# Resonant absorption at the vortex-core states in -wave superconductors

###### Abstract

We predict a resonant microwave absorption on collective vortex modes in a superclean -wave superconductor at low temperatures. Energies of the collective modes are multiples of the distance between the exact quantum levels of bound states in the vortex core at lower temperatures and involve delocalized states for higher temperatures. The characteristic resonant frequency is larger than the cyclotron frequency but lower than the Caroli-deGennes-Matricon minigap ; it has a dependence on the magnetic field and decreases down to with increasing temperature. We calculate the vortex mass as a response to a slow acceleration. This mass is equal, by the order of magnitude, to the mass of electrons inside the vortex “core” with dimensions by ; it increases with temperature. We discuss the universal flux-flow regime predicted in [1] and show that it exists in a broader temperature range than it has been originally found.

###### pacs:

PACS numbers: 74.60.Ge, 74.25.Jb, 74.25.Fy, 74.72.-h[

]

## I Introduction

The vortex dynamics in clean superconductors is determined by localized states in vortex cores. Motion of a vortex excites transitions between the states; a competition between the relaxation rate and the interlevel spacing controls the proportion between dissipative and reactive forces experienced by the vortex [2, 3]. In -wave superconductors, the vortex dynamics is expected to be more intricate due to a peculiar structure of the vortex-core states. The presence of gap nodes introduces the most important difference in the structure of core states compared to an -wave superconductor. As was shown in Refs. [1, 4], instead of a well-defined quasiclassical Caroli-deGennes-Matricon [5] interlevel spacing , the true quantum states in -wave superconductors have a much smaller separation between quantum levels, , which depends on the magnetic field. As a result, there appears another parameter which influences the vortex dynamics. According to Ref. [1], a new universal regime can be reached in superclean superconductors with longitudinal and transverse components of the conductivity tensor independent of the relaxation time. It is realized when the relaxation rate is smaller than the average distance between the quasiclassical energy levels but larger than the separation between the true quantum-mechanical states .

In the present paper, we study both steady and oscillatory motion of vortices at low temperatures and low magnetic fields in more detail using the microscopic theory outlined in Ref. [1]. For a steady flux flow, we find that the universal regime is realized in a considerably broader region of temperatures and magnetic fields than what was originally predicted in Ref. [1]. This conclusion agrees with the results of Ref. [6]. The relevant parameter which governs the vortex dynamics is shown to be the energy of “collective modes” . We calculate and show that it coincides with the true quantum mechanical interlevel spacing if the energy of excitations is , and involves delocalized states for excitations with higher energies . For an oscillatory motion of vortices which can be excited by a microwave irradiation, we predict a finite resonant absorption even for infinitely long relaxation time. The series of resonances occurs at multiples of with the resonant frequencies considerably lower than the average quasiclassical interlevel distance but higher than the cyclotron frequency . They have a magnetic field dependence at very low temperatures. For higher temperatures, the series of resonances compresses down to an absorption edge which approaches for .

## Ii Nonequilibrium excitations

### ii.1 Spectrum

We consider superconductors which have the coherence length much longer than the inverse Fermi momentum, . For these superconductors, a quasiclassical approximation is appropriate. A quasiclassical particle moves along a straight line parallel to its momentum which is conserved with the accuracy . The angular momentum is thus another approximately conserved quantity even if the order parameter does not have a cylindrical symmetry; it is now a continuous variable not directly related with labelling the quantum states in the vortex core. A quasiparticle passing at some distance near a vortex can become classically localized in the vortex core region. It will have an energy characterized by the momentum along the vortex axis , the momentum direction in the plane perpendicular to the vortex axis, and by the impact parameter coupled to the angular momentum .

For a -wave superconductor without admixture of other components, the order parameter is

(1) |

where is the angle between and the axis which is taken along one of the gap nodes. The modulus of the order parameter at large distances, . We take the axis along the vortex circulation, , and denote by the quasiparticle velocity in the plane. If and are the distance and the azimuthal angle in the cylindrical frame, the impact parameter of a quasiparticle moving through the point and the distance along the trajectory are

(2) |

The quasiparticle energy can be found using the quasiclassical Green function technique in the same way as it has been done for -wave superconductors in Ref.[7]. Here we summarize the most important properties of the energy spectrum of a quasiclassical particle in the vortex core. It is discussed in more detail in Appendix. The quasiclassical spectrum has several branches belonging to various values of the radial quantum number . The anomalous branch with crosses zero of energy as a function of the impact parameter. For small impact parameters ,

(3) |

where for and for ; the second term is the magnetic energy where is the cyclotron frequency with . Modulus of charge appears due to the choice of the axis. Note that the magnetic energy has the energy level spacing [8] equal to the Larmor frequency rather than just to as it would be the case for the Landau spectrum in the normal state. This is because the wave function is localized at distances of the general order of rather than at the magnetic radius which would be the case in the normal state. Near the gap nodes, i.e., when is small, the low-lying states with energies much below the gap at infinity, , correspond to . The derivative of the energy with respect to the angular momentum, , is the analogue of the Caroli-deGennes-Matricon minigap [5] in an -wave superconductor.

For large impact parameters, , the energy is

(4) |

The particles are localized on lines passing through the vortex perpendicular to the momentum direction. The localization length along the trajectory is . A similar spectrum was considered in Ref. [6].

Eqs. (3) and (4) can be combined into a single interpolation formula

(5) |

where is a constant. The fit to the large- region of energy gives ; the best fit for low- region is reached with . We define an angle at which the particle energy is equal to the gap ; the angle is close to one of the nodes if . It is a quasiparticle moving at an angle which is classically localized.

Other branches with nonzero radial quantum numbers are separated from by energies of the order of . The number of branches with energies between and is

(6) |

Eqs. (4) and (5) are quantitatively correct as long as where is the distance between vortices. Such distances are, in principle, accessible for a particle which moves near the direction of a gap node within the angle . We shall see, however, that the condition is fulfilled for energies .

A classically localized particle is not necessarily localized in a strict quantum-mechanical sense. This was also found in numerical calculations of Ref. [9]. The true quantum levels can be obtained from the semiclassical Bohr-Sommerfeld quantization rule [1, 10, 11]

(7) |

is an integer, appears because the single-particle wave function changes its sign after encircling a single-quantum vortex. The angular momentum is expressed through the quasiparticle energy according to Eq. (5). Consider first an energy . The integral in Eq. (7) converges according to Eq. (5) and is determined by angles . For this region of angles, the spectrum is actually given by the exact equation (3): where

(8) |

with

(9) |

The characteristic impact parameters are of the order of , i.e., . We obtain where

(10) |

A qualitative expression for the spectrum with the dependence on the magnetic field similar to Eq. (10) was found in Ref. [4] neglecting the magnetic energy. The argumentation was that the divergence of without the magnetic energy should be cut off at such angles that the characteristic distance of a qusiparticle from the vortex axis is of the order of the intervortex distance ; here the corrections to the quasiparticle energy induced by neighboring vortices is just of the order of . We see, however, that Eq. (10) is obtained exactly if one takes into account the magnetic quantization. We find simultaneously that the particles with energies are acually localized in the vortex core since ; thus the corrections from neighboring vortices are not important. For larger energies , the localization radius becomes of the order of . These particles can not be considered as localized any more because of the presence of other vortices. For higher energies , the part with extended trajectories in Eq. (7) dominates, and thus quasiparticles are no longer localized in the core even for an isolated vortex.

The average quasiclassical energy-level spacing determines the parameter which controls the vortex dynamics in clean -wave superconductors with . The value separates the so called moderately clean limit, , with a highly dissipative vortex dynamics from the superclean limit, , where dissipation is small.

In -wave superconductors, situation is more intriguing. It was predicted in Ref. [1] that a large dissipation persists for much longer . In the present paper we show that, for a steady vortex motion, the relevant parameter which marks the transition between dissipative and nondissipative dynamics is where is the temperature dependent characteristic energy of “collective modes” induced by moving vortex. For low temperatures, it coincides with the energy of “bound states” determined by Eqs. (7), (10): . The energy decreases down to when the temperature approaches . The parameter is much smaller than since . We have thus a hierarchy of energies . To get the parameter which controls the vortex dynamics, these energies should be compared either with the relaxation rate for a steady motion of vortices or with where is the characteristic frequency of vortex oscillations. Large values of require a very high purity of samples: the mean free path should be . Nevertheless, there are experimental evidences that such a regime can be realized in practice. [12]

### ii.2 Balance of forces on a moving vortex

The force on a vortex from the environment where it moves is the momentum transferred from the excitations. The exact microscopic expression for the force has been derived in Ref. [3]. It can equivalently be presented as a force produced by quasiclassical particles with a distribution , characterized by canonically conjugated coordinate and angular momentum [13]. The contribution from classically localized states is

(11) |

Using we get

(12) | |||||

We take the nonequilibrium distribution function in the form

(13) |

where is the equilibrium Fermi distribution, the factors and describe the longitudinal and transverse responses to the vortex velocity , and is the electron density; is the average over the whole Fermi surface which includes averaging over . We assume here a Fermi surface with not less than the tetragonal symmetry in the plane perpendicular to the vortex axis. For simplicity, we consider only the particle-like Fermi surface, the generalization for a surface with both particle-like and hole-like parts is given later.

The force from classically delocalized states is [3]

(14) | |||||

The force should be balanced by the Lorentz force where is the flux quantum. Since the average electric field is , the force balance gives a linear relation between the transport current and the electric field. The proportionality coefficients are the Ohmic and Hall conductivities. The Ohmic conductivity is

(15) | |||||

For the Hall conductivity one has the same expression with instead of and replaced with .

In Galilean invariant systems , and the force balance is sometimes presented in terms of the Magnus force from the superfluid component and the friction plus transverse forces from the normal component

In such representation, the Magnus force includes the Lorentz force and a part of the transverse force . The constants and are expressed through the conductivities as

## Iii Distribution function

### iii.1 Kinetic equation

The kinetic equation for the distribution function for a system of fermions characterized by canonically conjugated variables and has the form [13]

(16) |

A particle is classically localized if it has an energy and moves at an angle counted from one of the nodes. For localized quasiparticles, the derivative is the interlevel spacing. The collision integral is written in a -approximation. This equation can also be derived microscopically. [3]

If the vortex moves with a velocity with respect to the heat bath, the Doppler shift of the energy produces the “driving force” acting on quasiparticles. The third term in the l.h.s. of kinetic equation (16) contains this force multiplied by . We look for a solution in the form where is independent of . Eq. (16) gives

(17) |

where

(18) |

Here we assume that the vortex velocity has a form where .

Introducing the longitudinal and transverse responses to the vortex velocity according to Eq. (13) and taking into account that, for the tetragonal symmetry, the responses do not depend on the direction of the vortex motion with respect to the crystal lattice, one finds two coupled first-order differential equations for and :

(19) |

A quasiparticle is delocalized either for energies or for angles . For a homogeneous magnetic field the distribution function of delocalized electrons was shown [3] to satisfy Eq. (17) where is replaced with . Here is the number of states for a particle with given , , and at large distances from the vortex, i.e., the quasiclassical Green function :

The distribution function thus has the form of Eq. (13) with and satisfying Eqs. (19) where now

(20) |

Since and obey first-order differential equations they are continuous functions at . For , the potential is given by Eq. (20) in the whole region of angles.

### iii.2 Static response

Equations (19) can be easily solved. We have ; where

with

In the moderately clean limit, , the potential is always large, and we obtain the local solution as in an -wave superconductor [3]

(21) |

In the superclean limit , the potential is small almost everywhere except for vicinities of the gap nodes where it becomes large. Consider first energies and find the distribution function for the anomalous branch in the region of angles not specifically close to the gap nodes. It is this branch which is only excited at temperatures . The overall behavior of the distribution function is to the highest extent determined by what happens in a close vicinity of the gap nodes. It is this region which is responsible for the whole build-up of nonequilibrium distribution of excitations.

For angles larger than from the nodes one can neglect the potential . As a result, one has for where ,

(22) |

The constants and can be found by matching with the solution in the vicinity of the node, , where . This provides the boundary condition across the node at . Here . For such energies, the region of angles with delocalized trajectories is not important. The integral for converges and is determined by angles . This range of angles sets the width of the transition region near a gap node where the distribution function jumps from its value at to its value at . We obtain using Eq. (8)

(23) |

The solution for should be periodically continued to the rest of angles with the period since the response function has the same tetragonal symmetry as the underlying system: . Together with the above boundary condition, this gives

(24) |

We have after averaging over the azimuthal angle

(25) |

For energies , the region of angles with extended trajectories dominates; now it is which determines the width of the transition region where and jump. Compact expressions can be obtained if we assume that is independent of for . For simplicity, we replace with . This does not change the results qualitatively. We have for

(26) |

where

We can use Eq. (22) for angles . Since the contribution to form the region of angles is much smaller than that from the angles , we can extend Eq. (22) to the angles and . First, we match equations (22) and (26) at . Another condition is obtained by matching Eqs. (26) taken at with Eq. (22) taken at . We obtain four equations for four constants , , , and where now .

If is not considerably smaller than unity, we can neglect in the boundary conditions, and obtain

For values of and larger, one automatically has . As a result, both and are small, and we recover Eqs. (24) and (25) with the new definition for . The coefficients and are in this case

We see that the distribution function for energies is not small even for delocalized trajectories .

The contribution from the region of angles to Eq. (15) and to the corresponding equation for decreases together with as the parameter increases. For , the contribution from angles where the gap is of the order of temperature becomes important. In this case, we obtain from Eqs.(22) and (26)

As a result,

(27) |

We can thus use Eq. (25) within the whole range of and combine the two results for in different regions of energy into a single interpolation expression

(28) |

where has the meaning of a characteristic energy of “collective modes”

(29) |

which we discuss later.

For energies we have from Eqs. (26) and

(30) |

### iii.3 Frequency-dependent response

Assume that the external frequency . In this limit, the solution of the kinetic equation (16) has the form of Eqs. (25,28) where in Eq. (28) is replaced with . Consider first the case . The factors and are small. In the limit , the longitudinal response is

(31) |

where

(32) |

The response has poles at or at frequencies where and is given by Eq. (29). Harmonics with appear due to the absence of the axial symmetry. Note that . For low temperatures , the eigen frequencies are independent of temperature: . These poles are the collective modes of electrons involved into the vortex motion. For low energies , these modes coincide with multiples of the distance between the true quantum mechanical energy levels determined by Eq. (7). For higher energies, delocalized quasiparticles are involved into the vortex motion.

## Iv Conductivities

### iv.1 Steady motion

Since and do not depend on , the integration over in Eq. (15) can be reduced to the integration over . In the sums over , only the term with remains because all with are odd functions of . If the Fermi surface has electron-like and hole-like pockets, the conductivities are (compare with Ref. [3])

In the moderately clean case, the factors and are given by Eq. (21). The conductivities are

(33) |

where . This is similar to the results for an -wave superconductor [2, 3, 14].

In the superclean limit the factors and for low temperatures are determined by Eq. (25). The general expression for the conductivities are rather complicated. We consider two limits. The universal regime is reached when , i.e., when either for or when for . The region of the universal regime is thus much larger than it was predicted in Ref. [1]. This was pointed out by Makhlin [6]. The condition automatically implies that , therefore, both and are small. One has from Eq. (25) , which results in universal conductivities [1]

(34) |

In the limit we have from Eq. (27)

(35) |

where and

(36) | |||||

Here is determined by Eq. (10) with the logarithmic factor . The average is taken over the particle-like and hole-like parts of the Fermi surface. Eq. (35) also agrees with [6]. This regime is reached when either for or when for .

As the temperature approaches , the contribution from fully delocalized states with becomes more and more important. The delocalized states give the normal-state Ohmic and Hall conductivities in the limit . In the limit , the flux-flow parts of and start from the universal values and then first decrease as as long as remains large. The ratio measures the relative contribution from localized states. With further approaching , the moderately clean regime is reached, and becomes proportional to while is proportional to being small: (compare with Ref. [3]). In the limit , the universal regime is not realized, and the conductivity starts from Eq. (35) and saturates at while remains constant.

### iv.2 Dispersion of conductivity

#### iv.2.1 Low frequency: vortex mass

If both the frequency and relaxation rate are low such that the transverse response is . The transverse force is

(37) |

The longitudinal response,

gives the force

where is the friction coefficient and plays the role of the vortex “mass”. Indeed, the force balance takes the from of the Newton’s law

(38) |

Here the vortex acceleration is a result of action of the Lorentz, transverse, and friction forces. For a steady flux flow without a friction, we would have from Eq. (38) . In a system with the Galilean invariance, where and , the sum is the Magnus force. The friction coefficient and the mass per unit length are

The effective mass of a -wave vortex

is much larger than the mass of a conventional vortex [15] obtained in the same way. For temperatures , it is the mass [16] of electrons inside the “vortex core” with the dimensions by . The mass increases with temperature. Note that this mass is very large compared with what is usually obtained by calculating the corrections to the kinetic energy of a moving vortex proportional to the second power of its velocity including the electromagnetic energy. For example, the recent result of Ref. [17] is roughly by the factor of smaller! We have to stress that this mass appears as a response to a slow acceleration. With an increase in the characteristic frequency of the vortex motion, the response transforms into a highly dissipative resonant behavior where a mass is not an appropriate quantity.

#### iv.2.2 High frequency: resonances.

Due to the poles in , the dissipative component of the response has a real part even for . For , the absorption is concentrated near with the line width :

Averaging over the Fermi surface, i.e., integration over , gives rise to the van Hove singularities in conductivity at frequencies where , i.e., where .

For , we have

(39) |

The conductivity is exponentially small for low frequency . It starts to rise at the absorption edge . The average in the r.h.s. of Eq. (39) is equal to for large frequencies , when transitions between many resonance modes are excited. This is equivalent to the limit for a steady flow.

The absorption occurs at the collective modes of electrons involved into the vortex motion. For temperatures , these modes are transitions between the true quantum mechanical energy levels determined by Eq. (7). For higher temperatures, delocalized quasiparticles are involved into the vortex motion, and the resonances take place at collective modes with energies of the order of .

## V Summary

For a steady vortex motion, one can identify three regimes in the order of increasing : (i) The moderately clean regime with . In this case, the main response of vortices to the external current is dissipative with only a small Hall angle proportional to . The conductivities behave similarly to the -wave case, see Eq. (33). Next comes the superclean limit . However, in -wave superconductors, it further separates into two sub-regimes. The relevant parameter is the energy of collective modes excited by the moving vortex. The vortex dynamics in a -wave superconductor depends crucially on the behavior of excitations near the gap nodes. Due to the presence of gap nodes, only the excitations with very low energies are localized in a vortex core. The excitations with higher energies, coming into play for temperatures above , are actually the collective modes where both classically localized and delocalized particles participate. The interplay between the energy of these collective modes and the relaxation rate or the frequency of vortex oscillations determines which of the two sub-limits is realized: (ii) Intermediate or “universal” regime with but . Here both dissipative and Hall components of conductivity tensor are universal. They are independent of the relaxation time and are only determined by the magnetic field and by the number of electrons and holes under the Fermi surface in the normal state, Eq. (34). (iii) Extremely clean limit where the dissipative part of the vortex response is small, and the transverse Hall component of conductivity dominates.

Provided the superclean condition is satisfied, the universal regime can be realized for temperatures under the condition , i.e., . This condition for the universal regime was predicted in Ref. [1]. However, as was noticed in Ref. [6], the universal regime is not restricted to this temperature range. It can also be realized for higher temperatures if . For these temperatures, the dissipative conductivity vanishes only in the extreme clean limit .

The temperature and magnetic-field dependences of conductivities in the extremely clean limit are quite unusual. Extremely clean limit can be reached by increasing the magnetic field above for or above for temperatures . At the crossover from universal to extremely clean regime, the Hall conductivity grows by the factor while the Ohmic conductivity decreases, Eq. (35). For temperatures