Bridging atomic and mesoscopic length scales with Replica Scanning Tunneling Microscopy: Visualizing the intra-unit cell pair density modulation of superconducting FeSe at micron length scale

Abstract

Scanning Tunneling Microscopy (STM) is a cornerstone technique for visualizing the electronic density of states with atomic resolution (typically below 0.1 nm). While the field of view of most STM setups extends up to a few microns, obtaining atomic resolution over these large areas is often impractical and excessively time-consuming. This is due to the need to acquire maps with a point number reaching 107 or more with a full current or conductance vs voltage curve at each point. The standard procedure is to make large scale maps and then select small regions to zoom-in for high-resolution atomic scale analysis. However, this approach fails to address a question which is often critical: Does a specific atomic-scale modulation of the electronic density of states persist over much larger, mesoscopic length scales? Here we present a new method: Replica STM (R-STM), that overcomes this limitation, allowing the study of atomic-scale phenomena up to micron length scales. We obtained new large-area STM tunneling conductance maps in UTe2 and FeSe, spanning areas over 200 nm in size. In these large scale maps we discovered signals with wavelengths significantly exceeding interatomic distances. We show that these large-wavelength signals are replicas of the underlying atomic-scale density of states modulations. R-STM leverages these replica signals to efficiently track atomic-scale features over large areas. Using this novel technique, we show that the pair density modulation discovered recently in FeSe persists with the same characteristic wavelength up to hundreds of nm length scales. R-STM provides a powerful and practical new capability for STM to compare atomic scale with micrometer scale phenomena. The proof of principle of R-STM can be extended to any other scanning probe microscopy experiment where a periodic signal is traced as a function of position.

Figures (6)

• Figure 1: Simulated STM topography images. A sinusoidal square lattice with spacing of $0.38\,\mathrm{nm}$ is sampled on three different sampling grids. (a–d) Fine sampling of a 10 nm $\times$ 10 nm area with 128$\times$128 points. In this image, $k_{S,0}= 12.8\, \frac{1}{nm}\gg 2 k_{signal}=2 \times 2.6\, \frac{1}{nm}$. In (a) we show the sampled image. In (b) we show its Fourier transform. We find Bragg peaks at $k_{signal}$ (blue circles). In (c) we show a line profile along the $x$ axis. In blue we show the periodic signal. With black dots we show the points used in the image (a). The red line shows the wavefunction obtained from the black points. (d) Fourier transform of the black points (red) and of the signal (blue) of (c). (e–h) We now take the same square lattice and provide images using only 32$\times$32 points. Now $k_{S,1}= 3.2\, \frac{1}{nm} < 2 k_{signal}=2 \times 2.6\, \frac{1}{nm}$. In (e) we show the resulting pattern, which is quite obviously periodic. In the inset of (e) we show its Fourier transform. In (f) we copy the Fourier transform of (e) several times to reach the size in reciprocal space of (b). We mark with a blue circle the Bragg peaks of the original periodicity $k_{signal}$. We mark with a red circle the Bragg peaks of the pattern shown in (e), $k_{replica,1}$. Here, $k_{replica,1}=|k_{signal}-nk_{S,1}|$ with $n=1$. White arrows connect red with blue peaks, as explained in the text. In (g) we show a profile along the $x$-axis. With a blue line we show the original periodicity. Black points show the points used in (e) and the red line joins the latter black points. In (h) we show the Fourier transform of (g). Blue points are the Fourier transform of the original signal and red points of the replica. (i–l) Here we enlarge field of view to 100 nm lateral size and take 128$\times$128 points. Now $k_{S,2}= 1.28\, \frac{1}{nm} < 2k_{signal}=2 \times 2.6\, \frac{1}{nm}$. We observe again a periodic pattern (i) at a wavenumber which is smaller than the wavenumber of the generating function. Its Fourier transform is shown in the lower right inset of (i). In (j) we copy the Fourier transform of (i) to obtain a similarly sized reciprocal space as in (b). The Bragg peaks corresponding to the replica, $k_{replica,2}$ are marked by red circles. The original Bragg peaks as blue circles. In (k) we show as black points the points building up the image (i) along a line. The red dashed line shows the wavefunction obtained from the black points. In blue we show the generating periodic signal. In (l) we show the Fourier transform of (k). We now have $k_{replica,2}=|k_{signal}-nk_{S,2}|$ with $n=2$. We show as a grey shadow the so-called Nyquist band, given by $\frac{k_S}{2}$. We see that the original periodic signal at $k_{signal}$ has a replica within the Nyquist band. As a result, one can address short length scale phenomena over large scale fields of view with undersampling grids. The periodicity of the generating signal is recovered by copying the reciprocal space area as many times as needed to cover the area required to describe the original periodic signal (at $k_{signal}$) and up-folding the replica wavevector following the white arrows in (f,j; arrows are located, for clarity, slightly shifted with respect to the Bragg peaks). The white arrows are vectors $n_{x,y} \mathbf{k^{x,y}_S}$, with $\mathbf{k^{x,y}_S}$ the wave vectors of the sampling unit cell in reciprocal space along $x$ or $y$ axis.
• Figure 2: Replicas in STM topographic maps in UTe$_2$ of different sizes. (a) STM topography map with atomic resolution. We observe Te chains and several defects (dark spots in the image). The corresponding Fourier transform (b) shows six Bragg peaks. The two peaks with the highest intensity, marked in dark blue, correspond to the Te chains. The other Bragg peaks, in light blue, provide the distorted lattice of atoms formed by the surface cut to the bulk crystalline structure (the (011) plane, see Refs. talaveraCDW2025Aishwarya2023Gu2023Aishwarya2024LaFleur2024science.adk7219). The Fourier transform is shown as an inset on the larger images, with red arrows indicating the aliased Te chain peaks, and blue arrows indicating the atomic periodicity along the chains. (b,d,f) Extended Fourier transform of the images in (a,c,e), respectively. Black bar is $\mathrm{1 nm^{-1}}$ for (b,d,f). The black rectangle shows the boundaries of the downfolded reciprocal space of the image. Red and blue circles represent the true wavevectors of the Te chain and atomic lattice, respectively. Black arrows display the wavevectors used to translate the peaks from the aliased coordinates to their true positions. The white scale bar is 20nm for (a,c,e). Arrows are located, for clarity, slightly shifted with respect to the Bragg peaks ($\mathrm{V_{Bias} = -50~mV, I_{Set} = 200~pA, T = 4.7~K}$).
• Figure 3: (a,c,e) STM Topographic images in FeSe with different lateral sizes and the same number of points ($\mathrm{V_{Bias} = 6~mV, I_{Set} = 4~nA, T = 1~K}$). We show the Fourier transform of (a) in (b) and highlight with blue circles the position of the Bragg peaks of the atomic Se lattice. The Fourier transforms of (c) and (r) are shown in its upper right insets and provide also four Bragg peaks (red circles). These are located at wave vectors which are significantly smaller than the atomic Bragg peaks and with orientations which do not match directly the atomic Se lattice. In (d,f) we show the same Fourier transform, repeated in such a way as to recover an area in reciprocal space sufficiently large to represent the atomic Bragg peaks. We highlight the atomic Bragg peaks (the same as in (b)) with blue circles. The Fourier transforms of (c,r) are also marked by a black square in (d,f). Black arrows are ${\mathbf k_{sample,x,y}}$ used to up-fold the replica Bragg peaks (red circles) to the actual atomic lattice (blue circles).
• Figure 4: Symmetries in FeSe.(a) We show as orange and brown disks the Se lattice, with upper labels $^u$ and $^d$ to indicate the upper and lower Se planes, with respect to the surface. Blue disks represent the Fe atomic lattice. In the nematic phase, the in-plane crystalline axis are slightly different (black arrows). (b) Cartoon representation of the Fermi surface of the nematic phase of FeSe in recriprocal space. We show as a black square the Brillouin zone of the Fe lattice and as a black dashed line square the Brillouin zone of the Se lattice (the two-Fe lattice unit cell, also represented in real space by dashed arrows in (a)). In the nematic phase, there is a hole-like pocket at the center of the Brillouin zone and electron-pockets at the corners. Nematicity is shown by the in-plane symmetry breaking of the electronic properties. (c) When glide symmetry is broken close to the surface, the Fe atoms are no longer equivalent. Therefore, we use dark and light blue disks for the two Fe sites. (d) We show as a black square the two-Fe unit cell. The Fermi surface is now folded and there are unit cell sized wave vectors $q_1$ and $q_3$ which join the central Fermi surface hole pockets into each other.
• Figure 5: Pair density modulation observed with R-STM. (a) Atomically resolved STM topographic image in FeSe taken at 4.2K and zero field (bias voltage of 10 mV and current of 1.2 nA). White scale bar is 1nm long. Inset shows the average tunneling conductance on the area. The corresponding data form a matrix of 256$\times$256 points. Points in the matrix provide the color scale, which corresponds to changes in height by about 20 pm from black to white. (b) Fourier transform of (a). Simultaneously to (a) we took maps of the tunneling conductance as a function of the bias voltage at each point. For each tunneling conductance curve, we calculated the position of the quasiparticle peak. We built a map of the position in bias voltage of the quasiparticle peaks. In (c) we show the Fourier transform of the gap magnitude. Dark and light blue circles indicate Se and Fe true Bragg peaks, respectively. (d) STM topographic image taken in FeSe at a temperature of 1 K (tunneling current of 3 nA and bias voltage of 6 mV). White bar is now 40 nm large. The matrix of this image has 256$\times$256 points, which implies one point every two unit cells. Locations of profiles of the tunneling conductance shown in the Fig. \ref{['PDMProfiles']} are shown here in colored arrows. (e) Fourier transform of (d), showing clear Bragg peaks. We acquired simultaneously a tunneling conductance map. (f) Fourier transform of the gap magnitude. Dark and light orange circles indicate Se and Fe aliased Bragg peaks, respectively. (g) Same that as (f) but represented as the Extended Fourier transform. Orange circles indicate the observed aliased Bragg peaks, and blue circles their true counterparts. White arrows show the shifts used to connect both sets.