Criteria for Directly Detecting Topological Fermi Arcs in Weyl Semimetals

The recent discovery of the first Weyl semimetal in TaAs provides the first observation of a Weyl fermion in nature and demonstrates a novel type of anomalous surface state, the Fermi arc. Like topological insulators, the bulk topological invariants of a Weyl semimetal are uniquely fixed by the surface states of a bulk sample. Here we present a set of distinct conditions, accessible by angle-resolved photoemission spectroscopy (ARPES), each of which demonstrates topological Fermi arcs in a surface state band structure, with minimal reliance on calculation. We apply these results to TaAs and NbP. For the first time, we rigorously demonstrate a nonzero Chern number in TaAs by counting chiral edge modes on a closed loop. We further show that it is unreasonable to directly observe Fermi arcs in NbP by ARPES within available experimental resolution and spectral linewidth. Our results are general and apply to any new material to demonstrate a Weyl semimetal.

Figure 1

Four criteria for Fermi arcs. (a) A bulk (blue) and surface (red) Brillouin zone with four Weyl points ($\pm$) and various 2D manifolds with Chern number $n$. Sweeping a plane through a Weyl point changes $n$. Also, any closed loop on the surface hosts chiral edge modes protected by the enclosed bulk chiral charge. For instance, a closed loop enclosing a $+$ Weyl point will host one right-moving chiral edge mode [8, 9, 10]. (b) The criteria: (1) A disjoint contour. (2) A closed contour with a kink. (3) No kinks within experimental resolution, but an odd set of closed contours. (4) An even number of contours without kinks, but net nonzero chiral edge modes.

Figure 2

Surface states of NbP by ARPES. (a)–(d) Constant energy cut by vacuum ultraviolet APRES, at incident photon energy $h\nu =30\mathrm{eV}$, on the (001) surface of NbP. (e) Same as (a)–(d), at $E_B=0.05\mathrm{eV}$, with Weyl points marked and the two paths $\mathcal{C}$ and $\mathcal{P}$ on which we measure Chern numbers. (f) The difference of ARPES spectra at $E_B=0.05$ and $E_B=0.1\mathrm{eV}$, illustrating the direction of the Fermi velocity all around the lollipop and peanut pockets. The blue contour is always inside the red contour, indicating that the sign of the Fermi velocity is the same going around each contour. This result excludes the possibility that the lollipop actually consists of Fermi arcs attached to the $W_2$. (g) Cartoon summarizing the band structure. (h) Band structure along $\mathcal{C}$, with chiralities of edge modes marked by the arrows. We associate one arrow to each spinful crossing, even where we cannot observe spin splitting due to the weak spin-orbit coupling of NbP. There are the same number of arrows going up as down, so the Chern number is zero. (i) Same as (h) but along $\mathcal{P}$. Again, the Chern number is zero.

Figure 3

Numerical calculation of Fermi arcs in NbP. First-principles band structure calculation of the (001) surface states of NbP at (a) the energy of the $W_2$, above the Fermi level, (b) the Fermi level and (c) the energy of the $W_1$, below the Fermi level. In (a) we see (i) a small set of surface states near the $W_2$ and (ii) larger surface states near $\overline{X}$ and $\overline{Y}$. Near the energy of (b) there is a Lifshitz transition between the surface states at (i) and (ii), giving rise to lollipop and peanut-shaped pockets. At (c) we see that the lollipop and peanut pockets enlarge, so they are holelike. We also observe short Fermi arcs connecting the $W_1$. The numerical calculation shows excellent agreement with our ARPES spectra but also shows small spin splitting, suggesting that the weak spin-orbit coupling of NbP makes it challenging to directly observe Fermi arcs by ARPES. (d) Zoom-in of the surface states around the $W_2$, indicated by the white box in (a). We find two Fermi arcs and two trivial closed contours, illustrated in (e). (f) Zoom-in of the surface states around $W_1$, indicated by the white box at the top of (c). We find one Fermi arc, illustrated in (g).

Figure 4

Comparison of NbP and TaAs. (a) Energies of the $W_2$, their Lifshitz transition, and the Fermi level from the numerical calculation. One consequence of the small spin splitting is that the Lifshitz transition between the $W_2$ is only $\sim 0.007\mathrm{eV}$ below the energy of the $W_2$ and $\sim 0.019\mathrm{eV}$ above the Fermi level. (b) Because ${ϵ}_F<{ϵ}_L$, it makes no sense to calculate the Chern number on $\mathcal{C}$ and $\mathcal{P}$, because the system on that cut is not an insulator; see the interrupted dotted red line in the last row of (b). (c) Plot of the positions of the Weyl point projections in NbP and TaAs arising from one Dirac line (see Sec. I in the Supplemental Material [32]). We see that the separation of the Weyl points is $\sim 4$ times larger in TaAs due to the large spin-orbit coupling. (d) Fermi surface by vacuum ultraviolet APRES, at incident photon energy $h\nu =90\mathrm{eV}$, on the (001) surface of TaAs. (e) Same as (d), but with Weyl points and two paths $\mathcal{C}$ and $\mathcal{P}$ marked. (f) Band structure along $\mathcal{C}$, with chiralities of edge modes marked by the arrows. The net Chern number appears to be $+2$, inconsistent with the calculation. This can be explained by considering the small separation between the $W_1$. (g) Same as (f) but along $\mathcal{P}$. The path encloses only the well-spaced Weyl points and we find a Chern number $+2$, consistent with the calculation.