Measuring anisotropic resistivity of single crystals using the van der Pauw technique

Anisotropy in properties of materials is important in materials science and solid-state physics. Measurement of the full resistivity tensor of crystals using the standard four-point method with bar shaped samples requires many measurements and may be inaccurate due to misalignment of the bars along crystallographic directions. Here an approach to extracting the resistivity tensor using van der Pauw measurements is presented. This reduces the number of required measurements. The theory of the van der Pauw method is extended to extract the tensor from parallelogram shaped samples with known geometry. Methods to extract the tensor for both known and unknown principal axis orientation are presented for broad applicability to single crystals. Numerical simulations of errors are presented to quantify error sources. Several benchmark experiments are performed on isotropic graphite samples to verify the internal consistency of the developed theory, test experimental precision, and characterize error sources. The presented methods are applied to a $\mathrm{RuS}{\mathrm{b}}_2$ single crystal at room temperature and the results are discussed based on the error source analysis. Temperature resolved resistivities along the $a$ and $b$ directions are finally reported and briefly discussed.

Figure 1

Effect of the change of coordinates on the sample geometry and tensor. The sample (left) is anisotropic while the transformation results in an equivalent isotropic representation (right). $\phi$ is the angle between the tensor principal axis system and the laboratory axes. Before the transformation the laboratory axes are defined with $x$ along the AB side and $y$ perpendicular to this in the AD direction. $\chi$ is the angle in the anisotropic parallelogram while $\pi \alpha$ is the angle in the isotropic parallelogram.

Figure 2

Illustration of the conformal mapping with contact placements. Contact D corresponds to infinity in any direction on the upper complex half-plane. $k$ has to be in the range $0<k<1$ for contact C to be placed between contacts B and D.

Figure 3

Reshaping the sample between two measurements can provide enough information to determine $\alpha$. After the first measurement the middle sample is cut along either the dotted or dashed lines. In the right sample with the dotted line the tensor principal axes has been rotated relative to the lab frame, indicated by the angle $\theta$. In the left sample, the sample geometry has changed but the tensor principal axes have the same orientation. The sample can be reshaped in many more ways which will give the right result provided attention is paid to the rotation of the tensor principal axes.

Figure 4

Tensor element error relative to ${\rho }_{\mathrm{iso}}$ for displacement of the A (a) and B (b) contacts toward the adjacent corners. The displacement is relative to the $a$ and $b$ side lengths.

Figure 5

Relative change in tensor elements as a function of error in $\chi$. The simulation was carried out using the same sample as in Fig. 4, and normalized with ${\rho }_{\mathrm{iso}}$.

Figure 6

Tensor elements as a function of aspect ratio of the isotropic equivalent sample. The points are from the measured resistances and the lines are calculated from the aspect ratio and average resistivity. The graphite was shaped as a rectangle in (a) and as a parallelogram with $\pi \alpha$ corresponding to $63.{55}^{\circ }$ in (b).

Figure 7

Errors in tensor elements relative to ${\rho }_{\mathrm{iso}}$ for 20 repeated measurements. The graphite was shaped as a rectangle in (a) and as a parallelogram with $\pi \alpha$ corresponding to $63.{55}^{\circ }$ in (b) with aspects ratios $r=0.81$ and $r=0.86$, respectively.

Figure 8

Error in tensor elements relative to ${\rho }_{\mathrm{iso}}$ as a function of deviation from a parallelogram in degrees. The lines are guides to the eye.

Figure 9

Tensor elements divided by ${\rho }_{\mathrm{iso}}$ as a function of the change in $\chi$ in the parallelogram. $\alpha$ was extracted from Eq. (19). While ${\rho }_{xy}$ is not strictly 0 this is omitted since it is small and has a low variation.

Figure 10

Temperature resolved measurement of ${\rho }_a$ and ${\rho }_b$ of $\mathrm{RuS}{\mathrm{b}}_2$. The single crystal method and Eq. (17) with constant, temperature independent $\phi$ were used. The inset shows ${\rho }_a/{\rho }_b$.