Cavity phenomena in mesas of cuprate high-$T_c$ superconductors under voltage bias

Modeling a single crystal of cuprate high-$T_c$ superconductor, such as ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$, as a stack of intrinsic Josephson junctions, we formulate explicitly the cavity phenomenon of plasma oscillations and electromagnetic (EM) waves in mesas of cylindrical and annular shapes. The phase differences of the junctions are governed by the inductively coupled sine-Gordon equations, with the Neumann-type boundary condition for sample thickness much smaller than the EM wavelength, which renders the superconductor single crystal a cavity. Biasing a dc voltage in the $c$ direction, a state with $\pm \pi$ kinks in the superconductivity phase difference piled up alternatively along the $c$ axis is stabilized. The $\pm \pi$ phase kinks provide interlock between superconductivity phases in adjacent junctions, taking the advantage of huge inductive couplings inherent in the cuprate superconductors, which establishes the coherence across the whole system of more than $\sim 600$ junctions. They also permit a strong coupling between the lateral cavity mode of the transverse Josephson plasma and the $c$-axis bias, and enhance the plasma oscillation significantly at the cavity modes which radiates EM waves in the terahertz band when the lateral size of mesa is set to tens of micrometers. It is discussed that the cavity mode realized in a very recent experiment using a cylindrical mesa can be explained by the present theory. In order to overcome the heating effect, we propose to use annular geometry. The dependence of frequency on the radius ratio is analyzed, which reveals that the shape tailor is quite promising for improving the present technique of terahertz excitation. The annular geometry may be developed as a waveguide resonator, mimicking the fiber lasers for visible lights.

Figure 1

(Color online) Spatial distribution of (a) the static phase term $P^s$, (b) the supercurrent, and the standing (c) electric and (d) magnetic waves for the (1,1) mode of a cylindrical mesa. Here $Aq\zeta \text{cos}\phi =5000$ is taken for Eq. (5) in (a) and the kink is approximated as a step function in (b). The lateral coordinates are normalized by the radius of the cylinder. The quantities except the phase difference $P^s$ are up to the plasma amplitude $A$.

Figure 2

(Color online) Same as Fig. 1 for the (2,1) mode of a cylindrical mesa.

Figure 3

(Color online) Same as Fig. 1 for the (0,1) mode of a cylindrical mesa.

Figure 4

(Color online) Same as Fig. 1 for the (1,2) mode of a cylindrical mesa.

Figure 5

(Color online) $IV$ characteristics for the cylindrical mesa including the four lowest modes. The dimensionless voltage, or equivalently the wave number and frequency, is given by $\omega ={\chi }^c/a$ with ${\chi }^c=1.8412$, 3.0542, 3.8317, and 5.3314 for the (1,1), (2,1), (0,1), and (1,2) modes, respectively. The radius of cylinder is $a=0.4$ $\left(a=0.4{\lambda }_c\right)$.

Figure 6

(Color online) Radiation patterns for (a) (1,1) mode, (b) (2,1) mode, (c) (0,1) mode, and (d) (1,2) mode of cylindrical mesa at the respective resonance frequency. The radius of the mesa is $a=0.4$.

Figure 7

(Color online) Same as Fig. 1 for the (1,1) mode of an annular mesa except that the lateral coordinates are normalized by the outer radius and the ratio between the inner and outer radii is 1/2.

Figure 8

(Color online) Same as Fig. 7 for the (2,1) mode of an annular mesa.

Figure 9

(Color online) Same as Fig. 7 for the (0,1) mode of an annular mesa.

Figure 10

(Color online) Same as Fig. 7 for the (1,2) mode of an annular mesa.

Figure 11

(Color online) ${\chi }^a$ as a function of the radius ratio $a_{\text{i}}/a_{\text{o}}$ for the lowest four modes for the annular geometry.

Figure 12

(Color online) $IV$ characteristics for the annular mesa including the four lowest modes. The dimensionless voltage is $\omega ={\chi }^a/a_{\text{o}}$ with ${\chi }^a=1.3546$, 2.6812, 6.3932, and 6.5649 for the (1,1), (2,1), (0,1), and (1,2) modes, respectively. The radius ratio is $a_{\text{i}}/a_{\text{o}}=1/2$ and outer radius is $a_{\text{o}}=0.4$.

Figure 13

(Color online) Same as Fig. 6 for annular mesa. Multireflections due to the inner surface are neglected since the thickness of annular mesa is very small compared with the EM wavelength. The radii are $a_{\text{i}}=0.2$ and $a_{\text{o}}=0.4$.