# Gate Control of Superconductivity in Mesoscopic All-Metallic Devices

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## Abstract

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## 1. Introduction

## 2. Gate-Driven Supercurrent Suppression in Nb and V Nanojunctions

#### 2.1. Niobium Gate-Controlled Transistor

#### 2.1.1. Rectification Properties

#### 2.2. Vanadium Gate-Controlled Transistor

#### 2.2.1. Half-Wave Rectifier

#### 2.2.2. Amplification Properties

## 3. Nonthermal Origin of Supercurrent Suppression in Gated All-Metallic Superconducting Devices

#### 3.1. SCPDs in a Titanium Gate-Controlled Transistor

#### 3.2. Suspended Titanium Gate-Controlled Transistor

#### 3.3. Leakage Current Finite Element Method Simulations

#### 3.4. Heating through Single Cold-Electron Field Emission or Absorption

#### 3.5. Continuous Power Injection

#### 3.6. Unconventional Sum Rule

## 4. Summary and Further Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Bardeen, J. Critical Fields and Currents in Superconductors. Rev. Mod. Phys.
**1962**, 34, 667–681. [Google Scholar] [CrossRef] - De Simoni, G.; Paolucci, F.; Solinas, P.; Strambini, E.; Giazotto, F. Metallic supercurrent field-effect transistor. Nat. Nanotechnol.
**2018**, 13, 802–805. [Google Scholar] [CrossRef] [PubMed] - Paolucci, F.; De Simoni, G.; Strambini, E.; Solinas, P.; Giazotto, F. Ultra-Efficient Superconducting Dayem Bridge Field-Effect Transistor. Nano Lett.
**2018**, 18, 4195–4199. [Google Scholar] [CrossRef] [PubMed] - Paolucci, F.; De Simoni, G.; Solinas, P.; Strambini, E.; Ligato, N.; Virtanen, P.; Braggio, A.; Giazotto, F. Magnetotransport Experiments on Fully Metallic Superconducting Dayem-Bridge Field-Effect Transistors. Phys. Rev. Appl.
**2019**, 11, 024061. [Google Scholar] [CrossRef] [Green Version] - Paolucci, F.; Vischi, F.; De Simoni, G.; Guarcello, C.; Solinas, P.; Giazotto, F. Field-Effect Controllable Metallic Josephson Interferometer. Nano Lett.
**2019**, 19, 6263–6269. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Paolucci, F.; De Simoni, G.; Solinas, P.; Strambini, E.; Puglia, C.; Ligato, N.; Giazotto, F. Field-effect control of metallic superconducting systems. AVS Quantum Sci.
**2019**, 1, 016501. [Google Scholar] [CrossRef] [Green Version] - Bours, L.; Mercaldo, M.T.; Cuoco, M.; Strambini, E.; Giazotto, F. Unveiling mechanisms of electric field effects on superconductors by a magnetic field response. Phys. Rev. Res.
**2020**, 2, 033353. [Google Scholar] [CrossRef] - Rocci, M.; Suri, D.; Kamra, A.; Gilvânia, V.; Takamura, Y.; Nemes, N.M.; Martinez, J.L.; Hernandez, M.G.; Moodera, J.S. Large Enhancement of Critical Current in Superconducting Devices by Gate Voltage. Nano Lett.
**2021**, 21, 216–221. [Google Scholar] [CrossRef] - De Simoni, G.; Paolucci, F.; Puglia, C.; Giazotto, F. Josephson Field-Effect Transistors Based on All-Metallic Al/Cu/Al Proximity Nanojunctions. ACS Nano
**2019**, 13, 7871–7876. [Google Scholar] [CrossRef] - Virtanen, P.; Braggio, A.; Giazotto, F. Superconducting size effect in thin films under electric field: Mean-field self-consistent model. Phys. Rev. B
**2019**, 100, 224506. [Google Scholar] [CrossRef] [Green Version] - Shmidt, V.V.; Balkov, A.A. The Critical Current in Superconducting Films. Sov. Phys. JETP
**1970**, 6, 1137–1142. [Google Scholar] - Shmidt, V.V. Critical Currents in Superconductors. Sov. Phys. Uspekhi
**1970**, 13, 408–409. [Google Scholar] [CrossRef] - Aslamazov, L.G.; Larkin, A.I.; Landau, L.D. Josephson effect in wide superconducting bridges. Sov. Phys. JEPT
**1975**, 41, 381–386. [Google Scholar] - Kim, P.; Kang, K.T.; Go, G.; Han, J.H. Nature of orbital and spin Rashba coupling in the surface bands of SrTiO
_{3}and KTaO_{3}. Phys. Rev. B**2014**, 90, 205423. [Google Scholar] [CrossRef] [Green Version] - Park, J.H.; Kim, C.H.; Lee, H.W.; Han, J.H. Orbital chirality and Rashba interaction in magnetic bands. Phys. Rev. B
**2013**, 87, 041301. [Google Scholar] [CrossRef] [Green Version] - Petersen, L.; Hedegård, P. A simple tight-binding model of spin–orbit splitting of sp-derived surface states. Surf. Sci.
**2000**, 459, 49–56. [Google Scholar] [CrossRef] - Mercaldo, M.T.; Solinas, P.; Giazotto, F.; Cuoco, M. Electrically Tunable Superconductivity Through Surface Orbital Polarization. Phys. Rev. Appl.
**2020**, 14, 034041. [Google Scholar] [CrossRef] - Sauter, F. Uber das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs. Z. Phys.
**1931**, 69, 742–764. [Google Scholar] [CrossRef] - Heisenberg, W.; Euler, H. Folgerungen aus der Diracschen Theorie des Positrons. Z. Phys.
**1936**, 98, 714–732. [Google Scholar] [CrossRef] - Solinas, P.; Amoretti, A.; Giazotto, F. Schwinger effect in a Bardeen-Cooper-Schrieffer superconductor. arXiv
**2020**, arXiv:2007.08323. [Google Scholar] - Alegria, L.D.; Bøttcher, C.G.L.; Saydjari, A.K.; Pierce, A.T.; Lee, S.H.; Harvey, S.P.; Vool, U.; Yacoby, A. High-energy quasiparticle injection into mesoscopic superconductors. Nat. Nanotechnol.
**2021**. [Google Scholar] [CrossRef] [PubMed] - Ritter, M.F.; Fuhrer, A.; Haxell, D.Z.; Hart, S.; Gumann, P.; Riel, H.; Nichele, F. A superconducting switch actuated by injection of high energy electrons. arXiv
**2020**, arXiv:2005.00462. [Google Scholar] - Puglia, C.; De Simoni, G.; Giazotto, F. Electrostatic Control of Phase Slips in Ti Josephson Nanotransistors. Phys. Rev. Appl.
**2020**, 13, 054026. [Google Scholar] [CrossRef] - Rocci, M.; De Simoni, G.; Puglia, C.; Degli Esposti, D.; Strambini, E.; Zannier, V.; Sorba, L.; Giazotto, F. Gate-Controlled Suspended Titanium Nanobridge Supercurrent Transistor. ACS Nano
**2020**, 14, 12621–12628. [Google Scholar] [CrossRef] [PubMed] - Likharev, K.K. Superconductor digital electronics. Phys. C Supercond. Its Appl.
**2012**, 482, 6–18. [Google Scholar] [CrossRef] - De Simoni, G.; Puglia, C.; Giazotto, F. Niobium Dayem nano-bridge Josephson gate-controlled transistors. Appl. Phys. Lett.
**2020**, 116, 242601. [Google Scholar] [CrossRef] - Stromberg, T.F.; Swenson, C.A. Negative Surface Free-Energy Effects in Superconducting Niobium. Phys. Rev. Lett.
**1962**, 9, 370–374. [Google Scholar] [CrossRef] - Finnemore, D.K.; Stromberg, T.F.; Swenson, C.A. Superconducting Properties of High-Purity Niobium. Phys. Rev.
**1966**, 149, 231–243. [Google Scholar] [CrossRef] - Giazotto, F.; Heikkilä, T.T.; Luukanen, A.; Savin, A.M.; Pekola, J.P. Opportunities for mesoscopics in thermometry and refrigeration: Physics and applications. Rev. Mod. Phys.
**2006**, 78, 217–274. [Google Scholar] [CrossRef] [Green Version] - Tinkham, M. Introduction to Superconductivity; Dover Publications: Mineola, NY, USA, 2004; p. 454. [Google Scholar]
- Puglia, C.; De Simoni, G.; Ligato, N.; Giazotto, F. Vanadium gate-controlled Josephson half-wave nanorectifier. Appl. Phys. Lett.
**2020**, 116, 252601. [Google Scholar] [CrossRef] - Barone, A.; Paternò, G. Physics and Applications of the Josephson Effect; Wiley: Hoboken, NJ, USA, 1982. [Google Scholar] [CrossRef]
- Ullom, J.N.; Bennett, D.A. Review of superconducting transition-edge sensors for X-ray and gamma-ray spectroscopy. Supercond. Sci. Technol.
**2015**, 28, 084003. [Google Scholar] [CrossRef] - Paolucci, F.; Ligato, N.; Germanese, G.; Buccheri, V.; Giazotto, F. Fully Superconducting Josephson Bolometers for Gigahertz Astronomy. Appl. Sci.
**2021**, 11, 746. [Google Scholar] [CrossRef] - Gol’tsman, G.N.; Okunev, O.; Chulkova, G.; Lipatov, A.; Semenov, A.; Smirnov, K.; Voronov, B.; Dzardanov, A.; Williams, C.; Sobolewski, R. Picosecond superconducting single-photon optical detector. Appl. Phys. Lett.
**2001**, 79, 705–707. [Google Scholar] [CrossRef] - Ivanov, B.I.; Trgala, M.; Grajcar, M.; Il’ichev, E.; Meyer, H.G. Cryogenic ultra-low-noise SiGe transistor amplifier. Rev. Sci. Instrum.
**2011**, 82, 104705. [Google Scholar] [CrossRef] - Oukhanski, N.; Grajcar, M.; Il’ichev, E.; Meyer, H.G. Low noise, low power consumption high electron mobility transistors amplifier, for temperatures below 1 K. Rev. Sci. Instrum.
**2003**, 74, 1145–1146. [Google Scholar] [CrossRef] - Clarke, J.; Bragisnki, A.I. The SQUID Handbook; Wiley: Weinheim, Germany, 2004. [Google Scholar]
- Giazotto, F.; Peltonen, J.T.; Meschke, M.; Pekola, J.P. Superconducting quantum interference proximitytransistor. Nat. Phys.
**2010**, 6, 254–259. [Google Scholar] [CrossRef] [Green Version] - Giazotto, F.; Martínez-Pérez, M.J. The Josephson heat interferometer. Nature
**2012**, 492, 401–405. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fornieri, A.; Giazotto, F. Towards phase-coherent caloritronics in superconducting circuits. Nat. Nanotechnol.
**2017**, 12, 944–952. [Google Scholar] [CrossRef] [Green Version] - McCaughan, A.N. Readout architectures for superconducting nanowire single photon detectors. Supercond. Sci. Technol.
**2018**, 31, 040501. [Google Scholar] [CrossRef] [PubMed] - McCaughan, A.N.; Berggren, K.K. A Superconducting-Nanowire Three-Terminal Electrothermal Device. Nano Lett.
**2014**, 14, 5748–5753. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Morpurgo, A.F.; Klapwijk, T.M.; van Wees, B.J. Hot electron tunable supercurrent. Appl. Phys. Lett.
**1998**, 72, 966–968. [Google Scholar] [CrossRef] [Green Version] - Bezryadin, A. Superconductivity in Nanowires: Fabrication and Quantum Transport; Wiley-VCH: Weinheim, Germany, 2012; p. 282. [Google Scholar]
- Kurkijärvi, J. Intrinsic Fluctuations in a Superconducting Ring Closed with a Josephson Junction. Phys. Rev. B
**1972**, 6, 832–835. [Google Scholar] [CrossRef] - Bezryadin, A.; Goldbart, P.M. Superconducting Nanowires Fabricated Using Molecular Templates. Adv. Mater.
**2010**, 22, 1111–1121. [Google Scholar] [CrossRef] - Fulton, T.A.; Dunkleberger, L.N.; Dynes, R.C. Quantum Interference Properties of Double Josephson Junctions. Phys. Rev. B
**1972**, 6, 855–875. [Google Scholar] [CrossRef] - Fulton, T.; Dynes, R. Switching to zero voltage in Josephson tunnel junctions. Solid State Commun.
**1971**, 9, 1069–1073. [Google Scholar] [CrossRef] - Giordano, N. Evidence for Macroscopic Quantum Tunneling in One-Dimensional Superconductors. Phys. Rev. Lett.
**1988**, 61, 2137–2140. [Google Scholar] [CrossRef] [PubMed] - Zaikin, A.D.; Golubev, D.S.; van Otterlo, A.; Zimányi, G.T. Quantum Phase Slips and Transport in Ultrathin Superconducting Wires. Phys. Rev. Lett.
**1997**, 78, 1552–1555. [Google Scholar] [CrossRef] [Green Version] - Sahu, M.; Bae, M.H.; Rogachev, A.; Pekker, D.; Wei, T.C.; Shah, N.; Goldbart, P.M.; Bezryadin, A. Individual topological tunnelling events of a quantum field probed through their macroscopic consequences. Nat. Phys.
**2009**, 5, 503–508. [Google Scholar] [CrossRef] [Green Version] - Golubov, A.; Neurohr, K.; Schäpers, T.; Lüth, H.; Behet, M. Suppression of Josephson currents in ballistic junctions by an injection current. Superlattices Microstruct.
**1999**, 25, 1033–1040. [Google Scholar] [CrossRef] - Bezryadin, A.; Lau, C.N.; Tinkham, M. Quantum suppression of superconductivity in ultrathin nanowires. Nature
**2000**, 404, 971–974. [Google Scholar] [CrossRef] - Giordano, N.; Schuler, E. Macroscopic quantum tunneling and related effects in a one-dimensional superconductor. Phys. Rev. Lett.
**1989**, 63, 2417–2420. [Google Scholar] [CrossRef] [PubMed] - Fulton, T.A.; Dunkleberger, L.N. Lifetime of the zero-voltage state in Josephson tunnel junctions. Phys. Rev. B
**1974**, 9, 4760–4768. [Google Scholar] [CrossRef] - Ejrnaes, M.; Salvoni, D.; Parlato, L.; Massarotti, D.; Caruso, R.; Tafuri, F.; Yang, X.Y.; You, L.X.; Wang, Z.; Pepe, G.P.; et al. Superconductor to resistive state switching by multiple fluctuation events in NbTiN nanostrips. Sci. Rep.
**2019**, 9, 8053. [Google Scholar] [CrossRef] [Green Version] - Li, Q.; Huang, S.; Pan, D.; Wang, J.; Zhao, J.; Xu, H.Q. Suspended InAs nanowire gate-all-around field-effect transistors. Appl. Phys. Lett.
**2014**, 105, 113106. [Google Scholar] [CrossRef] - Iorio, A.; Rocci, M.; Bours, L.; Carrega, M.; Zannier, V.; Sorba, L.; Roddaro, S.; Giazotto, F.; Strambini, E. Vectorial Control of the Spin–Orbit Interaction in Suspended InAs Nanowires. Nano Lett.
**2019**, 19, 652–657. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Simmons, J.G. Generalized Formula for the Electric Tunnel Effect between Similar Electrodes Separated by a Thin Insulating Film. J. Appl. Phys.
**1963**, 34, 1793–1803. [Google Scholar] [CrossRef] [Green Version] - Fowler, R.H.; Nordheim, L. Electron emission in intense electric fields. Proc. R. Soc. Lond. Ser. A Contain. Pap. A Math. Phys. Character
**1928**, 119, 173–181. [Google Scholar] [CrossRef] - Wilson, R.G. Vacuum Thermionic Work Functions of Polycrystalline Be, Ti, Cr, Fe, Ni, Cu, Pt, and Type 304 Stainless Steel. J. Appl. Phys.
**1966**, 37, 2261–2267. [Google Scholar] [CrossRef] - Bhushan, B. Encyclopedia of Nanotechnology; Springer: Dordrecht, The Netherlands, 2012; pp. 824–837. [Google Scholar]
- Ummarino, G.A.; Piatti, E.; Daghero, D.; Gonnelli, R.S.; Sklyadneva, I.Y.; Chulkov, E.V.; Heid, R. Proximity Eliashberg theory of electrostatic field-effect doping in superconducting films. Phys. Rev. B
**2017**, 96, 064509. [Google Scholar] [CrossRef] [Green Version] - Zmuidzinas, J.; Richards, P. Superconducting detectors and mixers for millimeter and submillimeter astrophysics. Proc. IEEE
**2004**, 92, 1597–1616. [Google Scholar] [CrossRef] [Green Version] - Gousev, Y.P.; Gol’tsman, G.N.; Semenov, A.D.; Gershenzon, E.M.; Nebosis, R.S.; Heusinger, M.A.; Renk, K.F. Broadband ultrafast superconducting NbN detector for electromagnetic radiation. J. Appl. Phys.
**1994**, 75, 3695–3697. [Google Scholar] [CrossRef] - Lösch, S.; Alfonsov, A.; Dobrovolskiy, O.V.; Keil, R.; Engemaier, V.; Baunack, S.; Li, G.; Schmidt, O.G.; Bürger, D. Microwave Radiation Detection with an Ultrathin Free-Standing Superconducting Niobium Nanohelix. ACS Nano
**2019**, 13, 2948–2955. [Google Scholar] [CrossRef] [PubMed] - Mukhanov, O.A. Energy-Efficient Single Flux Quantum Technology. IEEE Trans. Appl. Supercond.
**2011**, 21, 760–769. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Pseudo-color scanning electron micrograph (SEM) of a typical niobium gated transistor with the bias scheme. The weak-link and the wire are in false-colored orange, and the gate is in blue. (

**b**) I vs. V curves for select gate voltages ${V}_{G}$ at a bath temperature of 20 mK. The curves are horizontally offset for clarity. Bipolar suppression of the ${I}_{S}$ is visible as $\left|{V}_{G}\right|$ increases. (

**c**) ${I}_{S}$ vs. ${V}_{G}$ for several bath temperatures T ranging between 20 mK and 3 K. ${I}_{S}$ values were collected by measuring 50 repetitions of the I vs. V characteristics.

**Figure 2.**(

**a**) Operation scheme of the niobium-based half-wave rectifier. The current bias is represented by the horizontal red dashed line in the ${I}_{S}\left({V}_{G}\right)$ graph. The time-dependent gate voltage (green to blues curve) is composed of an AC component ${V}_{AC}$ added to a DC bias ${V}_{DC}$. The effect of the gating provides a time-dependent switching current ${I}_{S}\left(t\right)$ (purple to yellow line) able to rectify the gate voltage signal. (

**b**) V vs. ${V}_{G}$ characteristic of the Josephson junction (JJ) measured with a four-probe technique with a lock-in amplifier. The reference signal of the lock-in is ${V}_{AC}$, and the bias current ${I}_{B}$ was set to $2.5\phantom{\rule{4pt}{0ex}}$μA. V. The signal is almost zero until ${I}_{S}\left({V}_{G}\right)<{I}_{B}$; then, a peak arises due to rectification of the ${V}_{G}$ signal.

**Figure 3.**(

**a**) Pseudo-color SEM of a representative vanadium-gated device. The weak-link and the wire are colored in orange, and the gate is in blue. (

**b**) ${I}_{S}$ vs. ${V}_{G}$ curves for different bath temperatures ranging from $2.0$ to $3.3$ K. The data were computed by averaging 25 acquisitions of ${I}_{S}$.

**Figure 4.**(

**a**) Bias scheme for AC measurements. The gate voltage is generated by adding DC ${V}_{DC}$ and AC ${V}_{AC}$ arbitrary waveform voltages. The ADC/DAC board that provides the AC signal performs real-time measurements of V. (

**b**) Voltage V vs. current I characteristics for different values of ${V}_{G}$ (yellow and purple curves). The dot couples show the operation points of the system for two different bias currents ${I}_{B}=18$, $71\phantom{\rule{4pt}{0ex}}$μA. ${V}_{G}$ vs. time t is the excitation signal (blue curve) that was realized by adding a DC voltage ${V}_{DC}=10$ V and an AC square-wave voltage with amplitude ${V}_{AC}=5$ V. Time-dependent V for different current biases are drawn in correspondence with the operation points. The measurements were performed at $T=3$ K.

**Figure 5.**(

**a**) Color-plot of V vs. ${V}_{G}$ (x-axis) and I (y-axis). From left to right, the three round symbols show the zero-resistance gate voltage value (light green), the super-to-normal transition (red), and the maxima of both ${V}_{G}$ and V (dark green). The dashed red curve represents the ${I}_{S}$ vs. ${V}_{G}$ characteristic. (

**b**) Time-dependent ${V}_{G}\left(t\right)$ obtained by adding a DC voltage ${V}_{DC}=11$ V and an AC sine wave voltage ${V}_{AC}$. (

**c**,

**d**) Time-dependent $V\left(t\right)$ for ${V}_{AC}=\phantom{\rule{4pt}{0ex}}3.5$ V (

**c**) and ${V}_{AC}=1.0$ V (

**d**). The color-map is the same as in panel (

**a**). All these measurements were performed at $T=3$ K.

**Figure 6.**(

**a**) Pseudo-color (SEM) and bias scheme of a representative Ti gate-controlled transistor. The superconducting wire and the Dayem bridge constriction are colored in orange, and the gate electrode is in blue. (

**b**) ${I}_{S}$ vs. ${V}_{G}$ characteristics at select bath temperatures ranging from 20 to 300 mK. Data are the result of the average of 50 acquisitions of ${I}_{S}$.

**Figure 7.**(

**a**) Switching current probability distributions (SCPDs) vs. I acquired at select bath temperatures from 20 to 90 mK in the Quantum Phase Slip (QPS) regime. The best fit curves are represented with dotted line. The inset shows $\sigma $ vs. T of the regime. (

**b**) SCPDs vs. I obtained at different temperatures from 120 to 150 mK in the Thermal Activated Phase Slip (TAPS) regime. The best fit curves are represented with a dotted line. The inset shows $\sigma $ vs. T of the regime. (

**c**) SCPDs vs. I obtained at different temperatures from 160 to 300 mK in the Multiple Phase Slip (MPS) regime. The inset shows $\sigma $ vs. T of the regime. For each SCPD, the total sampling number of ${I}_{S}$ is ${10}^{5}$. The crossover temperatures ${T}_{Q}\simeq 110$ mK and ${T}_{M}\simeq 160$ mK separate the QPS/TAPS and TAPS/MPS regimes, respectively. In all the panels, the temperature increases from right to left.

**Figure 8.**(

**a**) SCPDs vs. I at select gate voltages from 0 V to 21 V in the Electric Activated Phase Slip (EAPS) regime. The inset shows standard deviation $\sigma $ of SCPDs vs. gate voltage ${V}_{G}$ in the EAPS regime. (

**b**) SCPD vs. I at different gate voltage values from 24 V to 30 V in the MPS regime. The inset shows standard deviation $\sigma $ of SCPDs vs. gate voltage ${V}_{G}$ in the MPS regime. For each distribution, the total number of ${I}_{S}$ acquisitions is ${10}^{5}$. The curves are vertically offset for clarity. The crossover voltages are ${V}_{Q}\simeq 8$ V and ${V}_{E}\simeq 14$ V.

**Figure 9.**(

**a**) ${I}_{S}$-matched distributions. Red and orange distributions were acquired for a negligible electric field at ${V}_{G}=0$ V at select bath temperatures, whereas blue and green distributions were measured at $T=20$ mK for different gate voltage values. The values of ${I}_{S}$ are, respectively, from left to right $2.2,\phantom{\rule{4pt}{0ex}}2.8,$ and $4.0\phantom{\rule{4pt}{0ex}}$μA. (

**b**) Comparison between the $\sigma $ vs. ${I}_{S}$ characteristic obtained for thermal- and electric-driven distributions at ${V}_{G}=0$ V (lower curve) and $T=20$ mK (upper curve) respectively.

**Figure 10.**(

**a**,

**b**) SEMs of the suspended titanium transistor (original picture and pseudo-color). (

**c**) Back and forth current I vs. V characteristics for select values of ${V}_{G}$ measured at a bath temperature of $T=20$ mK. The characteristics are horizontally shifted for clarity. Grey colored regions highlight the gate-induced evolution of ${I}_{{S}_{1}}$, ${I}_{{S}_{2}}$, and ${I}_{{S}_{3}}$. (

**d**–

**f**) The ${V}_{G}$ dependence of the switching currents of ${I}_{{S}_{1}}$, ${I}_{{S}_{2}}$, and ${I}_{{S}_{3}}$, respectively.

**Figure 11.**$\left|\mathit{E}(x,y,z)\right|$ and streamlines on the XY (

**a**) and YZ (

**b**) planes. The simulations were performed with a gate voltage value of ${V}_{G}=-15$ V. The distribution of the electrostatic field shows that the field effect is confined upon constriction.

**Figure 12.**Current density module $\left|{\mathit{J}}_{FE}(x,y,z)\right|$ evaluated on XY (

**a**) and YZ (

**b**) planes. Data were obtained by analyzing the ballistic transport of the electrons through the vacuum from the gate electrode surfaces toward the titanium constriction (and vice versa for opposite values of gate voltage). Here, we set the gate voltage to ${V}_{G}=-15$ V and the work function equal to the literature value for titanium ${\varphi}_{0}=4.3$ eV. The spatial distribution of the electronic current highlights that the field emitted electrons influence a 500 nm section of the constriction.

**Figure 13.**Natural logarithm of ${I}_{L}$ between the gate electrodes and the constriction at a bath temperature of $T=20$ mK vs. the gate voltage ${V}_{G}$ measured on a titanium suspended device (Orange dots). Natural logarithm of ${I}_{FE}$ between the gate electrodes and the constriction vs. the gate voltage ${V}_{G}$ computed by integrating the Fowler–Nordheim (FN) current density (${\mathit{J}}_{FE}$) with ${\varphi}_{0}=4.3$ eV (blue dots).

**Figure 14.**Electronic temperature ${T}_{e}$ vs. time t of a mesoscopic superconducting weak-link that periodically absorbs electrons with an energy of the order of 10 eV. The red horizontal line represents the critical temperature of the superconductor. Each electron starkly increases the electronic temperature of the system, driving it in the normal state. $\tau $ is the measurement time.

**Figure 15.**(

**a**,

**b**) Combined effect of two electric fields on titanium Dayem bridges. Color plot of the normalized switching current as a function of ${V}_{{G}_{1}}$ (x-axis) and ${V}_{{G}_{2}}$ (y-axis) for two different devices (A and B).

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**MDPI and ACS Style**

Puglia, C.; De Simoni, G.; Giazotto, F.
Gate Control of Superconductivity in Mesoscopic All-Metallic Devices. *Materials* **2021**, *14*, 1243.
https://doi.org/10.3390/ma14051243

**AMA Style**

Puglia C, De Simoni G, Giazotto F.
Gate Control of Superconductivity in Mesoscopic All-Metallic Devices. *Materials*. 2021; 14(5):1243.
https://doi.org/10.3390/ma14051243

**Chicago/Turabian Style**

Puglia, Claudio, Giorgio De Simoni, and Francesco Giazotto.
2021. "Gate Control of Superconductivity in Mesoscopic All-Metallic Devices" *Materials* 14, no. 5: 1243.
https://doi.org/10.3390/ma14051243