doi: 10.1038/nphys4107 quasiparticle …...in the format proided b the authors and unedited. © 2017...

In the format provided by the authors and unedited.

© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

1

Quasiparticle Interference and Strong Electron-Boson Coupling in the Quasi-One-Dimensional Bands of Sr2RuO4

Zhenyu Wang, Daniel Walkup, Philip Derry, Thomas Scaffidi, Melinda Rak, Sean Vig, Anshul Kogar, Ilija Zeljkovic, Ali Husain, Luiz H. Santos, Yuxuan Wang, Andrea Damascelli, Yoshiteru Maeno, Peter Abbamonte, Eduardo Fradkin, and Vidya Madhavan

Supplementary Information part I

1. STM topography

In principle, cleavage in Sr2RuO4 can occur either between two SrO planes or between the RuO2– SrO planes. As one

can see from the crystal structure shown in Fig. 1b, these two processes should result in two different step heights

[1]. More than ten step edges were identified in the four samples studied in our work, and all of them correspond to

a step height of ~ 6.3Å, which is consistent with half the unit cell height and indicates a preferential cleave plane.

Figure S1a shows one of these step edges. Since cleaving along the SrO plane should be energetically favorable, we

believe that in the samples studied by us cleavage occurs between two SrO planes, exposing SrO terminated surfaces.

Figure S1. STM topographs of Sr2RuO4. (a) Typical step edge showing a step height of ~ 6.3Å. (b-d) Three kinds of

defects on the surface. The topographic lattice is indicated by black lines, and red circles denote those defects. The

inset of c shows a zoom-in of defect B with different color scale to highlight the core of this defect.

A square lattice with atomic spacing of 3.9 Å is observed on our cleaved surface. The STM topographs also reveal

three distinct defects (labeled as A, B and C in Fig. S1b-d). One can clearly observe that defect A sits between two

sites of the crystalline lattice. Defect B appears like a hole whose core is located between four lattice sites but shifted

1

Quasiparticle Interference and Strong Electron-Boson Coupling in the Quasi-One-Dimensional Bands of Sr2RuO4

Zhenyu Wang, Daniel Walkup, Philip Derry, Thomas Scaffidi, Melinda Rak, Sean Vig, Anshul Kogar, Ilija Zeljkovic, Ali Husain, Luiz H. Santos, Yuxuan Wang, Andrea Damascelli, Yoshiteru Maeno, Peter Abbamonte, Eduardo Fradkin, and Vidya Madhavan

Supplementary Information part I

1. STM topography

In principle, cleavage in Sr2RuO4 can occur either between two SrO planes or between the RuO2– SrO planes. As one

can see from the crystal structure shown in Fig. 1b, these two processes should result in two different step heights

[1]. More than ten step edges were identified in the four samples studied in our work, and all of them correspond to

a step height of ~ 6.3Å, which is consistent with half the unit cell height and indicates a preferential cleave plane.

Figure S1a shows one of these step edges. Since cleaving along the SrO plane should be energetically favorable, we

believe that in the samples studied by us cleavage occurs between two SrO planes, exposing SrO terminated surfaces.

Figure S1. STM topographs of Sr2RuO4. (a) Typical step edge showing a step height of ~ 6.3Å. (b-d) Three kinds of

defects on the surface. The topographic lattice is indicated by black lines, and red circles denote those defects. The

inset of c shows a zoom-in of defect B with different color scale to highlight the core of this defect.

A square lattice with atomic spacing of 3.9 Å is observed on our cleaved surface. The STM topographs also reveal

three distinct defects (labeled as A, B and C in Fig. S1b-d). One can clearly observe that defect A sits between two

sites of the crystalline lattice. Defect B appears like a hole whose core is located between four lattice sites but shifted

Quasiparticle interference and strong electron–mode coupling in the quasi-one-dimensional

bands of Sr2RuO4

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS4107

NATURE PHYSICS | www.nature.com/naturephysics 1

http://dx.doi.org/10.1038/nphys4107

http://www.nature.com/naturephysics


2

laterally towards one of those four sites (see inset of Fig. S1c). These two kinds of defects have been observed in

previous STM work [1]. In addition, we find a third kind of defect C, which sits right in between four sites of the lattice.

A recent study shows that the ruthenate surface has high chemical activity and residual carbon monoxide in the UHV

chamber may be adsorbed at the apical oxygen site [2]. From experiment and comparison with theory they [2] find

that when the carbon atom in CO replaces the apical oxygen atom at the surface, it forms a COO molecule which

shows a dark cross with one thicker arm and one thinner arm as we also see in our topography.

2. Energy integrated M- EELS data

Figure S2. Energy integrated M-EELS data taken along the (H, H) direction in reciprocal space shows a broad peak near (1/2, 1/2).






3

3. d2I/dV2 spectrum

Features at about ±40meV can be observed in the raw dI/dV curves. In order to show these features more clearly, we present the derivative of the spectrum in Fig. S3a. One can find that peak (or dip at negative bias side) are located at the same energy where the main kink appears in the dispersion. Its particle-hole symmetric nature suggests that it originates from coupling with a collective mode.

We also show the dI/dV spectrum in a large energy range in Fig. S3b. The background of the spectrum is quite asymmetric and shows a broad peak around -350 meV, which may be related to the bottom of the β band [3].

Figure S3. Additional information of dI/dV spectrum. (a) Derivative of dI/dV spectrum and comparing its energy scale

with the kinks in dispersion. (b) dI/dV spectrum in a large energy range.






4

Supplementary Information part II

4. QPI data analysis procedure

We perform three operations to enhance the signal to noise of the raw QPI images (Fig.S4e-h) and obtain the images shown in Fig.2. First, we use the drift-correction algorithm developed by Lawler-Fujita [4] to remove the effects of thermal and piezoelectric drift. Second, because the center peak around q = (0, 0) in the FFTs stems from randomly scattering defects and long range variations of the surface, we apply a small spatial mask (3-4 pixels, shown in Fig.S4c) at the positions of these defects in the conductance maps to suppress this effect. This process does not dramatically affect the quasiparticle scattering in our case. Finally, we symmetrize the QPI images. Comparison of the raw, symmetrized, and impurity-suppressed QPI images have been shown in Fig.S4. We note here that the QPI patterns along the qx and qy directions are similar, and that only their brightness differs.

Figure S4. QPI data analysis procedure. (a) Topography. (b) dI/dV conductance map taken at 3meV. (c) Mask created with the position of defects. (d) spatial-filtered conductance map with the mask at 3meV. (e-h)






5

conductance maps after drift-correction at four energies. (i-l) raw FFTs of the conductance map shown in e-h. (m-p) Symmetrized FFTs. (q-t) Symmetrized and impurity-suppressed FFTs.

5. Simulation of the QPI using T-matrix approach

In order to calculate the QPI for Sr2RuO4, we model the relevant conduction bands, derived from the Ru t2g orbitals, via an effective three-orbital tight binding model for electrons hopping on a 2D square lattice, given in [5] and Refs. therein:

𝐻𝐻 = ∑ 𝜓𝜓𝑠𝑠†(𝒌𝒌)𝑯𝑯𝑠𝑠(𝒌𝒌)𝜓𝜓𝑠𝑠(𝒌𝒌)

𝒌𝒌,𝑠𝑠 (1)

where 𝜓𝜓𝑠𝑠(𝒌𝒌) = (𝑐𝑐𝑎𝑎,𝑠𝑠,𝒌𝒌 , 𝑐𝑐𝑏𝑏,𝑠𝑠,𝒌𝒌 , 𝑐𝑐𝑐𝑐,−𝑠𝑠,𝒌𝒌)𝑇𝑇, representing the 4dxz, 4dyz, 4dxy Ru orbitals respectively for spin 𝑠𝑠 =

+1(−1) for up (down) spins. The Hamiltonian kernel is given by

𝑯𝑯𝑠𝑠(𝒌𝒌) = (

𝜖𝜖𝑎𝑎(𝒌𝒌) 𝑔𝑔(𝒌𝒌) − i 𝜂𝜂 i 𝜂𝜂𝑔𝑔(𝒌𝒌) + i 𝜂𝜂 𝜖𝜖𝑏𝑏(𝒌𝒌) − 𝑠𝑠 𝜂𝜂

− i 𝜂𝜂 − 𝑠𝑠 𝜂𝜂 𝜖𝜖𝑐𝑐(𝒌𝒌)) (2)

with 𝜖𝜖𝑖𝑖(𝒌𝒌) = −2𝑡𝑡𝑖𝑖,𝑥𝑥 cos(𝑎𝑎0𝑘𝑘𝑥𝑥) − 2𝑡𝑡𝑖𝑖,𝑦𝑦 cos(𝑎𝑎0𝑘𝑘𝑦𝑦) − 4𝑡𝑡𝑖𝑖,𝑥𝑥𝑦𝑦 cos(𝑎𝑎0𝑘𝑘𝑥𝑥) cos(𝑎𝑎0𝑘𝑘𝑦𝑦) − 𝜇𝜇𝑖𝑖 , and 𝑔𝑔(𝒌𝒌) =−4𝑡𝑡𝑎𝑎𝑏𝑏sin (𝑎𝑎0𝑘𝑘𝑥𝑥)sin (𝑎𝑎0𝑘𝑘𝑦𝑦). The spin-orbit coupling (SOC) is controlled by the parameter 𝜂𝜂, while 𝑔𝑔(𝒌𝒌) represents

the inter-orbital hopping between 𝑎𝑎 and 𝑏𝑏 (4dxz and 4dyz) orbitals. The parameters of this effective free model are chosen to reproduce the shape of the Fermi surfaces (FSs) and the ratio of effective masses of resultant bands as obtained by experiment [5], such that (𝑡𝑡𝑎𝑎,𝑥𝑥 = 𝑡𝑡𝑏𝑏,𝑦𝑦 , 𝑡𝑡𝑎𝑎,𝑦𝑦 = 𝑡𝑡𝑏𝑏,𝑥𝑥 , 𝑡𝑡𝑎𝑎,𝑥𝑥𝑦𝑦 = 𝑡𝑡𝑏𝑏,𝑥𝑥𝑦𝑦 , 𝑡𝑡𝑐𝑐,𝑥𝑥 = 𝑡𝑡𝑐𝑐,𝑦𝑦 , 𝑡𝑡𝑐𝑐,𝑥𝑥𝑦𝑦 , 𝑡𝑡𝑎𝑎𝑏𝑏 , 𝜇𝜇𝑎𝑎 = 𝜇𝜇𝑏𝑏 ,𝜇𝜇𝑐𝑐 , 𝜂𝜂) = (1.0, 0.1, 0.0, 0.8, 0.3, 0.01, 1.0, 1.1, 0.1) 𝑡𝑡, where 𝑡𝑡 = +150 meV to fit the MDC of the 𝛽𝛽 band around the Fermi level, in agreement with Ref. [6]. As noted in Ref. [6], both 𝑡𝑡𝑎𝑎𝑏𝑏 and 𝜂𝜂 produce repulsion between the bands, and as such there is some freedom in their choice. We adopt the value of 𝑡𝑡𝑎𝑎𝑏𝑏 used therein, which is smaller than in calculations without SOC (Ref. [7][8]) but in agreement with a recent fit to ARPES data that includes SOC [6].

Diagonalization yields three pairs of bands with pseudospin 𝜎𝜎 = ±1, labelled 𝛼𝛼, 𝛽𝛽 and 𝛾𝛾; the 𝛽𝛽 and 𝛾𝛾 bands are particle-like, while the 𝛼𝛼 band is hole-like. Throughout we use Roman letters to refer to orbital and spin space, while Greek letters refer to band and pseudospin. Figure 1a plots the FSs of these three bands across the first Brillouin zone (1BZ), which agree well with experimental results from ARPES and quantum oscillation studies [9] [10]. The orbital contributions to each band around the FS are depicted via the hue of the spectral function in Fig. 3a (same as Fig. S5a), with red, blue and green representing 𝑎𝑎, 𝑏𝑏 and 𝑐𝑐 respectively. The 𝛼𝛼 and 𝛽𝛽 bands are largely derived from the

𝑎𝑎 and 𝑏𝑏 (4dxz and 4dyz) orbitals, while the 𝛾𝛾 band is largely composed of the 𝑐𝑐 (4dxy) orbital.






6

The QPI due to each orbital is calculated as the power spectrum 𝑃𝑃(𝒒𝒒, 𝜔𝜔) = |𝑔𝑔𝑐𝑐(𝒒𝒒, 𝜔𝜔)|2 for the spatial interference pattern of the conductance measured via STM. We calculate the interference of scattered conduction electron states, treating the scattering in terms of the full T-matrix within this three-orbital model:

Δ𝜌𝜌𝑖𝑖(𝒒𝒒, 𝜔𝜔) = −𝜋𝜋−1Im ∑[𝑮𝑮𝑠𝑠0(𝒌𝒌, 𝜔𝜔)𝑻𝑻(𝒌𝒌, 𝒌𝒌 + 𝒒𝒒, 𝜔𝜔)𝑮𝑮𝑠𝑠

0(𝒌𝒌 + 𝒒𝒒, 𝜔𝜔)]𝑖𝑖𝑖𝑖𝑠𝑠,𝒌𝒌

, (3)

with 𝑮𝑮𝑠𝑠0(𝒌𝒌, 𝜔𝜔) = ((𝜔𝜔 + i 𝛿𝛿)𝑰𝑰 − 𝑯𝑯𝑠𝑠(𝒌𝒌))−1

the unperturbed (retarded) Green function matrix in orbital/spin space. Throughout we calculate the QPI over the 1BZ on a 2048 x 2048 grid in 𝒌𝒌-space, employing a broadening parameter δ = 5meV. The conductance is then given by tracing over the contribution from each orbital,

𝑔𝑔𝑐𝑐(𝒒𝒒, 𝜔𝜔) = 𝑔𝑔𝑐𝑐0(𝜔𝜔) + ∑ 𝐶𝐶𝑖𝑖Δ𝜌𝜌𝑖𝑖(𝒒𝒒, 𝜔𝜔),

𝑖𝑖=𝑎𝑎,𝑏𝑏,𝑐𝑐 (4)

such that the orbital-dependent coefficient 𝐶𝐶𝑖𝑖 reflects the magnitude of the tunneling matrix element between tip and orbital 𝑖𝑖. If equal contributions from all three orbitals is assumed, the calculated QPI at the Fermi level is given by Fig. S5b, with a strong signal from the 𝛾𝛾 band (𝑐𝑐 orbital), which is not observed in the experimental QPI. The spatial anisotropy of the 𝑎𝑎, 𝑏𝑏 and 𝑐𝑐 orbitals controls 𝐶𝐶𝑖𝑖 : experimental evidence suggests that the cleavage results in a surface SrO layer, so that there is an oxygen atom lying above each outermost Ru atom, as depicted in Fig. 1c. The 𝑝𝑝-orbitals

of this apical oxygen facilitate tunneling overlap between the 4dxz/4dyz (out-of-plane) orbitals and the tip, but not

the 4dxy (in-plane) orbital, and thus 𝐶𝐶𝑎𝑎 = 𝐶𝐶𝑏𝑏 ≫ 𝐶𝐶𝑐𝑐. Here we take 𝐶𝐶𝑎𝑎 = 𝐶𝐶𝑏𝑏 = 1 and 𝐶𝐶𝑐𝑐 = 0; this results in the effective spectral density visible via STM as plotted in Fig. S5c.

We assume scattering due a local perturbation in the potential of single site in the lattice; multiple independent, local scattering centers give rise to trivial moiré-type interference patterns in the QPI, as detailed in Ref. [11] and [12], but no qualitative effect on the observed QPI. The T-matrix is thus momentum-independent, 𝑻𝑻(𝒌𝒌, 𝒌𝒌 + 𝒒𝒒, 𝜔𝜔) =𝑻𝑻(𝜔𝜔) and diagonal in orbital/spin space, taking the form

𝑻𝑻(𝜔𝜔) = 𝑽𝑽 (1 + ∑ 𝑮𝑮0(𝒌𝒌, 𝜔𝜔)𝑽𝑽 (1 + ∑ 𝑮𝑮0(𝒌𝒌′, 𝜔𝜔)𝑽𝑽

𝒌𝒌′(1 + ⋯ ))

𝒌𝒌)

= 𝑽𝑽(𝑰𝑰 − 𝑮𝑮loc0 (𝜔𝜔)𝑽𝑽)−1

(5)

where 𝑮𝑮loc0 (𝜔𝜔) is the local Green function matrix in real space, 𝑽𝑽 = 𝑣𝑣0𝑰𝑰 the local scattering potential. Eq. (5)

represents the exact T-matrix for arbitrary scattering strength (controlled by 𝑣𝑣0); the Born series expansion is re-summed to infinite order, giving rise generically to a dynamic T-matrix, which reduces to 𝑻𝑻 ∼ 𝑽𝑽 in the weak-scattering (Born) limit, 𝑣𝑣0 ≪ 1. The resultant QPI at the Fermi level is plotted for the weak-scattering case in Fig. S5d (Fig. 3c in main text).

In addition to the bulk electronic structure described by Eqs. (1) & (2), Sr2RuO4 undergoes a well-known surface reconstruction upon cleavage at low temperature [13][14], resulting in a rotation of the RuO6 octahedra (of approx. 6°) and a doubling of the unit cell, see Fig. 1c. This reconstruction results in down-folding of the 𝛼𝛼, 𝛽𝛽 and 𝛾𝛾 bands into the reduced surface Brillouin zone (SBZ), and an accompanied adjustment of the strength of electronic correlations and FS topology [13][14]. In terms of the tight binding model described by Eqs. (1) & (2), this reconstruction of the surface band FSs is well described by a simple shift in the chemical potential, 𝜇𝜇𝑎𝑎/𝑏𝑏,surf = 𝜇𝜇𝑎𝑎/𝑏𝑏 − 25meV and 𝜇𝜇𝑐𝑐,surf =𝜇𝜇𝑐𝑐 + 25meV , as depicted in Fig. S7b (c.f. Figure 1 of Ref. [13], in which a detailed treatment of the surface reconstruction via DFT is carried out). The most significant effect of this reconstruction is a change in the FS topology of the 𝛾𝛾 band, which becomes hole-like rather than electron-like (as in the bulk). The van Hove point (at 𝑋𝑋 = (𝜋𝜋, 0)






7

in 𝒌𝒌-space), situated above the Fermi level for the bulk band, is thus shifted below the Fermi level for the surface band. By contrast, the 𝛼𝛼 and 𝛽𝛽 bands are only weakly affected.

Figure S5. Simulation of QPI. (a) Spectral density at the Fermi level for α, β and γ bulk bands. (b) Simulated QPI with all three bands. (c) Spectral density at the Fermi level for the out-of-plane orbitals in the bulk only, excluding the in-plane orbital, to which the STM tip couples only very weakly. (d) Simulated QPI with only out-of-plane orbitals. (e) Measured FT-QPI near the Fermi level. (f) Fermi-level QPI for out-of-plane bands considering the band folding effects.

Given the very poor tip-orbital overlap for the in-plane 𝑐𝑐 orbital (and thus the absence of a tunneling current due to the 𝛾𝛾 band), the effect of the surface band FS reconstruction on the calculated QPI is minimal. However, the surface structural distortion gives rise to a weak coupling between the folded and unfolded bands (as detailed in Ref. [15] for a similar distortion in the bilayer analogue Sr3Ru2O7), and thus scattering between the two sets of bands in the QPI. This results in folded features in the QPI, accounted for phenomenologically as

Δ𝜌𝜌obs(𝒒𝒒, 𝜔𝜔) = Δ𝜌𝜌(𝒒𝒒, 𝜔𝜔) + 𝑥𝑥 Δ𝜌𝜌(𝒒𝒒 + 𝑸𝑸, 𝜔𝜔) , (6)

where Δ𝜌𝜌(𝒒𝒒, 𝜔𝜔) is the undistorted interference pattern, 𝑎𝑎0𝐐𝐐 = (𝜋𝜋, 𝜋𝜋) and 𝑥𝑥 is a mixing parameter dependent on the strength of the coupling arising from the distortion. We note that not only the magnitude but also the sign of 𝑥𝑥 is






8

important as this is in effect the interference of two different QPI signals, i.e. 𝑃𝑃obs(𝒒𝒒, 𝜔𝜔) ≠ 𝑃𝑃(𝒒𝒒, 𝜔𝜔) + |𝑥𝑥|𝑃𝑃(𝒒𝒒 +𝑸𝑸, 𝜔𝜔). By setting 𝑥𝑥 = −0.5 we arrive at the calculated QPI plotted Fig. S5f (Fig. 3d in main text).

6. Intra-α-band scattering and α- β sheets nesting

On the basis of the experimentally observed curvature of QPI patterns, we believe that q2 represents the intra-band scattering of the surface β band while q1 represents that of the bulk β band. The question then arises: why is the intra-band scattering process of α band weak in FT-QPI experiments?

The apparent absence of scattering from the 𝛼𝛼 band in the experimental QPI is somewhat surprising given that the 𝛼𝛼

and 𝛽𝛽 bands are both largely derived from the 4dxz and 4dyz orbitals. However, adjusting the strength of the local potential scatterer in Eq. (5) gives rise to band-dependent scattering phase shifts, and thus the magnitude of the QPI signal due to each band varies, as shown in Fig. S6. The absence of the 𝛼𝛼 band signal is suggestive of intermediate-to-strong potential scattering, Figure S6c, in agreement with findings for Ti-doped Sr2RuO4 [16].

Figure S6. The evolution of simulated QPI at E = 0 meV with different strengths of scattering potential. QPI patterns at the Fermi-level, |𝑔𝑔𝑐𝑐(𝒒𝒒, 𝜔𝜔 = 0 𝑚𝑚𝑚𝑚𝑚𝑚)|2, due to a single potential scattering impurity of varying strength (𝑣𝑣0 = -225, +15, +225meV; Left to right), plotted over the range 𝑎𝑎0𝑘𝑘𝑥𝑥,𝑦𝑦 = (−2𝜋𝜋, +2𝜋𝜋), normalized by the magnitude of the scattering T-matrix trace. As the strength of the potential is varied, the relative intensities of the features attributed to intra-band scattering on the α and β bands changes due to constructive/destructive interference between different scattering events across the BZ.

We note here a possible origin of scattering vector q3 in our measurement is that it represents the scattering between and β bands. As shown by Mazin and Singh [17], the particular topologies of 1D α and β sheets suggest pronounced






9

nesting effects and give rise to a sizable antiferromagnetic (AFM) response at incommensurate wave vectors near (±2π/3a0, ±2π/3a0). Interestingly, the AFM instability has been confirmed by inelastic neutron scattering measurement, where the energy dependence of the imaginary part of the dynamical magnetic susceptibility shows a broad peak near 8meV in the normal state [18]. Although this scattering would be an interesting explanation for q3, this scattering process is likely to be heavily suppressed due to matrix element effects arising from the fact that the β band in this direction acquires dxy orbital character from the γ band [19]. q3 is therefore assigned to intra γ band scattering which can be seen by STM since the γ band acquires dxz/dyz character in this direction.

7. Scattering vector q4 arising from the band folding

The surface reconstruction produces two effects on the electron structures in Sr2RuO4. First, it gives rise to a set of surface-related Fermi surface sheets. Second, it reduces the first Brillouin Zone to half, leading to band folding with respect to the (π/a0, 0) - (0, π/a0) line (see Fig. S7a and b). To confirm q4 as a consequence of band folding, we check its energy evolution here. Our starting point is the geometric relationship between the origin band and its folded replica. In Fig. S7c, these solid blue sheets denote the origin band while dashed blue sheets denote the replicas after folding. It is obvious that q4 is not an independent vector and we have

|𝒒𝒒𝟐𝟐| = |𝒒𝒒𝟐𝟐′ | + 𝟐𝟐 × |𝒒𝒒𝟒𝟒| 𝐬𝐬𝐬𝐬𝐬𝐬 𝜽𝜽, 𝜽𝜽 = 𝐭𝐭𝐭𝐭𝐬𝐬−𝟏𝟏(𝒒𝒒𝒙𝒙 𝒒𝒒𝒚𝒚⁄ ). (7)

An easier way to get the value of 𝟐𝟐|𝒒𝒒𝟒𝟒| 𝐬𝐬𝐬𝐬𝐬𝐬 𝜽𝜽 is to measure the distance between the two parallel lines of q4 at qx, y = π/a0, then all three values can be directly extracted from our data. We plot the results in Fig. S7d, as one can see, the magnitude of q4 beautifully matches the predicted value from band folding.

Here we would like to discuss this interesting folding features. At the positive bias side, when we clearly see surface band scattering q2, the folding feature q4 is always present. At negative bias side where we mainly see bulk band q1, this feature is very weak and invisible at high energy. But near the Fermi energy, we can see folding features for both q1 and q2 (e.g., E = 13meV in Fig. 2), which means both the surface band and bulk band are folded. This is consistent with the ARPES measurement [13].






10

Figure S7. Features related to the band folding. (a, b) Fermi surfaces for the α (red), β (blue) and γ (green) bands

calculated with the tight binding model across the 1BZ. The reduced BZ due to surface reconstruction, and the

resultant surface bands Fermi surfaces are depicted by the dashed lines. (c) Schematic constant energy contours

(CECs) with band folding. Blue sheets denote the origin band while dashed blue sheets denote the replicas after

folding. (d) Energy evolution of q2, q’2 and 2|𝑞𝑞4| sin 𝜃𝜃. One can clearly see that |𝑞𝑞2| = |𝑞𝑞2′ | + 2 × |𝑞𝑞4| sin 𝜃𝜃.






11

8. Angle and energy dependence of intensity of QPI

Figure S8. Angle and energy dependence of QPI intensity. (a) Angle dependent intensity of q1 with θ defined in the inset. (b) Energy dependent intensity of q1 and q2 along ΓM (θ=0) direction.

The 1D nature of β band can be seen from the brightness of q1 and q2 scattering vectors along ΓM direction. In Fig. S8 we show the angle dependent intensity of q1 at three energies (-0.5meV, 3meV and 10meV), and pronounced peaks can be observed exactly along the high symmetric direction, which indicates that there is a singularity in the numbers of scattering processes with the same magnitude here. The intensity of q1/q2 reaches its maximum in an energy range from 10 to 15meV, which is the same energy scale as the gap-like feature in the dI/dV spectrum.






12

Supplementary Information part III

9. Extraction of the dispersion from data

Figure S9. Fitting procedure to extract the dispersion in Fig. 4a from line-cuts. (a, b) Line cuts along the ΓM direction (solid black curves). The solid red curves denote the fits with a linear background plus a Gaussian to the data to extract the position of the peaks. Data at different energy are normalized and offset vertically for clarity. Peaks related to other scattering vectors are marked with dashed ellipses. (c) High light the kinks at 38meV and 72meV (for a second sample we measured). (d) Similar procedures along ΓX direction.






13

10. Comparison with ARPES measurements along ΓM direction and dHvA data

Figure S10. Comparison with ARPES data along ΓM direction. (a) Dispersion of β band along ΓM direction extracted from ARPES measurements, QPI and dHvA data. (b, c) Fermi surface topology obtained by ARPES.

The comparison between our data and various ARPES measurements is now shown in Fig. S10a. Our QPI data reveal both bulk and surface beta bands which have already been observed by ARPES (ref. 33 and 34 in the main text). The Fermi surface sizes we find with QPI for the beta bands is in good agreement with that measured by ARPES (Fig. S10 b-c). We note however, that there is some variation in the ARPES data from various groups as shown in the figure. From dHvA measurements, the averaged Fermi velocity of the beta band is around 1*10-5 m/s (0.658 eV Å), which gives a mass enhancement factor about 3.5 (ref. 5 and 8 in the main text). Our QPI data gives a Fermi velocity of 0.46eV Å along ΓM direction, and a mass renormalization factor is ~5 which is a factor of 1.4 times of that obtained by dHvA measurements. One possible reason for this discrepancy is discussed in the main text.






14

11. Dispersion of all the Q-vectors and the repeatability of the kinks

We check the repeatability of the kinks observed in the dispersion by measuring four different samples. Samples #1, 3, 4 are from UBC and #2 is from Kyoto University. The results we obtained on these samples are plotted in Fig. S11a. Kink features can be observed at ~40meV and ~70meV for q1, ~35meV for q2 and ~10meV for q3. We also plot the dispersions of all Q-vectors obtained on sample #2 in Fig. S11b. Each pair of qi and q’i is mirror- symmetric with respect to the line q = π/a0, indicating that q’i originates from the Umklapp scattering process. Note the units using in these plots are different.

Figure S11. (a) Kink features on four samples. (b) Dispersion of all the Q-vectors seen in the sample discussed in the main text.






15

Supplementary References

1. Pennec, Y. et al. Phys. Rev. Lett. 101, 216103 (2008).

2. Stoger, B. et al. Phys. Rev. Lett. 113, 116101 (2014).

3. Iwasawa, H. et al. Phys. Rev. Lett. 109, 066404 (2012).

4. Lawler, M. J. et al. Nature 466, 347-351 (2010).

5. Scaffidi, T. et al. Phys. Rev. B 89, 220510(R) (2014).

6. Zabolotnyy, V. B. et al. J. Electron. Spectrosc. Relat. Phenom. 191, 48-53 (2012).

7. Nomura, T. and Yamada, K., J. Phys. Soc. Jpn. 71 404-407 (2002).

8. Wang, Q. H., et al. Euro. Phys. Lett. 104, 17013 (2013).

9. Mackenzie, A. P. et al. Phys. Rev. Lett. 76, 3786-3789 (1996).

10. Mackenzie, A. P. et al. Rev. Mod. Phys. 75, 657-712 (2003).

11. Mitchell, A. K. et al. Phys. Rev. B 91, 235127 (2015).

12. Capriotti, L. et al. Phys. Rev. B 68, 014508 (2003).

13. Veenstra, C. N. et al. Phys. Rev. Lett. 110, 097004 (2013).

14. Damascelli, A. et al. Phys. Rev. Lett. 85, 5194 (2000).

15. Lee, W. -C. et al. Phys. Rev. B 81, 184403 (2010).

16. Kikugawa, N. et al. J. Phys. Soc. Jpn. 72 237-240 (2003).

17. Mazin, I. & Singh, D. J. Phys. Rev. Lett. 82, 4324 (1999).

18. Sidis, Y. et al. Phys. Rev. Lett. 83, 3320 (1999).

19. Veenstra, C. N. et al. Phys. Rev. Lett. 112, 127002 (2014).





doi: 10.1038/nphys4107 quasiparticle …...in the format proided b the authors and unedited. © 2017...

Documents