Magnetic field measurements with Faraday active material added to the sample should give an answer to the question if our samples are localising or not. The idea, introduced in section2.3.9, is to break time reversal symmetry, thus travel-ling a closed path in opposite direction should not be the same any more.
First we want to start with the work of Lukas Schertel [216]. He used the setup described in sec.4.4.2to measure speckle correlations in transmission. The pur-pose of the work was to determine the strength of Faraday rotation that is needed to destroy time reversal symmetry. The idea is to measure speckle correlations as a function of the magnetic field. The claim is that a complete speckle decorrela-tion should be equal to destrucdecorrela-tion of time reversal symmetry. This should also be seen in the speckle distribution. In the case of localisation the speckle is dom-inated by a few bright spots [23], whereas in the diffusive case the speckle distri-bution is exponential. However, we do not expect to see any significant changes in the speckle pattern, since the ‘localisation’ signal represents only∼1% of the whole signal. Additionally breaking time reversal symmetry should lead to an increase of the transmitted intensity.
Since the Faraday effect of TiO2 is too small for our purpose we add CeF3 as Faraday active material to our powders. To increase the Verdet constant we ad-ditionally cool the samples down to liquid Helium temperature (see sec. 4.4).
As localising samples require a certain sample size, which implies very low trans-mission, the experiments are performed on thin samples in order to have enough transmission. On top of that a yet unknown effect causes an intensity loss even bigger thanI0·10−2while cooling the sample to 4 K. This effect is also observed for pure R700 and antase, but not for pure CeF3. Interestingly no change in ab-sorption is observed with time of flights. A big issue is the general instability of the speckle pattern, supposedly caused by thermal energy, which is reduced, but still present, when the sample is cooled. Thus a measurement of a deviation from a exponential speckle distribution is not possible. To overcome this prob-lem the sample is filled up with glycerol. The idea is to embed the powders into a solid matrix, thus preventing any movement. Indeed adding glycerol improved the stability to a suitable value. A careful study to characterise the Faraday ro-tation in dependence of the magnetic field, temperature and amount of CeF3 was performed. As an example fig. 6.19(left) shows the speckle decorrelation in dependence of the CeF3concentration. With all these data the amount of the needed CeF3 concentration could be estimated to be 5vol% or 15wt%, see fig.
6.19(right).
Knowing the suitable parameters we can create a sample that should have a feas-ible Faraday effect to destroy time reversal symmetry. The mixture with 15wt%
CeF3has the drawback thatl∗is reduced and because CeF3absorbs more than R700 the sample absorption is stronger. Nevertheless such a sample still shows a deviation in the long time tail in time of flights, despite the clearly visible higher
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Magnetic Field Measurements
0 2 4 6 8 10 12 14 16 18
B (Tesla) 0.0
0.2 0.4 0.6 0.8 1.0
g2(B)−1
Theory transmission slab TiO2+CeF3: L=1.1mm, kl∗=4.5, La=150µm
Veff=−135rad/Tm⇔2.34vol%
Veff=−380rad/Tm⇔5vol%
Figure 6.19: (left) Speckle correlation measurements for different concentrations of CeF3are shown, increasing from top to bottom. The solid lines are theory fits to estimate the effective Verdet constants. Figure taken from [216]. (right) Based on the results of correlation measurements the prediction can be made that we need 5vol% CeF3for a 1.1 mm thick sample. The correlation decreased belowg2(B)−1<0.1, which should be feasible to destroy time reversal symmetry. Figures taken from [216].
absorption (fig. 6.20(right)). Measuring transmission profiles was not possible in a first attempt, as the intensity was too low.
To exclude effects that are caused by the setup or the powder (R700), we first perform a measurement on the pure powder withL =0.94 mm. After recording a time of flight at room temperature and zero field the first step is to cool down the sample in the Oxford cryostat to∼4 K. We can observe a change in the pro-gression at longer times by cooling, as can be seen in fig. 6.20(left). The main effect is a reduction of absorption, visible in a lower slope afterτmax. This effect is not surprising and can be explained with the bandgap of TiO2. If the sample is cooled down the Fermi edge becomes sharper, since a low thermal energy re-duces states populated above the Fermi level. Thus leading to a reduction of ab-sorption. Indeed the effect is moderate and no big change was expected, since the wavelength we use (590 nm) is far enough away from the bandgap. The long time tail still persists. The next step is to apply the magnetic field. This is done in two steps, first to a field of 14 T. This is roughly the maximum field at which the photo multiplier does not lose count rate due to the magnetic field. Afterwards the field is increased to 18 T where the count rate is only half as high. The results are included in fig. 6.20(left). We see applying the magnetic field lowersτmax
slightly, although this could be within the measurement uncertainty. If we run down the field to zero Tesla and heat up the sample to approximately room tem-perature, we can see that the curve matches with the very first one. This result shows us that we do not change our sample by cooling and applying a magnetic field, the process is reversible.
Since we have not seen any unexpected effects, we can measure a mixture of R700
Chapter 6. Results
Figure 6.20: (left) A time of flight of a pure R700 powder is shown in dependence of the temperature and the magnetic field. Cooling down causes lower absorption and no further changes are visible. As expected the magnetic field has no effect. The legend represents the chronolo-gical order of the experiment. (right) The same measurement with R700 and 15% per mass CeF3 is shown. Absorption also decreases by cooling. Different to our expectation the magnetic field shows no influence on the long time tail.
with 15wt% CeF3of the sizeL=1.04 mm. Again we see a decrease in absorption due to cooling, see fig6.20(right). First, we also apply a magnetic field of 14 T and later increase it to 18 T. In this case we can observe a small shift ofτmax as well, assuming that it is a real feature. A possible explanation is that the magnetic field applies a force on the CeF3particles. The ‘pull’ can lead to a change in the sample, creating small holes which do not relax, thus leading to faster diffusion. What we do not see, against our expectation, is a decrease of the long time deviation. In the presented measurements the magnetic field has no measurable influence on the long time tail of our sample.
This observation is the hardest to interpret, since there can be many reasons that we see no change in the long time tail. We will discuss different possible reasons why we did not see any change. It shall be mentioned that the measurements were performed in a proof of principle setup. Therefore improvements are surely possible.
First, we could not find literature on the saturation magnetisation of CeF3and we have not the possibility to measure it ourelves. If CeF3saturates at values of some Tesla at 4 K we might not see any change, since the correlation is still quite high, see fig. 6.19. A way to measure the saturation magnetisation with our setup is to perform a speckle correlation experiment up to 18 T. The main problem is the setup stability, which is expected to be even worse with the longer Oxford cryo-stat. We can conclude from the experiments performed by Lukas Schertel, that no saturation is observable up to 7 T and 9 K [216]. Another possibility is that the
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Unresolved Issues
Faraday effect is simply not strong enough. There is no theory concerning the Faraday effect in a localising medium for a three dimensional system. Thus we can only assume that by destroying the speckle correlation, the rotation is strong enough to destroy time reversal symmetry. Indeed we do not know if this is feas-ible or if the Faraday effect can influence or destroy localisation in 3D media at all. We are also not able to transfer the predictions from one and two dimensions to three and we do not know if the experiments with weak localisation are com-parable with strong localisation. Another possible case is that we did not observe localisation at all in our experiments. It remains an open question we are not able to answer at this point.
We measured time of flight at low temperatures up to fields of 18 T. Destruction of time reversal symmetry could not be observed with the proof of concept setup.
More careful experiments have to be done to determine if or if not time reversal symmetry can be broken. A final statement is not possible at this point.