Angular Dependence

Figure 6.6: The two different orientations of the rotational axis at which measurements of the effective electron mass were carried out.

6.5.2 Angular Dependence of the Effective Electron Mass General Description

Measurements of the de Haas-van Alphen oscillation amplitude as function of the tem-perature were carried out at different tilting angles and for two different orientations of the rotational axis. The two different axes with respect to the crystal morphology are shown in figure 6.6. They will be referred to hereafter as orientation ”1”and ”2”in the way indicated in figure 6.6.

In the experimental results, the α-frequency Fα was present at all orientations. In many orientations, the harmonic frequency F_2α could also be observed. Occasionally, the β-frequency F_β was also present and sometimes their mixing products Fβ−α, F_β+α etc. as well. The latter, in general, however were of small amplitude and could only be seen in some measurements carried out at the lowest temperatures available in this series.

Sufficient data for an effective mass analysis were obtained for the frequenciesF_α andF_2α. The results obtained are shown in figure 6.7. All masses shown in this figure have been observed in orientation 2. The masses given for the harmonic frequency are obtained with a harmonic factorp= 2 inserted into the term (1.16). Other authors use the convention to keep this factor equal to one, regardless of the fact which harmonic is analysed. If results that are given within the framework of the latter convention are to be compared to the results presented in this work, the masses given in this work have to be multiplied with a factor of two to be consistent with the results to be compared with.

The error bars correspond to 95% confidence intervals as resulting from the Levenberg-Marquardt fitting algorithm.

The figure shows also two fits that have been obtained by fitting am₀/cos(θ)-model to the data points on the side of angles with a negative sign. Best fits are obtained for the following values of the zero angle massm0:

m^α₀ = (3.25±0.2)m_e (6.2)

m^2α₀ = 2.6me. (6.3)

Again, the error represents a 95% confidence interval for the fit parameters. For the 2α data, no confidence interval could be calculated because only two points of data went into the fit.

The fit was done without leaving a free parameter to account for a possible shift of the zero of the angular scale. This was done because the angles have been determined directly from the observed de Haas-van Alphen frequency.

The measurements concentrate on angles around 35^◦because previous results indicated an irregular behaviour there whereas the behaviour other angular ranges was found to be regular.

In addition, two masses were determined in orientation 1. At the position 38.4^◦, m^α₀ = (4.17±0.03me was found and at -41.2^◦, a mass of m^α₀ = (3.8±0.05)me could be observed.

The results obtained in orientation 2 are summarised in figure6.7.

As can be seen in figure6.7, the experimental results in the negative angle range follow the expected 1/cos(θ) behaviour, whereas this is not the case for the regime of positive angles. There is no symmetry around the zero position.

Moreover, the results for the harmonic frequencyF2α do not yield the same effective mass as the analysis of the data obtained from theF_alpha data.

In addition, in orientation 1, the following two masses could be determined:

θ m^α₀ 38.4^◦ 4.17m_e -41.2^◦ 3.9m_e

The excessive error at the angle 30.3^◦ is due to an anomalous dependence of the de Haas-van Alphen amplitude on the temperature which makes it difficult to evaluate for an effective electron mass as this position. Figure6.8shows this behaviour together with the fit curves. If the point at 0.4K is ignored in the fitting process, effective electron masses of m^α = (7.7±0.1)me and m^2α = (4.2±1.5) are obtained. A further measurement was carried out at an angle of 26.1^◦. The data of this measurement did not allow for any effective electron mass analysis at all. The temperature dependence found at this position is shown in figure 6.9.

The measurements mainly concentrate on angles |θ| > 30^o because the 1/cos(θ) de-pendence of the effective electron mass is not questioned by the previous results having stimulated these experiments. A comparison with these results will be given in the dis-cussion.

Some irregular behaviour of the effective electron mass on the side of positive angles is clearly visible.

Figure 6.7: Effective electron masses observed in κ-(ET)2Cu(NCS)2 as function of the tilting angle in orientation 2 extracted from the observed oscillations at the fundamental frequency F_α and the harmonic frequency F_2α. The lines show fits to the data obtained from the points on the side of negative angles to a 1/cos(θ) behaviour. The solid line represents a fit to the fundamental frequency mass m_α and the dashed one corresponds to the harmonic frequency mass m_2α.

0.4 0.6 0.8 1.0 1.2 1.4 1.6 Fit without point at 0.4K

Figure 6.8: Temperature dependence of the de Haas-van Alphen amplitude at 30.3^◦.

0.4 0.6 0.8 1.0 1.2 1.4

Figure 6.9: Temperature dependence of the de Haas-van Alphen amplitude at 26.1^◦.

The comparison of the points to the fit shows clearly the non 1/cos(θ) behaviour of the effective electron mass for angles θ > 25^◦. There is, however, no corresponding deviation from the 1/cosθ behaviour on the side of negative angles.

In orientation 1, an effective mass of 4.17me was measured at an angle of 38.4^o and one of 3.8m_e at -41.2^o.

6.6 Discussion

The observation of the apparent effective electron mass of κ-(ET)2Cu(NCS)2 as function of the tilting angleθwith respect to the magnetic field yielded several deviations from the behaviour that can be expected from a simple quasi-two dimensional system: the observed dependence does not follow a 1/cos(θ) law. Moreover, there is no symmetry around the zero position. The discussion of these experimental observations will be organised as follows: first, the findings will be discussed in the framework of known properties of the system. This includes a comparison with experimental findings made by other researchers in the field. After that, possible artefacts that could lead to these observations will be discussed and ruled out as far as possible. Then, an attempt is made to compare the findings with the predictions of theoretical models that consider the specific conditions of a two-dimensional electron system.

6.6.1 Range where Deviations from Lifshitz-Kosevich Behaviour Occur