Decagonal quasicrystals (QCs) are extremely tempting structures to study due to the unique combination of the periodic and QP orders in the same crystal. The lattice of such materials can be obtained by appropriate projection of the periodic lattice in five-dimensional space to the three-dimensional (3D) space1. The real and the reciprocal space (k-space) lattice vectors of decagonal QCs thus can be denoted by five vectors2 (Fig. 1a), spanning a reciprocal lattice in aperiodic plane and a set of discrete maxima with 2π/c spacing in the periodic (00001) direction, where c≈4 Å is a lattice parameter for d-AlNiCo in this direction3. The corresponding low-energy electron diffraction (LEED) pattern from the clean tenfold surface of d-AlNiCo measured at electron beam energy Ee=43 eV at room temperature is shown in Fig. 1b. The very existence of the decagonal QCs poses fundamental questions on how the duality between the periodic and QP orders manifests itself in electronic structure and how it is related to the stabilization of their unusual structure.

Figure 1: Quasicrystaline lattice basis and the model of electronic structure.
figure 1

(a) Set of reciprocal lattice base vectors. (b) LEED from the clean tenfold surface measured at electron beam energy Ee=43 eV. (c) Section of the theoretical FS model in k-space in the simplest case that implies only central FE-like sphere and set of tenfold surrounding spheres with strongest diffraction intensity.

These questions have been explored in a number of theoretical articles4,5,6,7,8,9,10,11 and experimental works12,13,14,15,16,17,18,19. Angle-resolved photoemission spectroscopy (ARPES) experiments utilizing vacuum ultraviolet (VUV) radiation established the general electronic structure of d-AlNiCo, showing that in the binding energy Eb range far from EF the Al sp-derived states have clear dispersive character and hence are effectively delocalized20,21. The observed dispersion character is close to free-electron like (FE like) with the effective mass ratio m*/me≈1. Moreover, k-space locations of the sp-bands observed in ARPES correspond to the reciprocal lattice vectors G dominant in LEED in both periodic and QP directions in agreement with theoretical calculations22,23. The survival of FE-like dispersion in the aperiodic QC media is similar in nature to the incommensurate periodicities, found, for example, in one-dimensional Peierls compounds, where the same phenomena was observed in ARPES experiment24. The spectral weight in such structures is redistributed in accordance with the strength of the crystal potential and the structure factor in k-space and concentrated mainly on the FE-like parabola. The corresponding q-Brillouin zone (q-BZ) in the tenfold plane could be defined as a boundary between the central diffraction maximum and the most intense reflections in diffraction pattern25,26. According to the Spot Profile Analysis (SPA)-LEED study27, these high-intensity spots are distributed over a circle with a radius k of 2.7 Å−1. Along the periodic direction, the unit cell includes two non-equivalent adjacent aperiodic planes, nevertheless the effective size of the q-BZ along the periodic axis in k-space is doubled and is equal to 4π/c due to the non-symmorphic space group symmetry of d-AlNiCo (analogous, for example, to graphite28). On the basis of these principles, Fig. 1c shows the section of the 3D Fermi surface (FS) model of d-AlNiCo, where for clarity only the strongest reciprocal lattice vectors with G||={0 Å−1, 2.7 Å−1} at GZ={−1.54 Å−1, 0 Å−1, −1.54 Å−1} were chosen as the centres of FE-like spheres. The real FS could be obtained by increasing the set of G-vectors and taking into account the structure factors as well as all possible Bragg scattering effects at the q-BZ boundaries. The latter open multiple gaps at the EF at certain points in k-space and transform the FS into a segmented FS that has a smaller surface area compared with the original one. Although the largest gaps are expected to appear at the first q-BZ boundary where the corresponding interaction potential Fourier components are the largest, due to the dense diffraction pattern the smaller gaps are expected to appear also in the higher q-BZs. Theoretical FSs were calculated ab initio for Y-AlNiCo structural model6 and for orthorhombic approximant to the decagonal phase10, both resulting in a highly anisotropic FS with a complex multiple-band structure.

However, all previously reported VUV-ARPES investigations did not succeed to establish the presence of dispersive states in the region near the EF, as well as the shape of the FS itself, which is important for understanding the mechanism of QC phase stabilization. Electronic structure calculations4,29,30 predicted that the density of states (DOSs) near EF should have two features: a pseudogap at EF with its characteristic energy scale of the order of 1 eV and much finer spikes and mini-pseudogaps with their characteristic scale of the order of tens of meV. The coexistence of these two types of the DOS features in decagonal QCs has been confirmed by various techniques13,15,16,31 including photoemission spectroscopy and scanning tunnel spectroscopy. The eV-scale pseudogap at EF is believed to result from the destructive interference of Fermi states at the boundaries of the first q-BZ, which is referred as the Hume–Rothery (HR) mechanism, and/or the hybridization between transition metal (TM) d-states and Al sp-states12 (sp-d hybridization), while the relative weight of these two mechanism has not been defined yet. In turn, the meV-scale pseudogaps could appear due to the scattering of the Al sp-states at the boundaries of the higher-order q-BZs or due to the localized character of the electron wave function near EF. Since the QC structure provides a higher symmetry of the q-BZ comparing with the periodic lattice, the HR mechanism could significantly contribute to the formation of this large pseudogap (HR gap), thus being a driving force of QC phase stabilization. The HR electron concentration rule recently was revised theoretically using full potential linearized augmented planewave method for QC approximants involving sufficient number of the plain waves expansion11. At the same time, although the sp-d hybridization may indeed form a pseudogap at EF, it is less likely to stabilize the QC structure because, with the d-states being localized, the interaction integral between sp- and d-states (where H is the Hamiltonian) that contributes to the system total energy, responds only smoothly to a variance of the atomic positions.

In this study, we endeavour to observe the 3D FS of a bulk QC and to understand the role of pseudogap in stabilization mechanism of QC structure. We demonstrate that FS consists of dispersive Al sp-states with large contribution from high-order bands. We show that FS–BZ interaction is highly anisotropic, which results in a different magnitudes of HR gaps along QC and periodic directions.


FS contours in QC planes

In ARPES, the HR gap should be clearly observed as bending of the band dispersions closer to EF32 at the q-BZ boundary, which leads to the formation of a pseudogap at EF. As a consequence, the FS should be not a sphere anymore but should break into a set of discrete pockets. To observe the 3D FS, we performed soft X-ray ARPES (SX-ARPES) experiments in the vicinity of EF on decagonal AlNiCo QCs. Increase of the photoelectron mean free path λ at higher photoelectron energies make the SX-ARPES experiment more bulk sensitive33, which is essential for the QC samples because their surface stoichiometry may be distorted by structural defects and surface segregation. In addition, SX-ARPES reduces the final-state effects distorting the photoemission response of the valence states, in particular because the increase of λ scales down electron momentum broadening in the surface normal direction34, thus allowing accurate resolution of the native 3D bulk dispersions E(k). The tenfold (00001) surface of the sample was oriented normal to the analyser axis. We varied the surface perpendicular momentum kz of the probed valence states through the photon energy hv and their k|| through the emission angle (refer also to the Methods section).

The entire set of our experimental results is presented in Fig. 2. The sketch in Fig. 2a shows the main features of the experimentally observed 3D FS. It differs from the one presented in Fig. 1c because of the specific structure factor due to the non-symmorphic space group symmetry of d-AlNiCo. In addition, previous theoretical simulations of the photoemission from QCs suggested that there is a certain threshold for the relative strength of the lattice potential35, which significantly reduces the number of observable bands in the typical photoemission experiment. Photoelectron intensity maps at constant Eb (constant energy maps, CEMs) were acquired in planes in k-space shown in the sketch in Fig. 2a. These CEMs are reported for two different Eb: corresponding to the FS (Fig. 2b–d) and halfway between EF and TM d-bands maximum (Fig. 2e–g). Surprisingly, in CEM measured parallel to the surface at kz in the Γ10-point (centre of the tenth BZ along the periodic axis, hv=875 eV), we observed a sharp and almost unaltered contour of FE-like Al sp-bands at EF (Fig. 2b) with the Fermi momentum kF0≈1.57 Å−1 (shown as a blue circle at lower Eb in Fig. 2e). Apart from the circle-like contour of dispersive Al sp-states centred at surface parallel momentum k||=0, there is a small hole-like pocket at k||=0 with kF≈0.3 Å−1 and a manifold of surrounding electron-like pockets with kF1≈0.7 Å−1 with tenfold symmetry (green circles in Fig. 2e). The centres of these latter e-pockets are distributed on a circle with k≈2.7 Å−1, which corresponds to the radius of the reflections in the LEED pattern27. These directions belong to the set of twofold Δ axes rotated by π/10 from the ∑ axes, both having different intensity distribution patterns in diffraction26.

Figure 2: SX-ARPES data.
figure 2

(a) Sketch of the 3D FS observed in the current experiment with the planes corresponding to the different cuts represented in bg. Red arrows denote the possible directions where eV-scale HR gap can appear in addition to the one observed along the periodical direction. (bd) Constant binding energy maps (CEMs) corresponding to the FS cut acquired along the planes parallel to the sample surface at kz=10·(2π/c) denoted as Γ10-point (b) kz=9·(2π/c) denoted as Γ9-point (c) and perpendicular to the sample surface at ky=0 (d). (eg) CEMs at Eb=−0.6 eV acquired at the same planes as (bd). The blue and green circles are the guide for the eye lines of Al sp-dispersion contours. (h,j,k) The ARPES intensity (logarithmic grayscale for h and j) maps I(Eb, k||) measured along the directions shown with red arrows in eg, respectively. (i) The 3D sketch of the sp-band dispersion parabola Eb (kx,ky) hybridized with the TM-d flat band.

The ARPES intensity map at Γ10-point as a function of Eb and k|| (I(Eb, k||)) along the axis, which belongs to the set of Δ axes is shown in Fig. 2h, where to enhance the figure contrast, we applied the background subtraction procedure (refer to the Methods section). The map appears to be quite complex in dispersion, especially in a region from EF to 3 eV in binding energy, due to the contribution of the higher-order dispersions that originates from other points of the dense reciprocal lattice. Nevertheless, one can recognize that the FE-like parabola of Al sp-band is effectively divided in two energetic regions as a result of the sp-d hybridization, which is schematically shown in Fig. 2i in 3D (see the Methods section and Supplementary Fig. 3). In this figure, the lower part of the sp-band paraboloid Eb (kx,ky) is shown in green and the upper part near the EF in blue. The high-Eb part of the Al sp-band in Fig. 2h appears to be more blurred than its near EF counterpart due to the effects of electron momentum broadening in the surface normal direction and increased DOSs of TM d-bands at higher Eb. In addition, the sp-d hybridization slightly reduces the value of kF of the Fermi states, but does not affect the FS continuity. The survival of the FE-like FS contour and the absence of dispersion bands bending near EF suggest that the energy scale of the pseudogap, induced by the HR mechanism in QP plane, is either vanishingly small or at least less than 70 meV, which is half of our actual experimental resolution. This value is significantly smaller than the reported value of the pseudogap of 1 eV order31.

Due to the non-symmorphic space group symmetry, completely different CEMs were observed at kz in the Γ9-point (hv=720 eV), which would otherwise be identical to Γ10-point. Figure 2c,f reveals very intense central ring with kF1 (green circle in Fig. 2f) and less pronounced tenfold large circles with radius kF0 surrounding this circle (blue circles in Fig. 2f). The former ring corresponds to an electron-like pocket with band minimum around Eb≈−2 eV as can be seen from I(Eb, k||) acquired parallel to kx through Γ9-point (Fig. 2j). Obviously, this electron-like pocket is a result of intersection of FE-like Fermi spheres formed by Al sp-states and centred at Γ-points at nearest even BZs along the periodic axis. The 2π/c shift of small and large e-pockets along the kz (blue and green circles in Fig. 2e,f) is consistent with the results of the SPA-LEED experiment27, where the same shift in the periodicity along kz of the full width at half maximum of the central and k≈2.7 Å−1 diffraction spots was observed. In addition, we note that the large circles that surround the central electron-like pocket in Fig. 2c do not show any significant gaps at EF at intersection points and one can see the crossing arcs of these circles. It again confirms that the energy scale of the HR gap in QP plane (if present) should be in sub-eV scale.

SX-ARPES along periodic direction

The FS character along the periodic direction is different however. In CEMs measured along the periodic axis (Fig. 2d,g), we observe sharp features at kz at the Γ10- and Γ9-point, whereas between them we find a continuum of states without any well-defined dispersions. From the I(Eb, k||) measured in the same plane along kz near the Γ9-point (Fig. 2k), it becomes clear that there are also no dispersive Al sp-states from EF to TM d-band at kx=0 near Γ9-point. This observation unveils the occurrence of the Bragg scattering of Al sp-states close to the EF at the effective q-BZ boundary between two even Γ-points in periodic direction since the Fermi momentum kF0 matches 2π/c=1.54 Å−1—the periodicity along kz. The observed gap of the eV-scale reduces the integrated DOS of the occupied states at EF, minimizing the total energy of the system. Formation of such a gap is a keystone of HR mechanism and should indeed electronically stabilize the sample structure, but unexpectedly this eV-scale gap was observed only in periodical direction. With this observation, we can conclude that the HR mechanism in d-AlNiCo exhibits strong anisotropy: the eV-scale HR gap appears along the periodic direction, while the energy scale of the HR gap in QP planes has to be in the sub-eV range. We emphasize that these experimental results are consistent with the scanning tunnel spectroscopy studies, which found a manifold of locally dependent sets of small (20–50 meV) gaps in the occupied DOS of d-AlNiCo16,31. The theoretical calculations of the spectral density along the Δ axis of the 2D Penrose tiling22 also revealed the gaps near centre of the bands due to the localized character of the states rather than scattering at the q-BZ boundary.

It is interesting as well to correlate the observed energy scale anisotropy with well-studied vibrational spectra of the QCs. A larger pseudogap between the optic and acoustic excitation was observed in 1/1 approximant comparing with the QC counterpart36, suggesting the stronger Bragg-plane reflection for the periodic structure. On the other hand, inelastic neutron scattering37 did not reveal a significant anisotropy between the vibrational modes along the QP and periodic directions in d-AlNiCo.

We note that the previous VUV-ARPES data21, presented in the high-Eb sector of electronic structure, have certain differences respective to our SX-ARPES data in the number of the bands observed. Contributing to these differences can be, first, the highly different degree of surface sensitivity. Second, VUV-ARPES poorly defines the kz component, which is intrinsically limited by non-FE character and kz broadening of low-energy final states34. Finally, the final-state wavevector kf in VUV-ARPES is much smaller than in SX-ARPES. If we treat the ARPES final state as a plane wave with weak damping towards the crystal interior, the photoemission intensity appears essentially as a matrix element between the initial state transformed by the dipole operator Hint and the final state . Therefore, the VUV- and SX-photoemission probe different low- and high-k Fourier components, respectively, of the transformed initial state.

Simulation of the FS contours in QC planes

To simulate the stronger in-QP-plane HR gap at the EF, we utilized a 3D theoretical model that consists of the single near-EF part of Al sp-d hybridized bands (blue part of the hybridized sp-d band in Fig. 2i) centred at the same spots in k-space as observed in the present experiment (see the Methods section). We simulated the FS cuts along k|| at kz in the even and odd Γ-points with and without Bragg scattering at the q-BZ boundaries in the QP and periodic directions (Fig. 3). These cuts reflect the measured CEMs shown in Fig. 2b,c with different parameters of the scattering strength at q-BZ boundaries taken into account. The value of the scattering strength was chosen to provide the HR gap of eV-order for the QP and periodic directions to match the experimentally observed value of the HR gap along the kz axis (Fig. 2k). Corresponding dispersion relations Eb (kx,ky=0) for different cases shown in Fig. 3a–f are presented in Supplementary Fig. 4a–f, respectively. From Fig. 3, it becomes clear that the experimentally observed FS (Fig. 2b,c) corresponds better to the case when the eV-scale HR gap is present along the periodic direction only (Fig. 3c,d), while the same HR gap present in both periodic and QC planes should significantly alter the FS contours (Fig. 3e,f).

Figure 3: Simulation of FS contours.
figure 3

(a,b) Simulation results of the CEMs at EF at kz in the Γ10- and Γ9-points, respectively, in (kx,ky) plane without the Bragg scattering at the q-BZ boundaries. (c,d) Same as a and b, but with the Bragg scattering allowed at the q-BZ boundaries only in periodical direction. (e,f) Same as a and b, but with the Bragg scattering allowed at the q-BZ boundaries in both QP and periodical directions.


It is commonly believed that there are two main alternative mechanisms for stabilization of the QP phase: the energy stabilization with the QC ground state and the entropy stabilization with a crystalline phase at low temperature38. The observed energy scale of the HR gap along the QC direction is significantly lower or even vanishing compared with the one along periodic direction, suggesting that the in-QC-plane (aperiodic) structure does not lead to a significant Peierls-type energy contribution to the structural stability. On the other hand, there could be an additional contribution to the 3D structure stabilization via the HR gaps along the ‘off-symmetry’ directions shown in Fig. 2a with red arrows. In fact, the distance in k-space between centres of FE-like sphere at k||=0 at Γ10-point and tenfold FE-like spheres observed on a radius k||=2.7 Å−1 at Γ9-point (Fig. 2a) matches the diameter of these spheres d=2·kF0, thus enabling the possibility for the eV-scale HR gap to appear. For the icosahedral QC, due to the presence of only aperiodic planes, we suppose that the energy scale of HR gaps is more likely to be uniform.

We note that the direct observation of the FS–BZ scattering anisotropy has only been possible by the virtue of accurate definition of 3D electron momentum achieved with SX-ARPES. This has been essential to disentangle the electron dispersions in the periodic and aperiodic planes of d-AlNiCo. With our results, we hope to stimulate the advanced theories capable of explaining the observed anisotropy from first principles.


Samples and SX-ARPES measurements

Single crystals of decagonal AlNiCo with the bulk composition Al70Ni20Co10 were grown by means of Czochralski method. Slow growth out of the melt over 300 h at the solidification temperature ensured high sample quality. The (00001)-oriented surface (tenfold) of AlNiCo was cleaned by cycles of Ar+ ion sputtering and subsequent annealing up to 750 °C leading to sharp tenfold symmetric LEED (Fig. 1b) and the single metallic component in Al2p core level photoemission spectrum. The measurements were conducted at the ADRESS beamline at Swiss Light Source synchrotron facility, Paul Scherrer Institute, Switzerland, using photon energy range hv=315–1,000 eV and different polarizations of incident X-rays33. The temperature of the samples during the measurements was around 12 K. The combined (beamline+analyser) energy resolution was down to 70 meV. The analyser entrance slit was oriented along the X-ray beam incident on the sample at a grazing incidence angle of 20° (for further experimental details see ref. 33).

Data processing

Due to the absence of the translational periodicity in the QC lattice and a small Al-sp cross-section at the given photon energies, the quasiparticle peaks in valence band maps are rather weak and broadened along the momentum axis. Nevertheless, the raw data itself allowed us to recognize immediately the Al sp-bands at main high-symmetry points (Supplementary Fig. 1a). To increase the quality of the observed dispersions, we applied several procedures to the measured ARPES intensity maps I (Eb, k||). First, the background from the charge-coupled device was taken into account and subtracted from the raw image. At the second step, the non-dispersive intensity from TM d-states was suppressed by subtraction of the substantially smoothed in momentum directions ARPES intensity map I (Eb, k||) from the original raw images. The resulting processed image is shown in Supplementary Fig. 1b. The subtraction of the strongly smoothed image was also applied for the contrast enhancement in constant binding energy maps (CEMs).

Polarization dependencies and sp-d hybridization

All the data presented in the paper were measured using the p-polarized light. The valence band maps measured at Γ9-point in s-polarized and p-polarized light are shown in Supplementary Fig. 2a,b, respectively. Interestingly, the photoemission intensity from the dispersive Al states vanishes in s-polarized light (anti-symmetric in the given experimental geometry) confirming the symmetric nature of these states with respect to the measurement plane. The degree of hybridization between Al sp- and TM d-states could be estimated from the angle integrated spectra measured at Γ-points at kz=6·(2π/c) (hv≈315 eV) and kz=10·(2π/c) (hv≈875 eV) shown in Supplementary Fig. 1c. Two spectra almost coincide however the relative cross-section of Al sp-states to TM d-states increases significantly with the photon energy39. Moreover, the sp-d hybridization is clearly observed in Fig. 2h as a bending of Al sp-bands closer to the original TM d-states to avoid crossing. This effectively divides the dispersion of Al sp-states into two regions with different Eb range (Fig. 2i). As a consequence, at Γ9-point, there is an additional spectral weight at Eb4.5 eV near kx=0 that comes from higher Eb part of the Al sp-dispersion at Γ10- and Γ8-points (Supplementary Fig. 2a).

The theoretical model and simulation results

The model to simulate the dispersion of the sp-bands in the d-AlNiCo structure is based on multiple Bragg reflections of the nearly free electrons on a set of the major components of the reciprocal quasi-lattice {Gi}. For a given k-vector, the corresponding tight binding Hamiltonian can be written as follows:

Here and are denoting the creation and annihilation operators, respectively, denotes the pairwise sum for and represents the valence electron potential in reciprocal space. The kinetic energy term of the model Hamiltonian was chosen to match the lower-Eb part of the hybridized sp-d band (blue curve in Supplementary Fig. 3) with kF0=1.57 Å−1, which was fitted with the polynomial equation:

where a0=−1.34, a1=0.25, a2=−0.03 and a3=0.06. The selection of a finite set of {Gi} for the calculation was made according to the most intense spots in the LEED. The set of {Gi} consists of an alternating stack in kz at (kx,ky)=(0,0) with kz=−3.08, 0 and 3.08 Å−1 and a tenfold rings with |k|=2.71 Å−1 at kz=−4.62, −1.54, 1.54 and 4.62 Å−1. This gives a total of 43 reciprocal vectors.

The strength of the potential was modelled as a Gaussian centred at k0:

where is the difference between pair of G-vectors, stands for the strength of the potential, k0=2.7 Å−1 and σ=0.84 Å−1. The parameter values were chosen to induce a gap of 1-eV order at EF at q-BZ boundaries. Figure 3a,b shows the simulation result for the CEMs at EF at Γ10- and Γ9-points along the kz, respectively, with the strength of the potential set to zero, which corresponds to the absence of the Bragg scattering at the q-BZ boundaries. Figure 3c,d shows the simulation result when the strength of the potential had a value of −0.7 eV for the periodic direction, while for the QP direction it was set to zero. This case corresponds to the occurrence of the Bragg scattering at the q-BZ boundaries along the periodic direction only. Finally, Fig. 3e,f shows the simulation result when the strength of the potential was set to −0.7 eV for both QP and periodic direction. The corresponding cuts Eb (kx,ky=0) for different cases shown in Fig. 3a–f are presented in Supplementary Fig. 4a–f, respectively.

Additional information

How to cite this article: Rogalev, V. A. et al. Fermi states and anisotropy of Brillouin zone scattering in the decagonal Al–Ni–Co quasicrystal. Nat. Commun. 6:8607 doi: 10.1038/ncomms9607 (2015).