All-Electrical Determination of Crystal Orientation in Anisotropic Two-Dimensional Materials

The crystal orientation of an exfoliated black phosphorous flake is determined by purely electrical means. A sequence of three resistance measurements on an arbitrarily shaped flake with five contacts determines the three independent components of the anisotropic in-plane resistivity tensor, thereby revealing the crystal axes. The resistivity anisotropy ratio decreases linearly with increasing temperature $T$ and carrier density reaching a maximum ratio of 3.0 at low temperatures and densities, while mobility indicates impurity scattering at low $T$ and acoustic phonon scattering at high $T$.

Figure 1

Conformal five-contact method on anisotropic BP sample $A$. (a) Polygon approximation of the sample in the $x\text{−}y$ lab frame. (b) Rotation by $\theta$ to the $c\text{−}a$ crystal axes. (c) Anisotropic scaling of crystal axes by $1/\alpha$ and $\alpha$, respectively, to equivalent isotropic resistivity sample in the $c^{\prime }\text{−}{a}^{\prime }$ basis. (d) Conformal mapping to the semi-infinite $u\text{−}v$ plane. Inset photo: sample $A$ in the lab frame overlaid with polygon approximation. All four-point resistances of the original anisotropic sample are identical to those of the semi-infinite plane. (e) C5C Variance minimization method: color plot of normalized variance $\sigma (\alpha ,\theta )$ from Eq. (3) on a $2\theta$ polar plot with minimum variance at ${\theta }_0=93°$ and ${\alpha }_0=0.85$ ($r=1.9$), where ${\rho }_{\mathrm{s},0}=11\mathrm{k}\mathrm{\Omega }$. (f) C5C Parametric intersection method: shown are various $\alpha (\theta )$ curves on a $2\theta$ polar plot. Variance minimization method results in (e) agree with parametric intersection results in (f).

Figure 2

Conformal five-contact method on the isotropic ${\mathrm{MoS}}_2$ sample $C$. (a) Inset photo: sample $C$ in the lab frame overlaid with polygon approximation; the main panel shows conformal mapping to the semi-infinite $u\text{−}v$ plane. (b) C5C Variance minimization method: color plot of normalized variance $\sigma (\alpha ,\theta )$ [Eq. (3)] on a $2\theta$ polar plot with minimum variance at ${\alpha }_0=0.99$ ($r=1.05$) at the center of polar plot, where ${\rho }_{\mathrm{s},0}=69\mathrm{k}\mathrm{\Omega }$. (c) C5C Parametric intersection method: shown are various $\alpha (\theta )$ curves on a $2\theta$ polar plot. Variance minimization results in (b) agree with parametric intersection results in (c).

Figure 3

Comparison between the anisotropy angle determined by the C5C method and polarized Raman spectrum on BP sample $A$. (a) Crystal orientation determined by the C5C method in Fig. 1 is ${\theta }_0=93°\pm 3°$. (b) Selected polarized Raman spectra on sample $A$ with polarization angle rotating from 0° to 360°. (c) Radiii of black dots on the polar plot represent peak amplitudes for polarized Raman intensities of the $A_{\mathrm{g}}^2$ peak, with the black curve representing a theoretical fit. Resulting Raman crystal orientation angle is ${\theta }_R=97°\pm 5°$ in agreement with more accurate ${\theta }_0$ from the C5C method.

Figure 4

Anisotropy analysis on BP sample $B$ ($d=28\mathrm{nm}$) as a function of density and temperature. Curves are guides to the eye. (a) Temperature dependence: at fixed density, carrier mobilities ${\mu }_{cc}$ and ${\mu }_{aa}$ along the armchair and zigzag directions, respectively, both increase with a power law in the low-temperature range, before saturating at high temperatures, indicating the onset of phonon scattering. (b) At fixed density, anisotropy angle ${\theta }_0$ (top panel) remains constant, and ratio $r$ (bottom panel) decreases linearly with temperature. (c) Density dependence: Carrier mobilities ${\mu }_{cc}$ and ${\mu }_{aa}$ increase with increasing carrier density. (d) Anisotropy angle ${\theta }_0$ (top panel) remains constant while the anisotropy ratio $r$ (bottom panel) decreases slightly with carrier density.