† Corresponding author. E-mail:
Project supported by the National Basic Research Program of China (Grant Nos. 2016YFA0300601, 2017YFA0304700, and 2015CB921402), the National Natural Science Foundation China (Grant Nos. 11527806, 91221203, 11174357, 91421303, and 11774405), and the Strategic Priority Research Program B of the Chinese Academy of Sciences (Grant Nos. XDB07010100 and XDB28000000), and the Beijing Municipal Science & Technology Commission, China (Grant No. Z191100007219008).
Recently, a contact-resistance-measurement method was developed to detect the minigap, hence the Andreev bound states (ABSs), in Josephson junctions constructed on the surface of three-dimensional topological insulators (3D TIs). In this work, we further generalize that method to the circumstance with radio frequency (rf) irradiation. We find that with the increase of the rf power, the measured minigap becomes broadened and extends to higher energies in a way similar to the rf power dependence of the outer border of the Shapiro step region. We show that the corresponding data of contact resistance under rf irradiation can be well interpreted by using the resistively shunted Josephson junction (RSJ) model and the Blonder–Tinkham–Klapwijk (BTK) theory. Our findings could be useful when using the contact-resistance-measurement method to study the Majorana-related physics in topological insulator-based Josephson junctions under rf irradiation.
In 2008, Fu and Kane proposed that the superconducting proximity effect between an s-wave superconductor (S) and the surface of a three-dimensional (3D) topological insulator (TI) can induce p-wave-like superconductivity and host Majorana bound states (MBSs) in the vortex cores or in the proximity-type S–TI–S Josephson trijunctions.[1] In more detail, the electron-like and hole-like Andreev bound states (ABSs) in single S–TI–S Josephson junctions are predicted to have 4π-period energy phase relations (EPRs), and the 1D Majorana edge modes are predicted to be fully decoupled when the phase difference reaches π, resulting in the complete close of minigap between the electron-like and hole-like ABSs. In Josephson trijunctions on 3D TIs, furthermore, MBS is predicted to exist at the center of the trijunction over extended regions in phase space. In the last years, experimental efforts have been paid to search for 4π-period current phase relations (CPRs), which are direct consequences of 4π-period EPRs. Some signatures, such as skewed CPR,[2–4] missing of odd Shapiro steps,[5–8] etc., have been discovered. Meanwhile, a contact-resistance-measurement method has also been developed for directly probing the EPRs in the junction area.[9–11] A linear closing behavior of minigap as a signature of 4π-period EPRs in single S-TI-S junctions and the complete close of minigap at the center of the S–TI–S trijunctions as an evidence of MBS have been observed.[10,11]
To further study the fractional Josephson effect in which the missing of odd Shapiro steps[5–8] is regarded as a signature of MBS, one would require to perform the contact-resistance-measurement method in the presence of radio-frequency (rf) irradiation in order to obtain the information of EPR and MBS. More generally, since MBSs suffer poisoning from quasiparticle fluctuations, one would ultimately require fast braiding/fusion operations and fast readout of MBSs at μs or ns time scales in the future, in order to test their non-Abelian statistics and to perform quantum computation. However, at such a short time scale and in the radio/micro frequency range, it is unclear whether the contact-resistance-measurement method can still be applied to probe the minigap, hence the MBSs, in the S–TI–S junctions. To clarify this issue, in this work we investigate the contact-resistance-measurement method in the presence of rf irradiation. We find that this method still works — the measured data of contact resistance under rf irradiation can be well understood and numerically simulated by using the resistively shunted Josephson junction (RSJ) model[12] and the Blonder–Tinkham–Klapwijk (BTK) theory.[13]
Figure
In the following, we will first present and discuss the IJ–VJ curves (where IJ is the current passing from one superconducting electrode to another, and VJ is the voltage drop across the Josephson junction), and then present and discuss the data of contact resistance between the Pd electrode and Bi2Te3 under rf irradiation.
In the presence of rf irradiation, it is known that there will be Shapiro steps on the IJ–VJ curves of a Josephson junction.[12] Figure
Usually, the Josephson supercurrent has a 2π-period CPR, which leads to the appearance of ordinary Shapiro steps on the IJ–VJ curves in the presence of rf irradiation. If the supercurrent contains a 4π-period component, however, the odd Shapiro steps might disappear, leading to the so called fractional Josephson effect.[14] In some S–TI–S junctions, the missing of the odd Shapiro steps has been observed.[5–7] In our devices, however, figure
The IJ–VJ curves under rf irradiation can be simulated by solving the following equation based on the RSJ model:[12]
With increasing rf power, the region containing Shapiro steps in Fig.
In the next, let us present and discuss the data of the contact resistance dVb/dIb across the Pd–Bi2Te3 interface. Figure
With increasing rf power Prf, the dip–peak–dip structure on the vertical line cuts of Fig.
Theoretically, shining the device with rf irradiation will cause two effects on the contact resistance measurement. First, it will generate a rf current Irf0sin(2πfrft) flowing mainly through the bulk of the junction, by which influencing the time-dependent phase difference ϕ(t) of the Josephson junction, thus influencing the minigap Δ(t). Second, the rf irradiation will also generate a rf current passing through the Pd–Bi2Te3 interface, by which modifying the contact resistance measurement. Combining these two effects, the contact resistance can be simulated as follows, still by using the RSJ model and the BTK theory.
In the presence of rf irradiation, the time-dependent minigap of the surface state should follow the 4π -period form[10] and can be expressed as Δ(t) = Δ0|cos(ϕ(t)/2)|, where the time-dependent phase difference ϕ(t) can be obtained by solving Eq. (
To conclude, we have examined and confirmed the validity of the contact-resistance-measurement method for detecting the minigap in S–TI–S Josephson junction under rf irradiation. Although both the phase difference across the Al–Bi2Te3–Al junction and the minigap measurement process across the Pd–Bi2Te3 interface are influenced by the rf irradiation, the measured contact resistance can still be interpreted by using the RSJ model and the BTK theory. These results might be useful when using the contact-resistance-measurement method to study the minigap of S–TI–S junctions under rf irradiation and Majorana-related physics.
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