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Evidence of zero point fluctuation of vortices in a very weakly pinned a-MoGe thin film

In a Type II superconductor, the vortex core behaves like a normal metal. Consequently, the single-particle density of states in the vortex core of a conventional Type II superconductor remains either flat or (for very clean single crystals) exhibits a peak at zero bias due to the formation of Caroli-de Gennes-Matricon bound state inside the core. Here we report an unusual observation from scanning tunneling spectroscopy measurements in a weakly pinned thin film of the conventional s-wave superconductor a-MoGe, namely, that a soft gap in the local density of states continues to exist even at the center of the vortex core. We ascribe this observation to rapid fluctuation of vortices about their mean position that blurs the boundary between the gapless normal core and the gapped superconducting region outside. Analyzing the data as a function of magnetic field we show that the variation of fluctuation amplitude as a function of magnetic field is consistent with quantum zero-point motion of vortices.


I. Introduction
In Type II superconductors, magnetic field penetrates inside the sample above the lower critical field (Hc1) in the form of quantized flux tubes called vortices 1 , each carrying a magnetic flux of Φ = = 2.07 × 10 . In the diamagnetic background, the magnetic flux in a vortex is sustained by a circulating supercurrent density (Js) that falls off inversely with distance from the center of the vortex, r, as ∝ 1/ . The divergence in as r  0 is cut off through the formation of a normal metal core at the center of the vortex of the size of the superconducting coherence length, ξ. In the presence of an external transport current, vortices are subjected to a Lorentz force which can cause them to move and cause dissipation in the system 2 . Since the dissipation caused by a moving vortex is contributed by its normal metal core, it is of practical interest to have a detailed understanding of the electronic structure inside the vortex core (VC).
Due to the confinement of normal electrons in the VC, Caroli-de Gennes-Matricon (CdGM) predicted that they will form bound states with energy with respect to the Fermi energy, EF, at = ± ∆ ( n = 1/2, 3/2, 5/2, …), where ∆ is the superconducting energy gap in zero field 3 . However, due to the smallness of ∆ , individual CdGM states are difficult to observe. So far, individual CdGM peaks have only been observed in some iron based superconductors where EF is small 4,5 . On the other hand, in conventional superconductors, the accumulated density of state from many CdGM states manifest in the form of a peak in the local density of states (LDOS) centered about EF, that smoothly decays when one moves away from the center of the vortex core 6 . However, the height of this peak depends on the cleanliness of the superconductor; disorder scattering can smear the CdGM peak, leaving a LDOS that is flat over energy scales few times ∆ in dirty samples 7,8 . Consequently, far from the VC, the LDOS exhibits a superconducting gap and coherence peak (partially broadened due to the orbital current), whereas inside the core the LDOS is either flat or exhibits a small peak at zero bias due to the formation of CdGM bound states in very clean samples. While the vast majority of superconductors follow the above description, a handful of superconductors, namely, High-Tc cuprates 9,10 and strongly disordered NbN thin films 11,12 exhibit a radically contrasting behavior. In these systems inside the VC the LDOS is neither flat nor shows a zero energy peak but rather displays a LDOS suppression over energy scale comparable to the superconducting energy gap.
While there is no consensus on the origin of this behavior, in high Tc cuprates it has been suggested that it is alternatively associated with preformed Cooper pairs 9 or to the presence of a competing order 10 . In strongly disordered NbN thin films it has been recently suggested 13 that it is associated with the formation of Josephson vortices due to the formation of superconducting puddles induced by strong disorder.
In this paper, we report spectroscopic measurements in the vortex state of a weakly pinned thin film of the amorphous superconductor Mo70Ge30 (a-MoGe) using a low temperature scanning tunneling microscope. We observe that a soft gap in the LDOS continues to exist in the VC right up to the center of the vortex. Interestingly, a-MoGe is a conventional s-wave Type II superconductor with no known competing order. Furthermore, the superconducting state for the 20 nm thick film used in this study is very homogeneous as confirmed from STS spectroscopic maps in zero field, forcing us to look for explanations that have not been considered so far. We show that the magnetic field dependence of the soft gap can be explained if we assume that the vortices are not static but spatially fluctuate rapidly about their mean position. The spatial fluctuation at our lowest 450 mK is consistent with quantum zero-point fluctuation of vortices 14,15 which has been theoretically predicted in 2-dimensional (2D) superconductors.

II. Sample and Experimental details
The sample used in this study consist of 20 nm thick a-MoGe film, similar to the ones used in refs. 16,17,18. The superconducting transition temperature is, Tc ~ 7.2 K. Earlier measurements showed that the very weakly pinned 2D vortex lattice undergoes two structural phase transitions with magnetic field, from a vortex solid to a hexatic vortex fluid (HVF) and from a HVF to an isotropic vortex liquid (IVL) 16 .
As shown before the films have extremely weak pinning as evidenced from a very low depinning frequency and the absence of any difference between the field cooled and zero field cooled states.
STS measurements were performed in a home-built scanning tunneling microscope (STM) down to 450 mK and fitted with 90 kOe superconducting solenoid 19 . For STS measurements, post deposition, the film was transferred in the scanning tunneling microscope using an ultrahigh vacuum suitcase without exposure to air. Measurements were performed using a normal metal Pt-Ir tip. The differential tunneling conductance, G(V) was measured by superposing a small a.c. voltage, Vac (150 µV, 2.11 kHz) on the d.c.
bias voltage Vdc and measuring the a.c. current Iac using a lock-in amplifier, so that, !" = " #$ % = &'/&"| )*) +, ≈ ' .$ /" .$ . To image the VL, G(V) maps were recorded with the bias voltage close to the superconducting coherence peak, such that each vortex appears as a local minimum in the conductance.
The full G(V)-V spectra were recorded by switching off the feedback at a given location, and recording G(V) while the bias voltage is swept from positive to negative bias. The time to acquire a complete spectrum is ~ 2 seconds. While acquiring full spectroscopic area maps each spectra was averaged over three sweeps at every point.

III. Results
We first characterize the zero-field properties of the a-MoGe film using STS. Fig. 1(a) 20 which accounts for all non-thermal sources of broadening. Fig. 1(b) shows the temperature variation of ∆ and Γ along with the sheet resistance, Rs. ∆(Τ) follows the expected BCS variation and goes to zero exactly at the temperature where resistance appears. To produce the conductance maps we normalize the conductance curve at each point with its value at high bias, i.e. 6 !"% = C!)% C!D E)% .In Fig. 1(c)-(d) we plot the normalized conductance maps at zero bias, 6 !0%, and at the bias voltage corresponding to the coherence peak ( V p ) averaged for positive and negative voltages, 6 F = GC 1 !) H %9C 1 ! ) H %I . Both the color plots as well as the very small spread in the conductance values observed from the histogram of 6 !0% and 6 F ( Fig. 1(e)) show the homogeneous nature of the superconducting state.
We now turn our attention to the spectroscopy of the vortex cores (Fig. 2). The top panels of

IV.B. Simulating the conductance maps for fluctuating vortices:
We can now explore if the abovementioned physical picture is consistent with our STS data. For that we need to simulate the conductance map with fluctuating vortices. We start with the simulation of a vortex lattice without spatial fluctuations.
The exact calculation of the LDOS in the presence of vortices is in-principle possible by solving the Usadel equations 22 , but this computation is very difficult 23 . Here, instead we adopt a phenomenological approach.
First, we simulate the regular VL in a superconductor where the vortices are static. Since in a-MoGe the electronic mean free path is very short (l ~ 0.15 nm), we assume that at the vortex core of an isolated vortex the CdGM peak is heavily smeared by disorder scattering 7 , such that, 6 !"% = 1. Far from the core we assume that the conductance is given by the usual BCS relation given by eqn. (1), 6 !"% = 6 JK4 !L, ", ∆, Γ%. 6 JK4 !L, ", ∆, Γ% is obtained by fitting the experimental zero field tunneling conductance spectra at the same temperature, with ∆ and Γ used as fitting parameters. To interpolate between the two, we use an empirical Gaussian weight factor, : ] (V is of the order of ξ ), such that, 6 !", O% = :!O% + [1 − :!O%] 6 JK4 !L, ", ∆, Γ% (2) For a VL, the resultant normalized conductance, X 6 !", O%, is assumed to be a linear superposition of the conductance from all vortices, i.e.
where ri is the position of the i-th vortex and sum runs over all vortices; the normalizing factor ensures that the X 6 !" = 0% = 1 at the center of each vortex. The consistency of this procedure has been checked by simulating the VL on an NbSe2 single crystal ( see Appendix A ).
where the index L runs over each of the 200 realizations of the distorted lattice. This procedure neglects any retardation effect, which in this context is equivalent to working in the Born-Openheimer approximation and is justified as long as the vibrational frequency of the vortices is much smaller than the gap frequency, Ω = c! % ℏ !~ 3.89 × 10 ij% . We will show later that it is the case here.
In Fig. 3(a)-(d), we show the outcome of these simulations at 10 kOe. Fig. 3(a) is for the case of no fluctuation, i.e. 6 !", O, ∆\ = 0% ≡ X 6 !", O%, whereas Fig. 3 larger Γ compared to its zero field value to account for the additional broadening from the orbital current around vortices. While we primarily analyze the variation of GN(0,r), we note that the same set of parameters reproduces the qualitative variation of GN(V p , r), but the conductance value is overestimated by 15-20%; this is most likely due to the fact that the phenomenological Γ parameter does not capture suppression of the coherence peak from orbital supercurrent accurately. In Fig. 5(b) we plot ∆\/\ as a function of magnetic field extracted at 450 mK. While ∆\ decreases with increasing field, ∆\/\ shows an overall increasing trend. There are two anomalies observed close between 5-7 kOe and around 70 kOe where we observe a sudden decrease. We will comment on this in the next subsection.

IV.C. Theoretical analysis of vortex fluctuation:
To quantitatively understand the magnetic field variation of ∆\/\, we now theoretically compute this quantity within harmonic approximation. The lattice vibration of the 2D vortex lattice is governed by two elastic moduli 25 : compression, C11, and shear C66. However C66 << C11, and therefore for small deformation the elastic energy of the VL is mainly controlled by C66. In the language of lattice vibrations this means that transverse lattice vibration modes are easier to excite than longitudinal ones. Since the vortex motion is overdamped we assume that each vortex oscillates individually. Therefore instead of focusing on the full dispersion relation we use a simple Einstein model of vortex lattice phonon as has been done in ref. 14. It is important to note that even though this analysis is formulated assuming a vortex solid, it will remain valid even in a vortex fluid as long as there are short range hexagonal coordination and the diffusive motion of the vortices is much slower compared to the Einstein frequency.
To start with let us assume that vortices in a row are displaced by a distance δ (<< a), from its equilibrium position as shown in Fig. 5(a). The shear strain in the lattice, The corresponding elastic energy per unit volume, So, the total elastic energy per vortex, 5 = t& = q DD r p .
s t&, where A is area of rhombus unit cell and d is the thickness of film. In the London limit, the shear modulus of an isotropic superconductor for small magnetic field is given by 25 , Using this expression, the elastic energy is, Since we have one vortex per unit cell, it = } , Therefore, motion of vortices about its equilibrium position, is analogous to a harmonic oscillator with Einstein frequency, where, " … is vortex mass and † = This expression is very similar to a more detailed calculation in ref. 15 and differs only by factor of 0.6 ( see Appendix C ). On the other hand if the motion is thermal in origin, " … € !∆\% ~ Ž J L. In this case, , which is independent of magnetic field.
In Fig. 5   kOe we note that these anomalies appear very close to the vortex solid to HVF, and HVF to IVL boundaries respectively. These anomalies most likely arise from the anharmonicity of the confining potential close to the phase boundaries, though a detailed understanding will require further investigations. It is interesting to note that the first anomaly appears when ∆. .~0 .14, which is in the ballpark value expected from the Lindemann criterion for melting.
Coming to the ∆.
. vs. H data at 2K, we note that the value is very close to the 450 mK value, for magnetic fields above 40 kOe but becomes larger than the 450 mK value at lower fields. To understand this behavior we need to account for the role of thermal fluctuation. The quantum to thermal crossover is interesting to explore this more carefully through transport and STS imaging down to lower temperatures.

Simulating the conductance map for a vortex lattice
To cross-check the validity of the phenomenological model used to simulate the conductance maps for a vortex lattice, we apply it on a conventional system, namely the vortex state in clean 2H-NbSe2 (single crystal), where we do not have any evidence that the vortices fluctuate about their mean positions. Fig. 6(a) shows the VL image on a NbSe2 single crystal at 5 kOe, 450 mK, keeping the bias voltage at 1.2 meV, close to the coherence peak. Fig. 6(b) shows the GN(V) vs V spectra along a line passing through the center of the vortex. Here, the core of vortex shows a zero bias conductance peak ( 6 !" = 0% > 1) resulting from the bound state of normal electrons inside the normal core, known as Caroli-de Gennes-Matricon bound state 3 .
On the other hand, spectra obtained at superconducting regions away from the vortex core has regular BCS characteristics, partially broadened by the circulating current around the vortex core (inset Fig. 6(c)). The experimental variation of 6 !0% along three lines passing through the center of the vortex as well as their average is shown in Fig. 6 (c).
The VL is simulated as follows. We assume that far away from the vortex core 6 !"% will be similar to the conductance spectra in zero field. This is obtained by fitting the zero field experimental tunneling conductance with BCS theory using superconducting energy gap, Δ and phenomenological broadening parameter, Γ as the fitting parameters; we call this resultant best fit spectrum as 6 JK4 !"%. For the spectra at the center at the vortex center, 6 $ ¡ S !"% we used the average of the experimental spectra obtained at the center of the several vortices. These two spectra are shown in Fig. 6 (d). One difference between the above simulation and the one in the case in a-MoGe is that in that case we take 6 $ ¡ S !"% = 1. The reason is that zero bias conductance peak at the center of the vortex core is very sensitive to disorder and very small amount of scattering (such as small amount of Co doping in NbSe2) destroys the peak giving a flat spectrum 32 . Therefore it is unlikely to be present even in the absence of fluctuations in a-MoGe where the electronic mean free path is very small due to the amorphous nature of the sample.

Appendix B
Fit of the GN(0) profile passing through the vortex core.
In Fig. 4 we showed two representative fits of our model with GN(0) profiles passing through the vortex core. In Fig. 7 (a)-(h) we show the fits at several other magnetic fields (at 450 mK), along with the best fit parameters. Fig. 7 (i) shows the value of Γ used to fit the profiles. Above 10 kOe we need to gradually increase the value of Γ to take into account the broadening arising from orbital current around the vortex core.

Appendix C
In this section, we will provide a detailed calculation to show that the expression of Δa/a, given by eqn.
(11) is similar to the expression derived in ref. 15. We start with eqn. 33 of reference 15, which in the harmonic limit reduces to, where av ≡ a is the separation between nearest neighbor vortices, © SE is the mean square displacement.