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 *y*exp[i*q*(**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 180^{o} 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 *J*_{s} flows (say, parallel to **z**), the
phase advances at a rate proportional to the magnitude of *J*_{s},
viz. d*q /*dz ~ *J*_{s}.
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 *J*_{s}, the phase advances in an anticlockwise sense
viewed from the left. (If *J*_{s} is absent, all the dials will
show the same value, selected spontaneously.) By increasing *J*_{s},
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 *J*_{s}.

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

*U*_{f} = (1/2) *E*_{J} *dq*^{2},

where *E*_{J} (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 *J*_{s}
(green arrow) to flow between the layers. From the discussion above, *J*_{s}
leads to a phase difference *dq*
between the layers (compare yellow dials), which produces the potential energy *U*_{f}.
Since it transfers Cooper pairs, *J*_{s} also leads to a charge
unbalance *dQ*, which increases
the electrostatic energy by

*U*_{Q} = (1/2)*dQ*^{2}/C

(*C* is the capacitance per unit area). The charging energy *U _{Q}*
oscillates out-of-phase with the potential energy

*H*_{J} = (2*ne*)^{2}/(2*C)*
+ *E*_{J} [ 1- cos(*dq *)
],

where we have written *dQ* = 2*ne*
as the number *n* of Cooper pairs transferred (with *e* the
elementary charge). The second term is the stiffness energy *U*_{f}valid
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 *E*_{J} 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 *w*_{J} is proportional to (*E*_{J}/*C*)^{1/2}.

Surprisingly, in the cuprate BiSrCaCuO, the phase spring-constant *E*_{J}
is so soft that *w*_{J}
occurs at microwave frequencies. Moreover, an external field *H*
weakens *E*_{J} (it introduces vortices into the sample).
Thus, *w*_{J} 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
*B*_{0}, when *w*_{J}
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 *B*_{0}
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.

See

Ophelia K.C. Tsui, N.P. Ong, Y. Matsuda, Y.F. Yan and J.B. Peterson,
“Sharp magnetoabsorption resonances in the vortex lattice of Bi_{2}Sr_{2}CaCu_{2}O_{8+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 Bi_{2}Sr_{2}CaCu_{2}O_{8+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 Bi_{2}Sr_{2}CaCu_{2}O_{8+d}
as a Probe of Vortex Correlation.” Europhys. Lett. **46**, 68-74 (1999).