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Dirac-like quasiparticles in a magnetic field

In the ground state of a superconductor (absolute temperature T = 0 K),  all the electrons pair up to form Cooper pairs, which collectively form the condensate.  At finite T, however, minimization of the free energy requires a small fraction of the electrons to be unpaired (this leads to a finite entropy).  The unpaired electrons are called quasiparticles. The energy of a quasiparticle (qp) created in the state with wavevector k equals E(k).  At temperature T, the density of quasiparticles is thermally activated and given by the expression

nk(T) ~ exp[-E(k)/kBT],

where kB is Boltzmannís constant.  In s-wave supercondcutors, the minimum qp energy is E(k) = D(T), where D(T) is the superconducting gap (D(T) is T dependent, but nominally independent of the direction of k).  Hence, the qp density is exponentially small at low T.

The d-wave symmetry of the superconducting gap in the cuprates presents a novel situation.  As we move around the 2D Fermi Surface (FS) (blue circle in left figure), the superconducting gap D(k) changes sign 4 times.  Hence it vanishes at 4 nodal points Q.   In the vicinity of each node, the contours of E(k) are elliptical (shown as red ellipses).  In terms of the wavevector q = k-Q measured relative to Q, E(q) has the form (the variation along z is insignificant)

E(q) = [ (vf q1)2  +  (vD q2)2 ]1/2,

where vf and vD are velocity parameters (with vf/vD = 8-10).  As indicated in the left figure, the principle axes q1 and q2 are normal and parallel, respectively, to the FS.


Because E(q) vanishes at the node, the qp density decreases as T2 (the number of states contained in the area of the ellipse shaded in yellow), which is much slower than the exponential decrease in s-wave superconductors.  The relatively large qp population presents an experimental opportunity to probe their properties at low T.

The expression for E(q) implies that, along a general direction q, the qp energy increases linearly with the magnitude q.  The right figure shows the Dirac cone described by E as the direction of q is varied (the slope of the cone is given by the velocity parameters vf and vD).  In a weak magnetic field, a quasiparticle (red ball) will move around the cone, staying on the same energy contour (this is what naive semiclassical theory would say).  Thus, the qpís in the cuprates present an unusual situation for studying the behavior of excitations that obey a Dirac-like dispersion, especially in an intense magnetic field.

Let us look at the nature of the quasiparticle state in more detail.  Although the qp may be regarded as simply an electron, it is actually a superposition of an electron and a hole state.  We represent a state occupied by an electron of momentum k and spin up by ck,up+ (the dagger symbol indicates electron creation).  Similarly, a hole state with the same momentum and spin is represented by dk,up+ (the hole excitation is equivalent to destroying a spin-down electron in the state Ėk; it is customary to write dk,up+ = c-k,down ).  The quasiparticle state, represented by gk,1+ is a quantum superposition of these two states with amplitudes uk and vk, respectively, viz.

gk,0+  =  uk ck,up+  + vk dk,up+.

A companion qp state gk,1+ of the same momentum k, but with spin down, is defined with uk and vk interchanged.

The quasiparticle gk,0+ has definite momentum k and spin up.  However, it has both particle-like and hole-like characteristics.  One or the other may dominate in a particular experiment.