TOTAL DE ENCUESTAS

Following the discussion of open questions in subsection 5.2.2, I performed low temperature resistivity measurements in the presence of magnetic fields, aiming for a comprehensive resistivity study of CeAuSb2 at low temperatures.

Two samples, cut from the same bigger sample of the hs006 batch, were measured with the ADR transport probe that was mentioned in the previous subsection and was described in more detail in Section 3.2.3. The two samples, which were named hs006-SAand hs006-SBrespectively, were both cut along the crystal principal axes (the a or b axis). Each sample had one pair of contacts for current injection and two pairs for voltage measurements. The two voltage contact pairs for hs006-SA and one pair for hs006-SB were connected to low temperature transformers while the other pair of hs006-SBwere connected to a room temperature transformer (SR554). (This arrangement was only due to the limited availability of low temperature transformers.) The gain was 100 for all transformers. The excitation current for

both samples was generated by a current source, composed of an AC voltage source (the AC voltage output of the lock-in amplifier SR830) and a 10 k resistor, whose impedance is much higher than the rest of the electric circuit. A summary of the measurement parameters and the sample dimensions is shown in Table 5.3.

Sample name W (µm) t (µm) L1 (µm) L2 (µm) Freqency (Hz) I (µA)

hs006-SA 160 130 250 280 71.13 100

hs006-SB 250 150 733 846 81.13 100

Table 5.3: The experimental parameters of the resistivity measurements on hs006 samples. W: sample width; t: sample thickness;L1: spacing between the first pair of voltage contacts; L2: spacing between the second pair of voltage

contacts; Freqency: measurement frequency;I: excitation current.

Figure 5.23 shows the primary results for fl(H) 50 _{as a function of magnetic field} (which was ramped up first and then ramped down), measured at 300 mK, with an excitation current of 100 µA. The main features of flab(H) are similar to those of Balicas et al.. Two abrupt changes of resistivity are observed atHMM = 2.8 T and Hc = 5.6 T, respectively, which is consistent with their report. By comparing our result with the previous report, we can identify that at this temperature, below HMM is the antiferromagnetic A phase, while between HMM and Hc is the anti- ferromagnetic B phase and above Hc is the paramagnetic phase (cf. Figure 5.5). In the paramagnetic phase, the field dependence of fl(H) for our samples is also similarly weak at H >Hc.

The hysteresis infl(H) of our sample showed distinct features. In Figure5.23, one can see that hysteresis in fl(H) is not only observed in the high resistivity phase (i.e. the antiferromagnetic B phase, betweenHMM andHc), but also in the antifer- romagnetic A phase (belowHMM). By carefully comparing Figure5.23and Figure 5.10 (from [7]), one can actually find that the hysteresis in the antiferromagnetic A phase could also be identified in the latter, although it is slightly less obvious. However Balicas et al. only recognised the hysteresis in the antiferromagnetic B phase and even stated that the other sample they measured showed no hysteresis at all (in either phase) [7]. In contrast, for our work, all the samples we measured showed the same hysteresis as that presented in Figure 5.23.

Figure 5.23: The resistivity of CeAuSb2 as a function of magnetic field, mea- sured at 300 mK. The red curve is for for field ramping-up and blue curve for field ramping down. The resistivity shows two abrupt changes, one at 2.8 T (in- dicated by the black arrow with label HMM) and the other at 5.6 T (indicated by the black arrow with labelHc), in agreement with the results of Balicaset al.

(cf. Figure 5.10). In addition, distinct hysteresis can been see in two regions, indicated by the red arrows in the figure. For more details, see the text.

Moreover, in the work of Balicas et al. (cf.Figure 5.10), some hysteresis at the two transition fields can be distinguished for the field increasing and decreasing curves, but it was ascribed by them to the time constant of their instruments. This in principle is possible, because in real measurements the field value provided by the magnet is usually obtained through calculation based on the excitation current of the magnet rather than through direct measurement. Due to the inductance of the magnet coils the set current cannot be established instantly, therefore the magnetic field estimated based on the set current may be slightly different from the real magnetic field in the centre of the coil where the sample sits. For field sweeps, the difference is further dependent on the field sweep rate. Although the argument of Balicaset al. may be true, it is not conclusive. To understand whether the phase transitions discussed above are of first or of second order though, it is desirable to make clear whether hysteresis of fl(H) at HMM and Hc really exists and (if it exists) how it evolves with temperature.

Figure5.24 (a) and (b) show the resistivityfl of CeAuSb2 as a function ofH close to HMM, and Hc respectively, both measured at 300 mK and at three different

Figure 5.24: fl(H) of CeAuSb2 close toHMM (panel (a)) and Hc(panel (b)), measured at three different field sweep rates. The difference of fl(H) between field increasing and decreasing curves is dependent on the sweep rate, as ex-

pected.

field sweep rates. Evidently for both transitions discrepancy is seen between the field increasing and decreasing curves at each sweep rate and the strength of the discrepancy (indicated by the arrows whose colour is in agreement with the corre- sponding field sweep rate) depends on the field sweep rate, as expected. As shown, faster field sweeps generate a stronger discrepancy between the field increasing and decreasing sweeps, which makes sense. The question is whether there is hysteresis left as the field sweep rate approaches zero. The Hall sensor, which was installed close to the samples, enabled me to answer this question, because (at a constant temperature) the Hall signal is only dependent on the real magnetic field, irre- spective of the field sweep rate. This essentially means that the Hall sensor can effectively work as a field indicator. The absolute value of the field cannot be read since the Hall sensor is not calibrated, however for our purpose it is already adequate.

The Hall signal close to HMM, is shown in the left panel of Figure 5.25. The dif- ference of magnetic field at a constant Hall voltage, between the field increasing and decreasing curves, is due to the time constant of the magnet. In this partic- ular measurement (shown in Figure 5.25), which was performed at 300 mK and at a field sweep rate of 0.15 T/min, the Hall signal suggests that a 11 mT shift of the magnetic field for the field decreasing sweep can result in good agreement

Figure 5.25: The Hall voltage close to HMM, measured by the Hall sensor at a sweep rate of 0.15 T/min. Left panel: Raw data (plus marks) and the linear fit (solid lines). Some difference of the Hall signal between field increasing and decreasing sweeps can be clearly seen, which reflects the time constant of the magnet. Right panel: Evaluating the field discrepancy between the field increasing and decreasing sweeps. As is shown, in this case a 11 mT shift can

adequately overcome the discrepancy.

with the field increasing sweep, thus the field discrepancy due to the instrument time constant can be evaluated as 11 mT. We can then remove the 11 mT field discrepancy from the original magnetic field difference in the resistivity measure- ment (close to HMM) and see if there is any residual hysteresis. By applying this method to the field sweeps shown in Figure 5.24, the resultant residual magnetic difference is obtained and is illustrated in Figure 5.26.

As shown in Figure 5.26, at a certain temperature, the residual difference of fl(H) in H for each transition is a field-sweep-rate independent constant. It means that both transitions in resistivity are accompanied by intrinsic hysteresis, although the hysteresis at the magnetic transition (at Hc) is weaker. The hysteresis can be equivalently demonstrated by showing the resistivity as a function of the Hall voltage (see Figure 5.27), since the Hall voltage is an effective way of showing the magnetic field. As shown in Figure 5.27, within experimental resolution the difference for each transition between difference field sweeps indeed has the same magnitude, as expected.

Figure 5.26: Residual difference offl(H) of CeAuSb2close toHMM(left panel) and Hc (right panel), measured at 300 mK and at three different field sweep rates. For both transitions the residual part becomes a constant, which corre-

sponds to the intrinsic hysteresis of the sample at both transitions.

Figure 5.27: The resistivity fl of CeAuSb2 as a function of the Hall voltage, close to the two transitions, measured at 300 mK and at three different field

sweep rates.

the measured temperature) although the Hall sensor is uncalibrated, because at least within the magnetic field range that covers the transition, the Hall signal is linearly proportional to the magnetic field (cf. Figure 5.25). For example, for the transition at HMM, the result in Figure 5.27 makes it possible to evaluate the hysteresis in units of the Hall voltage while the proportionality between the Hall voltage and the magnetic field (in unit of T) shown in Figure 5.25 further enables

us to convert this Hall voltage into magnetic field in T (or mT). I applied this procedure to obtain the hysteresis of both transitions quantitively and the results obtained for measurements at different temperatures are shown in Figure5.28.

Figure 5.28: The width of resistivity hysteresis of CeAuSb2 as a function of temperature.

The hysteresis of both transitions (shown in Figure 5.28) is strongly temperature dependent. Below 300 mK the hysteresis becomes saturated for both transitions within the experimental resolution, and the hysteresis at Hc is comparatively weaker. As the temperature increases the hysteresis for both transitions is sup- pressed quickly, and for the transition at Hc, above 3.5 K the hysteresis becomes unsolvable. For the transition at HMM though, the hysteresis persists to higher temperature but also disappears at close to 6 K where the two transitions merge. To carefully examine how the two phase transitions merge and to obtain a clear resistivity-field-temperature phase diagram is one of the tasks for this project. In line with the work for hysteresis analysis I performed isothermal magnetic field sweeps between 0 T and 7 T, with special attention paid to the region where the two phase transitions merge. The results are shown in Figure5.29and Figure5.30. The former shows the resistivity-field-temperature diagram below 7 K while the latter shows fl(H) at several representative temperatures and in addition shows how the two phase transitions merge in detail (the inset). From Figure 5.29 and

Figure 5.29: The resistivity- field-temperature phase diagram of

CeAuSb2

Figure 5.30: The resistivity of CeAuSb2 as a function of magnetic field at several selected tempera- tures. The inset shows how the two phase transitions merge into one. Figure5.30it is clear that the phase transition atHMM is only weakly temperature dependent while the one at Hcis strongly temperature dependent, consistent with the previous report ([7]). At approximately 6 K the two phase transitions merge into one and there is no hysteresis observed.

Field angle dependence of the resistivity

To understand the properties of CeAuSb2better, I performed isothermal resistivity measurements while ramping the magnetic field at a series of angles with respect to the c-axis direction, with a He-4 flow cryostat which is equipped with a single axis rotator. The magnetoresistivity flM, defined as flm = (fl(H)-fl(0))/fl(0), is shown in Figure 5.31 (based on the measurements with a sample that was cut from the hs006 batch and measured at 1.55 K.)

The magnetoresistivity in Figure5.31is interesting because it seems that the main outcome of tilting the magnetic field away from the c-axis is only to push the two transitions to higher fields, preserving the shape and magnitude of the response. To verify this, one can plot the magnetoresistivity as a function of the c-axis projection of the magnetic field, the outcome of which is presented in Figure 5.32.

Figure 5.31: The field angler de- pendence of the magneto-resistivity measured at 1.55 K. ◊ in the leg- end represents the field angle with

respect to the crystal c-axis.

Figure 5.32: The magnetoresis- tivity of CeAuSb2 as a function of the c-axis projection of the mag- netic field (measured at 1.55 K). From Figure5.31and Figure5.32, it is noticeable that at high angles (e.g. ◊Ø60¶), the magnetoresistivity at the first phase transition (at HMM) shows some fine structure, for example at ◊ = 70¶ a kink is seen at _{flM ¥} 0.6, above which the slope offlMis different. However this phenomenon was not clearly reproduced when another sample from the same batch was measured (in the LNCMI in Grenoble, France) with a dilution refrigerator in the presence of magnetic fields up to 35 T, whose direction can be tuned from parallel with the sample c-axis to parallel with the ab-plane.

Figure 5.33: The field angle dependence of the magnetoresistivity measured at temperatures below 100mK. ◊ in the legend represents the field angle with

Figure 5.33 shows flM as a function of magnetic field (panel (a)) and its c-axis projection (panel (b)), based on measurements on the second sample in fields up to 35 T and below 100 mK. The main features of flM in Figure5.33 are similar to those in Figure 5.31 and Figure 5.32. For example, the magnetic fields at which the two phase transitions occur, both increase as the field is tuned off the c-axis. In addition, the shape of the high resistivity phase is roughly preserved and sim- ilar collapse is seen when flM is plotted as a function of the c-axis projection of the field. On the other hand the results shown in Figure 5.33 also have evident differences from those shown in Figure 5.31 and Figure 5.32. For example, when the magnetic field is applied close to the ab-plane (e.g. ◊ _Ø 80¶), the width of the antiferromagnetic B phase deviates from the 1/cos(◊) scaling relationship ( i.e. the scaled width of the antiferromagnetic B phase is reduced) and the magne- toresistivity shows enhancement in each phase. Moreover the magnetoresistivity below Hc shows different field dependence at different field angles, which is dif- ferent from what was seen in Figure 5.31 and Figure 5.32, where only a kink of flM was seen at the first transition. So at the moment it is difficult to draw firm conclusions on the detailed evolution of flM close to the two transitions (or within the antiferromagnetic B phase), due to the sample dependence.

In Figure 5.33 (b), the scaled curve of flM with ◊ Ø 84¶ exhibits a noticeable difference from the rest. However, for◊close to 90¶a small deviation in◊can result in a relatively big change in the scaled field values, for example, a 1¶ reduction from 84¶ will result in a 0.4 T increase for the (Hcos(◊)) scaled metamagnetic transition field HMM and this would be enough to remedy the discrepancy in the scaling for this angle. Therefore the scaling deviation at high field angles is not absolutely clear.

In summary, the dependence of flM on ◊ provides evidence that the observed be- haviour is dominantly determined by the c-axis component of the applied magnetic field.

5.3.2 Torque magnetometry measurements