Nanoscale assembly of superconducting vortices with scanning tunnelling microscope tip

Vortices play a crucial role in determining the properties of superconductors as well as their applications. Therefore, characterization and manipulation of vortices, especially at the single-vortex level, is of great importance. Among many techniques to study single vortices, scanning tunnelling microscopy (STM) stands out as a powerful tool, due to its ability to detect the local electronic states and high spatial resolution. However, local control of superconductivity as well as the manipulation of individual vortices with the STM tip is still lacking. Here we report a new function of the STM, namely to control the local pinning in a superconductor through the heating effect. Such effect allows us to quench the superconducting state at nanoscale, and leads to the growth of vortex clusters whose size can be controlled by the bias voltage. We also demonstrate the use of an STM tip to assemble single-quantum vortices into desired nanoscale configurations.

/ 1 13 Penetration depth 4.2 K 4.5 K 5.0 K 5.5 K 6.0 K 6.5 K 6.7 K 6.9 K 7.0 K   Figure 9: Vortex cluster evolution in low temperature range. At low temperatures, the interaction between pinning centers and vortices dominate. As a result, the vortex cluster retains the same geometry up to T = 6.0 K. Only at higher temperatures, the vortex-vortex repulsion overcomes the attractive interactions between pinning centers and vortices, then the vortex cluster starts to decompose as illustrated in Fig. 3 of the manuscript. The scale bar equals 4 µm. are not really important when analyzing the local heating effect on vortex distributions, provided that ρ 0 is smaller than the vortex size.
As mentioned above, at a given q, the temperature distribution u in a metal reaches its steady distribution with a time constant ∼ 10 −14 s [3]. So, keeping in mind that the specific heat capacity at liquid helium temperatures is rather low, one can expect that the relaxation of the temperature profile is fast as compared to the other characteristic times for the problem under consideration. In other words, the temperature profile u(ρ) = T (ρ) − T 0 , which really affects the vortex distribution, corresponds to the steady solution of Eq. (2). This solution can be written in terms of the modified Bessel functions I ν (x) and K ν (x) [16]: Using the material and geometric characteristics of the system under consideration, which were discussed above, the parameters α and β can be estimated as α ≈ 0.1 to 20 mK and β ≈ 0.3 to 0.5 µm −1 .
As seen from Fig. 2a, the parameter ρ 0 affects the shape of the temperature distribution, given by the function f (ρ), only in the close vicinity of the center (at ρ ≤ ρ 0 ). Since the radius ρ 0 is assumed to be smaller than the size of a vortex core, the details of the temperature distribution at ρ ≤ ρ 0 and the precise value of ρ 0 cannot significantly affect the vortex cluster formation in the superconductor layer. In our simulations below we use ρ 0 = 100 nm. On a more relevant size scales, > 0.1 µm, the behaviour of f (ρ) is fully determined by the parameter β (see Fig. 2). In line with the estimate given above, in our simulations of vortex distributions we take β = 0.4 µm −1 .
As follows from Eq. (4) and Fig. 2, at α ≈ 0.1 to 20 mK the temperature elevation T − T 0 does not exceed 0.2 K even at the center of the hot spot. Although this value is significantly larger than that found for a semi-infinite Au sample, still it does not seem sufficient for the vortex cluster formation, observed in [1]. At the same time , we realize that -in view of uncertainties related to the material and measurement parameters in [1] -the actual values of α may considerably differ from those estimated above. For this reason, below we treat α as an adjustable parameter. Such an ad hoc approach seems quite appropriate taking into account that the main aim of the present note is to demonstrate qualitatively the possibility of the vortex cluster formation as a result of an STM-tip tunneling current pulse. The penetration depth is determined by using the monopole model to fit the vortex profile observed at various temperatures. Supplementary Figure 1a shows the SHPM images observed at various temperatures indicated above each image. By fitting the magnetic field profiles along the dashed line, e.g. for T = 6 K shown in Supplementary Figure 1b, we can get the value of +z0 at different temperatures, where is the penetration depth and z0 is the distance between the two-dimensional electron gas (TDEG) of our Hall cross and the sample surface (this distance is constant). According to the two-fluid model, the penetration depth follows a linear dependence on (1-t 4 ) -1/2 with t = T/Tc. As shown in Supplementary Figure 1c, from the slope of the linear fitting, we can get the penetration depth at zero temperature.

Supplementary Note 2: Determination of the coherence length and penetration field
To determine the coherence length, we measured the temperature dependence of the in-phase ac susceptibility at different magnetic fields, as shown in Supplementary Figure 2a  We have measured the penetration field at various temperatures using the criterion that the first vortex is seen to penetrate into the sample. The results are shown in Supplementary Figure 3. Here the critical field should be regarded as the penetration field Hp. Due to the sample geometry which leads to a large demagnetization effect, the real lower critical field should be considerably larger than the penetration field.

Supplementary Note 3: Determination of the number of vortices in a vortex cluster
To determine the number of trapped vortices in the cluster, two different ways are used: i) The applied magnetic field induces the nucleation of vortices in a superconducting area with the number N that can be determined as N = HS/Φ0, where H is the applied magnetic field, S is the superconducting area and Φ0 is the flux quantum. For the applied field value µ0H0 = 0.47 mT used in Fig. 2a of the manuscript, N = 58 single quantum vortices are expected in our scanning area of 16x16 µm 2 . The Supplementary Figure 4 shows three vortex patterns at the same area after performing a series of field cooling processes at µ0H0 = 0.47 mT. In each pattern the observed / 8 13 vortex number is N = 58±2, that is consistent with the expected number. The uncertainty is due to the vortices cut by the edges. To determine the number of vortices trapped in the cluster in Fig. 2a, we simply count the number (N1) of vortices outside the cluster. Then the number of vortices trapped can be determined as N-N1.
ii) The number of trapped vortices inside the cluster can also be roughly checked by first performing Fourier transform of the cluster pattern and then analyze the high frequency signal from the image (remove low frequency background). For example, Supplementary Figure 5 shows the vortex cluster pattern (corresponding to the cluster at Vbias=0.4 V in Fig. 2a) before (left) and after (right) filtering. From the right hand side image, we count 23 vortices in the cluster as indicated by the circles, which is fully consistent with the number (23) found by using method i). The results also suggest that there is no multi-quanta vortex in the cluster.

Supplementary Note 4: Monopole model
In the limit of (r 2 +z0 2 )>>λ 2 , where r=(x,y) is the distance from the vortex center, λ is the penetration depth, z0 is the distance from the sample surface to the two-dimensional electron gas of the Hall cross, the magnetic field profile of a vortex can be closely approximated by the monopole model with the following expression [1][2][3]: Here, Bz(r) is the magnetic field perpendicular to the sample surface and Φ is the total flux carried by a vortex. According to Ref. [4], the accuracy of the model can be enhanced by averaging over an area representative of the Hall probe active area to account for the convolution of the field over the probe. The integration of the equation above over a square active area of size s and divided by the area s 2 gives the following result: For our SHPM, a Hall probe with an active area of s 2 =0.4x0.4 µm 2 is used.
From supplementary Equation (2), it is clear that a doubly quantized vortex would double the magnetic field at each point. However, if the distance between the two adjacent pinning centers is much smaller than the spacial resolution of our scanning technique, it is rather difficult to distinguish between a giant 2Φ0 vortex and a vortex Φ0 + Φ0 cluster. As a result, the observed field distribution of a cluster with two Φ0-vortices could still be doubled as compared with a single quantum vortex, as is the case shown in Fig. 2b

I. Monopole model
In the limit of (r 2 +z 0 2 )>> λ 2 , where r=(x, y) is the distance from the vortex center, λ is the penetration depth, z 0 is the distance from the sample surface to the two dimensional electron gas (2DEG) of the Hall cross, the magnetic field profile of a vortex can be closely approximated by the monopole model with the following expression [1][2][3][4]: Here, B z (r) is the magnetic field perpendicular to the sample surface and Φ is the total flux carried by a vortex. According to Ref. [5], the accuracy of the model can be enhanced by averaging over an area representative of the Hall probe active area to account for the convolution of the field over the probe. The integration of the equation (1) over a square active area of size s and divided by the area s 2 gives the following result: With these values, the noise-equivalent magnetic field Bnoise=δVHall/ISI was determined for each bias current value I. Supplementary Figure 12c shows the noise-equivalent value, from which we can observe the optimal bias current (25 µA) for our Hall cross with the lowest noise level. The absolute sensitivity at I = 25 µA is 2.363×10 -2 VT -1 . Using the absolute sensitivity, the measured Hall voltage is directly converted to magnetic field by the SPM software.