Electron Mean Free Path Effect on Vortex Matter in a Superconducting Pb Island Grown on Si (111)

1. Materials and Methods

We simulated a superconducting Pb island grown on a silicon substrate (111) as illustrated in Figure 1, but also, the thickness change d and the mean free path $l_e$. The framework for our theoretical studies is the phenomenological Ginzburg-Landau (GL) theory [1]. We used the expressions for GL coeficients $\alpha$ and $\beta$ in the dirty limit, to include the variation $l_e$ the electron mean free path in the sample, i.e.

$$\begin{array}{ccc}\hfill \alpha \left(T\right)& =& -1.36\frac{\hslash }{2m^{*}{\xi }_0{l}_{\xi }}\left(1-\frac{T}{T_c}\right)=\frac{{\alpha }_0}{l_{{\xi }_0}}\left(1-\frac{T}{T_c}\right)\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill \beta & =& \frac{0.2}{N\left(0\right)}{\left(\frac{{\hslash }^2}{2m^{*}{\xi }_0{l}_e{k}_B{T}_c}\right)}^2=\frac{{\beta }_0}{l_{{\xi }_0}^2}\hfill \end{array}$$

(2)

Where $l_{\xi }=\frac{l_e}{{\xi }_0}$ is the ratio of the electron mean free path and BCS coherence length. The dimensionless form of the GL equations can be written as follow:

$$\begin{array}{ccc}\hfill {(-i\nabla -\mathbf{A})}^2\Psi & =& \left(\frac{1.367}{l_{{\xi }_0}}-\frac{1}{l_{{\xi }_0}^2}{|\Psi |}^2\right)\Psi \hfill \end{array}$$

(3)

Where lengths are scaled to ${\xi }_0$, penetration depth ${\lambda }_0$ is defined as ${\lambda }_0^2=m{c}^2{\beta }_0/16\pi \left|a_0\right|e^2$, the vector potential $\mathbf{A}$ is expressed ${\varphi }_0/2\pi {\xi }_0$, and the order parameter is in units of ${\Psi }_0=\sqrt{-\alpha /\beta }$.

2. Results and discussion

Process started calculating the full free-energy (F) spectrum and the corresponding vortex states as a function of applied magnetic field ($H_0$) for the island with a central hole and $l_{\xi }=1.0$ Figure 2. Initially, the magnetic field was kept perpendicular to the plane bottom of Pb island. The method for finding vortex states is multifold: First step, applied magnetic field is increased and decreased in the considered range with a kept history of the previously found states in the field sweep and the second step, considering each value of magnetic field, the calculation is initialize from the fluctuating normal state (randomized $|\Psi |<0.01$) and from the from the fully superconducting state ($|\Psi |\approx 1$). Following this steps, we construct the energy diagram of all stable vortex states, including those with higher energy. As a result, we found more than one stable vortex distribution, i.e. one with lowest energy (ground state) and several others with higher energy (usually called metastable).

We show lower energies and a higher stabilities represented in the wide ranges of magnetic fields in which the same number of vortices remains (see Figure 2, cases a.,b.,c.,d. and e.). This finding demonstrates the strong confinement imposed on vortices by Meissner currents on the edge of the island (see Figure 1), where the Meissner current is strongest, as we can notice in the snapshots of superconducting current (see Figure 2 (left) inset) and it confines vortices toward the interior of the sample. On the other hand, It can also be noted that stability decreases with the entry of a greater number of vortices when the magnetic field is increasing (see Figure 2, cases f.,h. and g.), due to vortices are basically parallel magnetic moments that destroy the superconductivity with increase of their number. Additionally, other evidence of the strong confinement on the superconducting island can be noticed that vortices repel each other, and this repulsion, in the absence of sample boundaries, leads to the formation of a triangular (Abrikosov) lattice, which is not the case in this research.

Figure 3a–d inset show some states where vortices have reached the center of superconductor island which can be verified by the circular diagram phase of the order parameter on the top of the island simulation. The vortex state is characterized by the total angular momentum L through $\Psi =\psi exp\left(iL\varphi \right)$, but, we can introduce an analog to the total angular momentum which is still a good quantum number. Choosing circular loops at the periphery of the center of the island, we find that the effective angular momentum $L=\Delta \varphi /2\pi$, and each time that in the path the color went from red to blue a vortex is found $L=1,2,3,...$. The superconducting order parameter $|\Psi |$, nucleated at the sample surface, traps then inside, in the island hole, carrying flux $L{\varphi }_0$ where ${\varphi }_0$ is the quantum flux . To check this ‘‘flux compression’’ model quantitatively, the self-consistent solution of the full Ginzburg-Landau (GL) equations is necessary Eq.3

The transition between the two quantum states $L=0$ (zero vortex) and $L=1$ (one vortex) can be used to calculate the field $H_1$, corresponding to the penetration of the first flux line into a superconducting island. One option to find the value of $H_0$ in which this transitions take place is considering every curve of Figure 3a–d where the peaks obtained at a lower value of applied magnetic fields in every curve of Figure 3 (F versus $H_0$), represent the first vortex entrance in the superconducting island, we can notice that it take place at different values of $H_0$. Figure 3 also can be used to calculate the field $H_0$ corresponding to the penetration of one flux line at different electron mean-free path $l_{\xi }$, and thus the transition between the two quantum states $L=0$ and $L=1$, as well as by using l we can tune the number of peaks or transitions.

The characteristics lengths of a superconductor, the coherence length and the penetration depth, take the effective values ${\lambda }_{eff}=0.65{\lambda }_0\sqrt{{\xi }_0/l}$ and ${\xi }_{eff}=0.85\sqrt{l{\xi }_0}$ considering $l_{\xi }=l_0/{\xi }_0$. Thus, the situation in the island is similar to that of a type II superconductor in the diffusive limit, in which correlation length $\xi$ and penetration depth $\lambda$ in a magnetic field depends on $l^{0.5}$ and $l^{-0.5}$ respectively. Therefore, the magnetic field in a superconductor is affected due to the increment of $\lambda$ with the increases of concentration of impurities, but also a strong variation of the number of superconducting electrons, i.e. electrons linked in Cooper pairs with the decrease of $\xi$. This behaviour is shown in Figure 2 and Figure 3. (inset) where variations of condensate are reached considering a spatial change of the electron mean free path.

In addition, the order parameter $|{\Psi }_0{|}^2$ is modified through the sample with the modification of l, as we expected, according to the proportionality between the order parameter and penetration depth which is $|{\Psi }_0{|}^2\sim 1/\lambda$. It implies changes on strong screening supercurrents (see Figure 2 inset) that may circulate in the island, where strong supercurrent carry on a strong diamagnetic (Meissner) effect as we can notice in Figure 3 where the $H_1$ (first vortex entry) is reached (see also Table A1 apendix A). It also can be noticed in the modification of l at different zones of sample (see Figure A1 apendix B) which allow to chose not only the location of vortex entrance, but also, the value of $H_0$.

Figure 4 (left) shows the proportionality of the green curve corresponding to the values of the second critical magnetic field where the normal state is reached ($H_2$) versus the mean free path (l), which shows a result not reported before for this type of heterostructures. The fine tune of $H_2$ shows great applications for the design of electronic devices, through the bombardment of the Pb islands, which can be controlled using high precision with current technologies. Thus, it is clear that $H_1$ can be tuned, which supposes a variation in the Meissner current and with it the repulsion of the applied magnetic field $H_0$ and entry of vortices in the superconducting island. We know that flux penetration in type-II superconductors occurs in form of quantized, flux-enclosing, supercurrent vortices. However, a more detailed analysis shows that, this flux penetration must first overcome an energy barrier at the surface which is called Bean-Livinstong barrier (BLB). Corresponding variations of the Gibbs free energy with l, for several values of $H_0$, are shown in Figure 3. One can see that $H_1$ take different values due to, in order to penetrate the superconductor island, vortex must first overcome BLB.

In order to identify the observed quantum phenomenon, we explored the evolution and stability of the superconductivity in the islands with the magnetic field (Figure 2, Figure 3 and Figure 4). At zero magnetic field, Figure 2 snapshot (a.) shows a spatially homogeneous superconducting condensate to exist in the entire nanoisland, evidencing the Meissner ($L=0$) state. As the field increases further, the snapshots in Figure 2 and Figure 3 reveal novel intriguing vortex configurations, evolving till the normal state is achieved in each the superconducting island, but in order to get a deeper insight into the field evolution of the condensate confined we need to focus on their phases.

In Figure 5 we present a color-coded diagram of (ZBC) vs magnetic field taken over a line crossing the island (depicted as a black line in the blue frame). This diagram shows a series of abrupt steplike transitions which are identified in a separating the states with different vorticity L, where color from red (superconducting state) to blue (normal state) corresponds to the local strength of superconducting condensate. Precisely, until $H_0\approx 27.5mT$ the island remain in the Meissner state ($L=0$) for $l=0.2$, in agreement with the color map in Figure 5. Also, we can notice that only one vortex penetrate due to only one steplike transition take place for $l=0.2$ until reach the total normal state at $H_0\approx 46.49mT$ (all region became blue). In order to get a deeper insight into these phases, we focused on the field evolution of the condensate confined in the superconducting island with $l=1.2$. At $H_0\approx 46.49mT$ T the first vortex appears in the sample; it is clearly identified by its normal state core (blue region). This single vortex state lasts until $H_0\approx 78.49mT$. The $L=2$ phase occurs at $H_0\approx 78.49mT<H_0<H_0\approx 110.49mT$, followed by the $L=3$ state, that also can be noticed in the CPD in the red snapshot located up of the $H_0\approx 110.49mT<H_0<H_0\approx 134.49mT$.

Remarkably, in $L=2$ for $l=0.8$ looks like a single round object located at the island center, instead of two individual vortices. The straightforward conclusion is that here one (two) individual vortex cores are merged to form a single giant vortex.

3. Conclusions

In conclusion, strong vortex confinement effects were studied in Pb island grown on Si (111), taking into account several changes of electron mean-free path, considering that experimentally to ion bombardment can be done over a superconducting sample, by using the expressions for Ginzburg-Landau (GL) coefficients $\alpha$ and $\beta$ in the dirty limit. In these type II superconducting island an unexpected behaviour of vortex configuration was studied, but also, their critical parameters which allow to control this heteroestructure in order to be applied in electronic devices.

Appendix A.1

Table A1. Stability range of vortex states for each value of the mean free path studied. Second critical field is included in the last column.

l/$nm$	$1st/mT$	$2nd/mT$	$3rd/mT$	$4th/mT$	$5th/mT$	$6th/mT$	$7th/mT$	$8th/mT$	$H_2/mT$
0.2	27.29								49.69
0.3	32.09								59.29
0.4	36.89	68,89							91.28
0.5	36.89	68,29							94.48
0.6	38,49	70,49	97.68						107.28
0.7	40,09	73,69	99.28						118.48
0.8	41,69	75,29	102.48						128.08
0.9	43,29	76,89	104.08						140.88
1.0	44,89	78,49	105.68						147.28
1.1	44,89	80,09	107.28	132.88					177.67
1.2	46,49	81,68	110.48	134.48	155.28				177.67
1.3	48,09	84,88	112.08	136.08	156.88				188.87
1.4	49,69	84,88	113.68	137.68	158.48				198.47
1.5	49,69	86,48	115.28	140.88	161.68	179.27			208.07
1.6	51,29	86,48	116.88	142.48	163.28	182.47			217.67
1.7	52,89	88,08	118.48	144.08	164.87	184.07	203.27		233.67
1.8	52,89	88,08	118.48	145.68	166.47	185.67	206.47		241.67
1.9	54,49	89,68	120.08	147.28	168.07	187.27	208.07	227.27	264.06
2.0	54,49	89,68	121.68	147.27	169.67	188.87	211.27	228.80	265.66

* Tables may have a footer.

Table A1. Stability range of vortex states for each value of the mean free path studied. Second critical field is included in the last column.

l/$nm$	$1st/mT$	$2nd/mT$	$3rd/mT$	$4th/mT$	$5th/mT$	$6th/mT$	$7th/mT$	$8th/mT$	$H_2/mT$
0.2	27.29								49.69
0.3	32.09								59.29
0.4	36.89	68,89							91.28
0.5	36.89	68,29							94.48
0.6	38,49	70,49	97.68						107.28
0.7	40,09	73,69	99.28						118.48
0.8	41,69	75,29	102.48						128.08
0.9	43,29	76,89	104.08						140.88
1.0	44,89	78,49	105.68						147.28
1.1	44,89	80,09	107.28	132.88					177.67
1.2	46,49	81,68	110.48	134.48	155.28				177.67
1.3	48,09	84,88	112.08	136.08	156.88				188.87
1.4	49,69	84,88	113.68	137.68	158.48				198.47
1.5	49,69	86,48	115.28	140.88	161.68	179.27			208.07
1.6	51,29	86,48	116.88	142.48	163.28	182.47			217.67
1.7	52,89	88,08	118.48	144.08	164.87	184.07	203.27		233.67
1.8	52,89	88,08	118.48	145.68	166.47	185.67	206.47		241.67
1.9	54,49	89,68	120.08	147.28	168.07	187.27	208.07	227.27	264.06
2.0	54,49	89,68	121.68	147.27	169.67	188.87	211.27	228.80	265.66

* Tables may have a footer.