Phase dynamics in intrinsic Josephson junctions and their electrodynamics

We present a theoretical description of the phase dynamics and its corresponding electrodynamics in a stack of inductively coupled intrinsic Josephson junctions of layered high-$T_c$ superconductors in the absence of an external magnetic field. Depending on the spatial structure of the gauge-invariant phase difference, the dynamic state is classified into: state with kink, state without kink, and state with solitons. It is revealed that in the state with phase kink, the plasma is coupled to the cavity and the plasma oscillation is enhanced. In contrast, in the state without kink, the plasma oscillation is weak. It points a way to enhance the radiation of electromagnetic from high-$T_c$ superconductors. We also perform numerical simulations to check the theory and good agreement is achieved. The radiation pattern of the state with and without kink is calculated, which may serve as a fingerprint of the dynamic state realized by the system. At last, the power radiation of the state with solitons is calculated by simulations. The possible state realized in the recent experiments is discussed in the viewpoint of the theoretical description. The state with kink is important for applications including terahertz generators and amplifiers.

Figure 1

(Color online) Schematic view of a stack of Josephson junctions based on the SIS model. The blue/light (green/dark) area denotes superconducting (insulating) layers.

Figure 2

(Color online) [(a) and (b)] Schematic view of two simplest configurations of the static phase $P_l^s\left(x\right)$. [(c), (e), (g) and (i)] $\left(2m+1\right)\pi$ phase kink for $m=0$, 1, 2, and 3, respectively. [(d), (f), (h), and (j)] Their corresponding unquantized static vortices.

Figure 3

(Color online) $IV$ characteristics calculated by numerical simulations (symbols) and the linearized theory (lines). The inset is an enlarged view.

Figure 4

(Color online) (a) Radiation power at the first current step obtained by simulations (symbol), the linear theory (solid line), and the power balance relation Eq. (33) (dashed line). The results are obtained with $C=0.000177$ and $R=707.1$. The arrow is the starting point of the first current step.

Figure 5

(Color online) Radiation power on current steps. The symbols are from the simulations and the lines are given by Eq. (33). The results are obtained with $C=0.000177$ and $R=707.1$.

Figure 6

(Color online) Configurations of $J_{\text{asym}}$, $J_{\text{sym}}$ and electromagnetic wave at successive time $t_1=0.06T_1$, $t_2=0.31T_1$, $t_3=0.83T_1$, and $t_4=0.94T_1$, where $T_1\equiv 2\pi /k_1$ is the period of plasma oscillation at the first cavity mode. We have subtracted the static part of $E^z$. The top figures are taken at the bottom of the first current step while the bottom figures are at the top of the first current step in Fig. 3.

Figure 7

(Color online) Radiation power from the state without kink and its corresponding $IV$ characteristics. Symbols are for simulations and lines are for theory. The vertical dashed line is the retrapping point obtained from the theory. The inset is the frequency spectrum at the strongest radiation. The results are obtained with $C=0.716$ and $R=10.0$.

Figure 8

(Color online) Configurations of phase $P$, magnetic field $B^y$, and electric field $E^z$ at successive time $t_1=0.07T_r$, $t_2=0.17T_r$, $t_3=0.33T_r$, and $t_4=0.4T_r$, where $T_r$ is the period at the retrapping point. The phase is normalized into $\left[0,2\pi \right]$ and we have subtracted the static part of $E^z$. The results are obtained by simulations with $C=0.716$ and $R=10.0$.

Figure 9

(Color online) Energy per junction stored in a thick stack of intrinsic Josephson junctions in the state with and without kink, (a) magnetic, (b) electric, (c) Josephson, (d) total energy. The results are obtained without radiation, and the results for the state with kink are obtained in the first current step shown in Fig. 3. Because the energy for different kink configurations is same, only the energy for the kink configuration ${\mathbf{P}}^{sT}=\left[-\pi ,+\pi ,-\pi ,+\pi \right]$ is shown in this figure.

Figure 10

(Color online) (a) Coordinate system for the calculation of radiation pattern from the mesa. (b) Radiation pattern of the mode (1,0). (c) Radiation pattern of the mode (1,1). Here $L_x=80\mu \text{m}$, $L_y=300\mu \text{m}$, and $L_z=1\mu \text{m}$.

Figure 11

(Color online) Radiation pattern of the state without kink biased at the retrapping point. Here $L_x=80\mu \text{m}$, $L_y=300\mu \text{m}$, and $L_z=1\mu \text{m}$. The anisotropy of the pattern in the $xy$ plane is due to the fact $L_y⪢L_x$.

Figure 12

(Color online) Radiation power from the zero-field steps caused by soliton motions. The inset is the $IV$ characteristics. The vertical dashed lines are the assignment of cavity mode according to $k_n=n\pi /L$ with $n$ being an integer and the ac Josephson relation. The results are obtained with $C=0.000177$ and $R=707.1$.

Figure 13

(Color online) Frequency spectrum at the first zero-field step at $J_{\text{ext}}=0.57$. The vertical dashed line is the frequency given by the ac Josephson relation with voltage $V=2\pi /L$. The fundamental peak does not come to the dashed line means that the ac Josephson relation is broken. The results are obtained with $C=0.000177$ and $R=707.1$.