Supporting Information of
Three-dimensional superconducting nanohelices
grown by He
+
-focused-ion-beam direct writing
Rosa Córdoba1,2,3*, Dominique Mailly4, Roman O. Rezaev5, 6, Ekaterina I. Smirnova5, Oliver G. Schmidt5,7, Vladimir M. Fomin5,8, Uli Zeitler9, Isabel Guillamón10, Hermann Suderow10, and José María De Teresa1,2,11*
1 Instituto de Ciencia de Materiales de Aragón (ICMA), Universidad de Zaragoza-CSIC,
E-50009 Zaragoza, Spain
2 Departamento de Física de la Materia Condensada, Universidad de Zaragoza, E-50009
Zaragoza, Spain
3 Instituto de Ciencia Molecular, Universitat de València, Catedrático José Beltrán 2, 46980
Paterna, Spain
5 Institute for Integrative Nanosciences, Leibniz IFW Dresden, Helmholtzstraße 20, D-01069
Dresden, Germany
6 Tomsk Polytechnic University, Lenin Ave. 30, 634050 Tomsk, Russia
7 Research Center for Materials, Architectures and Integration of Nanomembranes (MAIN),
Rosenbergstraße 6, TU Chemnitz, D-09126 Chemnitz, Germany
8 National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe
shosse 31, 115409 Moscow, Russia
9High Field Magnet Laboratory (HFML-EFML), Radboud University, Nijmegen, 6525 ED
Nijmegen, The Netherlands
10Laboratorio de Bajas Temperaturas y Altos Campos Magnéticos, Departamento de Física de
la Materia Condensada, Instituto de Ciencia de Materiales Nicolás Cabrera, Condensed Matter
Physics Center (IFIMAC), Universidad Autónoma de Madrid, 28049, Madrid, Spain
11 Laboratorio de Microscopías Avanzadas (LMA)-Instituto de Nanociencia de Aragón (INA),
Methods
Growth of 3D helical nanostructures
WC He+ FIBID helical nanostructures have been fabricated in a ZEISS ORION NanoFab instrument equipped with a Helium ion beam column and a single needle gas injection system (GIS) through which W(CO)6 gas is delivered to the process chamber.
The WC He+ FIBID helical nanostructures were deposited on top of the pre-patterned Ti pads (150 nm in thickness) to prevent charge effects on the insulator layer (250 nm thick of SiO2) thermally
grown on a silicon wafer 1. These chips were fabricated by using a routinely recipe of UV optical lithography using a lift-off method.
Typical deposition conditions used for the He+ FIBID process were: precursor material= tungsten hexacarbonyl, W(CO)6; Tprecursor= 55 ºC; GISneedle diameter ~ 500 m; GISz ~ 500 m; GISx,y ~ 60 m; Pbase ~ 3 × 10-7 mbar; Pprocess ~ 4 × 10-6 mbar; acceleration voltage = 30 kV; ion beam diameter:
1 nm; ion beam current= 1.24 pA; pattern shape: circle, frame; diameter of circular frame: 75 nm, 100 mn and 200 nm and ion beam spacing= 10 nm.
He+ FIBID is a technique based on a chemical vapor deposition process assisted by a He+ ion beam focused to 1 nm, using in this specific case W(CO)6 as starting material. These molecules in the
gas phase are adsorbed on a surface and are dissociated into non-volatile and volatile products (i.e. CO and CO2) by He+ FIB under the experimental conditions. The non-volatile ones become
described by using analytical modelling2 and Monte-Carlo based simulations3, the vertical growth of 3D nano-objects is mainly caused by secondary electrons of 1st order (SE-1) produced from the primary ion beam, the lateral growth is induced by the secondary electrons of 2nd order (SE-2) formed from scattered ions, whereas the direct contribution of the primary ion beam and the scattered ions are almost negligible. The resolution, the volume per dose and the throughput of 3D nano-objects are very sensitive to the growth conditions: ion beam energy, ion beam current, precursor flux, surface interactions with the beam, precursor molecules, etc. 2,3. To achieve high resolution and aspect ratio 3D nano-objects, one has to choose the highest voltage (30 kV) and the lowest current (≤ 1 pA). This method is highly recommended for direct-write nanofabrication of real 3D nano-objects with high resolution and aspect ratio 4–9.
Studies on the nanohelix geometry by modifying the diameter and beam dwell time were performed to find optimal process parameters. As an example, Figure S1 shows an array of twelve nano-helices fabricated by varying beam dwell time from 700 nm to 1800 ms, whereas beam current= 0.96 pA, nominal diameter= 75 nm and number of turns= 8 are fixed. Table 1 indicates the pitch values found as a function of the selected beam dwell time. Similar experiments were performed for nominal diameters of 100 nm and 200 nm. A linear dependence of the pitch as a function of the beam dwell time is found for the three series of experiments (Figure S2). Thus, to prepare 3D nanohelices on demand, the diameter of the circle, number of turns and dwell time are controlled and indicated in the main manuscript.
Figure S1. SEM images of WC nano-helices grown by He+-FIBID. I= 0.96 pA, number of turns= 8, beam spacing= 10 nm, external diameter= 75 nm, dwell time ranges from 700 ms in nanohelix 1 to 1800 ms in nanohelix 12. Left panel shows the top view, whereas the right panel the 52º tilted stage view.
Table S1. Measured pitch (LT) vs beam dwell time for WC nanohelices of Figure S1.
Nanohelix nº 1 2 3 4 5 6 7 8 9 10 11 12 Dwell time (ms) 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 Pitch (nm) 225 266 348 368 418 485 519 552 669 703 769 836 500 nm 500 nm 1 2 3 4 9 10 11 12 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11 12
Figure S2. Measured pitch (LT) as a function of beam dwell time for WC nanohelices with
different nominal helix diameter.
Nano-manipulation of 3D helical nanostructures
The nanomanipulation of 3D helical nanostructures was carried out inside an FEI Helios Nanolab 650 Dual Beam instrument equipped with a Schottky field-emission-gun electron column located along the vertical axis, and an Omniprobe nanomanipulator, model 200. The nanomanipulator is a sharp tungsten tip of 1 m in diameter which is inserted into the process chamber. The tip motion is motorized for the three dimensions of the space (x, y and z) with nanometer precision (maximum resolution of 10 nm).
Live SEM imaging is used to find nanohelices and the nanomanipulator. By using this tools, we pushed and placed nanowires horizontally flat on the SiO2 layer.
Growth of auxiliary Pt FIBID nanocontacts
Nano-helices were connected to the pre-patterned Ti pads by depositing four Pt FIBID nanocontacts using (CH3)3Pt(CpCH3) as precursor material. The growth of these nanocontacts were
600 800 1000 1200 1400 1600 1800 200 400 600 800 1000 75 nm 100 nm 200 nm Linear fit Linear fit Linear fit Pitch (n m) Dwell time (ms)
carried out inside of a FEI Helios Nanolab 650 Dual Beam instrument equipped with a Ga+ focused ion beam column and an individual GIS for each precursor.
Typical deposition conditions used for the PtFIBID process were: pattern dimensions: width~ 50 nm, thickness~ 50-100 nm, variable length; precursor material= methylcyclopentadienyl (trimethyl) platinum, (CH3)3(CH3CpPt); Tprecursor= 45 ºC; GISneedle diameter~ 500 m; GISz~ 150 m; GISx,y~ 50 m; Pbase~ 2.0 × 10-6 mbar; Pprocess~ 1.16 × 10-5 mbar; voltage= 30 kV; current= 1.7
pA; dwell time= 200 ns; volume per dose= 0.5 m3/nC.
Figure S3. SEM images of nanohelix type 1 (real nanohelix diameter= 100 nm, nanowire
diameter= 50 nm, pitch= 200 nm). (a) as-grown 3D nanohelix (52º tilted stage), (b) nanohelix situated on the substrate plane, (c) nanohelix contacted to four Pt FIBID wires for typical four probes magnetotransport measurements.
500 nm 500 nm 500 nm (a) (b) (c) I+ I -V -V+
Magnetotransport measurements
Magnetotransport measurements on the 3D nanohelices at low temperature (down to 0.5 K) and magnetic field applied perpendicularly to the substrate plane (up to +9 T) were performed using a ''Physical Properties Measurements System'' (PPMS), from Quantum Design equipped with a dilution refrigerator insert.
Tc is defined as the temperature at which the resistance drops to 0.5RN.
Jc is estimated from the critical current values, Ic (the value at which the resistance is finite)
extracted from the resistance versus current measurements.
Figure S4. Critical current density (Jc) as a function of the applied magnetic field at 0.5 K for
Figure S5. Experimental Resistance-vs-Current diagrams at 0.5 K for (a) nanohelix type 1, (b) nanohelix type 4, (c) nanohelix type 7.
Numerical simulations
To simulate the mechanism of jump-like dissipation in the helical WC structures we use the Ginzburg-Landau model coupled with the Poisson equation using link variable approach.
∂𝜓 ∂𝑡 = − ( 1 𝑖κ∇ − 𝐀) 2 𝜓 + (1 − |𝜓|2)𝜓 − 𝑖κ𝜑𝜓, (1) where 𝐀 the vector potential; 𝜑 the electric potential and κ = 𝜆/𝜉 the Ginzburg–Landau parameter with London penetration depth 𝜆 and coherence length 𝜉. Boundary conditions follow from the absence of normal component of the superconducting current at the free edges:
(𝐧, 1
𝑖κ∇ − 𝐀) 𝜓|∂𝐷𝑠
= 0; (𝐧,1
𝑖κ∇ − 𝐀) 𝜓|∂𝐷𝑦
= 0. (2)
The electric potential 𝜑 is found as a solution of the Poisson equation coupled with Eqs. (1), (2):
Δ𝜑 =1 𝜎(∇, 𝐣𝑠𝑐), (3) (a) (b) (c) 2 3 4 5 0 500 1000 1500 R ( ) I (A) 0 T 1 T 2 T 2 3 4 5 0 1000 2000 R ( ) I (A) 0 T 1 T 2 T 3 4 0 2000 4000 I (A) R ( ) 0 T 1 T 2 T
where superconducting current density is defined as 𝐣𝑠𝑐 = 1
2𝑖κ(𝜓
∗∇𝜓 − 𝜓∇𝜓∗) − 𝐀|𝜓|2 and 𝜎 is
the normal conductivity. The transport current density 𝑗𝑡𝑟(𝑦) = const = 𝑗𝑡𝑟 is imposed via the boundary conditions for Eq. (3) at the edges, to which electrodes are attached:
(𝐧, ∇)𝜑|∂𝐷𝑠 = −1
𝜎𝑗𝑡𝑟; (𝐧, ∇)𝜑|∂𝐷𝑦 = 0. (4)
Parameters in the aforementioned equations used for numerical simulation are presented in Tables S2, S3. The used geometrical denotations are previously reported in reference 10.
For simulation the dirty limit is used [1] and the corresponding parameters are calculated for the temperature T = 0.5 K (see Table S3) using the following expressions:
𝜉 = 0.855√ 𝜉0𝑙 1 − 𝑡 , 𝑡 = 𝑇 𝑇𝑐 𝜆 = 𝜆0√ 𝜉0 2(1 − 𝑡) × 1.33𝑙 𝐷 =𝑙 × 𝑣𝐹 3 𝜏𝐺𝐿 =𝜉 2 𝐷 𝐵𝐶2= Φ0 2𝜋𝜉2 𝐽𝑐𝑐𝑎𝑙𝑐,G= 𝑐𝐻𝑐 3√6𝜋𝜆 (Gaussian units),
𝐽𝑐𝑐𝑎𝑙𝑐,SI = 𝐽𝑐𝑐𝑎𝑙𝑐,G× 𝑘G→SI, 𝑘G→SI =
1
𝐻𝑐 =
𝐻𝑐2
√2𝜅, 𝜅 =
𝜆 𝜉
For the numerical simulation, we used a 2D approximation with the mesh 1725×25 nodes that provides the spatial resolution as small as 0.5. The discretization time is equal to 0.0005GL,
however, this parameter is dynamically adapted to the dynamics of the system: when there occurs only a small variation of the order parameter (phase and amplitude), the time discretization is increased and vice versa. With these parameters, the model is capable to reproduce the vortex dynamics in the presence of a transport current and a magnetic field. The number of iterations required to reach the specific order parameter pattern varies between 107 and 108. For the simulation, the supercomputer facilities (Taurus in ZIH at the Technical University of Dresden) were used.
Table S2. Calculated and evaluated parameters for the nano-helix type 6
Physical parameter and units Value
Coherence length 𝜉, nm 5.4
Magnetic field penetration length 𝜆, 𝑛m 345.0 Diffusion Coefficient 𝐷, m2⁄ s 8.5×10-6
Second critical magnetic field BC2, T 11.5
Critical current 𝐼C, µA (for magnetic field 1 T) 3.3 [evaluated from experimental data: the
value at which maximal resistance reached]
Critical current 𝐼C, µA (for magnetic field 2 T) 3.0 [evaluated from experimental data: the
value at which maximal resistance reached]
Table S3. Parameters taken for the particular simulation of the nanohelix type 6 in 2D
approximation
Geometrical and physical parameters and units
Value
Number of turns 2
Radius of the helical coil R, nm 138.5
Width W, nm (see Fig. 5) 58
Coherence length 𝜉0, nm 4.93 [averaged over all structures]
Magnetic field penetration length 𝜆0, nm 665 [averaged over all structures]
Conductivity in normal state 𝜎, 1/Ω ∙ m 4.75×105 [extracted from experimental data
and averaged over all structures]
Critical temperatures 𝑇c, K 6.63 [averaged over all structures]
Mean free electron path, nm 7.45 [evaluated from experimental data]
Fermi velocity, m/s 3419 [calculated from Drude model]
From an arbitrary initial state (for which a random distribution is taken), the order parameter evolves to one of the three following quasi-stationary patterns: (i) a pure vortex state (the number of vortices is in the range from 0 to N), (ii) a mixed one – vortices plus order parameter depression (phase slip) regions and (iii) pure order parameter depression regimes.
The top panels in Figure S6 represent the order parameter distributions resulting from the calculation. They reveal a number of phase slips occurring in some half-turns. We suggest that the whole half-turn, at which the order parameter depression appears, switches to the normal state due to the Joule heating as shown in the bottom panels in Figure S6. This is likely needed to account for the large change in resistance observed in experiment.
Figure S6. Simulated order parameter distributions plotted over the 2D surface of the helical
structure type 6 for 1 T. Each calculated state manifesting a number of phase slips in some half-turns (top panels) and the assumed phase slips extensions over the half-half-turns (bottom panels). The values of the applied transport current in units of the critical current are indicated in the panels.
I ~ 0.6IC I ~ 0.8IC
We have performed extensive simulations for structures of different helical radii (from 50 to 150 nm) and pitches (from 200 to 925 nm) under various applied transport currents (from 0 to 1.2Ic) and found no qualitative difference in order parameter patterns as a function of the applied
transport current. To unveil the mechanism of influence of the topology on the order parameter, we have simulated the planar structures with the same dimensions as the corresponding helical ones.
Results of this simulation for the fixed applied transport current (I ~ 0.6Ic) are shown in
Figure S7. For each number of turns, we also provide the order parameter for the corresponding planar structure (Figure S8). It is remarkable that for all planar structures the order-parameter pattern does not depend on the size, unlike that in helical structures.
Figure S7. The magnetic field and the order parameter distributions for the nanohelix type 1 with
the width W = 50 nm, the helical radius R = 50 nm, the pitch P = 200 nm and the number of turns
N = 1, N = 2, N = 3 and N = 4. The transport current I ~ 0.6Ic.
Figure S8. The order parameter distributions for planar structures of the width W = 50 nm and the
nanohelices with radius R = 50 nm, width W = 50 nm, the pitch P = 200 nm and different lengths corresponding to the numbers of turns N = 1, 2, 3, 4. The transport current I ~ 0.6Ic.
Finally, in Figure S9 we provide an evolution of the order parameter with increasing transport current for the helical structure with two turns and for the corresponding planar structure. The spatial regions occupied by the normal phase of the order parameter in a planar structure approximately linearly grow with the current, while the similar spatial regions in the helical structure remain almost unchanged. This qualitatively demonstrates how the helical geometry changes the resistive properties.
Figure S9. The order parameter distributions (panels 1 to 3) for the nanohelix type 1 with the W =
50 nm, R = 50 nm, pitch P = 200 nm and the number of turns N = 2. The three bottom panels (4 to 6) show the order-parameter distributions for the planar structures of the same dimensions. The values of the applied transport current in units of the critical current are indicated in the panels.
In addition, magnetotransport measurements on 3D nanohelix type 7 at low temperature and varying the applied magnetic field angle respect to the nanohelix long axis were performed at the High Magnetic Field Laboratory (HFML) in Nijmegen.
Figure S10. Experimental magnetic field angle dependence of the resistance at 4 K, bias current
of 0.15 A for the nanohelix type 7.
Explanation of the tilt-angle dependence of magnetic field
|𝐵𝑐(𝛽) sin 𝛽 𝐵𝑐⊥ | + ( 𝐵𝑐(𝛽) cos 𝛽 𝐵𝑐∥ ) 2 = 1, (1)
where 𝐵𝑐∥ and 𝐵𝑐⊥ are the in-plane and perpendicular critical magnetic induction values, correspondingly. Solving Eq. (1), we find the critical magnetic induction in the form
𝐵𝑐(𝛽) = 𝐵𝑐∥ 2 2 cos2𝛽(√ sin2𝛽 𝐵𝑐⊥2 + 4 cos2𝛽 𝐵𝑐∥2 − | sin 𝛽 𝐵𝑐⊥ |). (2) Let the applied magnetic field 𝐁 be tilted at the angle α relative to the helical coil axis and normal to the x-axis:
𝐁 = 𝐵𝟎(𝒆𝑦 sin 𝛼 + 𝒆𝑧 cos 𝛼) ≡ 𝐵𝟎𝒆𝛼, (3) where 𝒆𝑦, 𝒆𝑧 are the unit axes vectors (see Fig. 1 in Ref.11). The tangential unit vector for a centerline of a helical coil11 is
𝝉 = {− sin 𝜃 cos (2𝜋𝑠
𝑙 ) , sin 𝜃 sin ( 2𝜋𝑠
𝑙 ) , cos 𝜃}, (5)
where 𝑠 is the coordinate along the centerline,
𝑙 = √(2𝜋𝑅)2+ 𝑃2, sin 𝜃 =2𝜋𝑅
𝑙 , cos 𝜃 =
𝑃 𝑙,
𝑅 and 𝑃 are the radius and pitch of the centerline. The trigonometrical functions in Eq. (2) as functions of s acquire the form
The values 𝐵𝑐⊥ and 𝐵𝑐∥ = 6 T and 8.5 T are determined from the four-probe magnetotransport measurements of hollow wires9 at the lowest available for the in-plane case T = 5 K. The critical
magnetic induction (2) is calculated at P = 922 nm and R = 147 nm and averaged over 𝑠 on the whole length of the nano-helix [0, 𝑙].
Finally, we plot the averaged critical magnetic field 〈𝐵𝑐2〉 ≡ 𝜇0𝐻𝑐2(𝑇) as a function of the tilt
angle 𝛼 of the applied magnetic field in Figure S11 (left panel) and the experimental data are provided in the right panel for comparison. Numerical simulation qualitatively explains the observed dependence of the critical magnetic field on the tilt angle. A fully 3D simulation of superconducting-to-normal phase transition in a nano-helix is required to qualitatively interpret the tilt-angle impact on the transition.
Figure S12. SEM images of an array of WC nanohelices of type 1 (real nanohelix diameter= 100
nm, nanowire diameter= 50 nm, LT= 200 nm).
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