Quantum-well states at the surface of a heavy-fermion superconductor

ABSTRACT Two-dimensional electronic states at surfaces are often observed in simple wide-band metals such as Cu or Ag (refs. 1,2,3,4). Confinement by closed geometries at the nanometre

scale, such as surface terraces, leads to quantized energy levels formed from the surface band, in stark contrast to the continuous energy dependence of bulk electron bands2,5,6,7,8,9,10.

Their energy-level separation is typically hundreds of meV (refs. 3,6,11). In a distinct class of materials, strong electronic correlations lead to so-called heavy fermions with a strongly

reduced bandwidth and exotic bulk ground states12,13. Quantum-well states in two-dimensional heavy fermions (2DHFs) remain, however, notoriously difficult to observe because of their tiny

energy separation. Here we use millikelvin scanning tunnelling microscopy (STM) to study atomically flat terraces on U-terminated surfaces of the heavy-fermion superconductor URu2Si2, which

exhibits a mysterious hidden-order (HO) state below 17.5 K (ref. 14). We observe 2DHFs made of 5f electrons with an effective mass 17 times the free electron mass. The 2DHFs form quantized

states separated by a fraction of a meV and their level width is set by the interaction with correlated bulk states. Edge states on steps between terraces appear along one of the two

in-plane directions, suggesting electronic symmetry breaking at the surface. Our results propose a new route to realize quantum-well states in strongly correlated quantum materials and to

explore how these connect to the electronic environment. SIMILAR CONTENT BEING VIEWED BY OTHERS EVIDENCE OF A ROOM-TEMPERATURE QUANTUM SPIN HALL EDGE STATE IN A HIGHER-ORDER TOPOLOGICAL

INSULATOR Article 14 July 2022 TWO-DIMENSIONAL SHIBA LATTICES AS A POSSIBLE PLATFORM FOR CRYSTALLINE TOPOLOGICAL SUPERCONDUCTIVITY Article 10 July 2023 MEROHEDRAL DISORDER AND IMPURITY

IMPACTS ON SUPERCONDUCTIVITY OF FULLERENES Article Open access 03 June 2021 MAIN Heavy fermions form a unique class of quantum materials that exhibit exceptional properties related to their

narrow electronic-band dispersion12. Previous experiments on heavy fermions showed that reducing the dimensionality leads to enhanced electronic correlations and strong coupling

superconductivity15,16. Furthermore, narrow surface bands with a Dirac dispersion were found in a semiconducting heavy fermion17. In spite of intensive investigations, 2DHFs have not been

observed at surfaces of superconducting compounds and no quantum-well states owing to lateral confinement of heavy electron states have been realized. We investigate the heavy-fermion

superconductor URu2Si2, which exhibits correlated narrow electron bands crossing the Fermi level and undergoes a transition to the HO phase characterized by an as yet unknown order

parameter14. A partial gap opens in the electronic band structure below _T_HO = 17.5 K, out of which an unconventional superconducting state develops below _T_c = 1.5 K (refs. 14,18). The

surface electronic states observed until now mostly have small effective masses19,20,21,22. Using STM to investigate small-sized, atomically flat terraces on the U-terminated surface of

URu2Si2 at sub-kelvin temperatures, we observe clearly 2DHFs with an effective mass 17 times the free electron mass, as well as quantization owing to lateral confinement and uncover the

surface–bulk interaction. In Fig. 1a, we present a STM image of terraces of U-terminated surfaces on URu2Si2 (see Extended Data Fig. 1 for the surface termination). Our starting point is the

tunnelling conductance obtained in a small range of a few mV around the Fermi level and at a temperature of 0.1 K, well below the superconducting critical temperature, shown in Fig. 1b (see

Extended Data Fig. 2 for a larger bias voltage range). QUANTUM-WELL STATES BY CONFINEMENT We focus on the tunnelling conductance along the white line shown in Fig. 1a. Along this line, we

identify four terraces of different sizes, _L_1 ≈ 2 nm, _L_2 ≈ 5.5 nm, _L_3 ≈ 20 nm and _L_4 ≈ 38.5 nm (Fig. 1c). We observe a strong bias voltage and position dependence of the tunnelling

conductance, which is different for each terrace as illustrated in Fig. 1d. We show in Fig. 1e representative tunnelling conductance curves at each terrace, in which we identify a set of

regular peaks. Let us analyse the terrace _L_3 (dashed rectangle in Fig. 1a). We present a symmetrized map of the tunnelling conductance in Fig. 2a (see Methods for further details). We

identify a set of peaks in the tunnelling conductance, which evolve in both position and bias voltage. Subtracting the features at _ε_− and _ε_+ (details provided in Methods and in Extended

Data Figs. 3 and 4), we obtain the pattern shown in Fig. 2b, which shows the lateral quantization of 2DHFs. The quantization pattern for confined electrons resembles the Fabry–Pérot

expression for an interferometer made by partially reflecting mirrors23 (reflection coefficient _r_ and the phase shift _ϕ_ are the free parameters; details in Extended Data Fig. 4 and

results on different terraces in Extended Data Fig. 5). Lateral quantization results from interfering wavefunctions partially reflected at steps. Quantized levels obtained from the

Fabry–Pérot expression are the white dots in Fig. 2a,b, whose position coincides well with the peaks in the conductance pattern observed in the experiment. In Fig. 2c, we plot as points the

position of the peaks as a function of the wavevector _k_ and as a line the dispersion relation _E_ = _E_0 + _ħ_2_k_2/(2_m_*), in which _m_* is the effective mass and _E_0 the bottom of the

band. We obtain _E_0 = −2.3 meV and _m_* = 17_m_0, with _m_0 the free electron mass, that is, we find that the 2DHF is derived from a massive surface electron state. A detailed comparison of

the tunnelling conductance versus position with the square of wavefunctions confined by a lateral potential leads to excellent fits, shown in Fig. 2d. The phase shift _ϕ_ determines the

position and energy of the peaks (white dots in Fig. 2a,b) and the best account of our observations is obtained with _ϕ_ = −π. We find values around _r_ ≈ 0.2, which slightly increase when

approaching _E_0 (Fig. 2e). The low value of _r_ is also found in surface states of simple metals; for example, _r_ is between 0.2 and 0.4 in Ag and Cu (refs. 23,24,25). However, the energy

dependence (Fig. 2e) at the surface of URu2Si2 is completely different to that in usual metals. Although _r_ varies mildly in Cu or Ag in the range of a few eV (see dashed line in Fig. 2e

and refs. 23,24,25), here we observe instead that _r_ decreases markedly in a range of a few meV (points in Fig. 2e). We can reproduce the observed dependence of _r_ versus bias voltage

assuming a potential well23,24 (continuous line in Fig. 2e; details provided in Methods and Extended Data Figs. 3 and 4). It is also insightful to trace the tunnelling conductance as a

function of the bias voltage at the centre of the terrace in Fig. 2b (circles in Fig. 2f) and compare it to the expectation for _r_ ≈ 1 (dashed line in Fig. 2f) and for _r_ ≈ 0.4 (continuous

line in Fig. 2f). We see that, for a reduced _r_, both the periodicity and shape of the tunnelling conductance are well explained in an energy range of a few meV, that is, two orders of

magnitude below the energy range observed in conventional metals1,2,3,4,23,24,25,26. To further investigate the quantized levels, we have fitted each peak to a Lorentzian function, whose

width Γ (black arrow in Fig. 2b) provides the lifetime _τ_ (Fig. 2g) of the quantum-well states. Taking a two-dimensional electron gas, we expect \(\hbar /\tau ={\Gamma }_{0}+(| {E}_{0}|

/4\pi ){\left[(E+| {E}_{0}| )/{E}_{0}\right]}^{2}\)\(| {\rm{ln}}| (E+| {E}_{0}| )/{E}_{0}| -{\rm{ln}}(2{q}_{{\rm{TF}}}/{k}_{{\rm{F}}})-1/2| \), with _ħ_ the reduced Planck’s constant, _q_TF 

= 0.0906 Å−1 the Thomas–Fermi screening length and _k_F the Fermi wavevector27 (dashed line in Fig. 2g). Our data are not well reproduced by this expression. Taking instead \(\hbar /\tau

={\Gamma }_{0}+(| {E}_{0}| /4\pi ){\left[(E+| {E}_{0}| )/{E}_{0}\right]}^{2}\) with _E_0 = −2.3 meV and Γ0 ≈ 60 μeV (continuous line in Fig. 2g), we find a much better account of our

data27,28. The latter expression takes into account the connection between the 2DHF and bulk states, showing that, in our experiments, the lifetime _τ_ is set by the decay of the 2DHF into

heavy-fermion bulk states. Quantum-well states sense the bulk correlations, given by the quadratic energy term in _ħ_/_τ_. This has been observed in surface states of noble metals,

monolayers of Pb and in Sb. However, in those cases, the energy range was three orders of magnitude above the one we discuss here3,6,11,28. From the obtained value of Γ0 ≈ 60 μeV, we

estimate the lifetime of the ground state as _τ_0(_E_0 = −2.3 meV) = _ħ_/Γ0 ≈ 11 ps. Similarly, the lifetime of states close to the Fermi level is _τ_(_E_ = 0) = _ħ_/Γ(_E_ = 0) ≈ 3 ps. We

can also estimate a value for a mean free path, _ℓ_0 = _v_F_ħ_/Γ(0) ≈ 0.14 μm, with _v_F the Fermi velocity of URu2Si2. This value is on the same order of magnitude as those observed in

ultraclean URu2Si2 single crystals29. To vindicate the existence of a heavy-fermion surface state, we performed density functional theory calculations of the surface band structure of a slab

of URu2Si2 (Extended Data Fig. 6). We find a shallow, U-derived f-electron band with a flat dispersion relation compatible with our experiments around the _X_ point of the simple tetragonal

Brillouin zone. The bulk electronic spectrum is gapped in this part of the Brillouin zone21,22. The rest of the Brillouin zone provides surface states with much smaller effective masses.

ONE-DIMENSIONAL EDGE STATES In Fig. 3, we show that the 2DHF is peculiarly modified at the steps separating terraces at which one-dimensional edge states (1DESs) appear. 1DESs were

previously observed in simple metals, in which a gap opens at the step and is filled with a very large density of states at _E_1DES by the 1DES3,8,11,24,25,30. The width of the conductance

peak at _E_1DES, _η_1DES, results from inelastic scattering into bulk states24,25. Here we find that the features in the tunnelling conductance completely change at a step (Fig. 3a,b). We

find a high peak at _E_1DES ≈ −0.38 meV (Fig. 3b,c) in steps, but notably only along one of the two equivalent in-plane axes, that is, an in-plane symmetry breaking occurs. The HO is known

to cause breaking of the body-centred translation symmetry14, leading to inequivalent electronic properties in subsequent U layers, which is consistent with our measurements (details in

Methods and Extended Data Fig. 7). This should reduce inelastic scattering and favour the formation of a 1DES. Here, however, we observe a rather unique situation in which the edge state is

either observed or not, on two crystallographically equivalent in-plane axes. This indicates spontaneous symmetry breaking of the fourfold rotational symmetry close to the surface. Such

in-plane symmetry breaking has been proposed31 for bulk URu2Si2 but has been difficult to detect. This symmetry breaking would cause an orthorhombicity given by a tiny difference in the

basal-plane lattice constants _a_ and _b_ (\(\frac{| \,a-b\,| }{(a+b)}\approx 1{0}^{-5}\))32. Other experiments could, however, not confirm such in-plane symmetry breaking33, which could be

favoured by defects. Recent group theory considerations and nuclear magnetic resonance (NMR) data propose that the HO state can belong to four space groups, #126, #128, #134 and #136, all

having the same crystal structure as the high-temperature state34. NMR experiments suggest the presence of fourfold symmetry at Ru, Si and U sites, narrowing down the most probable choice to

#126 (refs. 34,35). Notably, although these measurements gave absence of fourfold symmetry breaking in bulk URu2Si2, our STM data sensitive to individual uranium layers clearly show an

in-plane symmetry breaking in the 1DES. We note that a possible source of changes in the electronic structure that could also affect the 1DES is modifications of the valence of U edge

atoms21. Nonetheless, the symmetry-breaking in the 1DES suggests a deeper origin, which pinpoints that the 1DES serves as a sensitive probe of fundamental electronic properties of the

near-surface U lattice. SUPERCONDUCTIVITY At the energy range below the superconducting gap, ΔSC ≈ 200 μeV, we observe that there is a large zero-bias conductance (see Fig. 4a). There are

indications for unconventional superconductivity in bulk URu2Si2, with a d-wave symmetry order parameter29. This can contribute to the suppression of the superconducting features in the

tunnelling conductance, but it hardly leads to the zero-bias conductance observed in our experiment. Similar small-sized superconducting features are found in other heavy-fermion

superconductors, such as CeCoIn5 or UTe2, and remain difficult to explain36,37,38. Most notably, macroscopic measurements such as specific heat or thermal conductivity provide, in all these

systems, a negligible zero-temperature extrapolation of the electronic density, suggesting that the superconducting density of states at the Fermi level is very small12. The 2DHF is strongly

coupled to the superconducting bulk states and the proximity effect from bulk superconductivity should provide only a small amount of states at low energies. However, it is important to

consider the coupling of the 2DHF to strongly energy-dependent resonant states giving peaks in the tunnelling conductance as well. The concomitant broadening then leads to a large zero-bias

tunnelling conductance. Using a model that takes this into account (see Methods and Extended Data Fig. 8), we can understand the main features of the tunnelling conductance and follow the

superconducting gap with temperature (Fig. 4b,c). This solves the discrepancy between macroscopic and surface experiments and shows the relevance of two-dimensional electronic states to

understanding the tunnelling conductance. In summary, we have observed 2DHFs in terraced surfaces inside the HO phase of URu2Si2. The 2DHF exhibits quantum-well states with energy separation

of fractions of a meV when confined between steps. The 2DHF is connected to the bulk heavy-fermion states. At steps, we observe a 1DES, which shows in-plane electronic symmetry breaking and

inequivalent electronic arrangement in subsequent U layers in the HO phase. The discovery of 2DHFs and related confined states opens new possibilities to study the interplay of quantized

heavy-fermion states and unconventional superconductivity, as several heavy-fermion materials show unconventional superconductivity in the bulk, often coexisting with other long-range

ordered phases. Apart from URu2Si2, there are other heavy fermions, such as CeCoIn5, UBe13, UPt3 or UTe2, in which the proposed superconducting states are spin-singlet d-wave or spin-triplet

p-wave and f-wave states. These could exhibit 2DHFs and the associated edge states could incorporate excitations with unique properties such as Majorana fermions following non-Abelian

statistics. Furthermore, because the source of quantization is lateral confinement, correlated quantum-confined states can be obtained in nanostructures built on the surface by manipulation

of adatoms or by controlling layer growth in thin films15,16,39,40. This opens new avenues to generate, isolate and manipulate excitations in unconventional superconductors. METHODS STM

EXPERIMENTS Single crystals of URu2Si2 were grown by the Czochralski technique in a Tetra Arc Furnace. We scanned samples for a low residual resistivity and a high superconducting critical

temperature, close to 1.5 K. Such samples were then cut in a bar shape with dimensions 4 × 1 × 1 mm3, with the long distance parallel to the _c_ axis. We mounted the samples on the sample

holder of a scanning tunnelling microscope. The scanning tunnelling microscope was mounted in a dilution refrigerator. The resolution in energy of the setup was tested by measuring the

superconducting tunnelling conductance with the tip and sample of s-wave superconductors Al and Pb down to 100 mK (ref. 41). Details of image-rendering software are provided in refs. 42,43.

The scanning tunnelling microscope head features a low-temperature movable sample holder, which is used to cleave the sample at cryogenic temperatures41,44. At the same time, and importantly

for this study, the sample holder allows modifying many times the scanning window. The terraces discussed in this work were found in three different samples, after studying hundreds of

fields of view. SURFACE TERMINATION IN URU2SI2 We focus on U-terminated surfaces. In Extended Data Fig. 1a, we show the URu2Si2 crystal unit cell highlighting the U, Si and Ru planes; their

inter-layer distances are indicated in units of the _c_-axis lattice parameter. In Extended Data Fig. 1b–d, we show STM images corresponding to different surface terminations. These surfaces

are all obtained after cryogenic cleaving. On the surfaces full of square-shaped terraces, we find the results obtained in the main text. An example is shown in Extended Data Fig. 1b. All

the observed terraces are separated by _c_/2 ≈ 4.84 Å. In Extended Data Fig. 1c,d, we show terraces with a triangular shape, in which we do not observe the phenomena discussed in the main

text. Here the distance between consecutive terraces is about 0.11_c_, about 0.39_c_ or about 0.61_c_, which correspond, respectively, to the three possible distances between U–Si planes

(coloured arrows in Extended Data Fig. 1a). Therefore, we see that the surfaces with terraces having a triangular shape correspond to Si layers, sometimes with a U layer in between. By

contrast, the surfaces with terraces having a square shape are U terminated. Atomically resolved images inside terraces (Extended Data Fig. 1e) provide the square atomic U lattice with an

in-plane constant lattice of _a_ = 4.12 Å . In Extended Data Fig. 1f, we show a typical atomic-sized image on Si-terminated surfaces. We do not observe atomic resolution and have sometimes

seen circular defects. Defects in the U-terminated surfaces are very different, as shown in Extended Data Fig. 1g–o. We distinguish two distinct types of defect. The defects can be either

point-like protrusions (Extended Data Fig. 1g,h) or troughs (Extended Data Fig. 1i). Sometimes, defects are arranged in small-sized square or rectangular structures (Extended Data Fig.

1j–o). Most of these defects are probably because of vacancies or interstitial atoms in layers below the U surface layer. TUNNELLING CONDUCTANCE IN THE HO STATE The tunnelling conductance of

the HO state has been discussed in refs. 45,46,47,48,49. We have reproduced the results, as shown in Extended Data Fig. 2a,b. The tunnelling conductance results from simultaneous tunnelling

into heavy and light bands, as in other heavy-fermion compounds50,51,52. The red line in Extended Data Fig. 2a for _T_ = 18 K follows a Fano function

$$g(E)=A\frac{{\left(q+\left(E-{E}_{{\rm{Fano}}}\right)/{\Gamma }_{{\rm{F}}}\right)}^{2}}{\left(E-{E}_{{\rm{Fano}}}\right)/{\Gamma }_{{\rm{F}}}+1},$$ (1) in which _A_ is a constant of

proportionality, _q_ is the ratio between two tunnelling paths and _E_Fano is the Fano resonance energy with width \({\Gamma }_{{\rm{F}}}=2\sqrt{{(\pi

{k}_{{\rm{B}}}T)}^{2}+2{({k}_{{\rm{B}}}{T}_{{\rm{K}}})}^{2}}\), _T_K being the Kondo temperature45,46. For the fit, we include an asymmetric linear background owing to the degree of

particle–hole asymmetry in the light conduction band45,53. To account for the thermal broadening, we convolute the result with the derivative of the Fermi–Dirac distribution. We find _q_ = 

0.8 ± 0.5, _E_Fano = 3 ± 1 mV, ΓF = 22 ± 1 mV and _T_K = 90 ± 5 K, consistent with previous reports45,46. Inside the HO phase (red line in Extended Data Fig. 2b), we use the same Fano

function, multiplied by an asymmetric BCS-like gap function with an offset _δ_E $${g}_{{\rm{HO}}}=(E-{\delta }_{{\rm{E}}}-i{\gamma }_{{\rm{HO}}})/\left[[\sqrt{{(E-{\delta

}_{{\rm{E}}}-i{\gamma }_{{\rm{HO}}})}^{2}-{\Delta }_{{\rm{HO}}}^{2}}]\right]$$ (2) The resulting function is convoluted with the derivative of the Fermi–Dirac distribution function. We find

_δ__E_(4.1 K) = 1.5 ± 0.5 meV and ΔHO(4.1 K) = 4.0 ± 0.5 meV, consistent with previous reports45,46. Note that we also observe further features at lower temperatures and smaller bias

voltages (Extended Data Fig. 2c). The red line in Extended Data Fig. 2c is a fit described below. The features above the superconducting gap can also be roughly obtained by using two

Lorentzians at _ε_− and _ε_+ and an asymmetric background. Probably, the peaks at _ε_− and _ε_+ are because of avoided crossings in the band structure of the 2DHF at very low energies. We

notice that the small feature at _ε_+ occurs at a very similar energy range as a kink in the band structure found at the surface of Th-doped URu2Si2 (refs. 48,49). In the calculations we

show below, we can identify features in surface f-derived bands that can be associated to such peaks in the tunnelling conductance. However, such features can form as a result of

correlations elsewhere in the Brillouin zone as well. QUANTUM-WELL STATES AT TERRACES BETWEEN STEPS In Extended Data Fig. 3a, we show the tunnelling conductance background subtracted from

Fig. 2a to obtain Fig. 2b. To carry out the background subtraction, we first identify the features at _ε_− and _ε_+ in the conductance map. These are the light-blue regions centred at _ε_+

and the red–yellow region centred at _ε_− in Extended Data Fig. 3a. We then identify the edge states occurring at the steps, given by the red areas at the sides of Extended Data Fig. 3a.

Similar peaks are obtained on steps separating different terraces. The nature and shape of the 1DESs is discussed below. We then model these features by a set of Lorentzians and obtain the

pattern shown in Extended Data Fig. 3a. We subtract this pattern from the experiment (Fig. 2a) to obtain the pattern shown in Extended Data Fig. 3b. To model the quantum-well states, we use

the Fabry–Pérot interferometer expression for the density of states _g_FP(_x_, _E_) given by $$\begin{array}{l}{g}_{{\rm{FP}}}(x,E)\propto {\int

}_{0}^{k}\frac{{\rm{d}}q}{\sqrt{{k}^{2}-{q}^{2}}}\\ \times \frac{(1-{r}^{2})[1+{r}^{2}+2r\cos (2q(x-L)-\phi )]+(1-{r}^{2})[1+{r}^{2}+2r\cos (2qx+\phi )]}{1+{r}^{4}-2{r}^{2}\cos (2qL+2\phi

)}\end{array}$$ (3) with \(k=\sqrt{2{m}^{* }(E-{E}_{0})/{\hbar }^{2}}\), _m_* the electronic effective mass, _r_ the reflection amplitude, _ϕ_ the phase and _L_ the width of the terrace23.

The Fabry–Pérot interferometer is an optical resonator made of semireflecting mirrors and provides a simple and insightful way to model electronic wavefunctions confined between two wells.

More information on surface band structure and on quantum-well states by confinement is provided in refs. 3,9,54,55,56,57,58,59,60. We assume a symmetric potential well with _L_ = 20 nm, _r_

 = 0.5 and _ϕ_ = −π. The pattern generated by equation (3) is shown in Extended Data Fig. 3c. White points provide the positions of quantized levels as in Fig. 2a,b. It is not difficult to

see that the structure of quantized levels is renormalized together with the electronic band structure. Smaller electronic effective masses imply larger quantized level width and separation,

and vice versa. However, the simple Fabry–Pérot model does not take into account relaxation by electron–electron interactions, which lead to the extra level broadening discussed in the main

text and in refs. 27,28,61. The black lines in Fig. 2d are fits to the equation (3). To account for the behaviour at the edges, we add the equation (5) for the 1DES. We use the parameters

extracted for the terrace _L_3, discussed in Extended Data Table 1. We show further examples in Extended Data Fig. 4. Note that, in Fig. 2d, we use equation (3) along with the contribution

from the 1DES and contributions for _ε_− and _ε_+ at the bias voltages at which these features are observed in the tunnelling conductance. The 2DHF quantization was observed on the surfaces

of different URu2Si2 samples. In Extended Data Fig. 5a–c, we show the result on another sample. Notice here that terraces have different sizes. We show in Extended Data Fig. 5a the STM

topography image. In Extended Data Fig. 5b, we show a height profile through the white line in Extended Data Fig. 5a. In Extended Data Fig. 5c, we represent the tunnelling conductance along

the central terrace (_L_ ≈ 57 nm) of this profile. We observe similar tunnelling conductance curves as those presented in the main text. Notice the features at _ε_− and _ε_+. The quantized

levels are also readily observed. These occur, however, at different energy values, as the size of the terrace _L_ is different to that of the terrace in the main text. In Extended Data Fig.

5d, we represent the values of the quantized levels found in terraces of different sizes _L_ by different colours; we show the dispersion relation of the 2DHF as a magenta line. In Extended

Data Fig. 5e, we show as coloured points the bias voltage dependence of the energy spacing Δ_E_ between consecutive quantized levels for terraces _L_3, _L_4, the terrace with length _L_ = 

57 nm (shown in Extended Data Fig. 5c) and a terrace with length _L_ = 27 nm (not shown). We can write that \(\Delta E={E}_{n+1}-{E}_{n}=\left(\frac{{\hbar }^{2}{\pi }^{2}}{2{m}^{*

}{L}^{2}}\right)\left({(n+1)}^{2}-{n}^{2}\right)\), with _n_ = 1, 2, 3,…. This gives a square-root dependence of Δ_E_ on the energy, \(\Delta E={E}_{n+1}-{E}_{n}=\left(\frac{{\hbar }^{2}{\pi

}^{2}}{2{m}^{* }{L}^{2}}\right)\left(2n+1\right)\propto \sqrt{E}\), shown in Extended Data Fig. 5e. In Extended Data Fig. 5f, we plot the average value of \(\frac{\Delta E}{2n+1}\) for each

terrace as a function of _L_. We find the expected \(\frac{1}{{L}^{2}}\) dependence. Note that in Extended Data Fig. 5 and Fig. 1d, we do not perform any symmetrization. We can see a

tendency of the quantized states to shift towards the sides of the terrace, giving intensity patterns that are slightly asymmetric. We have calculated the expected patterns for different

reflection coefficients at each side of the terrace. This produces asymmetric patterns similar to those observed experimentally. However, it is difficult to separate such an asymmetry from

the signal coming from the edge states in the tunnelling conductance. Lateral symmetrization thus remains the best way to analyse and understand quantum states in the terraces observed here

in URu2Si2. We use \(\hbar /\tau ={\Gamma }_{0}+\left(| {E}_{0}| /4\pi \right){\left[(E+| {E}_{0}| )/{E}_{0}\right]}^{2}\) to fit the energy dependence of the lifetime61. The parameter

|_E_0|/4π is a prefactor that fits our experiment well. The prefactor is sometimes provided as a number28 and has also been estimated as \(\frac{{e}^{2}{k}_{{\rm{F}}}^{2}\pi }{4\pi

{{\epsilon }}_{0}\,32\,\hbar {q}_{{\rm{TF}}}}\) or, equivalently, \(\frac{e{m}^{3/2}}{32\times {3}^{5/6}{\pi }^{2/3}{{\epsilon }}_{0}^{1/2}{\hbar }^{4}{n}^{5/6}}\) (with _ϵ_0 the dielectric

constant, _n_ the electron density and _q_TF the Thomas–Fermi screening length)27. These estimations provide similar values to |_E_0|/4π. To obtain the energy dependence of the reflection

coefficient, _r_(_E_), we used the model described in ref. 24. To this end, we consider a one-dimensional periodic array of scattering objects, each modelled by a square potential well of

width _b_ < _L_ (_L_ is the width of the terrace). A constant complex potential _W_ provides confinement and coupling to the bulk states. We can then write

$$R(E)=\frac{{e}^{iqb}-{e}^{-iqb}}{{e}^{iqb}\left(\frac{k-q}{k+q}\right)-{e}^{-iqb}\left(\frac{k+q}{k-q}\right)}{e}^{-ikb}$$ (4) with \(k=\sqrt{\frac{2{m}^{* }E}{{\hbar }^{2}}}\) and

\(q=\sqrt{\frac{2{m}^{* }(E-W)}{{\hbar }^{2}}}\). The reflection coefficient _r_(_E_) is given by _r_(_E_) = |_R_(_E_)|2. For the dashed line in Fig. 2e, we use typical parameters for Cu,

with _m_* = 0.46_m_0 and _W_ = (−2 − 1_i_) eV. We shift the obtained curve in energy to obtain a result within the energy range of our data. For the continuous line in Fig. 2e, we use _m_* =

 17_m_0 and _W_ = (−18 − 5_i_) meV. BAND-STRUCTURE CALCULATIONS AT U-TERMINATED SURFACES OF URU2SI2 The band structure of bulk URu2Si2 has been analysed previously in detail using DFT

calculations62,63,64. Relevant results coincide with angle-resolved photoemission, STM and quantum oscillation studies22,65,66,67,68,69,70,71. Several surface states have been observed by

angle-resolved photoemission spectroscopy19,20,21,22. The surface state discussed in refs. 19,20 is formed by a hole-like band with its maximum at −35 meV and is thus far from what we

observe here. At the _X_ point of the Brillouin zone, there are no bulk states. Angle-resolved photoemission spectroscopy measurements show hints of surface-like bands with two-dimensional

character at these points21. We have taken a closer look at the _X_ point through DFT calculations. To this end, we built a U-terminated supercell consisting of 37 atomic layers, giving a

total of ten U layers (Extended Data Fig. 6a). We performed DFT calculations using the full-potential linearized augmented plane-wave method with local orbitals as implemented in the WIEN2k

package72. Atomic spheres radii were set to 2.5, 2.5 and 1.9 Bohr radii for U, Ru and Si, respectively. We used a 19 × 19 × 1 mesh of _k_-points in the first Brillouin zone, reduced by

symmetry to 55 distinct _k_-points. The RKmax parameter was set to 6.5, resulting in a basis size of approximately 5,400 (more than 100 basis functions per atom). Spin–orbital coupling was

included in the second variational step73 and relativistic local orbitals were included for U 6_p_1/2 and Ru 4_p_1/2 states. The basis for calculations of the spin–orbital eigenvalue problem

consisted of scalar-relativistic valence states of energies up to about 5 Ry, resulting in a basis size of about 3,800. The local density approximation was used for the treatment of

exchange and correlation effects63,74. In Extended Data Fig. 6b, we highlight in particular the U spin-up character of the obtained surface-projected band structure. The spin-down character

is much less pronounced within the shown energy range. There are several bands inside gaps of the bulk band structure, but only those around the _X_ point of the simple tetragonal Brillouin

zone, _X_st (see Extended Data Fig. 6c), are sufficiently shallow to provide large effective masses. We find a surface state (upper inset of Extended Data Fig. 6b) that consists of two

hybridized hole bands, forming an M-shaped feature close to the Fermi level. The dispersion relation found in our experiment (magenta line in the upper inset of Extended Data Fig. 6b) is

compatible with the central part of the M-shaped feature. 1DES AND HO WITHIN U LAYERS To analyse the 1DES at the step between two terraces, we use a one-dimensional Dirac-function-like

potential at the step, _V_(_x_) = _U_0_δ_(_x_ − _x_1DES), in which _x_1DES is the position of the 1DES. We take _U_0 = _b_0_V_0, with _b_0 the width of the potential well and _V_0 the energy

depth (_V_0 < 0). We add a complex potential, _V_(_x_) → (_U_0 − _iU_1)_δ_(_x_ − _x_1DES) to simulate the coupling of the 1DES to the bulk of the crystal. A schematic representation of

this model is shown in Extended Data Fig. 7a. Solving the Schrödinger equation for _E_ < 0, we obtain the Green’s function of the states in the potential well $$G(E)=A\frac{{e}^{-|

x-{x}_{{\rm{1DES}}}| /{\lambda }_{x}}}{E-{E}_{{\rm{1DES}}}+i{\eta }_{{\rm{1DES}}}},$$ (5) in which \({\lambda }_{x}=\frac{{\hbar }^{2}}{{m}^{* }}| {U}_{0}{| }^{-1}\) is the decay length with

_m_* the effective mass. _E_1DES and _η_1DES are the energy position and the energy broadening of the 1DES given by $${E}_{{\rm{1DES}}}=\delta V+{E}_{1}=\delta V-\frac{{m}^{* }}{{\hbar

}^{2}}\left({U}_{0}^{2}-{U}_{1}^{2}\right)$$ (6) $${\eta }_{{\rm{1DES}}}=-\frac{{m}^{* }}{{\hbar }^{2}}{U}_{0}{U}_{1}$$ (7) in which _δ__V_ is the height of the potential barrier of the well

relative to the Fermi level. We can now fit the tunnelling conductance at the 1DES using $${g}_{{\rm{1DES}}}={A}_{0}\frac{{\eta

}_{{\rm{1DES}}}{e}^{-\frac{\left|x-{x}_{{\rm{1DES}}}\right|}{{\lambda }_{x}}}}{{\left(E-{E}_{{\rm{1DES}}}\right)}^{2}+{\eta }_{{\rm{1DES}}}^{2}}$$ (8) convoluted with the derivative of the

Fermi–Dirac distribution function. Extended Data Table 1 shows the extracted fitting parameters _E_1DES, _η_1DES, _λ__x_ and _x_1DES for the four different terraces _L_1 to _L_4 from Fig. 1.

From Extended Data Table 1, we see that the energy position and the energy broadening of the 1DES are independent of the terrace size, with average values of _E_1DES = −0.52 ± 0.14 meV and

_η_1DES = 0.45 ± 0.06 meV. We also see that all the spatial features are always at the same position with respect to the step, _x_1DES ≈ 4.0_a_0, _a_0 being the in-plane lattice constant,

with a decay length _λ__x_ ≈ 0.9 nm ≈ 2_a_0. The latter indicates that 1DESs and 2DHFs couple when the decay length reaches a few interatomic distances. With the extracted average values

from Extended Data Table 1 for _λ__x_, _η_1DES and _E_1DES, we obtain _U_0 = 5.4 meVÅ, _U_1 = 0.38 meVÅ and _δ__V_ = 3.1 meV. We can analyse the 1DES through the tunnelling conductance at a

step (Extended Data Fig. 7b,c). At low bias voltages, we find a dip in the tunnelling conductance of a few nanometres at the upper side of the step (blue lines in Extended Data Fig. 7b; for

example, at −1.2 mV). The dip fills with the 1DES at about _E_1DES (red lines in Extended Data Fig. 7b at −0.4 mV) and empties again at higher bias voltages. This shows that charge depletion

close to the step opens a gap in the band structure. The gap is filled at the resonant energy of the 1DES, as observed previously in metals30,75,76. By normalizing the tunnelling

conductance to its shape far from the step (Extended Data Fig. 7c), we can follow the decay of the 1DES into the quantum-well states of the 2DHF with the model described above (Extended Data

Fig. 7a). The decay length is on the order of the inverse of the wavevector of the 2DHF. Taking a closer look at the steps, we surprisingly find a notable in-plane anisotropy of the 1DES.

As we see in Extended Data Fig. 7b, the 1DES is observed when crossing steps along the dashed red line in Extended Data Fig. 7b but not along the dashed blue line, as discussed in the main

text. It is useful here to take a closer look at the HO as well. As discussed in refs. 14,34,64,77,78,79, there is no dipolar (magnetic) or structural order related to the HO phase. Instead,

the U lattice can present some sort of long-range electronic ordering, whose actual symmetry and shape is considered as a relevant and open mystery14. Previous NMR measurements indicated

the absence of fourfold in-plane symmetry breaking in bulk URu2Si2 (ref. 35), whereas our STM data clearly show an in-plane symmetry breaking in the 1DES. At the surface, there can be marked

changes in the electronic structure owing to a modification of the valence of uranium atoms21,80. Rather, the breaking of the in-plane symmetry observed here in the 1DES suggests that a

fundamental breaking of the near-surface electronic properties of the U lattice is at play in the HO phase. INTERPLAY BETWEEN SUPERCONDUCTIVITY AND THE 2DHF We consider several parallel

conduction channels between the tip and the surface. For simplicity, we take into account tunnelling into the 2DHF and into the feature of largest size at _ε_− (Extended Data Fig. 8a). The

first channel, _t_1, connects the tip with the 2DHF. The 2DHF is superconducting by proximity from the bulk superconductor, which we model using a coupling _t_s. With the second channel,

_t_2, we connect the tip to other surface features. We can write the retarded Green function \({\hat{G}}^{{\rm{r}}}\) as $${\hat{G}}^{{\rm{r}}}={\left(\begin{array}{cc}{\hat{M}}_{{\rm{2D}}}

& {\hat{t}}_{-}\\ {\hat{t}}_{-} & {\hat{M}}_{-}\end{array}\right)}^{-1}$$ (9) with

$$\begin{array}{l}{\hat{M}}_{{\rm{2D}}}\,=\,\left(\begin{array}{cc}E-{E}_{{\rm{2DHF}}}-\frac{{t}_{+}^{2}}{E-{E}_{+}}+\frac{E+i\eta }{\Omega }{\bar{t}}_{{\rm{s}}} & \frac{\Delta }{\Omega

}{\bar{t}}_{{\rm{s}}}\\ \frac{\Delta }{\Omega }{\bar{t}}_{{\rm{s}}} & E+{E}_{{\rm{2DHF}}}^{* }-\frac{{t}_{+}^{2}}{E+{E}_{+}^{* }}+\frac{E+i\eta }{\Omega

}{\bar{t}}_{{\rm{s}}}\end{array}\right)\\ \,\,{\hat{M}}_{-}\,=\,\left(\begin{array}{cc}E-{E}_{-} & 0\\ 0 & E+{E}_{-}^{* }\end{array}\right)\\

\,{\hat{t}}_{-}\,=\,\left(\begin{array}{cc}{t}_{-} & 0\\ 0 & -{t}_{-}\end{array}\right)\\ \,{E}_{j}\,=\,{\varepsilon }_{j}-i\frac{{\Gamma }_{j}}{2}\,(j={\rm{2DHF}},-,+)\\

\,{\bar{t}}_{{\rm{s}}}\,=\,\frac{{t}_{{\rm{s}}}^{2}}{W}\\ \,\Omega \,=\,\sqrt{{\Delta }^{2}-{(E+i\eta )}^{2}}\end{array}$$ (10) in which _E_2DHF is the energy associated to the 2DHF and

includes the shift of energy owing to HO and Fano resonance, _W_ is an energy scale related to the normal density of states at the Fermi level by _ρ_(_E_F) = 1/(π_W_), Δ is the

superconducting gap and _η_ is a small energy relaxation rate. We have added the self energies _i_Γ_j_/2, (_j_ = 2DHF, −, +), with Γ_j_ the width of the resonance _j_. The differential

conductance is calculated as $$\sigma (V)={\sigma }_{0}\int T(E)\left(-\frac{{\rm{d}}f(E-eV,T)}{{\rm{d}}\left(eV\right)}\right){\rm{d}}E$$ (11) with $$\begin{array}{l}T(E)=-\frac{1}{\pi

}{\rm{Im}}({G}_{11}^{{\rm{r}}}(E){t}_{1}^{2}+{G}_{33}^{{\rm{r}}}(E){t}_{2}^{2}\\ \,\,+{t}_{1}{t}_{2}({G}_{13}^{{\rm{r}}}(E)+{G}_{31}^{{\rm{r}}}(E)))\end{array}$$ (12) Here _f_(_E_, _T_) is

the Fermi–Dirac distribution at the energy _E_ and temperature _T_ and \({\sigma }_{0}=\frac{2{e}^{2}}{h}\) is the quantum of conductance (with _h_ being Planck’s constant and _e_ the

elementary charge). Notice that _T_ is the transmission, not the density of states often used to discuss STM measurements in superconductors. Notice also that we take into account tunnelling

into the 2DHF (\({G}_{11}^{{\rm{r}}}\)) and into _ε_− (\({G}_{33}^{{\rm{r}}}\)), with mixed contributions (\({G}_{31}^{{\rm{r}}}\) and \({G}_{13}^{{\rm{r}}}\)). To fit the tunnelling

conductance curves shown in Fig. 4a,b and Extended Data Fig. 8b, we subtracted an asymmetric background (see Extended Data Figs. 2 and 8b). In Extended Data Table 2, we show the parameters

used to obtain the tunnelling conductance with temperature (shown as black lines in Fig. 4a,b) from equation (11). We do not vary the parameters _η_ = 0.018 meV, _ε_2DHF = 12 meV, Γ2DHF = 1 

meV, Γ− = 0.55 meV and Γ+ = 0.14 meV with temperature. We see that the superconducting lifetime itself is practically negligible, _η_ = 0.018 meV ≪  Δ = 0.2 meV. The large zero-bias

conductance is not because of the incomplete coupling to the bulk, as _t_s is close to one. There is further smearing coming from features at _ε_+ and _ε_−. Γ+ and Γ− provide the smeared

superconducting density of states and a finite tunnelling conductance at zero bias. Notice that the superconducting gap vanishes at the critical temperature _T_c but that the strongly

bias-voltage-dependent tunnelling conductance remains up to higher temperatures (Extended Data Fig. 8). There are numerous evidences for d-wave or more complex superconducting properties in

URu2Si2. The differences in the superconducting density of states between these superconducting states and s-wave superconductivity are relatively small, particularly because the tunnelling

conductance obtained with the model described here provides smeared conductance curves. The same occurs for the temperature dependence of the superconducting gap. Instead, the shape and

asymmetry of the tunnelling conductance at defects might provide the connection between the properties of the superconducting 2DHF and the unconventional superconductivity of the bulk.

RESULTS AT POINT DEFECTS We analyse in more detail here the tunnelling conductance at defects. We plot the tunnelling conductance obtained on two different defects in Extended Data Fig. 9a

as blue and red lines. Notice the pronounced electron–hole asymmetry, which provides curves that widely differ from the curves far from defects (black line in Extended Data Fig. 9a). As

discussed above, we can identify two kinds of defects: protrusions with height increases of around 15 pm, probably because of interstitial atoms located beneath the surface (Extended Data

Fig. 9b,d,f,h), and troughs of around 15 pm in size, probably because of vacancies beneath the surface (Extended Data Fig. 9c,e,g,i). The defects visibly affect the tunnelling conductance.

We plot the tunnelling conductance at _ε_+, 0 mV and _ε_− for both types of defect in Extended Data Fig. 9f,g. In Extended Data Fig. 9h,i, we show the spatial dependence of the tunnelling

conductance along a crystalline axis for both types of defect. At the site of the defect, there is a pronounced electron–hole asymmetry, which is opposite for each kind of defect.

Protrusions provide a substantially enhanced tunnelling conductance for empty states above the Fermi level, whereas troughs provide the opposite (Extended Data Fig. 9a). At zero bias, the

troughs show a pronounced in-plane anisotropy and a reduction of the superconducting gap, whereas the protrusions are in-plane isotropic and show an opened superconducting gap. When leaving

the defect, the usual behaviour is recovered after several nanometres (Extended Data Fig. 9h,i). To explain the pronounced modification of the electron–hole asymmetry of the tunnelling

conductance, let us consider electron correlations creating a large Fermi surface scenario at some portion of the band structure. At low temperatures, correlations provide an avoided band

crossing owing to hybridized heavy and light bands. We can assume that it happens somewhere in the band structure of URu2Si2, for example, close to the Γ point, as suggested in refs. 

14,18,47. Then, we obtain at low temperatures a large Fermi surface with heavy electrons and a close-lying light band with smaller wavevectors above the Fermi level. We can then assume that

both kinds of defect couple to different parts of such a correlated band structure. Following our experiments, protrusions create 2DHF coupling to the small light band and a large density of

states for empty states above the Fermi level (red curve in Extended Data Fig. 9a). Troughs can instead favour coupling to the heavy band, giving the opposite behaviour (blue curve in

Extended Data Fig. 9a). The superconducting gap is disturbed at troughs. Generally, we expect all sorts of defects to be pair breaking, as URu2Si2 is not a s-wave superconductor. However,

two-dimensional quantized states screen pair-breaking interactions from the underlying superconducting bulk81. This suggests that the absence of in-gap states in protrusions is because of

screening by the 2DHF. Troughs, however, produce a strong coupling to the heavy bands that carry the heavy superconducting state and thus also contribute to pair breaking. In-gap states at

troughs have a certain in-plane anisotropy (Extended Data Fig. 9g, middle panel). The anisotropy of the superconducting gap has been analysed with macroscopic measurements82,83,84,85. There

are indications for gap nodes, for instance, from specific heat measurements82,83,85. Local vortex dynamics shows an in-plane fourfold or twofold pinning potential86. When measured as a

function of the angle, there is a pronounced out-of-plane anisotropy, which has been associated to nodes along the _c_ axis, but there is little in-plane variation85. Following such a nodal

structure, several experiments propose a chiral _k__z_(_k__x_ + _i__k__y_) superconducting wavefunction84,85,87. The surface states of a (_k__x_ ± _i__k__y_) superconductor are predicted to

be chiral arc states connecting the Weyl nodes88. It is so far unclear how these surface states of the superconducting phase adapt to the surface states of the normal phase, which appear as

a consequence of the surface-induced perturbation of the crystalline potential. As mentioned, we observe an asymmetry at zero bias (Extended Data Fig. 9g, middle panel). To increase the

signal-to-noise ratio, we were forced to apply a fourfold symmetrization. Thus, the anisotropy might be twofold or fourfold. The chiral _k__z_(_k__x_ + _i__k__y_) state is in-plane

isotropic. However, the shape of in-gap states is determined by the anisotropy of the normal-state wavefunctions, together with the symmetry of the superconducting wavefunction, so that the

observed in-plane anisotropy of the in-gap states probably reflects the normal-state in-plane anisotropy. DATA AVAILABILITY All data supporting the findings of this study are available from

the corresponding authors on request. CODE AVAILABILITY The simulation code WIEN2k used to compute the surface electronic band structure can be obtained from

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243201 (2015). Article  ADS  PubMed  Google Scholar  Download references ACKNOWLEDGEMENTS We acknowledge discussions on the symmetry of HO with H. Harima. This work was supported by the

Spanish State Research Agency (PID2020-114071RB-I00, PID2020-117671GB-I00 and CEX2018-000805-M), by the Comunidad de Madrid through the programme NanomagCOST-CM (programme no.

S2018/NMT-4321) and by EU (PNICTEYES ERC-StG-679080 and COST SUPERQUMAP CA21144). H.S., E.H. and I.G. acknowledge SEGAINVEX at UAM for the design and construction of the STM cryogenic

equipment. J.A.G. and E.H. acknowledge the support of the Ministerio de Ciencia, Tecnología e Innovación de Colombia (grants 122585271058 and 784(2017)). W.J.H. and E.H. acknowledge support

from the Universidad Nacional de Colombia (DIEB projects 48148, 57522 and 201010025979(2016)). J.P.B. and G.K. acknowledge support from the French National Agency for Research (ANR) within

the projects FRESCO no. ANR-20-CE30-0020 and FETTOM ANR-19-CE30-0037. J.R. and P.M.O. acknowledge support from the Swedish Research Council (VR), the Knut and Alice Wallenberg Foundation

(grant no. 2022.0079) and the Swedish National Infrastructure for Computing (SNIC), through grant no. 2018-05973. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Facultad de Ingeniería y

Ciencias Básicas, Universidad Central, Bogotá, Colombia Edwin Herrera & Jose Augusto Galvis * Departamento de Física, Universidad Nacional de Colombia, Bogotá, Colombia Edwin Herrera 

& William J. Herrera * Laboratorio de Bajas Temperaturas y Altos Campos Magnéticos, Unidad Asociada UAM/CSIC, Departamento de Física de la Materia Condensada, Instituto Nicolás Cabrera

and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, Madrid, Spain Edwin Herrera, Isabel Guillamón, Víctor Barrena & Hermann Suderow * School of Engineering,

Science and Technology, Universidad del Rosario, Bogotá, Colombia Jose Augusto Galvis * Departamento de Física Teórica de la Materia Condensada, Instituto Nicolás Cabrera and Condensed

Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, Madrid, Spain Alfredo Levy Yeyati * Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden Ján Rusz & 

Peter M. Oppeneer * University Grenoble Alpes, CEA, Grenoble-INP, IRIG, PHELIQS, Grenoble, France Georg Knebel, Jean Pascal Brison & Jacques Flouquet * Institute for Materials Research

(IMR), Tohoku University, Oarai, Japan Dai Aoki Authors * Edwin Herrera View author publications You can also search for this author inPubMed Google Scholar * Isabel Guillamón View author

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Herrera View author publications You can also search for this author inPubMed Google Scholar * Jose Augusto Galvis View author publications You can also search for this author inPubMed 

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author inPubMed Google Scholar * Peter M. Oppeneer View author publications You can also search for this author inPubMed Google Scholar * Georg Knebel View author publications You can also

search for this author inPubMed Google Scholar * Jean Pascal Brison View author publications You can also search for this author inPubMed Google Scholar * Jacques Flouquet View author

publications You can also search for this author inPubMed Google Scholar * Dai Aoki View author publications You can also search for this author inPubMed Google Scholar * Hermann Suderow

View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS E.H. performed the study and made all experiments, with the supervision of I.G. E.H.

analysed the data and compared with theory, with the supervision of W.H. and J.A.G. J.R. and P.M.O. performed the band-structure calculations and discussed the features of surface states.

The models to account for superconducting features and for the behaviour at the step were proposed by W.H. and A.L.Y. D.A. synthesized and G.K. and J.P.B. characterized the samples. D.A.,

J.F. and H.S. proposed the study. The manuscript was written by E.H., I.G., W.H., A.L.Y., P.M.O. and H.S., with contributions from all authors. CORRESPONDING AUTHORS Correspondence to Edwin

Herrera or Hermann Suderow. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing interests. PEER REVIEW PEER REVIEW INFORMATION _Nature_ thanks Yi-feng Yang and the

other, anonymous, reviewer(s) for their contribution to the peer review of this work. ADDITIONAL INFORMATION PUBLISHER’S NOTE Springer Nature remains neutral with regard to jurisdictional

claims in published maps and institutional affiliations. EXTENDED DATA FIGURES AND TABLES EXTENDED DATA FIG. 1 DIFFERENT SURFACE TERMINATIONS IN URU2SI2. A, URu2Si2 crystal structure. U

atoms are shown in green, Ru in magenta and Si in yellow. We show using the same colours the corresponding planes and indicate the distances between planes, normalized to the _c_-axis

lattice parameter. With coloured arrows, we highlight the distance between the planes observed in the STM images. B, STM topography image at a surface with terraces. As we discuss in the

text, these are all U-terminated terraces. The height scale is given by the bar on the left. In the upper-right inset, we show a height histogram (distances normalized to the _c_-axis

lattice constant). Notice that all peaks are located at an integer of _c_/2. The crystal axes are shown as white arrows. The small white square is the area shown in E. C,D, STM topography

images at surfaces showing terraces with a triangular shape. These are distinct terraces. The crystal axes are shown as arrows. In the upper-right inset, we show the height histogram, with

distances between terraces having different sizes. Coloured arrows are as in A and identify height differences between the U and Si terraces. The small white squares in C and D provide the

areas shown in F and G, respectively. We show other kinds of defect in H to O, with the colour scale given by the bars on the left of each image. All data were taken at 100 mK with a

tunnelling current _I_tunnel = 10 nA and a bias voltage _V_Bias = 10 mV. EXTENDED DATA FIG. 2 TUNNELLING CONDUCTANCE OF THE HO STATE. A, Tunnelling conductance versus bias voltage at a

temperature above the HO transition temperature (17.5 K) is shown as black points. The red line is a fit based on the Fano lineshape owing to parallel tunnelling paths into light and heavy

bands. B, Tunnelling conductance inside the HO state versus bias voltage is shown as black points. Notice the reduction of the bias voltage range, to highlight the features associated to HO

within the Fano lineshape. The HO gap is schematically marked by an arrow. The red line is a fit as described in the text. C, At the lowest temperatures and focusing on very low bias

voltages, we observe the tunnelling conductance represented as black points. The red line is a fit with a model described in Methods. We show by dashed lines the features at _ε_+, _ε_− and

at the superconducting gap value ΔSC. Temperatures at which the data were taken are provided in each panel. EXTENDED DATA FIG. 3 SUBTRACTED TUNNELLING CONDUCTANCE BACKGROUND AND QUANTIZED

DENSITY OF STATES PATTERN. A, Bias voltage dependence of the subtracted tunnelling conductance background versus distance. B, Bias voltage dependence of the tunnelling conductance obtained

after the subtraction. C, Fabry–Pérot calculation using the parameters discussed in the text. Quantized levels are represented by white points. Colour scale from blue (low conductance) to

red (high conductance) represents values given in the bars at the bottom left. EXTENDED DATA FIG. 4 FITS OF THE TUNNELLING CONDUCTANCE. We show as circles the tunnelling conductance as a

function of distance for the bias voltages indicated in the legend of each panel (taken at 0.1 K). Lines are fits described in Methods, with–in addition–a nearly constant background to

account for the features at _ε_− and _ε_+ and the features of the 1DES. EXTENDED DATA FIG. 5 DISPERSION RELATION AND QUANTIZATION ON DIFFERENT TERRACES. A, STM image on a field of view

containing a few U-terminated terraces (taken at 0.1 K). B, Height profile (normalized with the _c_-axis lattice constant) along the white line in A. C, Tunnelling conductance along the

central terrace (_L_ ≈ 57 nm) of the profile shown in B (colour scale from red, high conductance, to blue, low conductance). We mark the position of the features at _ε_+ and _ε_− with dashed

white lines. D, Dispersion relation of the 2DHF (magenta line). Points are the positions of the quantized levels obtained from different terraces as described in the text (size _L_ of each

terrace is 20 nm blue, 28 nm red, 38.5 nm orange and 57 nm green). E, separation between energy levels Δ_E_ as a function of the energy (coloured points following the colour code of D).

Lines are a square-root dependence, from \(\Delta E={E}_{n+1}-{E}_{n}=\left(\frac{{\hbar }^{2}{\pi }^{2}}{2{m}^{* }{L}^{2}}\right)\left(2n+1\right)\). F, Average of Δ_E_ (colour code as in

D) as a function of the inverse of _L_2. EXTENDED DATA FIG. 6 DFT CALCULATIONS OF THE SURFACE BAND STRUCTURE AROUND THE _X_ POINT OF THE SIMPLE TETRAGONAL BRILLOUIN ZONE. A, U-terminated

supercell structure of URu2Si2 used for DFT calculations. B, Band structure of URu2Si2 in a slab calculation (blue points), described in the text, along the high-symmetry directions of the

simple tetragonal Brillouin zone, Γst, _M_st and _X_st. The size of the points provides the U spin-up character of the bands. In the upper inset, we show a zoom-in around the _X_st point.

The magenta line provides the dispersion relation compatible with our experiments and the black points are the quantized levels we identified (from Fig. 2c). C, The usual Brillouin zone

construction of URu2Si2, with the tetragonal Brillouin zone (red lines) and the simple tetragonal (st) construction (yellow lines) used to describe the low-temperature HO phase. EXTENDED

DATA FIG. 7 1DES AND HO WITHIN U LAYERS. A, Schematic representation of the parameters used to describe the 1DES at a step between two consecutive terraces of length _L_. We represent the

quantized levels of the confined 2DHF on each terrace with pink colour. The dashed black (continuous red) line represents the exponential behaviour of the 1DES without (with) coupling to the

quantized levels. B, Topography image along a step between two consecutive terraces formed by U layers is shown in the top-left panel. The colour scale is shown as a bar on the left, in Å.

White arrows provide the in-plane crystalline axis. The other panels are tunnelling conductance images (colour scale provided at the left) in the same field of view for different values of

the bias voltage (taken at 0.1 K). Notice the different shape of the 1DES along steps that are located on the two in-plane directions of the U atom lattice. C, Circles in red and blue

represent the tunnelling conductance as a function of distance (referred to the tunnelling conductance far from the edge at _x_ = 50 nm) at the bias voltage at which the 1DES appears _V_ ≈ 

−0.4 mV, _σ_norm(−0.4 mV) = _σ_(_V_ = −0.4 mV,_x_) − _σ_(_V_ = −0.4 mV,_x_ = 50 nm) along the red and blue lines in the top-left panel of B. The continuous red line is the fit of the 1DES

(equation (8)) plus the quantized level model (equation (3)). The bottom panel shows (black line) the STM height profile along the red and blue lines in the top-left panel of B in units of

the _c_-axis lattice constant. EXTENDED DATA FIG. 8 MODEL AND TUNNELLING CONDUCTANCE FROM 0.1 K TO 4 K. A, Schematic representation of the model and its parameters described in Methods. B,

Tunnelling conductance as a function of temperature, also well above the superconducting transition temperature _T_c. Notice how the feature at _ε_+ vanishes because of thermal smearing,

whereas the more pronounced feature at _ε_− remains up to higher temperatures. Data are shown by coloured points and the black lines are fits, described in the text. EXTENDED DATA FIG. 9

TUNNELLING CONDUCTANCE AT ATOMIC-SIZED DEFECTS. A, Tunnelling conductance obtained far from defects (black line). We show by the vertical dashed lines the features at _ε_+ and _ε_−. Curves

are taken at 0.1 K. B,C, Topographic images of the U lattice showing two types of defect. White arrows provide the in-plane lattice constants. The white scale bar is also shown. The red and

blue triangles provide the positions at which the red and blue curves in A were taken. D,E, Profiles obtained along the dashed lines of B and C with the same colours. F,G, Tunnelling

conductance maps obtained at the bias voltages shown in each panel, using the colour scale provided on the left for protrusions and troughs, respectively. The scale bar is given in black. To

increase the signal-to-noise ratio, we have made a fourfold symmetrization in all these panels. H,I, Conductance across the defect site as a function of the distance from the centre of the

defects in B and C, respectively, along a crystalline direction. The colour scale is given by the bar on the left of F and G. RIGHTS AND PERMISSIONS OPEN ACCESS This article is licensed

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at the surface of a heavy-fermion superconductor. _Nature_ 616, 465–469 (2023). https://doi.org/10.1038/s41586-023-05830-1 Download citation * Received: 19 June 2022 * Accepted: 13 February

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