Mass of Abrikosov vortex in high-temperature superconductor YBaCuO

Mass of Abrikosov vortices defied experimental observation for more than four decades. We demonstrate a method of its detection in high-temperature superconductors. Similarly to electrons, fluxons circulate in the direction given by the magnetic field, causing circular dichroism. We report the magneto-transmittance of a nearly optimally doped thin YBaCuO film, measured using circularly polarized submillimeter waves. The circular dichroism emerges in the superconducting state and increases with dropping temperature. Our results confirm the dominant role of quasiparticle states in the vortex core and yield the diagonal fluxon mass of 2.2 x 10^8 electron masses per centimeter at 45 K and zero-frequency limit and even larger off-diagonal mass of 4.9 x 10^8 electron masses per centimeter.


Magneto-transmittance of circularly polarized light
We designed and developed a unique far-infrared (FIR) transmission experiment capable of probing the circular dichroism (see Fig. 1). The breakthrough that has eventually allowed us to conduct this research is a custom-made retarder 20 inserted in the optical path near the FIR-laser output aperture; it delays the horizontal polarization relative to the vertical one by an adjustable phase difference. A mutual phase shift of ±π/2 converts the THz beam of equal vertical and horizontal polarization components into the circularly polarized state. Since the phase delay introduced by the retarder is inversely proportional to the wavelength of the incoming light, each laser line requires a separate adjustment.
We measured the transmittance of the sample for several laser lines at wavelengths 119, 163, 312, 419, and 433 µm 21 , corresponding to the terahertz circular frequencies ω of 15.9, 11.6, 6.05, 4.50, and 4.35 ×10 12 rad/s, respectively. Our setup shown in Fig. 1a enables fast flips between clockwise (+) and anti-clockwise (−) circular polarizations, consequently, we probed the transmittance T + and T − of both polarizations under identical conditions. The transmittance, i.e., the fraction of laser energy transmitted through the sample, is evaluated as the bolometerto-pyrodetector signal ratio, effectively eliminating any possible time instability in the laser power. Identical profiles of both signals confirm that the transmittance is measured in the linear regime.
Our experimental protocol is as follows: in experimental runs, we apply a magnetic field at a temperature well above T c . As the temperature drops, vortices freeze into a regular Abrikosov lattice. The applied magnetic field B is kept constant since any change in B would result in inhomogeneous patterns of the vortex density. The temperature T is swept down and up at a steady sweep rate of 2.5 K/min, the instantaneous temperature of the sample being recorded together with the transmittance. Care is taken that the down and up sweeps do not show any hysteresis. Fig. 1c displays a typical temperature behavior of the transmittance observed in the external magnetic field of 10 T using a 312 µm laser line. The results obtained with different laser lines are similar. As expected, the transmittance measured in zero field does not show any dichroism, in absence of Abrikosov vortices to induce an asymmetry. In nonzero fields, however, the low-temperature circular dichroism is clearly observed and can be attributed to the formation of vortices threading the sample. The effect is enhanced in higher applied magnetic fields, thanks to the growing areal density of Abrikosov vortices (see Fig. 1d). We focus on the low-temperature region, T < 50 K, where the thermal quasiparticles play a negligible role and the response of the system is fully governed by the motion of fluxons.

Sample specification by time-domain terahertz spectroscopy
We chose the most common high-T c material YBa 2 Cu 3 O 7−δ in the form of a thin film with a thickness of L = 107 nm, and with CuO 2 planes parallel to the surface. The sample was prepared at National Chiao Tung University (Taiwan) using a pulsed laser deposition method from a stoichiometric target on a lanthanum aluminate substrate oriented in the (100) plane. The substrate dimensions are 10 × 10 × 0.5 mm 3 . Several measurements were performed to establish sample properties. Figure 2 summarizes some relevant results.
The critical temperature of the film, T c = 87.6 K, was determined from a dedicated measurement of dc resistivity ρ (Fig. 2a). For our slightly underdoped sample, this T c corresponds to the hole density 22 n 0 = 1.68 × 10 27 /m 3 . The linear slope of ρ extrapolated to zero temperature shows negligible residual resistivity, so that the relaxation time is inversely proportional to temperature. Additional film properties were established in a separate experiment using standard time-domain THz spectroscopy. Broadband linearly polarized THz pulses (0.3-2.5 THz) were generated by exciting an interdigitated LT-GaAs emitter with a Ti:sapphire femtosecond laser beam at 800 nm 23 . We measured the complex conductivity σ = σ + iσ for frequencies ω/2π in the range 0.5-2 THz at temperatures from 4 to 100 K (Figs. 2b,d,e). At the zero magnetic field and below T c , the two-fluid model fit confirms the dominant London contribution, σ 0 ≈ ine 2 /(mω), with a temperature-dependent condensate density n = n 0 (1 − T 4 /T 4 c ), the hole mass m = 3.3 m e , and the elementary charge e. Comparing the conductivities at temperatures of 4 K and T N = 100 K in Fig. 2d, we found the relaxation time τ N = 5 × 10 −14 s. Conductivities at all temperatures from 4 to 100 K are consistent with τ = τ N T N /T . Our sample is moderately clean. Its purity is given by the lifetime measured on the energy scale 18 as k 2 B T 2 c τ /hE F , where k B is the Boltzmann constant andh is the reduced Planck constant. The Fermi energy E F =h 2 k 2 F /2m depends on the hole doping. For the hole density of our sample, the Fermi surface is cylindrical rather than spherical, and the Fermi momentum follows from the 2D density of holes in the CuO 2 plane n 2D = n 0 c/2 = k 2 F /(2π), where c = 11.68Å is the YBaCuO lattice parameter in the z-direction. The resulting Fermi energy E F = 71 meV yields the value of k 2 B T 2 c τ /hE F ∼ 0.1, corresponding to the moderately clean sample.
According to Kopnin and Vinokur 18 , the moderately clean d-wave superconductor behaves as the s-wave one. In the absence of reliable formulas for angular frequency ω 0 of quasiparticles bounded in the vortex core in the d-wave superconductors, we usedhω for the conventional superconductors 24 . We deduced the coherence length ξ 0 from the upper critical field in the zero-temperature limit 25 , B c2 = 122 T = Φ 0 /2πξ 2 0 , where Φ 0 is the magnetic flux quantum. From the energy gap 2∆ 0 = 4.3k B T c , we obtained ω 0 = 4.4 × 10 12 rad/s, a value close to 4.5 × 10 12 rad/s of our 419 µm laser line.
To complete the sample characteristics, we assume pinning of vortices by layer imperfections, for example, the surface roughness. Figure 2e shows the conductivity in the magnetic field of 7 T applied perpendicularly to the film 26,27 . It was interpreted either with the theory specified below or with the model used by Parks 28 , both indicating that the vortex pinning is rather weak with the Labusch coefficient κ ≈ 2 × 10 5 N/m 2 .

Theoretical prediction
Our aim is to compare experimental values of T + /T − with a theoretical model. Using the above sample parameters, we can evaluate the film conductivity σ ± from which the transmittance T ± results. The theoretical prediction shown in Fig. 3 is based on the Yeh formalism 29 and covers interferences in the weakly birefringent substrate. For the purpose of discussion, we refer to an approximation T + /T − = |σ − | 2 /|σ + | 2 which differs from the exact theory by less then 4% as shown in the Supplementary Information 30 .
Kopnin and Vinokur 31 provided the theoretical conductivities σ ± derived under very general conditions. Our study allows for two simplifications. First, we focus on temperatures about 45 K, where we observe a large dichroic signal. At such low temperatures, extended quasiparticles are very dilute so that we can neglect their contribution to the electric current J as well as their effect on vortices. Second, the cyclotron frequency ω c = eB/m is small on the scale of the quasiparticle lifetime, ω c τ 1 for all experimental magnetic fields, which simplifies dynamics of quasiparticles in the vortex core. Under these conditions, the equation of motion for a fluxon of unitary length takes the form of Newton's law 31 with the time derivative of momentum p and the force F from the interaction of the vortex core with the crystal lattice. Unlike Kopnin and Vinokur, we include the pinning force with the Labusch parameter κ and vortex displacement u related to its velocity asu = v. The Magnus force, given by the vector product of the magnetic field direction z = B/B and the vortex velocity related to the condensate current, covers a force by which the flowing condensate acts on the fluxon. In Kopnin's model, the fluxon momentum p is a total momentum of quasiparticles in the vortex core which rotate about the vortex axis at an angular frequency ω 0 . For our moderately clean sample, the fluxon momentum at the low-temperature limit depends on its velocity as 18 The diagonal mass µ is complex at finite frequencies, which reflects a delay between a change of the vortex velocity and a change of the total momentum of quasiparticles in its core. The off-diagonal mass µ ⊥ describes a property common in anisotropic systems that the velocity of an excitation is not parallel with its momentum. Quasiparticles disturbed by the vortex motion and action of the FIR light eventually lose their momentum in collisions with impurities and phonons. Via these collisions, the crystal lattice acts on the fluxon by force F . With the collision integral approximated by a single relaxation time τ , the force and the momentum are simply related by reduces the Magnus force giving the Kopnin-Kravtsov force 19 in the dc limit.
The electric field in the film 31-33 where e ± = (x ± isy)/ √ 2 are vectors of the helical basis and s = sign(B · k) reflects the parallel or the anti-parallel orientation of the applied magnetic field B with respect to the wavevector k of incoming light. The eigen-vectors satisfy [e ± × z] = ±ise ± , therefore, each of the vector equations (1) and (3) splits into two independent scalar equations for (+) and (−) polarizations. One can easily eliminate the velocity v and write the conductivities needed for the theoretical prediction of the transmittance.

Experiment versus theoretical prediction
With the full set of sample parameters established from the time-domain spectroscopy, the simplified Kopnin-Vinokur conductivity (4) furnishes us with the theoretical prediction of the circular dichroism. Figure 3 compares this prediction and the experimetally observed dichroism for several values of the applied magnetic field and the THz-laser wavelength. The differences between theory and experiment are smaller than the experimental errors.
With increasing wavelength and field strength, the transmittance ratio T + /T − gradually deviates from unity. The observed trends can be understood in terms of Eq. (3). The field dependence arises from the dominant London contribution 1/σ 0 complemented by the Josephson-type resistivity, which is linear in B. The variation with wavelength has a similar cause: for lower frequencies, the London resistivity 1/σ 0 ≈ −iωm/(ne 2 ) is smaller so the Josephson part becomes dominant. Figure 3 documents that the theory of Kopnin and Vinokur is relevant for the THz dynamics of vortices. Since the observed frequency dependence of magneto-transmission agrees with the theoretical one, even with no adjustable parameters, the extrapolation of our results to low frequencies is justified. Based on this, we used their theory to find the vortex mass from our data.

Vortex mass from experiment
While the sample parameters n, σ 0 , and τ are sound, κ and ω 0 are less clear. The Labusch parameter κ has a minor effect on the dichroism, therefore we kept the value κ = 2 × 10 5 N/m 2 . The angular frequency ω 0 was established by the least-square fit of T + /T − data. The best-fit value ω 0 = 4.3 × 10 12 rad/s was very close to 4.4 × 10 12 rad/s estimated above. We believe that such close agreement of observed and estimated frequency is fortuitous.
At THz frequencies, where the circular dichroism was found, both diagonal and off-diagonal masses are complex. They become real in the low-frequency limit, as apparent from Eq. (2). Using the experimentally established values of ω 0 = 4.3 × 10 12 rad/s and other sample parameters, we evaluated the zero-frequency components of the fluxon mass. In YBa 2 Cu 3 O 7−δ at 45 K, the diagonal mass µ amounts to 2.2 × 10 8 m e /cm, while the off-diagonal mass µ ⊥ is more than twice larger, 4.9 × 10 8 m e /cm.
In summary, we have developed a reliable experimental method to measure the circular dichroism of superconducting films threaded by Abrikosov vortices. To interpret our data in terms of vortex dynamics, we have established all the essential material parameters from independent time-domain THz spectroscopy and dc-resistivity measurements. The observed dichroism is in good agreement with the theory of Kopnin and Vinokur based on the circular motion of quasiparticles in the vortex core. Their angular frequency was experimentally determined and used to extrapolate the vortex mass from THz frequencies to low-frequency motion. control the transmittance of the sample via the following mechanism: The electric field of the laser light drives the supercurrent. The Magnus force accelerates vortices in the direction perpendicular to the supercurrent; in reaction, the vortex motion affects the supercurrent and, thus, the transmittance. In the sketch, the electric field in the sample, as well as the vortices, rotate clockwise. If the light frequency is close to the cyclotron frequency of vortices, the motion of vortices is resonantly enhanced, leading to the observed dichroism. The extent of the cyclotron motion is strongly exaggerated; in fact, the fluxon circulates on a radius of less than 10 −12 m at the strongest laser line.

B B
(c) Transmittance of the YBa 2 Cu 3 O 7−δ superconducting sample, normalized to the normal-state transmittance T N at 100 K and plotted for two circular polarizations versus temperature. The dichroism is clearly visible below 70 K in a magnetic field of 10 T; in a zero field, no dichroism appears. (d) Transmittance ratio T + /T − measured in several applied magnetic fields plotted versus temperature. The data were obtained using a 312 µm laser line (6.1 × 10 12 rad/s). Above the critical temperature, the dichroism is absent, showing that the normal-state Hall component is negligible. For convenience, the x-axis was chosen parallel to the linear polarization of the ordinary ray. At a low temperature of 20 K, we established the substrate thickness D = 513.5 µm and dispersion of ordinary and extraordinary refractive indices shown in Figure S1. Measurements at temperatures up to 100 K did not reveal any appreciable deviation from the low-temperature values. When calculating the sample transmittance, it is necessary to combine two different approaches. Circular dichroism in YBaCuO is naturally described within a vector basis related to the circular polarization of the laser beam. On the other hand, the birefringence of the substrate is conveniently represented in a linear basis associated with the anisotropy axes of LAO. To match these two basis we employ Yeh's 4 × 4 matrix algebra [1][2][3][4] .
In each of the segments, vacuum|YBaCuO|LAO|vacuum, we write the resulting electric field E(z, t) = E(z)e −iωt as a sum of four partial waves 1, 3 where e j stand for eigen-polarization vectors and k j for wavevectors. Partial waves propagating forward are indexed by odd numbers j = 1, 3 and waves propagating backward by even numbers j = 2, 4. In vacuum and YBaCuO, the eigen-polarization vectors e j are given in the helical basis as e 1,3 = e 2,4 = (x ± iy)/ √ 2. In LAO, we use the linear basis e 1,2 = x and e 3,4 = y. Now we collect the amplitudes E j of partial waves into a four-component column vector and express the light propagation in a matrix form as connects the incident electric field with the transmitted one via a sequence of dynamical and propagation matrices. The propagation matrices are diagonal. For the YBaCuO film, we have (P film ) ij = δ ij e ik j L with k 1,2 = ±n + ω/c and k 3,4 = ±n − ω/c, where c is the vacuum speed of light and