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Linear and nonlinear low frequency electrodynamics of the surface superconducting states in an yttrium hexaboride a single crystal

We report the low-frequency and tunneling studies of yttrium hexaboride single crystal. Ac susceptibility at frequencies 10 - 1500 Hz has been measured in parallel to the crystal surface DC felds, H0. We found that in the DC feld H0>Hc2 DC magnetic moment completely disappears while the ac response exhibited the presence of superconductivity at the surface. Increasing of the DC field from Hc2 revealed the enlarging of losses with a maximum in the feld between Hc2 and Hc3. Losses at the maximum were considerably larger than in the mixed and in the normal states. The value of the DC field, where loss peak was observed, depends on the amplitude and frequency of the ac feld. Close to Tc this peak shifts below Hc2 which showed the coexistence of surface superconducting states and Abrikosov vortices. We observed a logarithmic frequency dependence of the in-phase component of the susceptibility. Such frequency dispersion of the inphase component resembles the response of spin-glass systems, but the out-of-phase component also exhibited frequency dispersion that is not a known feature of the classic spin-glass response. Analysis of the experimental data with Kramers-Kronig relations showed the possible existence of the loss peak at very low frequencies (<5 Hz). We found that the amplitude of the third harmonic was not a cubic function of the ac amplitude even at considerably weak ac felds. This does not leave any room for treating the nonlinear effects on the basis of perturbation theory. We show that the conception of surface vortices or surface critical currents could not adequately describe the existing experimental data. Consideration of a model of slow relaxing nonequilibrium order parameter permits one to explain the partial shielding and losses of weak ac field for H0>Hc2.


I. INTRODUCTION
There is a growing interest in exploring physical properties of materials involving boron-cluster compounds because of a wide variety of applications [1]. Due to sp 2 hybridization of valence electrons, large coordination number and short covalent radius, boron atoms prefer to form strong directional bonds with various elements. A large number of experimental and theoretical studies are concentrated on the families of compact B 12 icosahedrons and B 6 octahedrons with a large diversity of electrical and magnetic characteristics. The highest critical temperatures of the transition to the superconducting state in MB 6 and MB 12 compounds were found in YB 6 with T c ≤ 8.4 K and ZrB 12 with T c ≈ 6.0 K [2]. Both materials have a highly symmetrical crystal structure (CaB 6 type for YB 6 and UB 12 type for ZrB 12 ) that can be described as boron cages in which yttrium or zirconium atoms develop large vibrational amplitudes with an Einstein-like (nearly dispersionless) lattice mode. In spite of some common features, the two crystals have a few distinct physical characteristics: (i) while YB 6 is a classical type-II superconductor [3], ZrB 12 (at least, for temperatures above 4.5 K) may be regarded as a textbook example of type-I superconductor [4]; (ii) while the superconducting properties are enhanced at the ZrB 12 surface [4], they are suppressed in a YB 6 surface (see our tunneling data below). Therefore, ZrB 12 and YB 6 samples may serve as model systems for investigating surface-related superconducting effects.
Nucleation of a superconducting phase in a thin surface sheath, when the DC magnetic field, H 0 , parallel to the sample surface decreases, was predicted in 1963 by Saint-James and de Gennes in their seminal work [5]. They showed that the nucleation occurs for H 0 < H c3 = 2.39κH c , where H c is the thermodynamic critical field and κ is the Ginzburg-Landau(GL) parameter. Experimental measurements confirm this prediction [6,7,8,9,10,11,12], and it was found that at low frequencies a sample in a surface superconducting state (SSS) shows ac losses with a peak whose position with respect to the DC field depends on the ac amplitude. The peak magnitude exceeds the losses observed either in the normal state (H 0 > H c3 ) or in the bulk superconducting state(H 0 < H c2 ). It was also predicted that the H c3 /H c2 ratio, is temperature independent. In contrast, a decrease of this ratio was found in the vicinity of T c in several experiments [11,12]. This behavior was associated with the distribution of T c at the surface [13].
In the last few years the SSS has attracted renewed interest from various directions [4,14,15,16,17,18]. Stochastic resonance phenomena in Nb single-crystal were observed in the SSS [14]. In [17] it was assumed that at H 0 ≤ H c3 the sample surface consists of many disconnected superconducting clusters and subsequently the percolation transition takes place at H c c3 = 0.81H c3 . The paramagnetic Meissner effect is also related to the SSS [18]. Voltage noise and surface current fluctuations in Nb in the SSS have been investigated [15]. SSS' were found also in single crystals of ZrB 12 [16]. In agreement with the previous data [9] it was demonstrated [16] that the waveform of the surface current in an ac magnetic field has a non-sinusoidal character. A simple phenomenological relaxation model provides the good explanation of the experimental data for DC fields near H c2 only [16]. The relaxation rate in this model depends on the ac frequency and decreased with decreasing ω [16]. Detailed experimental study of the linear ac response in the SSS of single crystals Nb and ZrB 12 was published recently in our paper [4]. We showed that ac SSS losses in these materials could be considered in the achieved experimental accuracy as a linear ones and for several DC fields the real part of the ac magnetic susceptibility exhibited a logarithmic frequency dependence as for a spin-glass system.
In spite of the extensive studies, the origin of low frequency losses in SSS is not clear as yet. The critical state model developed for the SSS in [19] implies that if the amplitude of the ac field, h 0 , is smaller than some critical value the losses disappear. The authors of Ref. [9] claimed that the experiment on Pb-2%In alloy confirms this prediction. On the other hand, the observed response [17] for an excitation amplitude of 0.01 Oe that is considerably smaller than used in [9] showed losses in SSS in Nb sample at a frequency 10 Hz. Our measurement on Nb and ZrB 2 single crystals also have shown that the out-of-phase part of the ac susceptibility, χ ′′ 1 , was finite at low excitation level [4]. We consider these results as an indication of the inadequacy the critical state model for description of the ac response in SSS. If we assume that the reason for this discrepancy with experimental data, is the small value of the critical surface current, which is much smaller than the current amplitudes at the surface, then we have to expect a decreasing of the losses approximately as 1/h 0 when the amplitude of the applied ac field, h 0 , increases. On the contrary, χ ′′ 1 increases with h 0 . The ac investigation of YB 6 samples has some advantage due to actually ideal type II magnetization curves in this material that permits one to avoid possible difficulties in the interpretation of the experimental data. The experiment showed that near the transition temperatures SSS exist also in the fields below H c2 . We found that some features of nonlinear response took place at a very weak ac field with amplitude ≃ 0.005 Oe. The ac response at the third harmonic of the fundamental frequency did not leave any room for the perturbation theory. It was proposed that the losses in SSS are due to the slow relaxation of the order parameter at the surface and could not be ascribed to surface vortices. We found that for small h 0 in quasilinear approximation the integral equation with power dependent nuclear governed the time behavior of the magnetization in ac fields. Some features of the ac response resemble the ones of spin-glass system but one has to note that SSS present a different system with its unique properties.

A. Sample preparation
The yttrium hexaboride single crystal was grown by the inductive floating zone method of a powder sintered rod with an optimal composition YB 6.85 under 1.2 MPa of argon. According to the Y -B phase diagram, composition with the Y:B=1:6 ratio has undergoes peritectic melting [20] and irrespective of the Y/B ratio the YB 4 single crystal with preferential orientation [001] begins to grow. After enrichment of the melting zone by boron (flux method modification) the yttrium hexaboride single crystals with the [100] orientation grow with the composition of YB 5.79±0.02 (ESD). The total impurity concentration is less than 0.001 % in weight and the obtained lattice parameter is 4.1001(4)Å in accordance with published data [21]. These single crystals exhibited a sharp superconducting transition with T c ≈ 7.15 K.

B. DC and ac measurements
The magnetization curves were measured using a commercial SQUID magnetometer. In-phase and out-ofphase components of the ac susceptibility at the fundamental frequency, and the response at the third harmonic were measured using the pick-up coil method [9,22]. A home-made setup was adapted to the SQUID magnetometer, and the block diagram of the experimental setup was published in Ref. [16]. The crystal (10 × 3 × 1 mm 3 ) was inserted into one of a balanced pair of coils. The unbalanced signal and the third harmonic signal as a function of the external parameters such as temperature, DC magnetic field, frequency and amplitude of excitation, were measured by a lock-in amplifier. The experiment was carried out as follows. The sample was cooled down at zero magnetic field (ZFC). Then the DC magnetic field, H 0 , was applied. The amplitudes and the phases at all frequencies of both signals were measured in a given H 0 (including at zero field). The excitation amplitude,h 0 , was 0.0005 ÷ 0.5 Oe. It is assumed that in H 0 = 0, and at low temperature, the ac susceptibility equals to the DC susceptibility in the Meissner state with negligible losses. This permits us to find the absolute values of the in-phase and out-of-phase components of the ac susceptibility for all applied DC and ac fields and for all frequencies. Both H 0 and h 0 were parallel to the longest sample axis.

C. Tunneling measurements
Measurements of the tunneling spectra were carried out using a home made scanning tunneling microscope. The YB 6 single crystal was mounted inside the cryogenic scanning tunneling microscope and then cooled down to 4.3 K. The dI/dV vs. V tunneling spectra (proportional to the local density of states) were acquired using a conventional lock-in technique, while momentarily disconnecting the feedback loop.

A. Tunneling characteristics
Direct information about the energy gap value ∆ 0 = ∆(T = 0) at the surface of YB 6 was obtained from the tunneling spectra. The ratio 2∆ 0 /T c is a well known indicator of the electron-phonon coupling strength [24]. Two previous tunneling studies of YB 6 were performed on a single-crystal [3] and on thin films [23] and the ∆ 0 obtained were: 1.22 [3] and 1.24 [23] which yields the ratio 2∆ 0 /T c ≈ 4. In both cases the tunneling contacts were connected to underlying layers, and hence, monitored bulk properties. Therefore those values which signified a nearly strong coupling are attributed to the bulk characteristics. It was confirmed, in particular by Lortz et al. [25], who measured the deviation function 2 ) and found that the value of 2∆ 0 /T c is slightly above 4.0. Our tunneling spectroscopy results were obtained by STM and therefore better reflect the density of states at the surface. In contrast to our previous measurements on ZrB 12 single crystals that showed very high spatial homogeneity [4], the superconductivity in the present case appeared to be degraded on parts of the YB 6 sample surface, where nearly featureless tunneling spectra were observed. In other regions, however, reproducible ratios of differential conductances in superconducting and normal states (dI/dV ) s /(dI/dV ) n showing very clearly that BCS-like gap structures were acquired, such as presented in Fig. 1 (solid line). The spectra were compared with a temperature-smeared version of the Dynes formula [26] wich takes into account the effect of incoherent scattering events by introducing a damping parameter Γ into the conventional BCS expression for a quasiparticle density of states A very good fit to the experimental data (except for a small asymmetry in the normal resistance between negative and positive bias, the origin of which is not yet clear to us) was achieved with ∆(4.3K) = 1.0 meV and Γ = 0.10 meV. Recalling that the experimental spectrum was acquired at T = 4.3 K, which is about 0.6T c , with the BCS ∆(T ) dependence [27] we obtain the zerotemperature value ∆(T = 0) = 1.1 meV. With that, we find that 2∆ 0 /T c ≈ 3.59, very close to the BCS weak coupling value of 3.53. In contrast to ZrB 1 2, we assume that in YB 6 the electron-phonon strength is suppressed at the surface to a weak coupling state. From the hysteresis curve measured at 4.5 K, shown in Fig. 3, we are able to evaluate H c = 295 Oe, H c2 = 1500 Oe and GL parameter Fig. 3. The London penetration depth at T=0, λ L (0), can be estimated by using H c2 (T ) near T c , dH c2 /dT ≈ −560 29], and λ L (0) ≈ 1.4 × 10 −5 cm. Fourier analysis of the magnetization, M (t), under applied ac and DC fields, H(t) = H 0 + h 0 cos(ωt), yields an expression:

B. DC and ac magnetic characteristics
. In this paper we discuss the results for χ 1 and χ 3 susceptibilities. The field dependence of χ 1 (H 0 ) at T = 4.5 K and h 0 = 0.05 Oe for some frequencies is shown at Fig. 4. One can readily see that the curves shift toward higher DC fields with frequency. Decreasing the ac amplitude produces a similar effect. The curves shift to the higher field when h 0 −→ 0 ( Fig. 5). Similar effects were reported for a Pb-2%In sample in Ref. [9]. The typical magnetic field dependence of the nonlinear response, χ 3 , is presented at Fig. 6 for h 0 = 0.05 Oe and various frequencies. When the frequency increases the maximum in χ 3 moves toward larger DC fields as was observed for χ ′′  shielding, as well as Fig.  4). For H 0 > H c2 both χ ′′ 1 and χ 3 decrease as the frequency increases while for H 0 close to H c3 they increase. Fig. 8 shows the field dependence of χ 3 at ω/2π = 1465 Hz and various amplitudes of excitation, h 0 . The third harmonic cannot be adequately described in the frame of the perturbation theory which predicts that χ 3 ∝ h 2 0 . For example, at H 0 /H c2 = 1.3, χ 3 depends on h 0 strongly, while at H 0 /H c2 = 1.45, χ 3 is almost constant (see Fig. 8). We can discuss only the dependence of χ 3m (defined as the maximum value of the χ 3 (H 0 ) curve for any given frequency) on the ac amplitude h 0 . Fig. 9 demonstrates that χ 3m ≈ h 0.2 0 in contrast to what the perturbation theory predictions.
Below we consider the experimental results obtained   at higher temperatures. Fig. 10 demonstrates the field dependence of χ 1 at frequency ω/2π = 1065 Hz and h 0 = 0.05 Oe at various temperatures. The peak in χ ′′ 1 shifts toward H c2 with temperature and at 7 K this peak is located already below H c2 . One can see in the Fig. 10 that for T < 7 K full shielding (χ ′ 1 = −1/4π) is observed at low H 0 , whereas at 7 K only partial shielding is observed at low DC field. Also Fig. 11 shows that in the vicinity of T c χ 3m lies below H c2 . Because we did not observed any absorption peak and harmonic signal in the mixed state we consider that SSS are responsible for the experimental observations at T = 7 K too. Existence of the SSS below H c2 was predicted by H. Fink in 1965 [30].  Increasing the DC field we can reach the field at which χ 1 or χ 3 becomes zero. This field can be considered as the third critical magnetic field H c3 . Both conditions actually give the same value of H c3 . The experiment shows that the H c3 /H c2 ratio decreases with temperature.

IV. THEORETICAL MODEL
Let us consider a superconducting slab of thickness 2L in the parallel to its surface external DC and ac magnetic fields. Due to the considerably short relaxation time of the order parameter [27,28] one can use the stationary GL equations. We choose the coordinate system in which the x-axis is perpendicular to the slab sur-face, the plane x = 0 is in the center of the slab, and the external magnetic field is directed along the z-axis. Looking at the dimensionless order parameter in the form Ψ(x, y, t) = φ(x, t) exp(iky) the GL equation can be written as: Here a is a y-component of the dimensionless vector potential. The order parameter is normalized with respect to the absolute value of the order parameter in zero field, the distances with respect to the coherence length at zero temperature, ξ 0 , (x −→ x/ξ 0 , y −→ y/ξ 0 , l = L/ξ 0 ) and the vector potential with respect to c/2eξ 0 (a = A/( c/2eξ 0 )). The boundary conditions for calculation of surface states are φ(0, t) = dφ(±l, t)/dx = 0 and a(0, t) = 0, da(±l, t)/dx = h(t), where h(t) is the dimensionless applied magnetic field. These nonlinear equations can be solved by numerical methods. We add the time derivative ∂φ/∂t into the right side of Eq. (2) and seek the stationary solutions of the Eqs. (2,3). Replacing the space derivatives by finite differences on the grid with step dx = l/N Eq. (2) transforms into N first order differential equations. The solution of the obtained linear algebraic system can be found by regular method. The grid with N=1000 points was used. In the surface state the order parameter differs from zero only near the surface, at a scale of several coherence lengths, ξ(T ). Actually, the choice L = 5ξ(T ) ≡ D provides good accuracy for calculating φ. The real dimensions of the investigated samples, L, considerably exceed this scale by 3-5 orders of magnitude. Parameter k is not a gauge invariant quantity and we choose it using conditions a = 0 at x = 0. In SSS the magnetic field in the bulk is constant. So we can obtain k for a thick slab with L ≫ 5ξ(T ) from the solution of the problem for a thin slab with D ≥ 5ξ(T ) by gauge transformation and vector potential in the surface layer Here d ≡ D/ξ 0 , index s corresponds to the problem for a thin slab, and h zs is the z-component of magnetic field in the center of the thin slab. This note is important for numerical calculations.

V. DISCUSSION
It is well known (see, for example, [16]) that for a given external magnetic field there is a whole band of k for which surface solutions exist. These solutions describe the nonequilibrium states and only one solution corresponds to the equilibrium state, for which the magnetic field inside the bulk equals its external value and the total surface current, J s , equals zero. Parameter k is an integral constant of the nonstationary GL equations. That is, k is time independent, in contrast to φ, in the frame of the GL model. The relaxation time of the order parameter φ is considerably shorter than any ac period in our experiment. So when the external magnetic field is changing during the ac cycle, one may expect that φ follows the instantaneous value of the magnetic field and k remains approximately constant. Let assume that starting from an equilibrium state in some DC field, H 0 , we increase the external magnetic field but simultaneously hold k constant. In this case the surface current J s becomes different from zero. It is possible to consider two definitions of the surface critical current J s1 and J s2 [19,34]. The first definition of such a critical current is J s1 = (c/4π)dh s1 , where dh s1 = H 1 − H 0 and H 1 is the field for which the energy of the surface superconducting state equals the energy of the normal state [19]. The second definition is J s2 = (c/4π)dh s2 , where dh s2 = H 2 − H 0 and H 2 is the field for which SSS disappears. The quantities dh s1 and dh s2 have different values and different dependencies on the thickness of the sample, L. While dh s1 dramatically depends on L, dh s2 for L > 1000ξ actually does not. The value of dh s2 is considerably larger than dh s1 for large L. This difference is due to the large contribution of the magnetic field to the system energy, if the magnetic field in the bulk differs from the external field. These features are shown in Figs. 12a, 12b, where dh s1 and dh s2 are presented as a function of the DC magnetic field for different L's at T /T c = 0.9. In the reduced variables dh s1 /H c2 , dh s2 /H c2 , H 0 /H c2 the curves form is actually temperature independent. The assumption of slow relaxing k, permits one to understand qualitatively the effect of complete screening of a weak ac field with amplitude h 0 ≪ H 0 in SSS. Ac surface current J s (k, H) is a function of the instantaneous values of the external magnetic field and k. This function can be calculated for a thin slab of several coherence length thickness and then using the gauge transformation, Eqs.(4 and 5), to get a solution for a thick slab. As a function of k s and H, the J s (k s , H) is a slow function of H. For example, at T = 0.9T c numerical calculation gives ∂hzs(ks,H) Where H zs is magnetic field in the center of the slab. So for a thin slab, an almost complete penetration of the ac field inside the bulk takes place and the value of the surface current is very small. For a thick slab k = k s and the requirement of constant k during the ac cycle, implicitly means that k s also changed according to Eqs. (4 and 5). This leads to considerably large surface currents and to screening of the ac field. In reality, we have large dimensionless parameter L/ξ(T ) that increases ac field screening. Fig. 13 demonstrates the calculated (in the assumption of constant k) χ ′ = ∆M/dh s1 , as a function of the DC field when the external field was  increased by dh s1 . It is evident that for any macroscopically large sample, L ≥ 5000ξ, the complete screening, χ ′ = −1/4π, should be obtained for DC fields excluding fields close to H c3 . However our experiments (Fig. 4) do not confirm this conclusion. We see that χ ′ 1 in the field H 0 ≈ (H c2 + H c3 )/2 already differs from −1/4π. It means that slow relaxation of k takes place which leads to the losses and incomplete screening.
For a given ac amplitude, χ ′′ 1 has a maximum at some values of the DC field defined as H m (see Fig. 5). H m was considered in Ref. [31] as the DC field at which the amplitude of the ac surface current J 0s = (c/4π)h 0 equals approximately to the critical value J s1 . In order to test this in Fig. 14 we show J 0s as a function of H 0 and calculate a critical current J s1 for a slab of thickness L = 5 × 10 5 ξ. Theoretical data of the J s1 were arbitrarily normalized in order obtain the intersection with the experimental curve at H 0 /H c2 = 1.25. While the theoretical dependence of J s1 is almost a linear function of H 0 , the experimental  curve starts from H 0 /H c2 = 1.45 and is a nonlinear function of H 0 . One can conclude that losses observed in our experiment are not connected to the condition h 0 ≈ dh s1 for H 0 > H c2 . The maximal losses, χ ′′ m , at H m , as a function of h 0 is shown at Fig. 15. Inset to Fig. 15 shows that in the limit h 0 → 0 the losses do not disappear. In a linear system χ ′′ 1 should be amplitude independent. While our experiments show a linear dependence χ ′′ m on the ac amplitude (Fig. 15). It does not permit us to consider the response as a linear one even at very low amplitudes of excitation. Therefore more experimental measurements at low ac fields are needed.
In general, the magnetic moment can be presented by For h 0 ≃ 0.02 Oe the susceptibilities at higher harmonics are small and we can rewrite Eq. (6) as considering only the response at the fundamental frequency. Under this approximation, the response at fundamental frequency, matches the Kramers-Kronig relations (KKR): and then where ω 0 and ω m are the minimal and maximal available frequencies in our experiment, respectively, and 0 < ̟ < ω 0 . I(ω) can be calculated from the available experimental data and be presented in the form With c n > 0 we obtain χ ′′ 1 (̟) = πb. Coefficients a, b and c n could be found by least square fit. For ω 2 /ω 2 m << 1 it is sufficient to take into account only a few terms in Eq. (10). Results of this approach are presented at Fig. 16, where χ ′′ 1 (̟) and a as a function of the DC field are shown. The measured data in the frequency range 15-1460 Hz χ 1 at T = 4.5 K, was used for the calculation of I(ω) for 25 < ω/2π < 200 Hz with ω 0 /2π = 17.5 Hz and ω m /2π = 1455 Hz. The approximation of I(ω) by using expression Eq.(10) with n max = 1 produces χ ′′ 1 (̟) curve shown in Fig. 16. Because c n ≈ ω 2 c n−1 /ω m with ω 2 /ω 2 m ≈ 0.02 one could expect that expression (7) with n max = 1 gives the correct result. Taking into consideration the term c 2 gives unphysical result, because c 2 is very small and negative. It is due to the scattering of the experimental data and ignores in Eq. (10) the dependence of ̟ on ω. Fig. 16 shows that the calculated loss peak is approximately 3 times larger than the measured losses at ω/2π > 20 Hz, Fig. 4. Qualitatively this behavior can be explained as follows. Because χ ′′ 1 exhibits a weak frequency dispersion we can estimate integral in the left side of Eq. (9) by In Fig. 17 we showed the correspondence between the estimated R by Eq. (11)  We believe that the observed in SSS losses are the result of the relaxation k to its equilibrium value. This model can ascribe both the partial screening and losses for H 0 > H c2 . The other model assumes that the motion of the of 2D-vortices in the surface sheath [32] is responsible for the losses [33]. These vortices with surface density n s = H 0 sin(θ)/φ 0 appear if the applied field has a normal component to the sample surface H n = H 0 sin(θ), due to misalignment, or alternatively if the surface is not sufficiently smooth. One can estimate the conductivity of the surface layer σ = σ n H c2 /H 0 sin(θ) where σ n is the conductivity in the normal state. In our sample, σ n ≈ 10 17 CGS and the skin depth in the surface layer at frequency ω/2π = 10 Hz is considerably larger for any real angle( ≃ 10 −2 rad) to provide sufficient screening of the ac field by a layer with thickness 10 −5 ÷ 10 −6 cm.
The ac response of SSS resembles that of the spin-glass systems. Real and imaginary parts of χ 1 can be well represented by a polynomial of ln(ω) shown in Fig. 18 for H 0 = 2 kOe and T=4.5 K. In this figure, presentation of χ 1 ′ and χ 1 ′′ by polynomial a 0 + a 1 ln(ω) + a 2 ln 2 (ω) (12) are shown for a considerably wide frequency region 15 < ω/2π < 1465 Hz. For some DC fields the coefficient a 2 is small and one can get the spin-glass like χ ′ 1 . But χ ′′ 1 also exhibits the frequency dispersion that is not typical for spin-glass systems. The "π/2" rule [35], d ln(ω) , is not fulfilled in our data, Fig 19. The simple relaxation models of ac response is applicable only for a DC field near H c2 [16]. If in analogy with a spin-glass system we assume that the magnetization moment of the sample, M (t), can be found from the relaxation equation: with subsequent averaging over the relaxation rates, then where P (ν) is the distribution function of the relaxation rates. Using 1/(ν −iω) = ∞ 0 exp(−(ν −iω)t)dt we transform Eq. (14) to where P (t) = ∞ 0 P (ν) exp(−νt)dν. So, if Eq.(14) describes adequately the experimental data with some P (ν), then these two integrals should be equal each other Experimental data are available only for a finite frequency region 15 Hz < ω/2π < 1465 Hz, while integrals in Eq. (16) are expanded for all frequencies and we have to extrapolate our data to the entire frequency axis. This was done assuming that for ω/2π < 15 Hz and ω/2π > 1465 Hz χ 1 ′′ is a power function of frequency ω p . As a result, the sin-and cos-Fourier transformations in Eq. (16) give different values for P (t) as shown at Fig. 20 where the ac response in H 0 = 1.25H c2 was used. It is readily seen that the experimental data exclude the possibility consider the SSS as an analog of a spinglass system. Equation (7) shows that in the quasilinear approximation the magnetization of the samples satisfied an integral equation It is interesting to notice that the nuclear G(t, h 0 ) can be extracted by the Fourier transformation of 1/χ 1 (ω). Performing the same procedure as above, we obtained that sin-and cos-Fourier transformations in Eq.(11) yield different values for G(t, h 0 ) which is certainly due to the lack of experimental data for whole frequency axis. The extrapolation of the imaginary part of 1/χ 1 (ω) gives more accurate results and we consider only the G(t, h 0 ) that is obtained by the sin-Fourier transformation of 1/χ ′′ 1 (ω). Good approximation of G(t, h 0 ) provides the expression G(t, h 0 ) = A(t)/t q with slow function A(t) for t > π/1465 = t c . For t < t c function G(t, h 0 ) is singular, but integral tc 0 G(t, h 0 )dt has a finite value. The parameters q and A(t), depend on the DC field. For example, in field H 0 = 1.25H c2 q = 0.876 and A(t) = − exp(1.285 − 0.00842 ln 2 (2πt)). In Fig. 21 we show G(t, h 0 ) for some values of the DC magnetic field and the inset presents q versus H 0 . So, the dynamics of SSS is governed by an integral equation with retardation. This feature distinguishes SSS from other known systems.

VI. CONCLUSION
In this paper we have studied the low frequency linear and nonlinear dynamics of the SSS of a single crystal of yttrium hexaboride. The tunneling spectra were studied as well. Tunnel measurements allow us to make the assumption, that in this single crystal, unlike ZrB 12 , near the surface the electron-phonon interaction is suppressed and the situation of weak coupling is realized. We showed that the surface superconducting states define the peculiarities of the low frequency response. In spite of different behavior under magnetic fields (ZrB 12 is a type-I superconductor and YB 6 is that of type-II) and different surface properties the two materials exhibit very similar and universal ac characteristics reflecting the nature of the SSS. In both cases we observed a nonlinear response for very weak ac amplitudes (in experiments with YB 6 h 0 was as small as 0.005 Oe) and the question about the existence of a linear response is open. An extrapolation of the low-amplitude data did not reveal a linear regime. Similar to spin-glass systems (where finite losses at considerably low frequencies exist), the real part of the susceptibility exhibits a logarithmic frequency dependence at some DC magnetic field. But the out-of-phase component has a frequency dispersion. The frequency dispersion in SSS is different from that of the spin-glass systems. The slow relaxation of the phase of an order parameter leads to a frequency dispersion of the ac susceptibility. The analysis of the experimental data by means of Kramers-Kronig relations allow us to make the assumption of the presence of the loss peak at frequencies below 5 Hz.