Evidence of circulating charge density wave current: Shapiro interference in ${\text{NbSe}}_3$ topological crystals

We investigate the topological effect of charge density wave (CDW). We found a “beat” Shapiro step structure in ${\text{NbSe}}_3$ topological (ring) crystals. We interpreted it as an evidence of the circulating CDW current that flows in the rings and modifies the CDW fundamental frequency, namely, narrow-band-noise frequency. This result also implied that a precursor of Fröhlich’s CDW supercurrent emerged in the ring system.

Figure 1

(Color) (a) Optical microscopy image of a typical ring-shaped crystal of ${\text{NbSe}}_3$ equipped with two electrical contacts attached by silver paste. (b) Electrical circuit for measuring the Shapiro steps (ac-dc interference effect). The thick ring at the center of the circuit is the ring-shaped crystal of ${\text{NbSe}}_3$. Both dc and ac currents are induced into the crystal via the two electrical contacts. Inductive low-pass filters and capacitive high-pass filters separate the dc and ac lines. Differential resistance is measured as a function of dc current. The sample and filters are placed in an evacuated chamber and cooled to 77 K with liquid nitrogen.

Figure 2

(Color) dc bias current dependence of differential resistance $d{V}_{\text{dc}}/d{I}_{\text{dc}}$ with applied various ac voltages of $f_{\text{ex}}=100\text{MHz}$ at $T=120\text{K}$. The ac voltages $V_{\text{ac}}$ are 0, 12, 21, 36, 65, 73, and 82 mV from top to bottom. Harmonic and subharmonic Shapiro steps are indicated by arrows and index $p/q$, where $p$ and $q$ are integers. Moreover, there are small beat peaks ($p/q_{-}$ and $p/q_{+}$) on both sides of the main Shapiro peaks.

Figure 3

ac voltage amplitude dependence of Shapiro step width $\delta V$ of main peaks of (a) $p/q=1/1$, (b) 2/1, and (c) 3/1. The step width is estimated as the product of peak height and half width of half maxim of each peak of Fig. 2. Solid lines are calculation results of amplitudes of Bessel functions $\left|J_1\left(x\right)\right|$, $\left|J_2\left(x\right)\right|$, and $\left|J_3\left(x\right)\right|$, respectively.

Figure 4

(Color) ac voltage amplitude dependence of Shapiro step width $\delta V$ of negative and positive beat peaks of (a) $p/q=1/1$, (b) 2/1, and (c) 3/1. The calculation curves of amplitudes of Bessel functions $\left|J_1\left(x\right)\right|$, $\left|J_2\left(x\right)\right|$, and $\left|J_3\left(x\right)\right|$ are the rescaled ones in Fig. 3.

Figure 5

(Color) Differential resistance $d{V}_{\text{dc}}/d{I}_{\text{dc}}$ versus dc current $I_{\text{dc}}$ at $T=120\text{K}$ without ac voltage and with ac voltage of amplitude $V_{\text{ac}}=65\text{mV}$ and frequency $f_{\text{ex}}=40$, 80, 100, 200, and 300 MHz, respectively.

Figure 6

(Color) Frequency dependence of peak position $I_{\text{p}}=I_{\text{dc}}-I_{\text{th}}$ of each step. The peak positions of the harmonic, subharmonic, and beat peaks are proportional to frequency. The slopes of each line are shown as a function of the index $p/q$ in Fig. 7.

Figure 7

(Color) Slopes of each peak position versus frequency $I_{\text{p}}/f_{\text{ex}}$ as a function of index $p/q$. Circles, squares, and triangles indicate the main peaks (harmonic and subharmonic peaks), negative beat, and positive beat peaks, respectively. Each $I_{\text{p}}/f_{\text{ex}}$ is proportional to $p/q$. The coefficients of $p/q$ for each line are shown in the box. They were calculated by the least-squares method. The inset shows a model of the circulating current to explain the beat structure of the Shapiro step spectrum. If the circulating current is temporarily excited by an external electric field, it modifies the currents through the right and left arms as $I\pm \delta I$. As a result, fundamental frequency of each arm is modified and two beat peaks are produced around the main peaks.