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The thermally activated motion of vortex bundles and the anomalous Hall effect in type-II conventional and high-Tc superconductors

The anomalous Hall effect in type-II conventional and high-Tc superconductors is investigated based on the proposed novel theory of the thermally activated motion of vortex bundles over a directional-dependent energy barrier. Our calculations demonstrate clearly that anomalous Hall effect is induced by the competition between the Magnus force and the random collective pinning force of the vortex bundle. The Hall as well as the longitudinal resistivity for constant temperature and constant applied magnetic field of type-II superconducting films and bulk materials are calculated. The reentry phenomenon is also investigated. All the results are in good agreement with the experiments.


Introduction
The vortex dynamics in type-II superconductors, discovered by Abrikosov [1], has been focused tremendously theoretic and experimental efforts , especially the most appealing and confusing phenomenon-the anomalous Hall effect (AHE)-in type-II conventional and high-Tc superconductors [14][15][16][17][18][19][20][21][22][23][24][25], since the Hall resistivity was first measured by Niessen and Staas in 1965 [2], later the Hall anomaly was observed by Van Beelen et al. [3]. So far, there is no generally satisfactory explanation for this amazing anomaly, the origin of this Hall anomaly still remains unsolved. In this paper, we develop a novel theory for the thermally activated motion of vortex bundle over the directional-dependent energy barrier formed by the potential generated by the Magnus force and the collective pinning force, and the potential barrier generated by a myriad of randomly distributed strong pinning sites inside the bundle [4,[6][7][8][9]. Applying this theory, we successfully explain the remarkable Hall anomaly. In accordance with our theory, the AHE is indeed induced by the competition between the Magnus force and the random collective pinning force of the vortex bundle.
The rest of this paper is organized as follows. In Section 2, the theory of thermally activated motion of vortex-bundle is developed; the Hall and longitudinal resistivities are calculated. In Sections 3 and 4, the AHE for type-II superconducting bulk materials and thin films are investigated respectively. In Section 5, the reentry phenomenon for the AHE is discussed. In Sections 6 and 7, a general discussion and concluding remarks are conveyed.

The thermally activated motion of vortex bundles and the Hall and longitudinal resistivities
Realizing the fact that quenched disorder destroys the long-range order of the flux lattice (FLL), after which only short-range order, the vortex bundles, prevails [5][6]. In this section we develop a theory for thermally activated motion of vortex bundles over the directionaldependent energy barrier formed by the Magnus force, the random collective pinning force, and the potential barrier generated by the randomly distributed strong pinning sites inside the bundle. The energy barrier is directional-dependent due to the presence of Magnus force and random collective pinning force, namely, the energy barrier is different when the direction of activated motion is different. Our results demonstrate that the AHE is induced by the competition between Magnus force and random collective pinning force of the vortex-bundle.

Calculation of the coherent frequency of the vortex bundle by random walk theorem
It is well understood that [10] the vortex line oscillates inside the potential barrier due to thermal agitations. The thermal oscillation frequency of the individual vortex inside the potential barrier, by identifying the oscillating energy of the vortex line with the thermal energy, can easily be obtained as, , where A is the average amplitude of the oscillation, B k is the Boltzmann constant and m is the mass of the vortex line [11]. The oscillations of each individual vortex inside the vortex bundle are not coherent, namely, their oscillations are at random. In order to obtain the coherent oscillation frequency C ν of the vortex bundle as a whole, the frequency ν in Eq. (2.1) must be divided by the square root of the number of vortices in the bundle N , with 0 Φ is the unit flux, R is the transverse size of the vortex bundle, B is the value of the applied magnetic field. The essential property can be comprehended by considering the problem of random walk. Let 0 l be the length of each individual step, for a walk with totally N steps, if the walk were coherently in the same direction, the total length of N steps would be 0 l N L = ; however, these steps are not coherently in the same direction, they are at random.
With the aim of receiving the length after N steps it must be divided the above expression by the square root of N, which is the desired answer for the random walk with N steps.

Evaluation the root-mean-square of the angle between the random collective pinning force and positive y-direction
Let us consider the case for p-type superconductors with current in the positive x-direction and the magnetic field in the positive z-direction, if we assume that the angle between the random collective pinning force of a vortex line and the positive y-direction measured in counterclockwise sense is θ . This temperature and field dependent θ can be obtained as  [4,[6][7][8][9]. The temperature-and-field-dependentθ can now be expressed as where α ′ is a proportional constant. By considering the theorem of random walk, the direction of the random collective pinning force for the bundle is The root-mean-square of the angle between the random collective pinning force and the positive y-direction in counterclockwise sense for the vortex bundle can be obtained as, (2.6) Keeping in mind the fact that is usually very small in our theory, we obtain, (2.7)

Calculation of the Hall as well as longitudinal resistivity
To proceed let us calculate the energy barrier of the vortex bundle generated by the Magnus force, the collective pinning force, and the randomly distributed strong pinning sites inside the bundle. After some algebra, the energy barrier of the vortex bundle both in the positive and negative x-direction as well as y-direction are obtained respectively as, (2.11) in which the potential barrier U is generated by the randomly distributed strong pinning sites inside the bundle, is the magnetic field, V is the volume of the vortex bundle, R represents the transverse size of the vortex bundle, and the range of U is assumed to be of the order of R . The self-consistent equations for the velocity of the thermally activated motion for the vortex bundles over the directional-dependent energy barrier are therefore obtained in components as, ν presents the coherent oscillation frequency of the vortex bundle, and the distance between two adjacent minimum potentials of the vortex bundle is assumed to be of the order of R .
Observing the fact that , Eqs (2.12) and (2.13) can be approximately rewritten as, , and J E y xy = ρ , finally xx ρ and xy ρ can be obtained as, is the magnitude of the random collective pinning force per unit volume [6]. 8 It is well understood that when temperature (magnetic field) below ) ( P P B T , the quasiorderdisorder phase transition temperature (magnetic field), the vortex lines form large vortex bundles [6], both Hall and longitudinal resistivities approach to zero quickly as temperature (magnetic field) decreasing. When temperature (magnetic field) above .
Interesting to stress that, from the above equations, taking into account the fact that . As we have indicated before, the above equations give rise to the phenomenon of AHE, namely, as the applied magnetic field (temperature) decreases, the value of xx ρ decreases monotonically, and xy ρ decreases crossing over to negative value and reaches its minimum value then increases again. We will discuss the cases both for constant temperature and constant applied magnetic field in the following subsections separately.

The Hall and longitudinal resistivities for constant temperature
In this subsection we study the case when the temperature of the system is at a constant value T . As we have declared before, when applied magnetic field decreases, the value of xy ρ decreases, crossing over to negative value and reaches its minimum, then increases again. It is interesting to compare with the experiments for this case. When the temperature is kept at a From the above results we demonstrate that the value of xx ρ decreases monotonically and xy ρ indeed decreases crossing over from positive to negative and reaches its minimum value, then increases again with decreasing the applied field. If the applied field decreases beyond 1 Tesla, from our previous study, the quasiorder-disorder phase transition occurs [6]; this is the region of the thermally activated motion for the large bundles, both the Hall and longitudinal resistivties approach to zero quickly as the applied field decreasing. These results are in good consistent with experimental data on δ − 7 3 2

O Cu YBa
high-Tc bulk materials [20]. The regime crosses over to the usual flux-flow regime, when the field increases beyond 4 Tesla.

The Hall and longitudinal resistivities for constant applied magnetic field
Considering the case when the applied magnetic field is kept at a constant value  Shown from the above results, the value of xx ρ decreases monotonically, and xy ρ is genuinely crossing over from positive to negative value and reaches its minimum, then increases again with decreasing the temperature. As the temperature is less than K 87 , according to our previous study, the quasiorder-disorder phase transition occurs [6],

The anomalous Hall effect for type-II superconducting films
For type-II superconducting films, as we have detailed discussed in Sec 2.3, from Eqs (2.19) and (2.20), xx ρ and xy ρ now become, . As discussed in Section 3, the above equations sincerely signify the phenomenon of AHE for both the constant applied magnetic field as well as constant temperature. We will examine the cases in the following subsections.

The Hall and longitudinal resistivities for constant temperature
Studying the case when the temperature is kept at a constant value Analyzing above results, we see that the value of xx ρ decreases monotonically and the xy ρ decreases crossing over from positive to negative value and reaches its minimum, then increases again as the applied magnetic field decreasing, When the applied magnetic field decreasing below 6 Tesla, the quasiorder-disorder phase transition takes place [6], belonging to the region of thermally activated motion of large vortex bundles, both the values of xy ρ and xx ρ approach to zero rapidly as the applied magnetic field decreasing. The results are in good agreement with the experimental data on Si Mo 3 low-Tc thin films [21]. When the applied magnetic field greater than 25 . 7 Tesla, it belongs to the usual flux-flow regime.

The Hall and longitudinal resistivities for constant applied magnetic field
Next focusing the situation when the applied field B is kept at a constant value Recognizing the above results, the value of xx ρ decreases monotonically and xy ρ decreases crossing over from positive to negative value and reaches its minimum, then increases again as the temperature decreasing; the quasiorder-disorder phase transition occurs [6] when the temperature decreasing below
When the temperature greater than K 3 . 91 , it crosses over to the usual flux-flow region.

The reentry phenomenon for anomalous Hall effect
In this section we investigate the situation that if the random collective pinning force is not too large, the value of xx ρ decreases monotonically and xy ρ decreases crossing over from positive to negative value at a T , the onset temperature for sign reversal, reaching a minimum, then increases crossing over from negative back to positive value at R T , the reentry temperature, reaching a maximum value then decreases again with decreasing temperature. This is the fascinating reentry phenomenon. The condition for occurring the reentry is that if the random collective pinning force is not too large, such that when Our present results shown that the value of xx ρ decreases monotonically and the xy ρ decreases and crossing over from positive to negative value and reaches its minimum, then increases crossing over from negative back to positive value, researching a small local maximum then decreases again as the temperature decreasing. For temperature decreasing below K 72 , the quasiorder-disorder first-order phase transition happens [6]. high-Tc bulk materials [24]. While for the temperature higher than K 102 , the regime crosses over to the usual flux-flow regime.

Discussion
We would like to point out first of all that we have developed a novel theory for the thermally activated motion of vortex-bundle over the directional-dependent energy barrier formed by potential due to the Magnus force and the random collective pinning force and the potential U generated by the randomly distributed strong pinning sites inside the bundle.
We have successfully applied this theory and shown that the AHE is induced by the competition between the Magnus force and the random collective pinning force.
Secondly, our theory for AHE is indeed a microscopic one. Starting from the microscopic theory of BCS [26] by Gor'kov [27], one can get the Ginzgurg-Landau theory [28]. By the work of Abrikosov [1], the flux lines in mixed state type-II superconductors form a longrange order of the FLL. However, in the presence of quenched disorder, the long-range order of the FLL is destroyed, only short-range order prevails [5][6]. Based on the thermally activated motion of the vortex bundles over the directional-dependent potential barrier together with the Maxwell's equation and the effect of random collective pinning [6][7][8][9], we arrive at the theory of AHE. Therefore, our theory is a microscopic one.
Thirdly, we would like to emphasize that the quasiorder-disorder first-order phase transition or the peak effect [6] occurs inside the region of AHE. The phenomenon of AHE includes two interesting regions: the thermally activated motion of small vortex bundles, and the thermally activated motion of large vortex bundles; however, the flux flow region does not belong to it. It is worthwhile to stress that we only consider the systems that are in steady state, any time-dependent behavior of the system will not be discussed.
Fourthly, our theory is a very general theory, it can be applied to conventional and high-Tc superconductors. Although their mechanisms, the structure of vortex lattice, and even the method of pairing are entirely different, these do not affect the results of our theory.
Finally, the dimensional fluctuations have no effect on the essential structure of our theory. This is why we discuss the 2D and 3D anomalous Hall effect at the same time.

Conclusion
We have developed a theory for the thermally activated motion over the directionaldependent energy barrier of vortex bundles for the mixed state type-II superconductors, and applied this theory to investigate the AHE for conventional and high-Tc superconductors, bulk materials and thin films. Our calculations demonstrate that the Hall anomaly is indeed induced by the competition between the Magnus force and the random collective pinning force. The AHE for constant applied magnetic field, constant temperature, the reentry phenomenon are also discussed. All the results are in good agreement with the experiments.