«STUDENT RESEARCH PAPERS SUMMER 2011 VOLUME 22 REU DIRECTOR UMESH GARG, PH.D. REU Student Research Papers – Summer 2011 University of Notre Dame – ...»
Figure 3a shows that as the time increased, the rms velocity increased as well. Similarly, in Figure 3b the rms magnetic field increases as time increases, indicating that turbulence caused by supernovae explosions in a stratified disk does amplify the magnetic field. In further research, more runs should be done at an increased resolution after some existing issues with the boundary conditions are addressed.
Acknowledgements I would like to thank my advisor, Professor Dinshaw Balsara, for his guidance during the summer, as well as his graduate student, Christopher Matthews, for his work. I would also like to thank the 2011 NSF/REU Program and the University of Notre Dame for their support of this research
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The samples show interesting effects when subjected to the magnetic fields. In several of the samples, the magnetoresistance is proportional to square of the B field. The Hall Resistance has several features of interest that are explained in more depth in this paper. Low-Field Weak Anti-Localization also manifested itself in nearly every trial we did. We found that at high fields, the magnetoresistance of many samples would oscillate slightly. This ends up being the result of the Shubnikov-de Haas Effect.
The discovery of TIs was unusual for the field of condensed matter, in that they were predicted theoretically before they were observed in the lab. Several papers were published in 2005 and 2006 that predicted the existence of a type of material that would permit the movement of electrons only on its surface,, and TIs were observed later in 2007 .
We in the field are interested in Topological Insulators because of their unique conductive properties. The bulk of a TI acts as an insulator, while the surface allows the flow of electrons. You can therefore think of a TI as a piece of rubber or plastic with a thin layer of metal, only it achieves this conductive behavior by itself. You can achieve a similar effect through the Quantum Hall Effect. The (non-quantum) Hall Effect states that when you subject an electron current to a magnetic field, the carriers will tend toward one side of the
samples at low temperatures, you can start to measure quantum effects on the carriers. The Quantum Hall Effect states that, due to quantum effects, the carriers that have collected along the edge will move along the surface. This is similar to the behavior in TIs, but they are able to do this without the use of an external magnetic field.
TIs also have a unique effect on electron spin. The carriers that move along the surface of a TI are spin-locked. This means that the spin of each electron is perpendicular to its momentum. Figure 1a shows this as compared to a conventional insulator, and Figure 1b shows the band structure of a topological insulator.
As the electrons move to the right at the top of the material, their spins must be directed upwards. Those that move left have their spins reversed. These velocity-spin relationships would be switched for electrons at the bottom surface of the material. One effect of being spin-locked is that the electrons have a very high mobility, because scattering the carriers would require flipping their spin. This is impossible to do because that state is simply not available. These unique properties of TIs make them promising materials for research; there is great interest for applications in spintronics and quantum computing.
In order to learn some of the basic properties of our TIs, which were grown in the MBE lab at Notre Dame, we performed magnetotransport on Bi2Se3, Bi2Te3, and Bi2(TeSe)3. We accomplished this by soldering samples of various materials, size, and thickness to a sample holder. The sample holder was wired to instrumentation that would measure the magnetoresistance and the hall resistance of each sample. The sample holder, inserted to the top of the cryostat, was placed right next to a super conducting magnet. Using liquid helium and liquid nitrogen to cool the magnet, we were able to subject the samples to magnetic fields anywhere between 6 and -6 Tesla. By controlling the flow of liquid helium through the sample space, along with the help automatic internal heater, we were able to stabilize the temperature of the samples within a range of 2 K to about 200 K. The angle of the samples with respect to the magnetic field was easily adjusted by twisting the top of the sample holder. Thus for every sample we tested we measured a full sweep of the magnetic field for an assortment of angles and temperatures.
When subjected to a magnetic field, certain materials have a tendency to change their resistivity. This property is known as magnetoresistance. From Kohler’s Rule, the resistance
of a material is proportional to the square of the magnetic field. This is shown in Eq. 1:
we are able to determine the magnetoresistance. Figure 2 shows the resistivity of Bi2(TeSe)3.
This sample is a particularly good example of the resistance and B2 relationship.
Figure 2 – The resistivity of a sample of Bi2(TeSe)3 at 2.03K and its second order fit
As stated earlier, conductors in the presence of a magnetic field undergo the Hall Effect. Our topological insulators are no exception. Refer to Figure 3.
As the electrons flow longitudinally through the sample, they are pushed downward by the B field. This results in the expected potential difference, which in turn creates a Hall Resistivity. We measure this laterally according to the figure. For an example of a typical result, refer to Figure 4, which plots the Hall Resistivity versus the field strength for a sample of Bi2Se3 at various temperatures.
This equation predicts a linear function, which matches our shown results. However, linearity is not always the case. If an anomalous Hall Resistivity is introduced, then there will be a B3 term present. Notice too, the slope of the Hall Resistivity. The fact that it is a negative slope confirms that the carriers are electrons, as opposed to electron holes which have positive charge. From this data, it is possible to calculate the carrier concentrations and mobility in the sample. The carrier concentration is result of linear regression. The program we used calculated the concentration and mobility for each, and results are listed in Table 1.
Most if not all of the trials we did had an interesting dip in the graph near the zero Tesla mark. Sometimes these “features” were so small that you had to zoom in to see them, other times they dominated nearly the entire B field range. Figure 5 has an example of a sample with moderately sized features (Figure 2 also has an example of a very small feature).
These artifacts in the resistivity can be attributed to a phenomenon called Weak AntiLocalization (WAL) Effect. WAL is the suppression of backscattering due to impurities.
Less backscattering means more mobility, and less resistivity. This effect mainly manifests in the low field, which is why the features of low resistivity always center on the zero field point. Eq. 3 characterizes the WAL Effect.
If you were to zoom in on the graph in Figure 2, you would notice that the line does not follow a perfectly parabolic line, but instead wavers around it. Though it would be very easy to attribute this to slight error or imprecision, it is actually due to a phenomenon called the Shubnikov-de Haas Effect. Figure 6a shows the same data as in Figure 2, but limited to the range of 1T to 6T.
By taking a second order fit of that data, and subtracting that fit from the original data, we get Figure 6b. These oscillations, are not the result of error, but instead are the manifestation of the Shubnikov-de Haas Effect. This effect, which only occurs at low temperatures and high magnetic fields, is the manifestation of quantum mechanics at the macroscopic level.
The peaks in these oscillations occur at quantized Landau levels.
In this experiment we set out to measure the magnetoresistance and hall resistance of our Bismuth Selenide and Bismuth Telluride samples. In doing so we were able to confirm properties that are expected in TIs. For example, our hall resistance data supported the idea that electrons were the carriers involved and not holes. We were also able to identify several interesting (though already known) phenomena in our samples. These include the oscillations caused by the Shubnikov-de Haas Effect in some of our samples, and the features created by the Weak Anti-Localization Effect in nearly all of our samples. Overall, we’ve collected a lot of data for various sample sizes, temperatures, and angles which we think will be beneficial as research on Topological Insulators progresses.
 Kane, C. L., and Mele, E.J., 2005, “Z2 Topological Order and the Quantum Spin Hall Effect”, Physical Review Letters, v. 95, Issue 14  Bernevig, B. Andrei, et al., 2006, “Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells”, Science Magazine, v. 314 no. 5806 pp. 1757-1761  König, Markus, et al., 2007, “Quantum Spin Hall Insulator State in HgTe Quantum Wells”, Science Magazine, v. 318, no. 5851 pp. 766-770  Krumrain, J., et al., 2011, “MBE growth optimization of topological insulator Bi2Te3 films”, Journal of Crystal Growth, 324 (2011) 115-118  Hong-Tao He, et al., 2011, “Impurity Effect on Weak Antilocalization in the Topological Insulator Bi2Te3”, Physical Review Letters, 106, 166805 (2011)  Eto, Kazuma, et al., 2010, “Angular-dependent oscillations of the magnetoresistance in Bi2Se3 due to the three-dimensional bulk Fermi surface”, Physical Review B 81, 195309 (2010)
Using the SQUID magnetometer, we studied the magnetic properties of La0.7Sr0.3Mn1-xNixO3 (x=0, 0.05, 0.10, 0.20 and 0.30) perovskites. Both temperature dependent and field dependent magnetic properties show that Curie temperature decreases as a function of doping, ranging from 380K to 300K. Both magnetic measurements and neutron diffraction data hint at long-range magnetic ordering in samples at low temperature and transition phases from paramagnetic to ferromagnetic to antiferromagnetic ordering in samples at room temperature.
where g is the Lagende g-factor, μB = 9.274*10-21 erg/Oe is the Bohr magneton, and J is the angular momentum. All atoms have 2J+1 energy levels. For an electron g is usually approximated to be 2.00, which is the spectroscopic splitting factor. Magnetization is the magnetic dipole moment per volume.
Using the methods of statistical mechanics, the paramagnetic magnetization of a solid is given by
where N is the number of ions per unit volume, H is the applied magnetic field, and k is the Boltzmann constant. BJ, the Brillouin function, is defined as
If we study the equation for magnetization, we study using Pierre Curie's approximation, which deals with high temperatures and low magnetic fields, that when we increase the temperature and decrease the magnetic field. When we set x=μB/kBT 1, in the Brillouin function, the magnetization
Ferromagnetism is the spontaneous magnetization of magnetic moments lining up with each other due to the influence they have on each other. The mean-field approximation derives from the
assumption that every ion experiences a field proportional to the magnetization:
where λ is a material-specific constant that is independent of temperature. The mean-field approximation is temperature dependent. BE is assumed to be the equivalent of an exchange field, which is an internal interaction lining up the magnetic moments parallel to each other.
Above the Curie temperature, TC, the spontaneous magnetization vanishes and the material is paramagnetic. Below the Curie temperature, the material is ferromagnetic. The magnetization is a complicated function of temperature and applied field. We can find TC by observing λ from the meanfield approximation.
Looking at the paramagnetic phase, we have
where χp is the magnetic susceptibility, and BA is an applied magnetic field which causes a finite magnetization. Assuming that the material is in the paramagnetic phase, we can take Curie's Law for paramagnetic susceptibility to be
We see that the denominator is zero when T=Cλ, in the above equation. At that temperature and below, the magnetization is spontaneous, and because of that we can have a finite magnetization for Ba=0.
which describes the susceptibility variation in the paramagnetic region above the Curie point.
With this information we can define λ to be