Supplementary information for Probing atomic structure and Majorana wavefunctions in mono-atomic Fe chains on superconducting Pb surface Rémy Pawlak 1, Marcin Kisiel 1, Jelena Klinovaja 1, Tobias Meier 1, Shigeki Kawai 1, Thilo Glatzel 1, Daniel Loss 1, and Ernst Meyer 1 1 Department of Physics, University of Basel, Klingelbergstr. 82, 4056 Basel, Switzerland. Keywords: Majorana fermions, scanning tunneling microscopy, atomic force microscopy, superconductor, mono-atomic wire 1
I. SAMPLE PREPARATION AND STM CHARACTERIZATION FIG. S1: Pb(110) surface. a, Topographic STM image of the Pb(110) after few preparation cycles. b, Atomic-scale STM image of the surface, (I =75 pa, V = -50 mv). c, di/dv point spectra (red) and corresponding I/V (black) of the Pb(110) surface obtained with a metallic tip showing a gap of 1.1 mev. d, Temperature-dependency of the superconducting gap fitted with the BCS density of states. The blue curves corresponds to a temperature of 0.8 K and the red curve to 4.5. The Pb(110) sample was chemically etched ex-situ with a solution of hydrogen peroxide H 2 0 2 and acetic acid C 2 H 4 O 2, leading to a drastic change of the surface color from dark to bright gray. Immediately after, the sample was introduced in ultrahigh vacuum (UHV) and prepared by several cycles of sputtering (1 kev) and annealing at 430 K. STM topographies (Fig. S1a) reveal a flat and atomically-clean surface with large mono-atomic terraces extending up to 200 nm. Figure S1b shows the atomic resolution obtained on the Pb(110) surface by STM revealing a row structure. The inter-atomic distances along the rows is 2
0.35 nm, while the distance between rows is 0.5 nm in agreement with the (110) face of the face-centered cubic structure of lead and as previously reported by Nadj-Perge et al. [1]. FIG. S2: STM topographic images after Fe deposition on Pb(110). a-c, Collection of STM topographic images of the Pb(110) surface after deposition. Numerous clusters are always present at the surface accompanied with the mono-atomic chains. The close-up views of b-c shows that two type of end-state are always observed by STM. The chain ends having an end-state are marked with white arrows, (I =60 pa, V = -30 mv). Figure S1c reports the superconducting gap of about 1.1 mev measured with a normal metal tip on the bare Pb(110) surface. The important broadening of our di/dv spectra is induced by the high temperature of our experiments ( 5 K) compared to previous works [1, 2] as well as the normal metal nature of our STM probes. Indeed, the employed tips are made of Tungsten and were always first gently indented into an atomically-clean Cu(111) surface to sharpen their apexes. Thereafter, we exchange the Cu(111) sample to the Pb(110) where we proceeded to further gentle indentations. According to the literature (Methods of [3]), our preparation procedure is not sufficient to obtain a substantial amount of lead material at the apex in order to obtain a superconducting state at the apex. The preparation leading to superconducting tips requires much stronger experimental conditions, i.e. deeper indentations with high voltage applied ( 100 V). To further confirm the normal metal nature of our tips, Figure S1d shows an approximation of the superconducting gap fitted with the Bardeen-Cooper-Schrieffer density of states at 0.8 K (blue) and 4.5 K (red), respectively. The theoretical spectra obtained at 4.5 K remarkably coincides with our experimental data considering a normal metal tip probing the superconducting lead. Finally, it is well-established that superconducting tips probing superconductors induce characteristic 3
STS spectra due to the convolution of the superconducting gaps of tip and sample [2, 3, 8]. Such spectra were never obtained in our study confirming the normal metal nature of our STM tips. The deposition of Fe atoms was conducted on slightly annealed Pb(110) samples leading to the formation of clusters and chains as shown in the STM overview Fig. S2. The mono-atomic chains always show two types of termination that we defined in the main manuscript. Chains having end-states according to the STM topographic images Fig. S2b and c are marked with white arrows. Figure S3a shows a constant-current STM image of a chain with an end state. The FIG. S3: STM characterization of the Fe chains a, STM image of a chain with end-state, (I =60 pa, V = -10 mv). b-c, STM profiles taken along the dashed lines of a across and along the chain. d, Comparison of STM profiles between chains with (red) and without (blue) end-states, (I =60 pa, V = -11 mv). dashed lines on the image report the positions where profiles were taken across and along and shown in Figs. S2b and c, respectively. The chains have an apparent STM height of 0.2 nm. As described in the main manuscript, two STM signatures are visible at their extremity that we assigned to chains with and without an end-state. The apparent height between those chains differs by 0.1 nm as shown in Fig. 1d. Above the chain, two faint periodicities can be observed by STM. The short periodicity is a 1 0.4 nm while the second period is larger a 2 1.7 nm. For comparison, Figure S5 shows an AFM profile extracted 4
along an Fe chain (Fig. 2c of the main manuscript) revealing its lattice periodicity. Because AFM is not sensitive to the local density of states between tip and sample, the inter-atomic distance along the chain is unambiguously resolved, exempt of electronic contributions and equal to 0.37 nm. FIG. S4: Characterization of the chain without end-state. a, STM image of a chain without end-state. b, Constant-height zero-bias AFM map of the chain end. Although a double-tip effect is present, the atomic periodicity is similar to the one of the main manuscript (Fig. 2). The end of the chain however does not show the halo in the force channel observed when the MBS is present. c, LDOS of the chain without end-state revealing the absence of MBS. d, Profile of the AFM image taken along the chain, the interatomic distance is 0.37 nm. Figure S4 shows a cross-comparison of AFM and conductance data of a chain which does not provide an end-state according to the STM topography (Fig. S4a) as described in the main manuscript. Figure S4b shows the corresponding AFM image obtained at constantheight, zero-bias and 5 K. Despite of a double tip effect, the atomic lattice is observable and identical to chains hosting a MBS described in the main manuscript ( 0.37 nm). In contrast, the force signature at the extremity of the chain is not present (see black arrow) due to the absence of MBS. Indeed, LDOS mapping on such chain is reported in Fig. S4c which reveals a homogeneous density of states at the Fermi level all along the chain without 5
a zero-bias conductance peak at its end. In contrast to the data shown in the main text, these chains thus suggest the absence of the MBS at their ends. Further experimental work with higher spectral resolutions are, however, required to elucidate the physical origin of this particular case. Figure S8 shows an additional data set of a chain exhibiting an end-state FIG. S5: AFM profile of the mono-atomic Fe chain. in the STM topography (Fig. S8a). The corresponding constant-height LDOS maps along the chain (Fig. S8b) in analogy to Fig. 4 of the main manuscript. The orange curve is the fit obtained from the theoretical MBS wave-function including the tip size approximation. II. PROBABILITY DENSITY OF A MAJORANA BOUND STATE Following Refs. [1,2], we obtain the probability density of the MBS wavefunction Ψ(x) for a semi-infinite chain as Ψ(x) 2 = 1 N N = υ F m 4 [e 2 υ x F + e [ m 2 υ F ] x 2e m υ x F cos(2k F x) 1 ( m ) 4 (2 υ F k F ) 2 + 2 m, (1) ], (2) where υ F is the Fermi velocity and k F the Fermi momentum. The topological phase for an RKKY chain corresponds to m > [2], where m is the effective field of the helix (which itself depends on the exchange coupling constant, see [2]) and the proximity gap. Here, N is the normalization factor of the wavefunction, obtained by imposing 0 Ψ(x) 2 dx = 2. Note that Ψ(x) 2 has its minimum at x = 0 ( Ψ(x = 0) 2 = 0), followed by a maximum at around λ F /4 (for m 2 ) before it decays. However, if m, the maximum is close 6
FIG. S6: STM topographies of the chain end at 5K and 10K, respectively. a, Constantcurrent STM image of the Fe Chains on Pb(110) at 5K and 10K, respectively. The STM parameters are identical, i.e. V = -10 mv and I =100 pa. b, Corresponding topographic profiles taken along the chain. The black curve correspond to 5K and the red curve to 10K. FIG. S7: LDOS along the wire. a, LDOS map taken along the wire of Fig. S8, a defect-free wire like in Fig. 3 of the main manuscript. b, Profile taken along the wire of Fig. S8b showing the long decay of the MBS wavefunction. The contrast has been enhanced for clarity. c, Typical Fourier Transform of the LDOS map of a wire employed to extract the kf parameter, kx corresponds to the direction along the wire and ky perpendicular to the wire. to x = 0. In other words, the maximum should be shifted somewhat away from the left end of the chain at x = 0. Alternatively, one can perform the fitting of the experimental data with respect to the 7
FIG. S8: MBS Localization lengths. a, STM topography of a chain with end-state, (I = 70 pa, V = -10 mv). b, constant-height zero-bias LDOS maps. c, LDOS profile along the chain (black) and theoretical fit with the tip approximation (orange). Fit Parameters are : ξ 1 = 20 nm, ξ 2 = 0.8 nm, d = 0.15 nm. localization lengths ξ 1 and ξ 2 and to the Fermi wavevector k F, using the form Ψ(x) 2 = 1 N [ ] e 2x/ξ 1 + e 2x/ξ 2 2e x/ξ 1 e x/ξ 2 cos(2k F x), (3) ξ 1 = υ F, ξ 2 = υ F m, (4) where N is arbitrary. It seems best to fit first to the longer of the two localization lengths, ξ 1 ξ 2, in the regime x ξ 2 where Ψ(x) 2 1 N e 2x/ξ 1, or ln( Ψ(x) 2 ) 2x/ξ 1 ln N, x, ξ 1 ξ 2. (5) In a second step, one fits ξ 2 and k F in the small x regime. From the fit values for ξ 1, ξ 2, and k F, and from the proximity gap we get M and v F and thus the Fermi energy E F k F v F /2 in the Fe chain. We note that there are two choices for fitting the data with Eq. 3: either ξ 2 > ξ 1 or ξ 2 < ξ 1. We use the first choice, ξ 2 > ξ 1, since only this choice turns out to be self-consistent, giving typically m and values for the Fermi energy E F much larger than m, which corresponds to the strong spin orbit interaction regime assumed in this work. The spatial oscillations with the period λ F /2 could be less pronounced in the experimental data if the signal is picked up not from a single point x but from the area of the size 2d 8
P(x,d) FIG. S9: The probability density P (x, d) of the MBS wavefunction for a semi-infinite chain without [blue curve, d = 0] and with [yellow curve, d/λ F = 0.2] averaging of the probability density, see Eq. (7). The localization lengths are chosen to be ξ 1 /λ F = 1 and ξ 2 /λ F = 10. We note that Eq. (7) is valid only for x > d, i.e. x/λ F > 0.2, that determines the range of the plot. around the point x. This means that, in the simplest model, the probe measures the averaged probability density P (x, d) defined as P (x, d) = 1 2d x+d x d dy Ψ(y) 2, (6) for x > d. Using Eq. (3), we calculate the integral analytically arriving at P (x, d) = 1 [ e 2x/ξ sinh(2d/ξ 1 1) + e 2x/ξ sinh(2d/ξ 2 2) 2e x/ξ 1e x/ξ 2 N 2d/ξ 1 2d/ξ 2 (1/ξ 1 + 1/ξ 2 ) 2 + 4kF 2 ( [ cos(2k F x) (1/ξ 1 + 1/ξ 2 ) 2 sinh[d(1/ξ ] 1 + 1/ξ 2 )] cos(2k F d) + 4k 2 cosh[d(1/ξ 1 + 1/ξ 2 )] sin(2k F d) F d(1/ξ 1 + 1/ξ 2 ) 2k F d [ sinh[d(1/ξ1 + 1/ξ 2 )] cos(2k F d) 2k F (1/ξ 1 + 1/ξ 2 ) sin(2k F x) cosh[d(1/ξ ] 1 + 1/ξ 2 )] sin(2k F d) )], (7) d(1/ξ 1 + 1/ξ 2 ) 2k F d P (x, d) = 1 [ e 2x/ξ sinh(2d/ξ 1 1) + e 2x/ξ sinh(2d/ξ 2 2) 2e x/ξ 1 e x/ξ 2 N 2d/ξ 1 2d/ξ 2 ( sinh[d(1/ξ1 + 1/ξ 2 )] cos(2k F d) [ (1/ξ 1 + 1/ξ 2 ) 2k ] F cos(2k F x) d (1/ξ 1 + 1/ξ 2 ) 2 + 4kF 2 sin(2kf x) (1/ξ 1 + 1/ξ 2 ) 2 + 4kF 2 cosh[d(1/ξ 1 + 1/ξ 2 )] sin(2k F d) [ 2k F (1/ξ 1 + 1/ξ 2 ) ])] cos(2k F x) d (1/ξ 1 + 1/ξ 2 ) 2 + 4kF 2 + sin(2k F x) (1/ξ 1 + 1/ξ 2 ) 2 + 4kF 2. (8) For an illustrative plot of P (x, d) see Figure S9. For fixed d and k F, this expression can be used to fit the data and to deduce the localization lengths ξ 1 and ξ 2. III. BOUNDARY EFFECTS In this section we demonstrate that the shape of the wavefunction depends on the imposed boundary conditions. However, the characteristic scales of the wavefunctions such 9
as the two localization lengths and oscillations with half of the Fermi wavelength remains unchanged. As an instructive example, we consider a 1D superconducting chain divided into the topological and non-topological sections in a tight-binding approach. In the topological section, the helical magnetic Zeeman field of period λ F /2 and strength M dominates over the superconducting order parameter, i.e., M >. The corresponding tight-binding model with hopping amplitude t is written as H = t c nσc mσ + sc (c n1c + H.c.) + n 1 M <n,m>,σ n n>n B (e i2kf na c n1c n 1 + H.c.), (9) where the Fermi wavevector k F is determined by the chemical potential µ, a is the lattice constant, and the operator c nσ is an annihilation operator acting on electron at the site n with spin σ. The first sum runs over neighbouring sites and the last sum runs over sites in the topological section on the chain. We numerically diagonalize this Hamiltonian exactly and find the MBS wavefunctions. In Fig. S10, we demonstrate that the oscillations of the MBS wavefunctions are much less pronounced at the boundary between two sections (topological and non-topological superconducting phases) than at the boundary to the vacuum (where the MBS wavefunction cannot leak out). This can provide a qualitative explanation why the 2k F oscillations are suppressed in the experimental data. However, fitting results in a quantitative way would require a more sophisticated modeling (which is beyond the present scope) where the chain is placed on a 2D or even 3D substrate into which the MBS can leak out. The same leakage effect occurs when the tunneling rate between STM tip and chain is large [8]. 10
P(n) 0.030 0.025 0.020 0.015 0.010 0.005 100 200 300 400 500 600 n FIG. S10: The probability density P (n) of the MBS wavefunction as function of lattice site n for a finite chain divided at n B = 200 into two sections. The system is modeled as tight binding chain, Eq. (9), and numerically evaluated with the chosen parameters /t = 0.01, M /t = 0.06, k F a = π/10, and µ/t = 2 cos(k F a). The chain is composed of N = 600 sites. The oscillations of the MBS wavefunction is much more pronounced at the boundary to the vacuum (right end of the chain with hard-wall boundary condition). At the boundary at n B = 200 between topological ( M /t = 0.06) and non-topological phases ( M = 0), the oscillations of the MBS are strongly suppressed but still present in the tail inside the topological section. [1] Nadj-Perge, S. et al. Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. Science 346, 602 (2014). [2] Ruby, M. et al. End States and Subgap Structure in Proximity-Coupled Chains of Magnetic Adatoms. Phys. Rev. Lett. 115, 197204 (2014). [3] Franke, K. J., Schulze, G. & Pascual, J. I. Competition of Superconducting Phenomena and Kondo Screening at the Nanoscale. Science 332, 940 944 (2011). [4] Heinrich, B. W., Braun, L., Pascual, J. I., Franke, K. J. Protection of excited spin states by a superconducting energy gap. Nature Phys. 9, 765 768 (2013). [5] Bauer, J., Pascual, J. I. & Franke, K. J. Microscopic resolution of the interplay of Kondo screen- 11
ing and superconducting pairing: Mn-phthalocyanine molecules adsorbed on superconducting Pb(111). Phys. Rev. B 87, 075125 (2013). [6] Klinovaja J. & Loss, D. Composite Majorana fermion wave functions in nanowires. Phys. Rev. B 86, 085408 (2012). [7] Klinovaja, J., Stano, P., Yazdani, A. & Loss, D. Topological superconductivity and Majorana fermions in RKKY Systems. Phys. Rev. Lett. 111, 186805 (2013). [8] Chevalier, D. & Klinovaja, J. Tomography of Majorana Fermions with STM Tips. Phys. Rev. B 94, 035417 (2016). 12