Abstract
In a hybrid system of topological insulator (TI)/superconductor (SC), the proximityinduced topological superconductivity is expected to appear at the interface. Here we propose and demonstrate that a TI/SC hybrid Bi_{2}Te_{3}/PdTe_{2} heterostructure serves as a platform for exploring topological superconductivity with various features: all made of tellurium compounds, epitaxial growth, and a small charge transfer interface. In the Bi_{2}Te_{3}/PdTe_{2} heterostructure films, we observe large nonreciprocal charge transport near the superconducting transition temperature under a transverse inplane magnetic field. The observation indicates the interplay between the topological surface state and superconductivity, suggesting that the Bi_{2}Te_{3}/PdTe_{2} heterostructure is a candidate for a topological superconductor. Also observed is an unexpected sign reversal of the nonreciprocal coefficient when the inplane magnetic field is slightly tilted toward the outofplane direction. The analysis reveals that the sign reversal occurs with the change of dominant vortex type, that is, the change from spontaneous vortices to externalfield induced ones.
Similar content being viewed by others
Introduction
Realization and utilization of topological superconductors hosting Majorana fermions have recently been attracting enormous interest^{1}. One promising way to realize topological superconductivity is to fabricate a hybrid system of topological insulator (TI)/swave superconductor (SC), where topological superconductivity of the spinless chiral pwave can be induced by the proximity effect at the spinmomentum locked surface of the TI^{2}. So far, the TI/SC hybrid systems have been explored in a wide range of materials, mostly classified into two groups according to their structures. One is a system where a thin film of TI is grown on a single crystal of an SC^{3,4,5}. Spectroscopic measurements such as angleresolved photoemission spectroscopy^{5} or scanning tunneling microscopy/spectroscopy^{3,4} have been carried out to characterize the system while fabrication of elaborate devices utilizing the Majorana fermions would be difficult in this type of system. The other is a system where elemental SC, such as Nb or Al, is deposited on a thin film of TI^{6,7,8,9}. Selective area growth or etching technique enables the fabrication of devices such as Josephson junctions. Using these devices, nontrivial transport properties of proximityinduced topological superconductivity have been demonstrated. To realize innovative topological SC devices with chiral Majorana fermion modes, it is necessary to tune the chemical potential into the bulk bandgap of TI. However, tuning is difficult in these systems due to a large amount of charge transfer between elemental SC and TI. Thus, it is highly desired to establish a materials platform of TI/SC hybrid compatible with the device fabrication and the chemicalpotential tuning^{10}.
Nonreciprocal charge transport, or socalled diode effect, has been investigated as a sensitive probe to the inversion symmetry breaking. The nonreciprocal charge transport originally studied in noncentrosymmetric crystals or interfaces under magnetic fields^{11} has been recently extended to the twodimensional (2D) superconductors without inversion symmetry^{12,13,14}. In the TI/SC hybrid systems, the interplay between the inversionsymmetrybroken surface state of the TI and the superconductor is expected to give rise to the nonreciprocal charge transport^{14,15}. A recent experimental study on a TI/SC hybrid system Bi_{2}Te_{3}/NbSe_{2} has revealed the effect of the external inplane magnetic field on the normal state Fermi surface by scanning tunneling microscopy/spectroscopy^{16}.
In this paper, we propose and demonstrate that Bi_{2}Te_{3}/PdTe_{2} heterostructure serves as a platform for exploring topological superconductivity. We chose Bi_{2}Te_{3} and PdTe_{2} as TI and SC, respectively, and fabricated a Bi_{2}Te_{3}/PdTe_{2} thinfilm heterostructure. The Bi_{2}Te_{3}/PdTe_{2} heterostructure is a TI/SC hybrid system all made of epitaxially grown tellurium compounds. The firstprinciples calculation of the electronic band structure shows a minimal charge transfer through the interface owing to the close electronegativity in the two tellurium compounds. Large nonreciprocal resistance is observed around the superconducting transition temperature under an inplane magnetic field perpendicular to the current direction, indicating inversion symmetry breaking of the electronic state at the interface. We further observe the sign reversal of the nonreciprocal coefficient when the magnetic field is tilted to the outofplane direction. The sign reversal is associated with the crossover in the type of the dominant vortex from the spontaneous to the fieldinduced one.
Results
Electronic structure in Bi_{2}Te_{3}/PdTe_{2} heterostructure
A superconductor 1TPdTe_{2} with a superconducting transition temperature T_{c} = 1.7 K is a member of transitionmetal dichalcogenides^{17}. PdTe_{2} shares many common features in the crystal structure with Bi_{2}Te_{3}; both have layered van der Waals structures consisting of Teterminated triangular lattices with comparable lattice constants (Fig. 1a). The chemical and structural similarities between Bi_{2}Te_{3} and PdTe_{2} suggest that they can be an ideal combination to form a TI/SC heterostructure^{18,19}. We fabricated a thin film of Bi_{2}Te_{3}/PdTe_{2} on an InP(111)A substrate by molecular beam epitaxy (see the “Methods” section). The epitaxial growth of thin films and heterostructures of Bi_{2}Te_{3} and PdTe_{2} are confirmed by the Xray diffraction (see Supplementary Fig. 1 in Supplementary Information), the crosssection transmission electron microscopy (TEM), and energydispersive Xray (EDX) spectroscopy (Supplementary Fig. 2). The heterostructure has a sharp interface and the elemental intermixing at the interface is negligibly small. In a relatively thick film (e.g. 45 nm) of PdTe_{2} on InP(111)A substrate, a sharp superconducting transition is observed at 1.7 K, which is consistent with bulk crystals^{17} and previously reported thin films^{20}. As the thickness is reduced, e.g. down to 6 nm, T_{c} becomes low, and the superconducting transition becomes broad (see Supplementary Fig. 3 for the details).
To investigate the electronic state in Bi_{2}Te_{3}/PdTe_{2} heterostructures, we have performed the firstprinciples calculations of the electronic band structure (see the “Methods” section for details). Figure 1b shows the band structure for PdTe_{2} (1 layer)/Bi_{2}Te_{3} (6 layers)/PdTe_{2} (1 layer). Here, the thickness of the TI Bi_{2}Te_{3} layer, 6 layers, is chosen to be thick enough to avoid the formation of a hybridization gap between the top and bottom surface states. Around the Γ point, the topological surface states originating from Bi_{2}Te_{3} are formed. Small carrier pockets from bulk bands indicate the small charge transfer between Bi_{2}Te_{3} and PdTe_{2}. The common use of tellurium with close electronegativity results in reduced charge transfer, in contrast to the case of, e.g. Nb/Bi_{2}Te_{3} with large charge transfer, making the Bi_{2}Te_{3}/PdTe_{2} heterostructure an ideal platform of TI/SC hybrid. Most importantly, the Fermi level of the heterostructure appears to cross the interface Dirac dispersion, strongly suggesting the formation of topological superconductivity.
Twodimensional superconductivity in Bi_{2}Te_{3}/PdTe_{2}
We first investigate the superconducting transport properties in the heterostructure film of Bi_{2}Te_{3} (10 nm)/PdTe_{2} (6 nm) (Fig. 1c) in a Hallbarshaped sample (see the “Methods” section). Figure 1e shows the temperature T dependence of the first harmonic (Ohmic) resistance R^{ω} around the superconducting transition. Due to the thin PdTe_{2} layer (6 nm), it is expected to realize a twodimensional superconducting state. In fact, the R^{ω}T curve can be described with the Aslamazov–Larkin formula on a hightemperature side^{21} and with the Halperin–Nelson formula on a lowtemperature side^{22}. Red curve in Fig. 1e is the fit by the Aslamazov–Larkin formula of the paraconductivity^{21}, \({{\Delta }}G = \frac{{e^2}}{{16\hbar }}\frac{{T_{{{{\mathrm{c}}}}0}}}{{T  T_{{{{\mathrm{c}}}}0}}}\), where, \({{\Delta }}G = \left( {R^\omega } \right)^{  1}  R_{{{\mathrm{N}}}}^{  1}\) is the excess conductance, R_{N} is the normal state resistance, and T_{c0} is the meanfield critical temperature. The fitting parameters are R_{N} = 153 Ω and T_{c0} = 1.174 K. Note that R_{N} is dominated by the 6nmthick PdTe_{2} layer and that the resistance of the 10nmthick Bi_{2}Te_{3} layer is approximately 3 times larger than that (see Supplementary Fig. 4). Green curve in Fig. 1e is the fit by the Halperin–Nelson formula of twodimensional superconducting transition^{21}, \(R^\omega = R_{{{\mathrm{c}}}}\exp \left( {  2\sqrt {\frac{{b(T_{{{{\mathrm{c}}}}0}  T)}}{{T  T_{{{{\mathrm{BKT}}}}}}}} } \right)\), where R_{c} = 457 Ω and b = 3.50 are materials parameters and T_{BKT} = 1.060 K is the Beresinskii–Kosterlitz–Thouless (BKT) transition temperature estimated from the current–voltage characteristics (see Supplementary Fig. 5). The temperature dependence of resistivity and the current–voltage characteristics verify the twodimensionality of the superconductivity with the BKT transition^{23,24}. Because of the nature of 2D superconductivity, free vortices and antivortices are spontaneously formed below T_{c0}. In the temperature regime T_{BKT} < T < T_{c0} represented by a green area in Fig. 1e, the resistance is generated by the currentinduced motion of the vortices. Formation of vortex–antivortex pairs results in the zeroresistance state below T_{BKT}. In Fig. 1d, the temperature dependence of the inplane and outofplane upper critical fields, \(B_{{{{\mathrm{c}}}}2}^\parallel\) and \(B_{{{{\mathrm{c}}}}2}^ \bot\) are shown, which are defined here by the midpoints of the normal state resistance. \(B_{{{{\mathrm{c}}}}2}^\parallel\) is larger than \(B_{{{{\mathrm{c}}}}2}^ \bot\), which is typical of 2D superconductors^{23,24}.
Nonreciprocal transport under inplane magnetic field
Next, we investigated the nonreciprocal charge transport near the superconducting transition. Figure 2a schematically illustrates the experimental configuration for the nonreciprocal transport measurement. We applied current I along the xaxis and magnetic field B along the yaxis (inplane). In Bi_{2}Te_{3}/PdTe_{2}, the mirror symmetry with respect to the xyplane is broken at the interface (polar axis P  z, where z is a unit vector normal to the film plane). The resistance R in a polar system under a magnetic field B is phenomenologically described^{11} as \(R = R_0[1 + \gamma \left( {{{{\mathbf{B}}}} \times {{{\mathbf{z}}}}} \right) \cdot {{{\mathbf{I}}}}]\), where R_{0} is the resistance at zero magnetic fields and γ is the nonreciprocal coefficient. Therefore, the nonreciprocal transport (the second term) is expected to emerge in the configuration shown in Fig. 2a. The voltage–current characteristic is represented by \(V = R_0I + \gamma R_0BI^2\) including the Ilinear (Ohmic) and Isquare (nonreciprocal) terms. Under the application of an AC current \(I(t) = \sqrt 2 I_0{{{\mathrm{sin}}}}(\omega t)\), in addition to the firstharmonic voltage \(V^\omega = \sqrt 2 R_0I_0{{{\mathrm{sin}}}}(\omega t)\), the secondharmonic voltage \(V^{2\omega } =  \gamma R_0BI_0^2{{{\mathrm{cos}}}}(2\omega t)\) appears reflecting the Isquare term. From the measured V^{ω} and V^{2ω}, we obtained the first harmonic (Ohmic) resistance R^{ω}(B) as well as the nonreciprocal resistance \(R^{2\omega }(B) = \frac{1}{{\sqrt 2 }}\gamma R_0BI_0\). Considering the symmetry of the system, we symmetrized (antisymmetrized) R^{ω} (R^{2ω}) with respect to B.
Figures 2b, c respectively, show the magnetic field dependence of R^{ω} and R^{2ω} at various temperatures measured in the configuration in Fig. 2a. As the temperature decreases below 1.22 K, the nonreciprocal resistance R^{2ω} appears under the inplane magnetic fields. The peak position of R^{2ω} shifts to the large magnetic fields as the temperature is lowered. The emergence of nonzero R^{2ω} in Fig. 2c indicates the realization of fluctuating superconductivity without inversion symmetry. We confirmed that R^{2ω} almost vanishes for B   I as expected from the symmetry (see Supplementary Fig. 6). R^{2ω} in Fig. 2c is enhanced in the magnetic field range where R^{ω} transits between 0 and the normal resistance. This trend can be clearly seen in Fig. 2d, the color plot of nonreciprocal coefficient \(\gamma = \frac{{\sqrt 2 R^{2\omega }}}{{R_0BI_0}}\) in the magnetic field–temperature plane. In the region between the two gray curves of R^{ω} = 120 and 1 Ω, which respectively, correspond to the normal states and the zeroresistance states, γ is remarkably enhanced. Above the upper gray curve, where superconductivity is broken, γ is significantly suppressed. Note that γ is difficult to evaluate for R^{ω} < 1 Ω due to the smallness of both R^{ω} and R^{2ω}. The largest value of γ in this plane is 6.8 × 10^{4} T^{−1} A^{−1} at B = 0.02 T and T = 1.10 K. Table 1 compares the observed nonreciprocal coefficient in the present study with those reported in the previous studies on noncentrosymmetric 2D superconductors^{12,14,25}. In the table, γ’ is the normalized value of the nonreciprocal coefficient γ by the Hall bar width w (γ’ = γw, w = 100 μm in the present study). Large values of γ’ tend to appear in materials with small T_{c}. In the case of metals/semiconductors, the nonreciprocal coefficient is known to be inversely proportional to the Fermi energy^{11}. The observed trend can be understood by substituting the Fermi energy with the superconducting gap.
As a check, we have also explored the nonreciprocal transport in ZnTe (10 nm)/PdTe_{2} (6 nm), where a trivial insulator ZnTe is used instead of a topological insulator Bi_{2}Te_{3}. The observed nonreciprocal signal in ZnTe/PdTe_{2} is much smaller than Bi_{2}Te_{3}/PdTe_{2} (see Supplementary Fig. 6), indicating that the proximity effect of SC to the surface state of TI is necessary for the emergence of the large nonreciprocal response.
Figure 2e shows the temperature dependence of γ (black curve) at B = 0.02 T obtained from the temperature dependence of R^{ω} and R^{2ω}. γ increases steeply with decreasing temperature below T_{c0}. In T_{BKT} < T < T_{c0}, the resistance R is generated by the motion of spontaneously formed free vortices^{22}, and it can be written as \(R = \frac{{\phi _0^2}}{\eta }n_{{{{\mathrm{free}}}}}\). Here, \(\phi _0 = h/2e\) is the magnetic flux quantum, η is the friction coefficient of vortex motion, and n_{free} is the density of free vortices. The temperature dependence of free vortex density, \(n_{{{{\mathrm{free}}}}} \propto \exp \left( {  2\sqrt {\frac{{b\left( {T_{{{{\mathrm{c}}}}0}  T} \right)}}{{T  T_{{{{\mathrm{BKT}}}}}}}} } \right)\) (valid for \(\frac{{T  T_{{{{\mathrm{BKT}}}}}}}{{T_{{{{\mathrm{c}}}}0}  T_{{{{\mathrm{BKT}}}}}}} \ll 1\)), is the origin of the characteristic exponential temperature dependence of R in the BKT transition^{22}. A recent theory considering a topological surface state with a superconducting gap shows that external current renormalizes the superfluid density^{15}. As a result, T_{BKT} becomes current dependent, \(T_{{{{\mathrm{BKT}}}}} = T_{{{{\mathrm{BKT}}}}}^0(1 + {{\Lambda }}B_yI)\), with \({{\Lambda }}\) a material parameter. Accordingly, this renormalizes n_{free} as \(n_{{{{\mathrm{free}}}}}\left( I \right) = n_{{{{\mathrm{free}}}}}\left( 0 \right)\left[ {1  \frac{{T_{{{{\mathrm{BKT}}}}}^0\sqrt {b\left( {T_{{{{\mathrm{c}}}}0}  T_{{{{\mathrm{BKT}}}}}^0} \right)} }}{{\left( {T  T_{{{{\mathrm{BKT}}}}}^0} \right)^{\frac{3}{2}}}}{{\Lambda }}B_yI} \right]\) to the first order of \({{\Lambda }}B_yI\). Because the resistance R is proportional to n_{free}, the second term gives rise to the nonreciprocal resistance. Hence, γ is proportional to \(\left( {T  T_{{{{\mathrm{BKT}}}}}^0} \right)^{  \frac{3}{2}}\) at temperatures near \(T_{{{{\mathrm{BKT}}}}}^0\) and thus expected to diverge at \(T_{{{{\mathrm{BKT}}}}}^0\). The observed temperature dependence of γ is compared with the formula \(\gamma = \gamma _0\left( {T  T_{{{{\mathrm{BKT}}}}}^0} \right)^{  \frac{3}{2}}\) with γ_{0} = 380 T^{−1} A^{−1} K^{1.5} as shown in Fig. 2e. The increasing trend in γ with decreasing T roughly agrees with the theoretical curve. The consistency supports that the observed nonreciprocal transport can be understood in terms of the currentinduced modulation of the superfluidity density in the superconducting proximitycoupled TI surface state.
Nonreciprocal transport under tilted magnetic field
Finally, we discuss the nonreciprocal transport when the magnetic field is slightly tilted toward the outofplane (z) direction, as illustrated in Fig. 3a. Figure 3b–d depict representative magnetic field dependence of R^{2ω} at 1.06, 1.10, and 1.16 K for θ = 4.5° (see the “Methods” section for the measurement of θ). Similarly to the case of an inplane magnetic field (θ = 0°), R^{2ω} appears, but the sign of R^{2ω} under B > 0 varies depending on temperature and magnetic field strength. This makes a stark contrast to the result for θ = 0° where R^{2ω} under B > 0 is always positive. To demonstrate the sign reversal of the nonreciprocal response more clearly, we show in Fig. 3e the color contour plot of γ in the B_{z}−T plane, where B_{z} = B sinθ is the outofplane component of the magnetic field. At low B_{z} and high temperature, positive γ is observed as in the case of the genuine inplane magnetic field (Fig. 2d). On the other hand, at high B_{z} and low temperature, negative γ shows up. It is noteworthy that, when the magnetic field is tilted toward the −z direction (θ < 0), a similar signreversal of γ is observed (see Supplementary Fig. 7 for the full data).
To obtain further insight into the sign reversal of γ, we consider a model involving spontaneous free vortices and vortices induced by B_{z} (Fig. 3f). When B_{z} is weak enough, free vortices and antivortices with the density n_{free} are thermally excited (Fig. 3f left). On the other hand, when B_{z} is strong enough but still smaller than \(B_{{{{\mathrm{c}}}}2}^ \bot\), vortices induced by B_{z} become dominant (Fig. 3f right), and their density is expressed by \(n_{{{{\mathrm{ind}}}}} = \left {B_z} \right/\phi _0\). We suggest in our model that the crossover between dominant vortex types is related to the signreversal of γ, which is observed only under the tilted magnetic field. Therefore, we introduce the ratio \(\beta = n_{{{{\mathrm{ind}}}}}/n_{{{{\mathrm{free}}}}}\). To experimentally estimate the value of β, we assume that the Ohmic resistance R^{ω} is proportional to the total number of vortices: \(R^\omega = \frac{{\phi _0^2}}{\eta }\left( {n_{{{{\mathrm{free}}}}} + n_{{{{\mathrm{ind}}}}}} \right)\). This relationship can be rewritten as \(\frac{{R^\omega }}{{\left {B_z} \right}} = \frac{{\phi _0}}{\eta }\left( {1 + \frac{1}{\beta }} \right)\) to give a onetoone correspondence between β and the experimentally measured value of R^{ω}/B_{z}. Thus, using the experimental data, we produce in Fig. 3g the contour plot of R^{ω}/B_{z} in the B_{z}−T plane for θ = 4.5°. The increase in B_{z} or decrease in T leads to the reduction in R^{ω}/B_{z}, reflecting the increase in the population of the induced vortices and hence in β. By comparing the color plots of γ in Fig. 3e and R^{ω}/B_{z} in Fig. 3g, one can find that the boundary between regions of γ > 0 and γ < 0 coincides qualitatively with the contour line of R^{ω}/B_{z} = const. (dotted and solid black lines). Such correspondence between the two contour plots indicates that the sign of γ is determined by R^{ω}/B_{z}, or in other words, by β, supporting the validity of our model. γ is positive on the freevortices dominant side with small β and changes its sign on the inducedvortices dominant side with large β.
Figures 4a–c show the color plots of γ in the B_{z}−T plane for θ = 1.5°, 4.5°, and 24.8°. For each θ, we have estimated the critical ratio β_{c} such that the curve \(\frac{{R^\omega }}{{\left {B_z} \right}} = \frac{{\phi _0}}{\eta }\left( {1 + \frac{1}{{\beta _{{{\mathrm{c}}}}}}} \right)\) gives the best fitting to the boundary between the regions with γ > 0 and γ < 0. The fitting results are shown by green curves. In all cases, the boundary is well fit by the curves with comparable values of β_{c} close to unity, as shown in Fig. 4d. This analysis indicates that β determines the sign of γ regardless of θ. In this fitting analysis, we used a constant value of η ~ 0.68 nm^{2} T^{2} Ω^{−1} at T = T_{BKT} (see Supplementary Note 1 for the estimation of η).
Discussion
The above analysis has shown that both spontaneous and fieldinduced vortices give rise to nonreciprocal transport in Bi_{2}Te_{3}/PdTe_{2}, while the sign of γ is defined by the dominant vortex type. Besides our system, the nonreciprocal charge transport originating from vortex dynamics has been studied in various 2D superconductors^{12,13,14,25}, which can be classified into two groups according to the orientation of the polar axis P: outofplane polar ones (polar axis Pzaxis) and inplane polar ones (Pxyplane). Reflecting the orientation of P, the direction of the magnetic field necessary for the emergence of nonreciprocity is different between these two groups, leading to the difference in the type of vortices. In the outofplane polar systems, which are represented by Bi_{2}Te_{3}/FeTe^{14} or gateinduced superconductivity in SrTiO_{3}^{13}, the nonreciprocal transport governed by the motion of spontaneous free vortices has been investigated under inplane magnetic fields. In contrast, in the systems with the inplane breaking of inversion symmetry, as exemplified by gateinduced superconductivity in MoS_{2}^{12}, the nonreciprocity due to the dynamics of vortices induced by outofplane magnetic fields has been verified. The present Bi_{2}Te_{3}/PdTe_{2} system with the outofplane polarity shows a different symmetry from the inplane polarity case for MoS_{2}. Nevertheless, the motion of induced vortices governs the nonreciprocal charge transport with negative γ; this is an unconventional situation of the polar 2D superconductors or possibly of the interface topological superconductors. In this case, the coexistence of B_{y} and B_{z} is essential; B_{z} induces vortices, while the breaking of both timereversal symmetry and mirror symmetry by B_{y} is a prerequisite for the emergence of nonreciprocity.
The sign change in γ depending on the type of vortices indicates that the different microscopic mechanisms of nonreciprocal response are at work. The nonreciprocal transport with the positive γ in the freevortices dominant situation has been elucidated by the currentinduced renormalization of superfluid density, which results in the current direction dependence of n_{free}^{15}. On the other hand, when the number of fieldinduced vortices is fixed by B_{z}, the asymmetry in the motion of vortices should be the main mechanism of nonreciprocal resistivity. One of the most likely candidates is the ratchettype potential for the vortex motion, although the sign of γ depends on the details of the potential in the material^{26}. Further theoretical and experimental investigation is required to fully reveal the origin of negative γ.
We also note that the superconducting diode effect^{27}, which is the nonreciprocity in the superconducting critical current, has recently been reported in a multilayer superconductor system. Several theoretical studies predict the nonreciprocity in the superconducting critical current in the presence of Rashbatype spin–orbit interaction^{28,29,30}, while these theories are not directly applicable to the BKT regime of the present study.
In summary, we have fabricated a Bi_{2}Te_{3}/PdTe_{2} thinfilm heterostructure as a TI/SC hybrid system and investigated the nonreciprocal charge transport arising from the motion of vortices. Under an inplane magnetic field, large nonreciprocal signals have been observed near the superconducting transition temperature. Moreover, we have observed the sign reversal of γ by tilting the magnetic field toward the outofplane direction. The observed sign reversal of γ has been consistently explained by the scenario that γ is positive (negative) when the motion of spontaneous vortices (fieldinduced vortices) generates resistance. The appearance of nonreciprocal charge transport is clear evidence for the inversion symmetry broken superconductivity in the Bi_{2}Te_{3}/PdTe_{2} heterostructure, suggesting that the superconducting proximity effect on the TI surface state is effective. Our study suggests that Bi_{2}Te_{3}/PdTe_{2} can be a good materials platform to explore proximityinduced topological superconductivity. Fabrication of Bi_{2}Te_{3}/PdTe_{2}based thinfilm devices will lead to the direct detection of topological superconductivity or Majorana fermions.
Methods
Thinfilm growth and device fabrication
We grew the heterostructure film by molecular beam epitaxy (MBE). The base pressure of the MBE growth chamber was on the order of 10^{−7} Pa. Thin film samples were grown on semiinsulating InP(111)A substrates. Molecular beam fluxes of Bi and Te were supplied from standard Knudsen cells and that of Pd was from a special Knudsen cell customized for hightemperature operation. The beam equivalent pressures of Pd, Bi, and Te were set at 3.3 × 10^{−7}, 5.0 × 10^{−6}, and 1.0 × 10^{−4} Pa, respectively. The substrate temperature was set at 340 °C throughout the growth. The beam fluxes of Pd and Bi were turned on/off by controlling the shutters. The growth rates were 0.25 and 0.29 nm/min for PdTe_{2} and Bi_{2}Te_{3}, respectively. The substrate was rotated during the growth to improve the homogeneity of the film. After the growth, the substrate heater was turned off and the substrate was cooled to room temperature and taken out of the growth chamber. Then, the grown film was transferred to the chamber of atomic layer deposition and covered with a 3nmthick AlO_{x} deposited at room temperature for the passivation.
The samples were characterized by Xray diffraction (XRD) with Cu Kα1 radiation. After the thinfilm growth, Hall bar devices were defined using ultraviolet photolithography and Ar ion milling. Both the width w and the distance between the voltage probeslwere 100 μm. The electrodes Au (25 nm)/Ti (5 nm) were formed by electron beam deposition.
Transport measurements
Transport measurement was conducted using a ^{3}He cooling system of a physical property measurement system (PPMS, Quantum Design). In the nonreciprocal transport measurement, we used copper plates bent at different angles, and the sample was mounted on a copper plate fixed on a sample puck. The tilt angle of magnetic field θ was estimated from the Hall effect at 300 K. We applied AC current using an AC current source (Keithley, model 6221) and measured the first and second harmonic voltages using digital lockin amplifiers (Stanford Research Systems, model SR830). The measurement current and frequency were 10 μA (root mean square) and 37 Hz, respectively.
Firstprinciples calculation for the electronic structure of Bi_{2}Te_{3}/PdTe_{2} heterostructure
We employed the Vienna Ab initio Simulation Package (VASP)^{31} for the ab initio calculation based on the density functional theory (DFT). We used the projector augmented wave (PAW)^{32} potential sets recommended by VASP and set the kineticenergy cutoff to 350 eV. In the calculations of the lattice optimization and the electronic band structure, we employed the generalizedgradient approximation (GGA) with the Perdew–Burke–Ernzerhof parametrization for solid (PBEsol)^{33} without the spin–orbit interaction and the GGA of the Perdew–Burke–Ernzerhof (PBE)^{34} with the spin–orbit interaction, respectively. We employed the 6 × 6 × 1 k mesh for the lattice optimization and the 8 × 8 × 1 k mesh for the calculation of the electronic band structure. We used the inplane lattice constant of Bi_{2}Te_{3} as that of the interface system. We have confirmed that there is no qualitative difference in the band structure when using the lattice constant of PdTe_{2} (Supplementary Fig. 8).
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Sato, M. & Ando, Y. Topological superconductors: a review. Rep. Prog. Phys. 80, 076501 (2017).
Fu, L. & Kane, C. L. Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys. Rev. Lett. 100, 096407 (2008).
Xu, J. P. et al. Experimental detection of a Majorana mode in the core of a magnetic vortex inside a topological insulator–superconductor Bi_{2}Te_{3}/NbSe_{2} heterostructure. Phys. Rev. Lett. 114, 017001 (2015).
Zhao, H. et al. Superconducting proximity effect in a topological insulator using Fe(Te,Se). Phys. Rev. B 97, 224504 (2018).
Wang, E. et al. Fully gapped topological surface states in Bi_{2}Se_{3} films induced by a dwave hightemperature superconductor. Nat. Phys. 9, 621–625 (2013).
Wiedenmann, J. et al. 4πperiodic Josephson supercurrent in HgTebased topological Josephson junctions. Nat. Commun. 7, 10303 (2016).
Schüffelgen, P. et al. Selective area growth and stencil lithography for in situ fabricated quantum devices. Nat. Nanotechnol. 14, 825–831 (2019).
Kayyalha, M. et al. Absence of evidence for chiral Majorana modes in quantum anomalous Hallsuperconductor devices. Science 367, 64–67 (2020).
Williams, J. R. et al. Unconventional Josephson effect in hybrid superconductortopological insulator devices. Phys. Rev. Lett. 109, 056803 (2012).
Qi, X. L., Hughes, T. L. & Zhang, S. C. Chiral topological superconductor from the quantum Hall state. Phys. Rev. B 82, 184516 (2010).
Tokura, Y. & Nagaosa, N. Nonreciprocal responses from noncentrosymmetric quantum materials. Nat. Commun. 9, 3740 (2018).
Wakatsuki, R. et al. Nonreciprocal charge transport in noncentrosymmetric superconductors. Sci. Adv. 3, e1602390 (2017).
Itahashi, Y. et al. Nonreciprocal transport in gateinduced polar superconductor SrTiO_{3}. Sci. Adv. 6, eaay9120 (2020).
Yasuda, K. et al. Nonreciprocal charge transport at topological insulator/superconductor interface. Nat. Commun. 10, 2734 (2019).
Hoshino, S., Wakatsuki, R., Hamamoto, K. & Nagaosa, N. Nonreciprocal charge transport in twodimensional noncentrosymmetric superconductors. Phys. Rev. B 98, 054510 (2018).
Zhu, Z. et al. Science 374, 1381–1385 (2021).
Teknowijoyo, S. et al. Nodeless superconductivity in the typeII Dirac semimetal PdTe_{2}: London penetration depth and pairingsymmetry analysis. Phys. Rev. B 98, 024508 (2018).
Xue, H. Y. et al. Molecular beam epitaxy of superconducting PdTe_{2} films on topological insulator Bi_{2}Te_{3}. Sci. China Phys. Mech. Astron. 62, 76801 (2019).
Bai, M. et al. Novel selfepitaxy for inducing superconductivity in the topological insulator (Bi_{1x}Sb_{x})_{2}Te_{3}. Phys. Rev. Mater. 4, 094801 (2020).
Liu, C. et al. Twodimensional superconductivity and topological states in PdTe_{2} thin films. Phys. Rev. Mater. 2, 094001 (2018).
Aslamasov, L. G. & Larkin, A. I. The influence of fluctuation pairing of electrons on the conductivity of normal metal. Phys. Lett. A 26, 238–239 (1968).
Halperin, B. I. & Nelson, D. R. Resistive transition in superconducting films. J. Low Temp. Phys. 36, 599–616 (1979).
Saito, Y., Nojima, T. & Iwasa, Y. Highly crystalline 2D superconductors. Nat. Rev. Mater. 2, 16094 (2016).
Tinkham, M. Introduction to Superconductivity 2nd edn (Dover, New York, 2004).
Zhang, E. et al. Nonreciprocal superconducting NbSe_{2} antenna. Nat. Commun. 11, 5634 (2020).
Hamamoto, K., Park, T., Ishizuka, H. & Nagaosa, N. Scaling theory of a quantum ratchet. Phys. Rev. B 99, 064307 (2019).
Ando, F. et al. Observation of superconducting diode effect. Nature 584, 373–376 (2020).
Daido, A., Ikeda, Y. & Yanase, Y. Intrinsic superconducting diode effect. Phys. Rev. Lett. 128, 037001 (2022).
Yuan, N. F. Q. & Fu, L. Supercurrent diode effect and finite momentum superconductors. Proc. Natl Acad. Sci. USA 119, e2119548119 (2022).
He, J. J., Tanaka, Y. & Nagaosa, N. A phenomenological theory of superconductor diodes. N. J. Phys. 24, 053014 (2022).
Kresse, G. & Furthmuller, J. Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set. Phys. Rev. B 54, 11169 (1996).
Blöchl, P. E. Projector augmentedwave method. Phys. Rev. B 50, 17953 (1994).
Perdew, J. P. et al. Restoring the densitygradient expansion for exchange in solids and surfaces. Phys. Rev. Lett. 100, 136406 (2008).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
Acknowledgements
We thank Y. Iwasa for the fruitful discussions. M.M. is supported by the Japan Society for the Promotion of Science (JSPS) through a research fellowship for young scientists (No. 19J22547). This research was supported by the Japan Society for the Promotion of Science through JSPS/MEXT GrantinAid for Scientific Research (Nos. 18H01155, 18H03676, 20K14390), and JST CREST (Nos. JPMJCR16F1, JPMJCR1874).
Author information
Authors and Affiliations
Contributions
Y.T. conceived the project. M.M., R.Y. and Y.I. grew the thin films with the help of A.T., K.S.T. and M.Kawasaki. M.M. and Y.I. performed the measurement with help from M. Kawamura, R.Y., R.W. and D.M. M.H. performed the band calculation. M.M., M. Kawamura, R.Y., M.H., J.J.H., N.N. and Y.T. analyzed and interpreted the data. M.M., M. Kawamura, and Y.T. jointly wrote the manuscript with contributions from all authors.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Masuko, M., Kawamura, M., Yoshimi, R. et al. Nonreciprocal charge transport in topological superconductor candidate Bi_{2}Te_{3}/PdTe_{2} heterostructure. npj Quantum Mater. 7, 104 (2022). https://doi.org/10.1038/s4153502200514x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s4153502200514x
This article is cited by

Lightinduced giant enhancement of nonreciprocal transport at KTaO3based interfaces
Nature Communications (2024)

Superconducting diode effect sign change in epitaxial AlInAs Josephson junctions
Communications Physics (2024)

Primitive to conventional geometry projection for efficient phonon transport calculations
npj Computational Materials (2023)

The supercurrent diode effect and nonreciprocal paraconductivity due to the chiral structure of nanotubes
Nature Communications (2023)

The superconducting diode effect
Nature Reviews Physics (2023)