Magnetic Vortex Properties of the MgB 2 Superconductor. Tommy O Brien 2009 NSF/REU Program Physics Department, University of Notre Dame

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1 Magnetic Vortex Properties of the MgB 2 Superconductor Tommy O Brien 2009 NSF/REU Program Physics Department, University of Notre Dame Advisor: Prof. Morten Ring Eskildsen August 5, 2009

2 Magnesium diboride (MgB 2 ) is a well known Type II superconductor. The main purpose of this paper is to present results on the exploration of some of the vortex properties of this superconductor, including its flux quantization, metastable phases, and phase diagram. The results were obtained during a Small Angle Neutron Scattering (SANS) experiment conducted at the Institut Laue Langevin in Grenoble, France, in June of I Basic Superconductivity A superconductor is a compound which possesses no resistance when cooled below its critical temperature, T c. That is, current is free to pass unhindered through a compound in the superconducting state. As a superconductor is cooled below T c in a low magnetic field, the magnetic field lines are forced out of the sample, leaving the interior with a magnetic induction of B = 0. A superconductor behaving in this manner is said to be in the Meissner State [4]. Superconductors that transition directly from the normal state to the Meissner State are Type I superconductors. Type II Superconductor Phase Diagram Figure 1: Phase Diagram for Type II Superconductor [2] For our experimental purposes, we are interested in the behavior of MgB 2, which is a Type II superconductor. As shown in Fig. 1, Type II superconductors display the characteristics of a normal metal above an upper critical field of H c2 and the characteristics of a Type I superconductor in the Meissner state when below the lower critical field H c1. When the sample is between H c1 and H c2, it is in the so-called vortex state. That is, the superconducting and normal states exist in equilbrium. It is important to remember, though, that even though there are normal regions in the material in the vortex state, the sample is still superconducting (i.e. resistance in zero even though B is not). Fig. 2 shows a schematic of a superconductor

3 in the vortex state. The gray shaded area is in the Meissner State and the darker rings represent the normal state vortices with magnetic flux lines passing through. Figure 2: Schematic of Type II Superconductor in Mixed State These vortices interact, and in a static equilibrium they will fall into the most energetically favorable alignment. In the case of MgB 2, the magnetic vortices form a hexagonal lattice which, at low temperature and field, is arranged in such a way that the peaks are aligned with the sample s crystalline a axis. However, the vortex lattice changes its orientation as either temperature or applied field changes. II Experimental Setup Small angle neutron scattering experiments require that the neutron has a known wavelength upon interaction with the magnetic vortices. Because of this, the experiment is arranged in the following fashion: 1. The first step is to pass the neutrons through a velocity selector which absorbs the range of neutrons outside of the selector s setting. In addition to selecting a particular velocity the system also selects a particular wavelength for the passed neutrons. This experiment used a range of 5 Å to 10 Å. Since the diffracted peaks move further out as the field increases and further toward the beam center as field decreases, it is necessary to use smaller wavelengths at higher fields and larger wavelengths at lower fields to guarantee the lattice is easily seen on the detector. 2. The next section is the collimation section which is responsible for the width of the beam as it hits the sample. This section has two apertures, a source aperture and a sample aperture to minimize beam divergence. The sample aperture also ensures that the neutrons actually contact the sample. For this experiment, we used source apertures ranging from 10 mm to 30 mm and sample apertures from 0.5 mm to 2 mm. 3. After passing through the sample aperture, the beam enters the cryomagnet and interacts with the sample where it is diffracted by the vortex lattice.

4 4. The final step for the neutrons is to strike the detector located in the evacuated tank behind the sample. The detector tracks the number of neutrons to strike a particular spatial position. A diagram of the experimental setup is shown in Fig. 3. Then, using a SANS-specific piece of analysis Figure 3: SANS experimental setup [2] software called Grasp, we are able to analyze the size and shape of the vortex lattice. III Results a. Flux Quantization Magnetic flux through each vortex in a mixed-state superconductor is quantized to integer values of the flux quantum, φ 0. MgB 2 is thought to posses unique physical properties that may allow a non-integer values of φ 0 for induced magnetic field. Fractional vortex quantization has been proposed theoretically by Babaev in [1]. The first task of this experiment, then, is to verify the quantization of flux in the vortices displayed by MgB 2 while in the vortex state. Figure 4: Reciprocal Space SANS Image

5 Fig. 4 shows an example of a vortex lattice (VL) reciprocal-space image. Using this image we can then calculate the induced field B by making a series of equation substitutions. The unit cell for the vortex configuration in 4 is spanned by the vectors q 1 and q 2 and has an area of q 1 q 2. We know from electricity Figure 5: SANS schematic showing the detector distance D and the distance from the vortex peak to center of vortex lattice, R. and magnetism that B is defined as flux per unit area. We know φ 0 to be the flux in each vortex, so we have B = φ 0 4π 2 q 1 q 2 (1) in reciprocal space. To return to real space, we start with q = 2ksin2θ and insert the well known formula for k, k = 2π λ. Applying the small angle approximation of sinθ θ. Then, we are left with q = 2( 2π λ )θ where 2θ is the angle between the incident neutron beam and the reflected beam, as shown in Fig.??. Using the small angle approximation again, this time yielding sin(2θ) R D, where R is the distance from the center of the vortex to the center of the lattice, and D is the distance from the sample to the detector shown in Fig.??. Resubstituting q into our original equation, we are left with the final result, B = φ 0 λd R 1 R 2 (2) Using the SANS software, one can easily fit a Gaussian to the diffraction peak, calculate the distance R to the center of the lattice, and apply the equation above. Performing this same analysis on both the MgB 2 and the LuNi 2 B 2 C (which is known to be in a 1:1 ratio), it is possible to compare the two to see if there is any worthy difference between the slopes of the B vs H plots of either sample. The results from the experiment are shown in Fig. 6. The plot clearly shows that there is no extra flux or induced field in the MgB 2 sample; vortex quantization holds.

6 1 2 8 B (L u ) (k G ) B (M g B 2 ) (k G ) L in e a r F it B (L u ) L in e a r F it B (M g B 2 ) B (k G ) A p p lie d F ie ld (k G ) Figure 6: Magnetic Induction (B) versus Applied Field (H) b. Metastable Phases A metastable phase is the existence of a sample in a particular VL configuration when it is energetically unfavorable to be in that phase. More precisely, a superconductor is said to be in a metastable phase when it fails to transition upon crossing a phase boundary in either temperature or magnetic field. The phase of MgB 2 at a particular temperature and field has a strong history dependence which marks the existence of metastable states, but the purpose of this experiment was to further explore metastable states generated by different cooling techniques. Assuming that a superconductor starts in the normal state, there are two distinct paths to force it into its superconducting state. The first is a Field Cool, which means that the magnetic field is ramped while in the normal state and the temperature is then dropped below T c. The other is a Zero Field Cool in which the steps are reversed. That is, the sample is cooled first from the normal state to a temperature below T c and then the field is ramped. The MgB 2 superconductor itself has three VL phases, each of which is identified by its relation to the crystalline a axis. The crystalline axis for this experiment was determined by X-ray diffraction. The first phase is shown in Fig. 7(a) in which opposite peaks are aligned along the crystalline a axis. Fig. 7(b) shows the phase in which opposite peaks are exactly misaligned with the crystalline a axis. The transition phase is shown in Fig. 7(c). Notice that during a transition, each of the six peaks in the aligned phase has split to make a total of twelve peaks. Previous MgB 2 vortex phase in [3] using the Field Cool method shows that measurements taken at 0.7 T and base temperature ( 2 K) give the transition phase, but that measurements at 0.9 T give the misaligned

7 a (a) a Aligned Phase (b) a Misaligned Phase (c) Transition Phase Figure 7: MgB 2 Phases phase. Our experiment, which used the Zero Field Cool method, consistently gave both the 0.7 T and the 0.9 T measurements to be in the transition phase. The explanation for the discrepancy between the two experiments, then, is that the Field Cool method actually freezes in the magnetic behavior of high temperatures until that configuration is so energetically unfavorable that the vortices transition to the aligned phase. That is, the Field Cool alters the history of the superconductor which leads to a metastable state. To prove this explanation, we first measure at 0.9 T and base temperature, using the Field Cool method. Then, once the misaligned phase has been affirmed, we apply a damped field oscillation of 250 G to try to force the vortices into the more energetically favorable phase. After this, the 0.9 T wiggled measurement demonstrated the transition phase that we saw originally in the Zero Field Cool measurement. This was repeated for many such discrepancies between Field Cool and Zero Field Cool measurements, thus affirming the hypothesis that Field Cooling can produce metastability in the vortex lattice phase. c. Phase Diagram Now that the effectiveness of the 250 G wiggle has been demonstrated, it is possible to systematically

8 construct the vortex phase diagram for MgB 2. The most pertinent information in a phase diagram is the transition lines, thus the experiment primarily focused on tracing these lines. In the interest of time, a systematic allocation of all points in between the transitions was left for later exploration. The results of this process can be seen in Fig. 8. Figure 8: MgB 2 Vortex Phase Diagram As the diagram shows, the transition between the aligned phase (long hexagonal axis aligned with crystalline a axis) and the misaligned phase (30 o offset from crystalline a axis) depends drastically on temperature. More specifically, as the temperature increases, the difference in field between the two phases decreases. The gray shaded region in Fig. 8 represents the area covered by this experiment. The transition line for low temperatures and fields is shown clearly, but both higher fields and temperatures still need to be explored. It is unclear so far if the upper and lower transition lines meet before T c or if they asymptotically approach a predetermined applied magnetic field value. Currently, the experiment is restricted because field magnitiude and temperature have such a drastic impact on intensity.

9 References [1] Egor Babaev. Vortices with fractional flux in two-gap superconductors and in extended faddeev model. Phys. Rev. Lett., 89(6):067001, [2] Morten Ring Eskildsen. Small Angle Neutron Scattering Studies of the Flux Line Lattices in the Borocarbide Superconductors. PhD thesis, RisøNational Laboratory, December [3] Cubitt, R. et. al. Effects of two-band superconductivity on the flux-line lattice in magnesium diboride. Phys. Rev. Lett., 91(4):047002, [4] Charles Kittel. Introduction to Solid State Physics. John Wiley & Sons, Inc., seventh edition, p