Fermi surface tomography

Fermi surfaces are essential for predicting, characterizing and controlling the properties of crystalline metals and semiconductors. Angle-resolved photoemission spectroscopy (ARPES) is the only technique directly probing the Fermi surface by measuring the Fermi momenta (kF) from energy- and angular distribution of photoelectrons dislodged by monochromatic light. Existing apparatus is able to determine a number of kF -vectors simultaneously, but direct high-resolution 3D Fermi surface mapping remains problematic. As a result, no such datasets exist, strongly limiting our knowledge about the Fermi surfaces. Here we show that using a simpler instrumentation it is possible to perform 3D-mapping within a very short time interval and with very high resolution. We present the first detailed experimental 3D Fermi surface as well as other experimental results featuring advantages of our technique. In combination with various light sources our methodology and instrumentation offer new opportunities for high-resolution ARPES in the physical and life sciences.


Introduction
One hundred years ago, the explanation of the photoelectric effect was awarded the Nobel Prize.Since then, studies based on the detection of photoelectrons have led to many important discoveries, most recently topological insulators [3], as well as the establishment of one of the most informative tools in condensed matter physics -angleresolved photoemission spectroscopy (ARPES).The technique itself is relatively simpleone counts the outgoing electrons as a function of their kinetic energy, direction and photon energy -four variables in all.However, the instrumentation remains quite demanding: variable photon energies require sophisticated light sources including synchrotron radiation, while atomically clean surfaces require an ultra-high vacuum, which severely limits the detection equipment.As a consequence, the cost of a relatively ordinary ARPES setup easily exceeds six figures.Currently, acquiring a single highresolution 2D cut through the most important and succinct feature of the electronic structure -the Fermi surface (FS) -takes up a significant portion of the typically allocated synchrotron beamtime thus leaving 3D Fermi surfaces of the materials only partially explored.

Results and Discussion
In Fig. 1a we show the 3D FS obtained with our novel technique.One hundred highresolution 2D FS maps were taken within one hundred minutes using the synchrotron light with variable photon energies.The experimental data are compared with those calculated in the framework of density functional theory FS (Fig. 1b).Despite the general qualitative agreement, the differences in the size and curvatures of the individual layers emphasise the need to perform such measurements.It is the exact 3D shape of the Fermi surface that defines physical properties of the material, such as electrical or optical conductivities, elasticity, thermal expansion, magnetic susceptibility, etc.The data set underlying the 3D FS shown in Fig. 1a consists of more than 17 million data points, allowing the corresponding intensity distribution to be viewed in any two-or one-FERMI SURFACE TOMOGRAPHY dimensional section of this volume in k-space, revealing even smallest details of the 3D FS.
Why was it possible to record such a data set?The aim of the ARPES experiment is to obtain the intensity distribution of the photoelectrons flying out of the surface in different directions and with different kinetic energies with the best possible angular and energy resolution.In general, applying an electric or magnetic field to the photoelectrons would reduce the resolution because the elements inducing these fields are never geometrically perfect.To keep the experimental setup minimal and assume a point source of electrons, the goal can be achieved by simply placing a hypothetical detector, sensitive to both position and energy, in front of the sample under investigation.The first reality is that the photon beam always leads to a finite region on the sample that emits the photoelectrons.Even with such an advanced detector, the angular resolution is limited by the size of this area and the distance to the sample, as the same spot on the detector can now be reached by electrons with different emission angles.
We overcome this problem by placing an electron lens between the sample and the detector (Fig. 2).Such a lens works similarly to an optical lens, which is a textbook example of a Fourier transform engine.In other words, all electrons leaving the sample at the same angle can be focused on a single point in the back focal plane of the lens.In Fig. 2 we show schematically how the focusing works for two angles: normal emission (green rays) and 10° (blue rays).As a position-sensitive detector, we use a micro-channel plate (MCP) consisting of millions of channels, each of which is capable of detecting the incoming electrons.Although in this arrangement the angular distribution of the photoelectrons can be detected, each channel of the MCP would detect all electrons flying in the same direction, regardless of their kinetic energy, i.e. so far we only have positionbut not energy-sensitive detection system.
There are many ways to sort electrons according to their kinetic energy.Again, we try to keep the design minimal and use probably the simplest method.We apply a negative potential to the front surface of the MCP itself, which repels all electrons that have a lower energy than that set by this retarding potential (red beams in Fig. 2).This method is particularly advantageous when measuring the distributions of electrons from the Fermi level -precisely because there are no photoelectrons with the kinetic energy greater than that defined by Einstein's equation for the photoelectric effect.By setting the threshold energy, say a few meV lower than this cutoff, we are able to obtain the desired angular distribution of photoelectrons from the Fermi level.It is important to mention that the retarding field of the MCP obviously modifies the focusing field of the lens and a FERMI SURFACE TOMOGRAPHY special solution should be found for the potential pattern of all elements involved to get the desired result.An interactive animation [4] allows a better understanding of our concept.
In the following, we show some examples of the angular maps taken with the FeSuMa (Fermi Surface Mapper) spectrometer, designed and constructed as explained above [5], in Fig. 3. Due to the almost instantaneous result -one can see the FS contour live on the monitor (see video in the Supplementary Information section) -our technique offers numerous possibilities to vary individual experimental parameters while keeping all others constant.In Fig. 3a we start with presenting the photon energy dependence in TiTe 2 .This is perhaps the most important ARPES dataset as it provides a direct window into 3D k-space.We have tested FeSuMa at two synchrotron light sources in the broad photon energy range between 14 and 105 eV and always observed high-resolution angular distributions.The maps taken at lower photon energies shown in the left panels are similar to those underlying the 3D FS from Fig. 1a.Higher photon energy maps in Fig. 3a indicate the presence of further FS sheets that adopt the trigonal symmetry of the system.
To fully capture these pockets, one can rotate the sample, as its angular position exceeds the lens acceptance of FeSuMa.Fig. 3b presents the results of a typical polarization-dependent experiment in topological insulator Bi 2 Te 3 .In this case, only the polarisation of the incident laser light was switched from right-circular to left-circular.To emphasize the difference in the maps we present also the dichroism -the difference between the data taken using the light of opposite helicities.A classical picture caused by the spin-momentum locking of the topological surface states is observed (inset to Fig. 3, left panel).The duration of the whole experiment was less than a minute.
Recently, it was demonstrated that iron-based superconductor LiFeAs ( T c = 18 K ) shows signatures of the nematic order upon entering the superconducting state [6].One of them is the deformation of the Fermi surface.The data from Fig. 3c clearly show that also the smallest FS sheet, which just fits into the angular window of FeSuMa when using the laser radiation, is deformed when the temperature changes from 20 K to 3 K, i.e. when the sample enters the superconducting state.The variations in intensity are best seen after subtracting the maps (inset to Fig. 3, right panel).
Two other parameters, the elapsed time after the cleavage of the single crystal and the position of the photon beam on the sample surface, are often crucial for understanding the ARPES spectra (Fig. 3 d, e).Also in this case, the maps can be recorded successively while keeping all other conditions unchanged.This particular example shows the FERMI SURFACE TOMOGRAPHY temporal evolution of the spectra in Bi 2 Te 3 as well as a significant dependence of the signal on the probed region.Finally, the bottom row of panels (Fig. 3 f) concludes the presentation of angular distributions showing the FS maps of various single crystals, ranging from high-temperature superconductors to topological insulators.
As mentioned earlier, there are four basic variables of ARPES: the kinetic energy, the photon energy and two angles that define the direction.So far, we have limited ourselves to the electrons with the largest energies in the material -those near the Fermi level.The vast majority of phenomena in condensed matter physics depend crucially on the underlying dispersion of the electronic states, i.e. on how the momentum distribution changes as a function of the binding energy.With our setup, this information is also available.This is the third new aspect of our technique and it is as simple as the previous two.We record a series of 2D angular distributions, each corresponding to the threshold energies lower than the one used to record FS map, thus scanning the energy interval of interest.The difference between two adjacent measurements would represent the angular distribution corresponding to the particular binding energy.Some of the most representative 2D momentum-energy slices from such 3D differential scans are shown in Fig. 4. Panels a) and b) display the intensity distributions corresponding to the horizontal cuts close to the centers of the angular maps from Fig. 3d and Fig. 3b respectively.The former clearly show the presence of the quantum well states in Bi 2 Te 3 not only in the previously observed region of binding energy near the Dirac point (~ 400meV), but also near the Fermi level, which explains the presence of multiple concentric FS from Fig. 3d.Fig. 4c and Fig. 4d represent the vertical cuts from Fig. 3f and Fig. 3c for BSCCO and LiFeAs samples respectively.We have clearly observed the opening of the anisotropic dwave superconducting gap along the FS portions seen in Fig. 3f.One is able to track changes in the spectra caused by the lowering of the temperature either directly from the energy distribution curves ( EDC ) in Fig. 4f or as a shift of the leading edge midpoints in Fig. 4i.Even in the superconductor with much lower critical temperature of 18 K, LiFeAs, we were able to detect the corresponding shift of the leading edge (Fig. 4j) which indicated the opening of the superconducting gap of the order of 3 meV.These observations are not surprising in view of the achieved overall energy and momentum resolutions.As the EDC shown in Fig. 4g demonstrates, the overall energy resolution achieved so far is of the order of 12 meV.We believe this result can still be improved.Since we do not accelerate the photoelectrons in the lens and only minimally change their initial trajectories, the theoretical resolution exceeds that of conventional FERMI SURFACE TOMOGRAPHY analyzers.The full width at half-maximum of the momentum distribution curve in Fig. 4h suggests that features in the momentum space can be resolved with better than a hundredth of a Brillouin zone precision.
The presented data sets imply that FeSuMa is a promising ARPES tool despite its simple design.Compared to existing hemispherical, time-of-flight and grating/display analyzers [7][8][9], it has a number of advantages.The absence of the entrance slit not only allows the angular distribution to be recorded in 2D (and thus much faster), but also guarantees isotropic angular resolution.The latter is anisotropic in hemispherical analyzers -the sampling of angles perpendicular to the slit is defined by the slit size itself, within which the signal must be integrated.This inevitably leads to "stripy" FS maps, the recording of which requires further measurements involving either rotation of the sample or deflection of the electron beam within the lens by additional electron optical elements or by movement of the lens itself [10,11].Such methods distort the original 2D angular distribution so that only a narrow strip of it can be recorded in each case.Since the hemispheres cannot be placed in the immediate vicinity of the sample, the entrance slit is usually situated in the "second" focal plane of the lens, which inevitably causes the complexity of the lens itself.Time-of-flight technique needs a pulsed light source, which results in longer measuring times and considerable space-charge effects, as well as sophisticated delay-line detectors, significantly complicating the operation of the instrument.Finally, the display analyzers typically involve a number of grids acting as energy filters, which results in a significantly worse momentum and energy resolution.Absence of the detailed and highresolution experimental 3D Fermi surfaces in the literature [12] can be considered as an indicator of limited possibilities offered by the state-of-the-art ARPES equipment.
Among the further advantages of FeSuMa are the easiness of adjustment and operation.The open MCP allows to detect also the photons, which are scattered from the sample edges or imperfections, thus guaranteeing the detection of electrons from the mirror-like portion of the studied surface -a requisite for high quality ARPES experiment.Such portion of the surface specularly reflects the photons and they do not enter the analyzer.The analyzer can work in a direct spatially-resolving mode imaging the source of electrons in two dimensions.This makes positioning of the source directly into the focus of the analyzer a matter of several seconds.Finally, a very compact and simple design -as is sketched in Fig. 2, detection of the photoelectrons can occur ~10 cm away from the source, -makes it possible to carry out high-resolution ARPES experiments at FERMI SURFACE TOMOGRAPHY very low temperatures and at a fraction of usual costs.FeSuMa without cables would easily fit on this page.
Our technique also has fundamental implications.A few decades ago, anglemultiplexing analyzers enabled the transition from 1D EDCs to 2D momentum-energy maps, turning photoemission from a mere band-mapping tool into a sophisticated manybody spectroscopy.We hope that the addition of another dimension to ARPES will make the 3D spectral function accessible and reveal previously unknown aspects of electronic interactions and correlations.We believe that the increased efficiency of our instrument will provide the basis for high-throughput measurements necessary to match global computational efforts for material design, enable a new level of quality control of thin films and single crystals by complementing the usual crystal structure analysis with electronic structure analysis, and enable measurements of systems sensitive to radiation damage, such as organic crystals.Finally, the rapidly developing time-resolved and nanoderivatives of ARPES, which are beginning to successfully probe the electronic structure of quantum materials subjected to gate fields, currents, screening and strain in operando [13], should certainly benefit from the high informativity of Fermi surface tomography.

Methods
We used single crystals of TiTe 2 , BiTeI, Bi 2 Te 3 , ZrTe 3 and (Pb,Bi) 2 Sr 2 CaCu 2 O 8+δ grown by Helmuth Berger.Growth procedure for each of them can be found elsewhere.LiFeAs single crystals were grown by self-flux using the standard method [14].
Self-consistent band structure calculations of TiTe 2 were performed using the linear muffin-tin orbital method in the atomic sphere approximation as implemented in PY LMTO computer code.
ARPES data have been collected using FeSuMa electron analyzer and synchrotron radiation from BESSY and ASTRID2 storage rings at beamlines U125-2_NIM ( 14 -34 eV ) and AU-SGM3 ( 20 -105 eV) as well as laboratory-based 6 eV laser source.suggestion to implement the direct (spatial) mode in FeSuMa, Uwe Siegel, who successfully guided us through the technology transfer process as well as Roland Hübel, Falk Sander and Frauke Thunig for invaluable technical support.

Fig. 1 .
Fig.1.Three-dimensional Fermi surface of TiTe 2 a) A voxelgram of photoemission intensity recorded using FeSuMa analyzer and synchrotron light.Photon energies between 14 and 34 eV are scanned with 0.2 eV step.Voxels within normalized intensity interval between 0.012 and 0.020 are shown.Total number of points of the underlying dataset is 1.71924e+07.Average intensity is 0.007, absolute maximum is 0.041.b) Calculated Fermi surface of TiTe 2 within the same momentum volume.

Fig. 2 .
Fig. 2. Simple ARPES Schematics of the method is shown.Electrons originate at the sample surface.Green rays correspond to normal emission.Blue rays represent electrons emitted at 10° angle in the vertical plane.Red rays are the electrons having lower kinetic energies from both beams, deflected by the retarding field.Three cylinders represent the electron lens.MCP is the position-sensitive detector.

Fig. 3 .
Fig. 3. Two-dimensional Fermi surface maps Angular distributions, typically 500x500 pixels, of the photoemission intensity within the solid angle of ~30° and kinetic energy interval 5-20 meV in the vicinity of the Fermi level.a) Angular distributions from TiTe 2 recorded at two synchrotron light sources.Lower energies -BESSY (6 K), higher energies -ASTRID2 (35 K). b) Off-normal (~5°) emission laser (6 eV) data from Bi 2 Te 3 .recorded using left-and right-hand circularly polarized light, as well as their difference.T = 3 K. c) Temperature-dependent data from LiFeAs and their difference.d) Off-normal (~3°) emission data from Bi 2 Te 3 recorded at different time after cleavage.T = 3 K. e) The same as d), but taken earlier (after ~5h) and from different positions at the cleaved surface.f) Angular distributions from different materials.First four maps are recorded at BESSY at approximately 6 K. Two other maps -in the laboratory using the laser at 3 K.

Fig. 4 .
Fig. 4. Energy direction and resolution a) Momentum-energy cuts extracted from the differential scans -3D datasets giving the intensity vs. two component of momentum and kinetic energy.Corresponding Fermi surface maps are shown in Fig. 3d.Laser, 6 eV.T = 3 K.Energy step -10 meV.Acquisition time of the whole 3D dataset ~30 min.The asymmetry of the dispersions stems from the finite angle between the sample normal and forward direction to the analyzer.b) Same as a), but taken earlier after cleavage and with energy step of 5 meV.c) Dispersion along the cut through the antinodes in BSCCO above and below superconducting transition ( T c ~ 65 K ).Synchrotron, 24 eV.Energy step -5 meV.