D$. The location of these cuts is shown by red lines in (a). The photoelectrons form a cone in the energy-angle space, with the cone bottom at the normal emission. k, Integrated photoemission spectra (energy distribution curves, EDCs) obtained from (f-j) by integrating the spectral intensity over all the detector angle $\varphi^D$. The total intensity is also presented (black line) which is obtained by integrating all photoelectrons over the detector angles $\varphi^D$ and $\vartheta^D$. In this case, the high binding energy cutoffs of the secondary electrons are broad and not well-defined. l, EDCs obtained from (f-j) at the detector angle $\varphi^D=0$. In this case, the high binding energy cutoffs of the secondary electrons are sharp and well-defined. m, The first derivative of the EDC at the normal emission (the black curve in (l) for Cut 3). The left peak corresponds to the secondary electron cutoff while the right peak represents the Fermi level cutoff. The precise position of these two peaks are used to determine the sample work function. n-q, Normal emission EDCs measured on some representative samples, including $Bi_2Se_3$(n), overdoped Bi2212 with a Tc of 67 K (o), LiFeAs(p) and $CsV_3Sb_5$ (q). The horizontal axis is plotted as E-E$_F$+h$\nu$ so the sample work function can be directly obtained from the position of the secondary electron cutoff.