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Phase Rigidity and the Josephson Plasma Resonance

The defining property of a solid object is its rigidity, or resistance to mechanical deformation.  A superconductor is characterized by an analogous rigidity, but in a more abstract quantity, namely the phase of its wave function.  The superconducting state is well-described by a complex wave function yexp[iq(r)] that exists throughout the superconducting region.  (The amplitude y, which measures the density of superconducting of electrons in the sample, is not important for this discussion.)  In the absence of currents, the phase q(r) strongly resists deformation (it prefers to be uniform throughout).  Experiments that manipulate the phase, however, can reveal many interesting phenomena that are highly specific to the superconducting state.


In a solid, rigidity implies that potential energy is stored when the solid is bent.  A dramatic consequence is the mechanical resonance in which the potential energy oscillates 180o out-of-phase with the kinetic energy of the vibrating sub-components.  In a tuning fork or a gong, the resonant oscillation can persist for a very long time (high Q-factor).  Likewise, in a superconductor, potential energy may be stored by ‘winding’ the phase.  A very important property of the superconducting state is that, when a supercurrent Js flows (say, parallel to z), the phase advances at a rate proportional to the magnitude of Js, viz. dq /dz ~ Js. In the figure above, the value of q(r) at each point r is indicated by the arrow on the dial.  In the direction of Js, the phase advances in an anticlockwise sense viewed from the left. (If Js is absent, all the dials will show the same value, selected spontaneously.) By increasing Js, one may increase the pitch of the phase winding (and the energy stored), much as one winds up a watch.  How may we cause this stored energy to oscillate resonantly?  To produce an oscillation, we need to discuss the charging energy caused by Js.

The cuprate superconductors grow as a stack of copper-oxide layers.  In this layered structure, the phase stiffness is highly anisotropic.  Within each layer, the phase stiffness is so large that the phase may be assumed to be uniform.  Between layers, however, the phase stiffness is much weaker, so q(r) may fluctuate between layers with less cost.  (The solid analog is a stack of plates connected by soft springs.)  If the phase difference between adjacent layers is dq, the cost incurred (per unit area) is written as

Uf = (1/2) EJ dq2,

where EJ (the Josephson energy) is an intrinsic parameter that describes the stiffness of the ‘spring constant’ between adjacent layers.


For simplicity, we consider just two layers (blue squares in figure).  We have a parallel-plate capacitor that allows a weak supercurrent Js (green arrow) to flow between the layers.  From the discussion above, Js leads to a phase difference dq between the layers (compare yellow dials), which produces the potential energy Uf.  Since it transfers Cooper pairs, Js also leads to a charge unbalance dQ, which increases the electrostatic energy by

UQ = (1/2)dQ2/C

(C is the capacitance per unit area).  The charging energy UQ oscillates out-of-phase with the potential energy Uf.  The total energy (Hamiltonian) of the system is

HJ =   (2ne)2/(2C)  +  EJ [ 1- cos(dq ) ],

where we have written dQ = 2ne as the number n of Cooper pairs transferred (with e the elementary charge).  The second term is the stiffness energy Ufvalid for large phase deviation.

The Hamiltonian is just that of a simple pendulum, with dq representing the pendulum angle, and C the rotational inertia I of the pendulum bob (red circle on right).  The phase stiffness spring constant EJ is analogous to the gravitational acceleration g.  Since the pendulum frequency equals (mgL/I)1/2 (L is the cord length), we infer that the Josephson Plasma resonance frequency wJ is proportional to (EJ/C)1/2.

Surprisingly, in the cuprate BiSrCaCuO, the phase spring-constant EJ is so soft that wJ occurs at microwave frequencies.  Moreover, an external field H weakens EJ (it introduces vortices into the sample).  Thus, wJ decreases monotonically as H increases.  To observe this resonance, we expose the sample to microwave radiation at fixed frequency w, and slowly increase H.  At the resonant field B0, when wJ coincides with w, the sample strongly absorbs the microwave radiation, causing a slight rise in its temperature.  Hence, a trace of sample temperature versus H displays a sharp resonance (see traces in inset to Fig. 1).  The sharp absorpion resonance in a field was first observed by Ophelia Tsui et al. in 1994, and shown to be the Josephson Plasma Resonance in 1996 (an independent demonstration was obtained by Yuji Matsuda’s group in Hokkaido).

Figure 1  (Main Panel)  The variation of the resonant field B0 versus tilt angle q between the field H and the copper-oxide layers.  Each solid line connects data taken at a fixed microwave frequency (as indicated).  The solid lines are fits to a model by Koshelev (slightly modified).  Broken lines continue the fit to smaller q where deviation from theory occurs..  The inset shows the raw trace of microwave absorption versus applied field H with the sample exposed to microwave radiation at 48 GHz.  Three runs are displayed with q fixed as indicated.  The resonance signals excitation of the Josephson Plasma Resonance.  Experiment by Sibel Bayrakci et al. [Europhys. Lett. 46, 68-74 (1999).]

Examples of the resonant absorption are shown in the inset of Fig. 1 with the applied field fairly close to alignment with the copper-oxide layers (the resonance is much sharper when H is aligned with c).  Recently, the Josephson Plasma Resonance interpretation has been tested in detail by Bayrakci et al. who compared a model by Koshelev with measurements over a broad range of resonance frequencies and orientation of the applied field.  Fits are shown in the main panel of Fig. 1.

Ophelia K.C. Tsui, N.P. Ong, Y. Matsuda, Y.F. Yan and J.B. Peterson, “Sharp magnetoabsorption resonances in the vortex lattice of Bi2Sr2CaCu2O8+d”. Phys. Rev. Lett. 73, 724  (1994).
Ophelia K. C. Tsui, N. P. Ong, and J. B. Peterson, “Excitation of the Josephson Plasma Mode in Bi2Sr2CaCu2O8+d in Oblique Field.”, Phys. Rev. Lett. 76, 819 (1996).
Sibel P. Bayrakci, Ophelia K. C. Tsui, N. P. Ong, K. Kishio and S. Watauchi, “The Josephson Plasma Resonance in Bi2Sr2CaCu2O8+d as a Probe of Vortex Correlation.” Europhys. Lett. 46, 68-74 (1999).