Dynamical vortex phase diagram of two-dimensional superconductivity in gated $\mathrm{Mo}{\mathrm{S}}_2$

Recent discoveries of two-dimensional (2D) superconductors have uncovered various aspects of physical properties including vortex matter. In this paper, we report transport properties and a dynamical phase diagram at zero magnetic field in ion-gated $\mathrm{Mo}{\mathrm{S}}_2$. In addition to the universal jump in the current-voltage characteristic showing unambiguous evidence of the Berezinskii-Kosterlitz-Thouless transition, we observed multiple peaks in the temperature and current derivative of the electrical resistance, based on which a dynamical phase diagram in the current-temperature plane was constructed. We found current-induced dynamical states of vortex-antivortex pairs, containing that with the phase slip line. Also, we present a global phase diagram of vortices in gated $\mathrm{Mo}{\mathrm{S}}_2$ which captures the nature of vortex matter of clean 2D superconductors.

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Figure 1

Berezinskii-Kosterlitz-Thouless (BKT) transition in ion-gated $\mathrm{Mo}{\mathrm{S}}_2$. (a) A schematic image of electric-double-layer transistor structure. (b) Superconducting transition in ion-gated $\mathrm{Mo}{\mathrm{S}}_2$. The effective thickness of gate-induced superconductivity is around monolayer. Inset: A magnified view near the superconducting fluctuation regime. Black dashed curve shows Aslamazov-Larkin (AL) and Maki-Thompson (MT) fit. The superconducting transition temperature ($T_{\mathrm{c}0}$) defined by AL and MT fit is 7.6 K. (c) Current-voltage curves on a logarithmic scale at various temperatures varying in 1-K steps from 2 to 4 K, in 0.2 K-K steps from 5 to 5.8 K; in 0.1 K steps from 5.9 to 7.2 K, and 4.2, 4.5, 5.7, 7.5, 8 and 10 K. The short black dashed lines are the log-log fit to the data in the low-current limit (below critical region) at each temperature. Red dashed line shows the same fit at the BKT transition temperature ($T_{\mathrm{BKT}}$) (d) Temperature dependence of the power-law exponent α, as deduced from the fits shown in (c). From this plot, we defined the $T_{\mathrm{BKT}}$ is 6.0 K, where the α  discretely becomes 3. (e) Plots of $R_{\mathrm{sheet}}$ as a function of ${\left[\left(T_{\mathrm{c}0}-T\right)/\left(T-T_{\mathrm{BKT}}\right)\right]}^{1/2}$ at 5 μA. The black dashed line indicates the fitting of the Halperin-Nelson equation, $R=R_{0}exp\left\{-2b{\left(\frac{T_{\mathrm{c}0}-T}{T-T_{\mathrm{BKT}}}\right)}^{1/2}\right\}$. Here, $R_0$ and $b$, which are materials-dependent parameters, are 508 Ω and 1.06, respectively.

Figure 2

Longitudinal nonlinear resistance $R_{\mathrm{NL}}$ and its derivative $\mathrm{d}{R}_{\mathrm{NL}}/\mathrm{d}T$ versus temperature at various currents. (a) $R_{\mathrm{NL}}$ at various bias currents varying in 5-μA steps from 5 and 90 μA; in 20-μA steps from 110 and 170 μA; and 100, 180, 200 μA. The black and white circles show $T_1$ and $T_2$, respectively, which are defined by white and black triangles in (b). Also, $T_3$ is defined by the temperatures where the resistance becomes 95% of the normal state resistance, which is indicated by white diamonds. (b) Temperature derivative of $R_{\mathrm{NL}}$ versus temperatures at various currents. Black and white triangles show the double (35–65 μA) and single (5–30 and 70–200 μA) peak positions of $\mathrm{d}{R}_{\mathrm{NL}}/\mathrm{d}T$ versus $T$ curve, respectively.

Figure 3

Differential resistance $\mathrm{d}lnV/\mathrm{d}lnI$ at various temperatures. (a) $\mathrm{d}lnV/\mathrm{d}lnI$ as a function of $I$ at various temperatures from 2 to 10 K. Here, we provided data at representative temperatures. We choose data at 2, 3.2, 3.7, 4.2, 4.7, 5.2, 5.6, and 6.2–10 K. Black triangles show the strongest peak positions below 60 μA. We defined $I_{\mathrm{c}1}$ of each $\mathrm{d}lnV/\mathrm{d}lnI$ curve as the current where the $\mathrm{d}lnV/\mathrm{d}lnI$ curves show these strong peaks indicated by black triangles. (b) A magnification of (a) around 50–76 μA. Here, we provide additional data at 2.5 and 3.0 K. White diamonds show the second strong peaks around 50–60 μA. We defined $I_{\mathrm{c}2}$ of each $\mathrm{d}lnV/\mathrm{d}lnI$ curve as the current where the $\mathrm{d}lnV/\mathrm{d}lnI$ curves show these peaks indicated by white diamonds. (c) A magnification of (a) at the high-current regime. White triangles show the broad peak positions at the high-current regime above 50 μA. We defined $I_{\mathrm{c}3}$ of each $\mathrm{d}lnV/\mathrm{d}lnI$ curve as the current where the $\mathrm{d}lnV/\mathrm{d}lnI$ curves show these broad peaks indicated by white triangles.

Figure 4

Current-temperature phase diagram of ion-gated $\mathrm{Mo}{\mathrm{S}}_2$ at zero magnetic field. Schematic images in the regime of the BKT state and vortex flow state show binding vortex-antivortex pairs (region A), and free vortices and antivortices in the current (region C). Red and white triangles show peaks of $\mathrm{d}R/\mathrm{d}T-T$ curves at lower ($T_1$) and higher ($T_2$) temperatures in Fig. 2, respectively. Green triangles, which are defined by the temperatures where the resistance becomes 95% of the normal state resistance, show a boundary between fluctuation regime and normal state. Light blue, red, and white circles show $I_{\mathrm{c}1}, I_{\mathrm{c}2}$, and $I_{\mathrm{c}3}$, respectively. The vertical error bars for $I_{\mathrm{c}1}, I_{\mathrm{c}2}$, and $I_{\mathrm{c}3}$ and the horizontal error bars for $T_1, T_2$, and $T_3$ represent uncertainties in determining each data point. The color map of $\mathrm{d}{R}_{\mathrm{NL}}/\mathrm{d}T$ values as a function of $I$ and $T$ is displayed in the measurement regime of $T>2$ K and $T>5$ μA. $T_{\mathrm{c}0}$ and $T_{\mathrm{BKT}}$ are determined from AL and MT fit and I-V characteristic in Fig. 1, respectively. Region A is the BKT state with zero resistance where the correlated vortices and antivortices are binding. Region B ($I<I_{\mathrm{c}1},T_{\mathrm{BKT}}<T<T_2$) shows the BKT critical state with the finite V-AV correlation length. Region C ($I_{\mathrm{c}2}<I<I_{\mathrm{c}3},T_1<T<T_2$) shows the flux-flow region of kinematic vortices, which results in the intermediate state in nonequilibrium. Region D ($I_{\mathrm{c}1}<I<I_{\mathrm{c}2},T<T_1$) shows another intermediate state possibly because of the fast flow of Abrikosov vortices called as the vortex street. Region E ($T_2<T<T_3,I>I_{\mathrm{c}3}$) shows the fluctuation state.

Figure 5

The velocity of a single vortex versus current. (a) Current-voltage curves on a logarithmic scale at 4.5, 4.7, 5.0, 5.2, 5.4, 5.6, 5.8, and 10 K. Dashed lines show regions following $V\propto I^{\alpha }$ below and above $I_{\mathrm{c}1}$. (b) Velocity of the single vortex as a function of the current calculated by $v_f=\frac{V/L}{C\left(T\right)I^{\alpha \left(T\right)-1}{\varphi }_0}$, where $L$ is the length between the four terminal contacts, ${\varphi }_0$ is a magnetic flux quantum. $C$($T$) is a determined the fit of the $\mathit{IV}$ curve at the lowest-current limit using the relation $V=2\pi {\xi }^2\left(T\right)C\left(T\right)R_N{I}^{\alpha \left(T\right)}$, where ξ is the GL coherence length; $R_{\mathrm{N}}$ is the normal state resistance of 220 Ω. Black triangles show the position of $I_{\mathrm{c}1}$.

Figure 6

Dissipated power. Temperature dependence of the dissipated power at $I_{\mathrm{c}1}$ and $I_{\mathrm{c}3}$ calculated by $P=IV$, where $I$ is $I_{\mathrm{c}1}$ (red) or $I_{\mathrm{c}3}$ (blue), and $V$ is the value at $I_{\mathrm{c}1}$ or $I_{\mathrm{c}3}$ at each temperature. The vertical error bars of $P$ at $I_{\mathrm{c}1}$ represent uncertainties in determining the voltage at $I_{\mathrm{c}1}$ from the measured I-V characteristic.

Figure 7

Comprehensive B-I-T phase diagram in single-crystal-based clean 2D superconductors with weak pinning. The whole phase diagram is based on the results in Fig. 4 and previous works in Refs. [16, 21]. In the B-T plane, white, blue, yellow, green, and orange regions show the zero resistance (ZR), the quantum metal, the thermally activated vortex creep, the vortex free flow, and the quantum Griffiths states, respectively. Here, $T_{\mathrm{c}0}$ is the transition temperature determined by the thermal fluctuation theories (Aslamazov-Larkin and Maki-Thompson models), and $T_{\mathrm{BKT}}$ is the BKT transition temperature. $T_{\mathrm{cross}}$ is the crossover temperature between the thermal activated vortex creep regime and the quantum creep regime, which are determined by the activation plot. $B_{\mathrm{SM}}$ is the hypothetical transition magnetic field from the ZR state to the quantum metal (vortex liquid). $B_{\mathrm{c}2}^{\mathrm{MF}}$ is the mean-field upper critical field. With increasing magnetic field, the system goes to the quantum Griffiths state up to the characteristic critical magnetic field $B_{\mathrm{c}}^{*}$ through the free flow state (unpinned vortex). In the B-I plane, $B_1$ and $B_2$ are low- and high-threshold magnetic fields for nonreciprocal response [21], and $B_{\mathrm{c}}$ is the crossover magnetic field between free flow states and the fluctuation regime. In the I-T plane, white, yellow, green, and red regions correspond to A, B, C, and D states in Fig. 4, respectively. A is the BKT state, and B, C, and D are attributed to the thermally activated V-AV, the kinematic vortex states, and the vortex street (plastic flow), respectively. Here, $I_{\mathrm{c}1}, I_{\mathrm{c}2}$, and $I_{\mathrm{c}3}$ are determined in Fig. 3. The outermost region of these states is governed by the amplitude fluctuation.