Intrinsic atomic-scale modulations of the superconducting gap of $2H\text{−}{\text{NbSe}}_2$

We present scanning tunneling microscopy and spectroscopy measurements at 100 mK in the superconducting material $2H\text{−}{\text{NbSe}}_2$ that show well defined features in the superconducting density of states, with changes that follow atomic periodicity. Our experiment demonstrates that the intrinsic superconducting density of states can show atomic size modulations, which reflect the reciprocal space structure of the superconducting gap. In particular, we obtain that the superconducting gap of $2H\text{−}{\text{NbSe}}_2$ has sixfold modulated components at 0.75 and 1.2 mV. Moreover, we also find related atomic size modulations inside vortices, demonstrating that the much discussed star shape vortex structure produced by localized states inside the vortex cores has a superposed atomic size modulation, which has hitherto been undetected. The tip substrate interaction in an anisotropic superconductor has been calculated, giving position dependent changes related to the observed gap anisotropy.

Figure 1

(Color online) In the lower right inset of (a), we show typical topographic images obtained in $2H\text{−}{\text{NbSe}}_2$, where the atomic Se lattice is clearly viewed, as well as the charge density wave, which produces additional modulation in topography for each of the three Se atoms. We show $\sigma \left(V\right)$ curves obtained on top of Se atoms and in between. STS images showing small groups of atoms constructed from these data at $V_1=\pm 0.75\text{mV}$ are shown in the top panels. In (b), we show $d\sigma /dV\left(V\right)$ at the same positions. The panels in the figure show STS images constructed from $d\sigma /dV$ at $V_1$. The full range in contrast in the STS panels from black to white represents, respectively, a difference in (a) $\sigma$ of $0.3\mu \text{S}$ and (b) $d\sigma /dV$ of 2 mS/V. The circles in the images give the Se atomic positions, as determined through topography.

Figure 2

(Color online) In the inset, we show the Fourier transform of an STS image constructed from $\sigma \left(V=0.75\text{mV}\right)$ in $2H\text{−}{\text{NbSe}}_2$ at 100 mK. The red circles represent the position of peaks in the Fourier transform of the corresponding topography image. The Fourier transform of STS images at other voltages is either flat, close to the Fermi level and at high voltages, or has the same form as the one shown. This is seen in the figure, where we represent the height of the sixfold peaks of the Fourier transform shown in the inset as a function of the bias voltage (the line is a guide for the eyes). It is important to remark that no other peaks appear in the Fourier transform when changing the voltage and that the features discussed in this paper fully disappear at voltages well above the superconducting gap.

Figure 3

(Color online) In the inset of (a) (lower left corner), we show the STS image made at zero bias of a single vortex at 0.03 T. The color scale represents the value of the LDOS at the Fermi level, following previous representations of a vortex as measured with STS (Refs. 10, 30) (the whole color change corresponds to differences in the conductance from 0 to about $2\mu \text{S}$). A zoom-up of this image close to the vortex core, at the small red square of the inset, gives STS images (a)–(e) made at the bias voltages represented in each panel. The color scale in these panels, from green to blue, represents differences in the conductance of $0.2\mu \text{S}$, i.e., changes not higher than 10% of the high bias conductance. The bias dependence of the height of the Fourier transform peaks of the images is shown in (f). The points that have been obtained from the Fourier transform of images (a)–(e) are highlighted as larger circles. The inset shows the Fourier transform of the STS image at 0.2 mV, with red circles signaling the position of the peaks from the Fourier transform of the topography. Again, the effect of the bias voltage is to change the height of the peaks shown in the inset, becoming featureless at high bias voltages, as shown by the decrease of the Fourier amplitude in the figure. No new significantly remarked peaks appear when changing the voltage.

Figure 4

(Color online) In (a) we show a Fermi surface obtained from the model discussed here ($a$ is the lattice spacing). High symmetry points are labeled as $\Gamma$, $K$, and $K^{\prime }$. In (b) we represent schematically the anisotropy of the tip orbitals. The ellipse in the figure graphically shows the anisotropy of the tip-substrate interaction. The $\text{Nb}{d}_{x^2-y^2}$ and $d_{xy}$ orbitals are represented as in plane lobes (green and red, respectively). ${\theta }_0$ is the angle between the long axis of the anisotropy of the tip and the straightest line connecting the center of the tip to the nearest Nb atom, schematically represented here at the center of the two-dimensional real space coordinate system.

Figure 5

(Color online) Plot of the LDOS for different angles when the tip describes a circle around a Nb atom (${\theta }_0=0°$, black; 10°, red; and 20°, green). We show the result near the $K$ point. The parameters used for the superconducting gap are ${\Delta }_0=1\text{meV}$ and ${\Delta }_1=0.3\text{meV}$.