Direct optical detection of Weyl fermion chirality in a ... · NATURE PHYSICS | 8 8 DOI: 10.1038/NPHYS4146 SUPPLEMENTARY INFORMATION © 2017 Macmillan …

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Supplemental Information for

Direct optical detection of Weyl fermion chirality in a topological

semimetal

(Dated: March 10, 2017)

1

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS4146

NATURE PHYSICS | www.nature.com/naturephysics 1

http://dx.doi.org/10.1038/nphys4146

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This file includes:

I. Symmetry relation of the Weyl nodes in TaAs

II. Temperature and power dependences of the photocurrent

III. Materials and methods

III.1. Mid-infrared photocurrent microscopy setup

III.2. Single crystal growth and magneto-transport measurements

III.3. First-principles band structure calculations

III.4. Methods for photocurrent calculations

III.5. Single crystal x-ray diffraction methods

IV. Details of the photocurrent calculations

IV.1. An intuitive explanation for the non-vanishing photocurrent

in an inversion breaking Weyl semimetal

IV.2. Symmetry analyses of the photocurrent

IV.3. Calculated photocurrents from each Weyl fermion

IV.4. Estimation of the photocurrent amplitude

V. Exclusion of other CPGE mechanisms

VI. Additional photocurrent measurements

VII. Previous transport and ARPES measurements on Weyl semimetals

2NATURE PHYSICS | www.nature.com/naturephysics 2





I. SYMMETRY RELATION BETWEEN WEYL NODES IN TaAs

TaAs crystalizes in a body-centered tetragonal lattice system. The lattice constants are

a = b = 3.437 A and c = 11.646 A, and the space group is I41md (#109, C4v) (Fig. S1).

The important symmetries are the time-reversal symmetry (T ), the four-fold rotational

symmetry around the z axis (C4z) and two mirror reflections about x = 0 and y = 0 (Mx

and My). Interestingly, the chirality flips under a mirror reflection M, whereas it remains

unchanged under time-reversal T and the C4z rotation. More specifically, the transformation

under symmetries can be written as

|χ = ±1, kx, ky, kzMx=⇒ |χ = ∓1,−kx, ky, kz (S1)

|χ = ±1, kx, ky, kzT

=⇒ |χ = ±1,−kx,−ky,−kz (S2)

|χ = ±1, kx, ky, kzC4z=⇒ |χ = ±1,−ky, kx, kz. (S3)

We use these symmetries to understand the Weyl node configuration in TaAs. If we pick

one Weyl node on the kz = 0 plane (W1) and assign its chirality, then C4z, Mx and My

will generate another 7 W1 Weyl nodes on the same kz = 0 plane (Fig. S2a-c). Similarly,

if we fix one Weyl node at a finite kz (kz = k0 = 0, labeled W2), the T , C4z, Mx and My

symmetries can give rise to another 15 W2 Weyl nodes at both kz = ±k0 planes (Fig. S2d-g).

Therefore, among the total 24 Weyl nodes in TaAs, only two are independent Weyl nodes

and the others can be automatically related by symmetries.

We define Jn as the photocurrent from the nth Weyl node. We explain the following issue:

If two Weyl nodes (i, j) are related by certain symmetry, whether their currents ( J i, J j) can

also be related by the symmetry.

• If Weyl nodes (i, j) are related by a 90 rotation around z (C4z), then J i, J j can only

be related by C4z if the light propagates along z (Figs. S3a,b). Specifically, if the

propagation direction of the light is along z, we have J jx = −J i

y, Jjy = J i

x, Jjz = J i

z.

• If Weyl nodes (i, j) are related by the mirror plane Mx, then J i, J j can only be related

by Mx if the light propagates along x (Figs. S3c,d). Specifically, if the propagation

direction of the light is along x, we have J jx = −J i

x, Jjy = J i

y, Jjz = J i

z.






c

a

y

a

FIG. S1: Crystal structure of TaAs and relevant symmetries. a, A unit cell of TaAs lattice. b, No

mirror symmetry along c. c, Mirror symmetries along a(b) and the four-fold rotational symmetry

around z axis (C4z).






a

Mx My

kz

kx

kyW2

kz

kx

ky

kz

kx

ky

e

g

T

kz

b c

dkz

kz

f

kz

kx

ky

kz

Γ

C4z

kx

ky

kx

ky

kx

ky

Γ

C4z Mx My

W1

FIG. S2: a-c, If we assign the W1 highlighted by the dashed circle with a definite chirality, then

C4z, Mx and My will generate another 7 W1 Weyl nodes on the same kz = 0 plane. d-g, If we

assign the W2 highlighted by the dashed circle with a definite chirality, the T , C4z, Mx and My

symmetries can give rise to another 15 W2 Weyl nodes.






FIG. S3: The propagation direction and electric field of the light in the presence of the C4z

rotational symmetry (panels (a,b)) or the mirror symmetry Mx (panels (c,d)).






II. TEMPERATURE AND POWER DEPENDENCE OF THE PHOTOCURRENT

FIG. S4: a, The photocurrent as a function of light polarization measured at different temperatures.

b, Temperature dependence of both the circularly polarized light induced photocurrent and the

photo-thermal current. c, Nearly linear power dependence of the circularly polarized light induced

photocurrent. The beam-size is 50 µm × 50 µm. Thus a 10 mW power corresponds to a power

density of about 4× 106 W·m−2.

As one increases temperature from T = 10 K to T = 300 K, the circularly polarized

light induced photocurrent reduces significantly (Fig. S4a). This is quite intuitive since

the photocurrent depends on the relaxation time (see equation S7) and we expect the elec-

tron mobility and relaxation time decrease significantly with increasing temperature [8]. By

contrast, the photo-thermal current near the contact (Fig. S4b) increases with increasing






temperature. This is also expected because the Seebeck coefficient of metals is larger at

higher temperatures [9]. The temperature dependent data, especially the opposite tem-

perature trends seen in Fig. S4b allow us to further separate these two effects.The power

dependence in Fig. S4c shows that our measurements stay in the linear regime.






III MATERIALS AND METHODS

III.1. Mid-infrared photocurrent microscopy setup

c 200

-200

I (nA

)

b

y

z

a a

cb

FIG. S5: a, Mid-infrared photocurrent microscopy setup. b, Optical photograph of the TaAs

device. Scale bar: 300 µm. c, Reflection image of the entire device. d, Photocurrent image of the

device.

In our experiment, the sample is contacted with metal wires and placed in an optical

scanning microscope setup (Figs. S5a,b) that combines electronic transport measurements

with light illumination [1, 2]. The laser source is a temperature-stablized CO2 laser with

a wavelength λ = 10.6 µm (ω ≃ 120 meV). A focused beam spot (diameter d ≈ 50

µm) is scanned (using a two axis piezo-controlled scanning mirror) over the entire sample

and the current is recorded at the same time to form a colormap of photocurrent as a

function of spatial positions (Fig. S5d). Reflected light from the sample is collected to form

a simultaneous reflection image of the sample (Fig. S5c). The absolute location of the photo-

induced signal is therefore found by comparing the photocurrent map to the reflection image.

The light is first polarized by a polarizer and the chirality of light is further modulated by

a rotatable quarter-wave plate characterized by an angle θ.






FIG. S6: a, Hall resistivity measurement of the TaAs sample. b, Shubnikov-de Haas (SdH)

oscillation of the longitudinal resistivity of the TaAs sample used for photocurrent measurements.

The non-oscillatory-background is removed. The quantum oscillation is described by ρxx = ρ0[1 +

A(B,T ) cos(2π(F/B+γ))] [4], where A(B,T ) is the amplitude of the oscillations, F is the frequency

of the oscillation, and γ is the Onsager phase. The frequency of the oscillation is obtained from

this data is F = 7.7 T. c, First-principles calculated extremal area of the W1 Fermi surface as

a function of the energy from the W1 Weyl node. d, Schematic band structure of the W1 and

W2 WFs. According to the first-principles calculations, the W2 Weyl node is 13 meV above the

W1 Weyl node. By matching the measured extremal area of the Fermi surface to our calculations

(panel (c)), we obtain that the Fermi level is about 17.8 meV above the W1 Weyl nodes.

III.2. Single crystal growth and magneto-transport measurements

Single crystals of TaAs were prepared by the standard chemical vapor transfer (CVT)

method. The polycrystalline samples were prepared by heating up the stoichiometric mix-






tures of high quality Ta (99.98%) and As (99.999%) powders in an evacuated quartz am-

poule. Then the powder of TaAs (300 mg) and the transport agent were sealed in a long

evacuated quartz ampoule (30 cm). The end of the sealed ampoule was placed horizontally

at the center of a single-zone furnace. The central zone of the furnace was slowly heated

up to 1273 K and kept at the temperature for 5 days, while the cold end was less than

973 K. Magneto-transport measurements were performed using a Quantum Design Physical

Property Measurement System.

Figure S6a shows the Hall resistivity measurement at T = 2 K. The negative slope

of ρyx − B data shows that the sample’s magneto-transport is dominated by electron-like

carriers at low-temperatures. The oscillatory part of the longitudinal resistivity (Fig. S6b)

is periodic in 1B. The observed quantum oscillations are described by the standard formula

ρxx = ρ0[1+A(B, T ) cos(2π(F/B+γ))] [4], where A(B, T ) is the amplitude of the oscillations,

F is the frequency of the oscillation, and γ is the Onsager phase. By Fourier analysis of

the ρxx(1B) data, we obtain a frequency F = 7.7 T. The extremal cross-section area of the

Fermi surface (AF) can be obtained from the SdH oscillation frequency by AF = 2πeF =

7.3 × 10−4 A−2. These numbers extracted from the SdH oscillation is very close to our

previous measurements on similar TaAs samples [3]. In Fig. S6c, we show the calculated

extremal area of W1 Fermi surface. By matching the measured value to our calculation, we

obtain that the Fermi level is about 17.8 meV above the W1 Weyl nodes. Since calculations

show that the W2 Weyl node is 13 meV higher than the W1 node, the Fermi level is about

4.8 meV above the W2 Weyl node. Therefore, EF is very close to the W2 Weyl nodes

but relatively far from the W1 Weyl nodes. This explains the electron-like Hall resistivity

observed in Fig. S6a. Moreover, as shown in Ref.[5], the photocurrent vanishes when EF is

located at the Weyl node. Therefore, we expect that our observed photocurrent is dominated

by the contribution from the W1 WFs.






III.3. First-principles band structure calculations

FIG. S7: a, The 24 WFs in the first bulk Brillouin zone (BZ) of TaAs. The (100), (010), (001)

directions are defined. b, The E − kx dispersion of the W1 and W2 WFs highlighted by dotted

circles.

First-principles calculations were performed by the OPENMX code within the framework

of the generalized gradient approximation of density functional theory [7]. Experimental

lattice parameters were used, and the details for the computations can be found in our pre-

vious work in Ref. [6]. A real-space tight-binding Hamiltonian was obtained by constructing

symmetry-respecting Wannier functions for the As p and Ta d orbitals without performing

the procedure for maximizing localization. The calculated k space positions of the Weyl

nodes are shown in Table. S1






kx (πa ) ky (πa ) kz (πc )

W1 0.014 0.949 0

W2 0.037 0.520 0.592

TABLE S1: The calculated k space positions of the Weyl nodes. The positions of other WFs can

be related by symmetries.

Direction v+ (eV·A) v− (eV·A)

(100) +2.03 -3.55

(010) +0.70 -2.48

(001) +0.17 -0.17

(110) +0.80 -3.14

(101) +1.42 -2.52

(011) +0.49 -1.78

TABLE S2: Calculated Fermi velocities of the W1 WF near at (kx, ky, ky) = (0.0072πa , 0.4752π

a , 0)

(highlighted by the dotted circle). The Fermi velocities of other W1 WFs can be related by

symmetries.

We use the first-principles calculated band structure to obtain the degree of tilt needed

for our photocurrent calculation shown by Tables S2, S3. We see that v+ = −v−, which

means that the WFs in TaAs are indeed tilted.

The low-energy effective Hamiltonian expanded around each Weyl node can be generally

written as:

HWeyl(q) = vtqtσ0 + vFvi,jqiσj , (S4)

where vF is a velocity parameter without tilt, σj are Pauli matrices, vi,j represents anisotropy,

vt gives the tilt velocity and qt = t · q with t being the tilt direction. These parameters (vF,

vi,j, vt, qt) are obtained by fitting the dispersion from this effective Hamiltonian to that

of from the first-principles calculated results. In fact, along most directions away from the

Weyl node, linear band structure is a quite good approximation to the DFT band structure.

As an example, we study the dispersion of the W1 Weyl cone in Fig. S8 along the ky and kz






Direction v+ (eV·A) v− (eV·A)

(100) +1.57 -2.81

(010) +2.40 -1.01

(001) +0.92 -2.68

(110) +2.52 -2.41

(101) +0.30 -2.43

(011) +0.81 -1.06

TABLE S3: Calculated Fermi velocities of the W2 WF near at (kx, ky, ky) =

(0.01852πa , 0.28312π

a , 0.60002πc ) (highlighted by the dotted circle). The Fermi velocities of other

W2 WFs can be related by symmetries.

FIG. S8: a, The 24 Weyl nodes of TaAs in the BZ. The green arrows represent the k paths for the

dispersions shown in panels (b,c). b,c, Band structure of the W1 Weyl node along ky and kz as

indicated by the green arrows in panel (a).

directions. As shown in Figs. S8b,c, the DFT dispersions do not deviate significantly from

linearity.

On the other hand, we do find one exception, i.e the direction connecting two Weyl nodes.

The dispersion along this direction is heavily affected by nonlinear corrections, because one

Weyl cone “feels” the influence from its neighbour, as shown in Fig. S9. Most significantly,

the energy gap at k points between the two W1 nodes is always smaller than 120 meV

(Figs. S9a,b), which prohibits a 120 meV optical transition (The same issue does not exist






kx

ky

kz = 0

kx

ky

kz = +0.6 × /c

a b

c d

W2 cut x

W2 cut x

E(e

V)

kx

(Å-1)

-0 .12

-0.08

-0.04

0.00

0.04

0.08

-60 -40 -20 0 20 40 60x10

-3

-0 .08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

-30 -20 -10 0 10 20 30x10

-3

E(e

V)

W1 cut x

W1 cut x

FIG. S9: a,b, The dispersion across a pair of nearby Weyl nodes. The energy gap at k points

between the two W1 nodes is always smaller than 120 meV, which prohibits a 120 meV optical

transition. c,d, The same as panels (a,b) but for a pair of W2 nodes. The energy gap at k points

between two W2 nodes can be larger than 120 meV as shown in panel d.

for W2 because the gap goes above 120 meV, Figs. S9c,d) This was certainly not captured

in our linear approximation. In order to estimate how much this affects our result, we study

the dispersion as a function of θ (θ defined in Fig. S10a,b). In the kx, ky plane, we see

that the energy gap at kx = 0 increase with θ(Fig. S10e-g). At θ = 26, the energy gap

reaches 120 meV. Doing this analysis in 3D k space yields a solid angle (Figs. S10c,d) in the

“bridge” region connecting two nodes. The gap between the two W1 Weyl bands is smaller

(larger) than 120 meV inside (outside) this solid angle. For W1 Weyl cones, this solid angle

is 13% of the 4π solid angle. Therefore, the phase area that is strongly affected by this

nonlinear correction is quite small.

We further consider whether a 120 meV optical transition may involve a 3rd band.

Fig. S11b shows the band structure of the W1 Weyl cone. Within a ±1 eV energy win-






FIG. S10: a, The 8 W1 Weyl nodes on kz = 0. b, A zoom-in view of a pair of nearby W1 nodes.

We study its dispersion as a function of θ in panels (d-f). e-g, Dispersions across the W1 Weyl

node a function of θ. The gap increases with θ. At θ = 26, the gap reaches 120 meV. c,d, Doing

the same analysis in 3D k space yields a solid angle in the ”bridge” region connecting two nodes.

dow, we find two additional bands (bands T1 and T2). Along this k path, the energy of

the optical transition between a Weyl band and band T1 is at least 182 meV. Similarly, the

energy of the optical transition between any Weyl band and band T2 is at least 391 meV.

We have investigated the band structure throughout the BZ. The k point that allows

the optical transition between a Weyl band and band T1 with the lowest photon energy is

located very close to the W2 Weyl node at (0.282πa, 0, 0.2752π

c) (along Cut 2 in Figs. S11a,c).

The lowest photon energy is 139 meV. Similarly, for band T2, lowest photon energy is 360

meV. Therefore, we conclude that for our sample (EF 18 meV above W1 Weyl nodes), the

120 meV transition is exclusively within the two bands that cross to form the Weyl nodes.






FIG. S11: a, The 24 Weyl nodes of TaAs in the first BZ. The momentum space paths for the

dispersions shown in panels (b,c) are indicated by the green arrows. b, The dispersion of the

W1 Weyl node along Cut1. c, The dispersion along Cut2. Along this path, the optical transition

between Weyl bands and trivial band T1 has the lowest energy, 139 meV.






III.4. Methods for photocurrent calculations

We elaborate on the methods of photocurrent calculation. For a single Weyl node, the

photocurrent density due to a deviation of the electronic distribution δnl(q) = nl(q)− n0l (q)

is:

J = (−e)∑

q,l=±

vl(q)δnl(q) = (−e)∑

q

∆v(q)δn+(q), (S5)

where l = +,− represents the upper part (+) and lower part (−) of the Weyl cone, ∆v(q) =

v+(q)− v−(q) is the velocity of each particle-hole pair. Then, based on the relaxation time

approximation and Fermi’s golden rule, we have:

J = −2πe

∫

d3q

(2π)3∆v(q)|q+|V |q−|

2δ(∆E(q)− ω)(n0−(q)− n0

+(q)), (S6)

where A is the amplitude of the vector potential, V is the electron-light coupling responsible

for photoexcitations from |q− to |q+ and ∆E(q) = E+(q) − E−(q). We first numerically

extract an effective k · p Weyl Hamiltonian H(k) from ab-initio band structure calculation.

Then, by the Peierls substitution, the coupling Hamiltonian is calculated as V = H(k +

A(t))−H( A). Rewriting the integrand in dimensionless quantities,

J = C J

=

(

e3τI

16π22ǫ0c

)[∫

d3(

−vF q

ω

) ∆v(q)

vF|q+|

V

vFA/2|q−|

2δ

(

∆E(q)

ω− 1

)

(n0−(q)− n0

+(q))

]

.

(S7)

C determines the order of magnitude of the current. The dimensionless vector J , which

is of order one, determines the direction of the current. We note that terms in J depends on

system details. For example, δ(

∆E(q)ω

− 1)

and n0−(q)−n0

+(q) depend on the band structure

of the sample; q+|V

vFA/2|q− depends on both the wavefunctions of the WFs in the sample

and the properties of the light.

We first only focus on the dimensionless J . In Table S4, we show calculation results

of J in TaAs under different conditions measured in experiments. Indeed, we see that the

current along b is finite for a RCP along a. In the latter three cases, the current is zero due

to cancellation.






Properties of the light J

RCP along a Jb = −0.10932

Jc = 0

RCP along c Ja = 0

Jb = 0

TABLE S4: Calculated J in TaAs under different conditions measured in experiments.

FIG. S12: Our calculations show JTHY = χW1klight × c.

In section IV.3 of this document, we further show detailed calculation for the photocur-

rents from each Weyl fermion.

III.5. Single crystal x-ray diffraction methods

Single crystal diffraction is employed to extract elemental distributions and accurate de-

termination of interatomic distances, coordination environments, and eventually determin-

ing the three dimensional structural features. The single crystal was mounted on the tips

of sample holder using nitrocellulose in ethyl acetate solvent, which can not be diffracted

with x-ray. Room temperature intensity data were collected on a Bruker Smart Apex II

diffractometer using Mo Kα radiation (λ = 0.71073 A) with the crystal to detector distance

fixed at 100 mm. Data were collected over a full sphere of reciprocal space with 0.5 scans






with an exposure time of 5 s per frame to avoid strong absorption on Ta atoms. The 2θ

range extended from 4 to 70. The face-index was conducted based on the single crystal

reflections. Figure S13 illustrates the single-crystal XRD precession image of the (0kl) plane.

The crystalline directions determined by these XRD data are shown in Fig. S14.

FIG. S13: Single-crystal XRD precession image of the (0kl) plane in the reciprocal lattice of TaAs.

No diffuse scattering is seen. All the resolved spots have fitted the crystal lattice structure of

tetragonal I41md TaAs.






FIG. S14: Crystalline directions of our TaAs sample determined by single crystal XRD.






IV. DETAILS OF THE PHOTOCURRENT CALCULATIONS

IV.1. Intuitive explanations for the non-vanishing photocurrent in an inversion

breaking Weyl semimetal

IV.1.1. The chirality selection rule and asymmetric Pauli blockade

We first elaborate on the chirality selection rule and asymmetric Pauli blockade.

The chirality selection rule is a result of the conservation of angular momentum. The

simplest Weyl Hamitonian can be written as

HWeyl =∑

i=x,y,z

vikiσi (S8)

From this equation, we see that the two bands that cross to form the Weyl node has

angular momentum of ±12, respectively. Recall that right(left) circularly polarized light has

an angular momentum of ±1. Therefore, the optical transition via a circularly polarized

photon is allowed on one side of the Weyl cone but prohibited on the other depending on

the chirality of the WF and that of the photons.

Due to the Pauli exclusion principle, electrons must be excited from filled states to empty

states. Therefore, if the chemical potential is at the Weyl node (Figs. S15c,e), the excitation

is always allowed irrespective of the tilt of the cone. By contrast, if the chemical potential

is away from the Weyl point (Figs. S15d,f), Pauli exclusion occurs. Without the tilting

(Fig. S15d), the Pauli blockade is symmetric about the Weyl node. Finally, with tilting

(Fig. S15f), excitation becomes asymmetric because of the combination of Pauli exclusion

and energy conservation.






E

μ

v+v-

f

E

μ

c dE

μ

E

μ

v+v-

e

k

k k

k

E

kz

Optical selection rule

μ μ

χ =+1 χ = 1-a b

Pauli Blockade

FIG. S15: a,b, The Chirality selection rule of a WF arises from the conservation of angular

momentum. It happens at opposite k points for WFs of opposite chirality. c,e, If the chemical

potential is at the Weyl node, the excitation is always allowed irrespective of the tilt of the cone.

d,f, By contrast, if the chemical potential is away from the Weyl point, Pauli exclusion occurs.

Without the tilting (panel d), the Pauli blockade is symmetric about the Weyl node. Finally, with

tilting (panel f), the Pauli blockade becomes asymmetric, i.e., only one side of the Weyl cone is

excited because of the combination of Pauli exclusion and energy conservation.






IV.1.2. Intuitive Explanation 1

We consider four Weyl nodes on the kz = 0 plane that are related by the mirror sym-

metries Mx and My and RCP light propagating along +kx (Figs. S16a,b). The chirality

selection rule means that, for a χ = +1 WF, the optical transition is allowed on the +kx side

but forbidden on the −kx side. For a χ = −1 WF, the selection rule is reversed (Fig. S16b).

We show that the total current along y does not cancel in the presence of tilt. We

start from the black dotted line 1, which cuts across the top-right Weyl cone. The mirror

symmetries Mx and My will relate the black dotted line 1 to black dotted lines 2-4, which

cut across the other three Weyl cones . Because these dotted lines do not cut across the

Weyl node, the band structures along them are gapped (Fig. S16d,e). Moreover, because

lines 1 and 2 are symmetric about the mirror plane Mx, the energy dispersion along 1 and 2

are identical (Fig. S16d,e). We also know that chirality selection rule allows the transitions

along both 1 and 2. Now we need to further consider the tilt of the WF. A finite tilt

means that the dispersions in Fig. S16d,e are not symmetric about the top (bottom) of the

valence (conduction) band. Thus, as seen in Fig. S16d,e, the Pauli blockade also becomes

asymmetric, and a finite current is obtained when summing up the contribution from the

black dotted lines 1 and 2. By contrast, the current from the black dotted lines 3 and 4

is zero because the transition there is forbidden by the chirality selection rule (Fig. S16c ).

Therefore, the total current from all black dotted lines 1-4 is nonzero. In order to fully cover

the Weyl cones, we sweep the black dotted line 1 along the kx direction. Because these four

lines are related by Mx and My, it is evident that the other 3 lines will be synchronized

to line 1 during the sweeping and hence all four Weyl cones are simultaneously covered.

Therefore, we argue that the total current along y is nonzero in the present of tilt.

By contrast, we show that the total current along x vanishes due to cancellation. The

arguments are similar to those above. We consider the four symmetry-related red dotted

lines along ky in Fig. S16c. Their band structures (Figs. S16f-i) are related by Mx and My.

For each line (e.g., red line 1 in Fig. S16f), the optical transition is only allowed on one side

because of chirality selection rule. Due to mirror symmetry Mx, we see that the currents

from red lines 1 and 2 (3 and 4) cancel each other. Therefore, the total current along x

vanishes due to cancellation.






FIG. S16: An intuitive explanation for the non-vanishing photocurrent in an inversion breaking

Weyl semimetal. Circles represent the 24 WFs of TaAs. Green and Blue represent the positive

and negative chirality. Pink shows the area allowed by the chirality selection rule.






IV.1.3. Cancellation of CPGE along c

In order to explain why the current along z vanishes, we first prove the following facts:

(1) the band structure of any W1 node has no tilt along kz (2) the band structure of W2

nodes on the +kz side and those on the −kz side has the opposite tilt.

Proof: For each Weyl node, the tilt can be described by a velocity vt = (vtx, vty, vtz) [5].

C4c rotation does not change vtz , while time-reversal T changes vtz to −vtz.

For a W1 node (e.g. W1#1), rotating by 180 (C4c) translates it to W1#4, which indicates

that vtz(W1#1) = vtz(W1#4). Time-reversal also takes W1#1 to W1#4, which indicates

that vtz(W1#1) = −vtz(W1#4). Therefore, we have vtz(W1) = 0.

For a W2 node, C4c relates W2(#9 − #16) and W2(#17 − #24). Time-reversal takes

W2#9 to W2#17. Therefore, we have vtz[W2(#9−#16)] = −vtz [W2(#17−#24)]

We now prove the cancellation for light along x (Figs. S17a-c,g-i). A light along x excites

either +x or −x side of the Weyl cone depending on its chirality. For W1 WFs, no current

is produced from a single node because of no tilt along z (Figs. S17a-c). For W2 WFs, the

currents from time-reversal partners cancel because they have opposite tilt (Figs. S17g-i).

We also prove the cancellation for light along z (Figs. S17d-f,j-l). A light along z excites

either +z or −z side of the Weyl cone depending on its chirality. For W1 WFs, the currents

from opposite chirality cancel because their chirality selection rule is opposite (Figs. S17d-f).

The same applies to W2 WFs.






FIG. S17: Cancellation of CPGE along c. Circles represent the 24 WFs of TaAs. Green and Blue

represent the positive and negative chirality. Pink shows the area allowed by the chirality selection

rule.






IV.1.4. Current reversal upon rotating the sample

kc

kc

kc

ka

kb

kakb

kb

ka

d

f

e

ky

kx

kz

bc a

yz x C2z

c

C2xC2y

bc

a

b

ca

C2z

C2xC2y

ac

FIG. S18: Current reversal upon rotating the sample

We further explain why the CPGE reverses upon rotating the sample by 180 around a

or b while remaining the same upon rotating around c. This can be understood in both real

space and reciprocal space.

In real space, the +c and −c directions of the TaAs crystal are nonequivalent. Hence

rotating the sample by 180 around a or b would take +c to −c, while the system remains

invariant by a rotation around c.

Figs. S18d-f draw the k space defined by the laboratory coordinate (kx, ky, kz), rather

than the sample coordinate (ka, kb, kc). The laboratory coordinate corresponds to the WF

chirality seen by the light because the light direction remains unchanged in the lab. We see

that rotating the sample by 180 around a or b reverses the WF chirality seen by the light.

Therefore, both real space and reciprocal space analyses show that the CPGE reverses

upon rotating the sample by 180 around a or b while remaining the same upon rotating

around c.






IV.2. Symmetry analyses of the photocurrent

The second order photocurrent response tensor ηαβγ is defined through [11–13]:

Jα(ω = 0) = ηαβγ(ω,−ω)Eβ(ω)E∗

γ(ω), (S9)

where E∗γ(ω) = Eγ(−ω). Since J is real, the response tensor has to satisfy:

ηαβγ(ω,−ω) = η∗αγβ(ω,−ω). (S10)

Physically, the real and imaginary parts of ηαβγ correspond to the linear and circular pho-

togalvanic effects, respectively.

The 27 elements of ηαβγ have to obey the space group symmetry of the system. For a

circularly polarized light along a, the currents along b and c are given by

Jb = ηbbcEbE∗

c + ηbcbEcE∗

b (S11)

Jc = ηcbcEbE∗

c + ηccbEcE∗

b . (S12)

For a circularly polarized light along c, the currents along a and b are given by

Ja = ηaabEaE∗

b + ηabaEbE∗

a

Jb = ηbabEaE∗b + ηbbaEbE

∗a . (S13)

In TaAs, the presence of Ma and Mb forces ηαβγ to vanish when it contains an odd

number of momentum index a or b. Therefore, we see that symmetry does not force ηbbc

and ηbcb to be zero, which means nonzero Jb is symmetry allowed for light propagating along

a. ηcbc = ηccb = ηaab = ηaba = ηbab = ηbba = 0, which in turn proves Jc = 0 for light along

a (Eq. S12) and Ja = Jb = 0 for light propagating along c (Eq. S13). The above analyses

demonstrate the observed cancellations. As for the sign reversal in Fig. 3 in the main text,

since both ηbbc and ηbcb in Eq. S11 contains an odd number of c index, Jb should switch sign

if one changes from +c to −c. Indeed, this is achieved by a 180 rotation around a or b but

not around c.

IV.3. Calculated photocurrents from each Weyl fermion

In this subsection, we present the calculated photocurrent from each WF. We show the

dimensionless J in Eq. S7. This shows the direction of the current from each WF. It also






shows the relative contribution from the W1 and W2 WFs. The absolute amplitude of the

total photocurrent also depends on the constant C in Eq. S7 For clarity, in Fig. S19, we

assign a specific number to each Weyl node.

FIG. S19: Labeling of Weyl nodes in TaAs, used to calculate the current generated from each Weyl

node. The green dot is χ = +1 node and the blue dot is χ = −1 node.

Table S5 show the calculated currents Jn = (Jnx , J

ny , J

nz ) with right circularly polarized

(RCP) light along x (n denotes the nth WF as defined in Fig. S19). We see that the

photocurrents from W1 WFs are twenty times larger than those from W2. This is because

the chemical potential is close to the W2 Weyl nodes.

Table S6 show the calculated currents Jn = (Jnx , J

ny , J

nz ) with right circularly polarized

(RCP) light along z.






W1(kz = 0) Jx Jy Jz

1 -1.993 +0.100 0

2 +1.993 +0.100 0

3 +2.151 -0.0526 0

4 -2.151 -0.0526 0

5 +2.528 +0.342 0

6 -2.528 +0.342 0

7 -1.553 -0.447 0

8 +1.553 -0.447 0

Total (1-8) 0 -0.115 0

W2(kz > 0) Jx Jy Jz

9 +2.095 -0.000551 +0.000708

10 -2.095 -0.000551 +0.000708

11 -2.090 +0.00451 +0.00569

12 +2.090 +0.00451 +0.00569

13 -2.097 +0.00206 +0.00292

14 +2.097 +0.00206 +0.00292

15 +2.089 -0.00460 +0.00653

16 -2.089 -0.00460 +0.00653

W2(kz < 0) Jx Jy Jz

17 +2.090 +0.00451 -0.00569

18 -2.090 +0.00451 -0.00569

19 -2.095 -0.000551 -0.000708

20 +2.095 -0.000551 -0.000708

21 -2.089 -0.00460 -0.00653

22 +2.089 -0.00460 -0.00653

23 +2.097 +0.00206 -0.00292

24 -2.097 +0.00206 -0.00292

Total (9-24) 0 +0.00568 0

TABLE S5: Current components of W1 and W2 nodes when RCP is propagating along +a.






TABLE S5: The chemical potential is taken to be 17.8 meV aboveW1 and 4.8 meV aboveW2, which

is obtained from the magneto-transport measurement described in Section III. 2. The direction

refers to the hole current, which is the same as the measured current direction.






W1(kz = 0) Jx Jy Jz

1 +0.470 +0.570 -2.092

2 -0.470 +0.570 +2.092

3 +0.470 -0.570 +2.092

4 -0.470 -0.570 -2.092

5 +0.570 +0.470 +2.092

6 -0.570 +0.470 -2.092

7 +0.570 -0.470 -2.092

8 -0.570 -0.470 +2.092

Total (1-8) 0 0 0

W2(kz > 0) Jx Jy Jz

9 +0.00190 -0.00217 +2.097

10 -0.00473 -0.00537 -2.088

11 +0.00473 +0.00537 -2.088

12 -0.00190 +0.00217 +2.097

13 -0.00537 +0.00473 -2.088

14 +0.00217 +0.00190 +2.097

15 -0.00217 -0.00190 +2.097

16 +0.00537 -0.00473 -2.088

W2(kz < 0) Jx Jy Jz

17 -0.00473 +0.00537 +2.088

18 +0.00190 +0.00217 -2.097

19 -0.00190 -0.00217 -2.097

20 +0.00473 -0.00537 +2.088

21 +0.00217 -0.00190 -2.097

22 -0.00537 -0.00473 +2.088

23 +0.00537 +0.00473 +2.088

24 -0.00217 +0.00190 -2.097

Total (9-24) 0 0 0

TABLE S6: Current components of W1 and W2 nodes when RCP is propagating along +c.






TABLE S6: The chemical potential is taken to be 17.8 meV aboveW1 and 4.8 meV aboveW2, which

is obtained from the magneto-transport measurement described in Section III. 2. The direction

refers to the hole current, which is the same as the measured current direction.






IV.4. Theoretical estimation of the photocurrent amplitude in TaAs

As we have demonstrated above, the dimensionless vector J can be calculated based on

the properties of the light and the TaAs band structure.

In order to calculate the amplitude, we still need to know several other quantities. We

show below that, some of them are known whereas others are not. We discuss the major un-

known factors that prevent us to theoretically predict the photocurrent amplitude measured

in experiments.

Below is a list of quantities required to calculate the amplitude of the photocurrent

• Absorption coefficient α: We know α ≃ 30% at λ = 10.6 µm from Ref. [10].

• Skin-depth δ: We know δ ≃ 360 nm at λ = 10.6 µm by the formula δ = 1πσfµ

, where

f is the frequency of the light, µ is the absolute magnetic permeability, and σ is the

optical conductivity at λ = 10.6 µm which is obtained from Ref. [10].

• Relaxation time τ : We have J = C J , where C = e3τI16π22ǫ0c

. The relaxation time τ

measures the time scale for the following process: Exactly at the time when a laser-

pulse hits the sample, the selection rule requires that only one side of a Weyl node is

excited. At a finite time-delay after the laser,the photo-excited quasi-particles can be

relaxed to the other side of the Weyl node due to electron-electron scattering.

• Current distribution: We need to know whether all current generated by the light

are experimentally collected by the current meter. For example, a fraction of the

current may flow internally within the sample, which will not be detected by the

current meter.

We now discuss the major unknown factors.

• The thickness of the sample is 260 µm whereas the penetration depth is only 360 nm.

Hence the majority of the sample is not driven by light. This undriven part of the

sample has a resistance Rundriven sample ∼ 5× 10−4 Ω. On the other hand, the external

load (contacts, wires, etc) has a resistance Rexternal load ∼ 5 Ω. The total current

generated by light are distributed by the ratioIundriven sample

Iexternal load= Rexternal load

Rundriven sample∼ 104.






The measured current is Iexternal load, which is subject to a factor of 104 reduction by

considering this fact that most of the current is bypassed through the very metallic,

undriven part of the sample.

• The relaxation time has not directly measured.

• We assumed that all photons absorbed by the sample interact with electrons. On the

other hand, it is possible that a fraction of the photons directly interact with phonons.

Because of the above unknown factors, it is difficult to theoretically estimate the ampli-

tude of the photocurrent. By making the following crude estimation, we can calculate an

amplitude

• The relaxation time has been inferred from transport data (45 ps) [8] and not directly

measured.

• We assumed that all photons absorbed by the sample interact with electrons.

• We include this 104 reduction of current amplitude as discussed above.

By doing these approximations, and using laser power of 10 mW and lateral beam-size of

50 µm×50 µm, we obtain that the calculated current is 1.015×10−4 A. This is about 4 orders

of magnitude higher than of the experimentally measured value 40×10−9 A, which matches

theIundriven sample

Iexternal load= Rexternal load

Rundriven sample∼ 104 factor discussed above. While this geometrical factor

may play a major role, other unknown factors described above can also contribute to the

difference between the data and calculations in terms of the photocurrent amplitude.






V. Exclusion of other CPGE mechanisms

Below we discuss and rule out alternative CPGE mechanisms including its excitation in

trivial systems with strong spin-orbit and surface Rashba terms [11, 14, 15] and the photon

drag originating in trivial bulk bands [15, 16].

The CPGE due to the Rashba effect has been observed in semiconductor (such as

GaAs/AlGaAs quantum well samples). As explained in Refs. [11, 14] (Fig. S20), it re-

quires the following two key conditions: (1) Both the conduction and valence bands are

Rashba-split states; (2) The energy gap between conduction and valence equals the photon

energy (120 meV in our case). In our case, the lowest-lying two bands are the two bands

that cross and form Weyl nodes. They do not feature Rashba-split states. In addition, the

surface state band structure (as shown by ARPES) also does not feature Rashba-split states.

In fact, the surface states are localized on the top 1− 2 nm whereas our penetration depth

is 360 nm. Thus contribution from bulk states dominate over that from the surface states.

Therefore, we can rule out the possibility of the Rashba-origin for our CPGE.

The photon drag effect arises due to a momentum transfer from photons to free carriers.

As explained in Refs. [15, 16], photons with an oblique incidence has a finite in-plane

momentum, which can be transferred to electrons (Fig. S21), leading to a finite photocurrent

along the momentum-transfer direction. We can rule out this possibility because, in our

experiments, we shine light along c and observe a photocurrent along b. Since the photons

travelling along c do not have a finite momentum along b, the photon drag effect can be

excluded.

More quantitatively, the second order photocurrent due to the photon drag effect is given

by [15]:

Jdragλ = TλδµνqδEµE

∗ν , (S14)

with the condition:

Tλδµν = T ∗λδνµ. (S15)

Here qδ is the photon momentum responsible for the electron momentum transfer during

the interband optical excitation. The fourth rank response tensor Tλδµν also obeys the

space group symmetry of the system. Out of the 81 elements, 21 of them can be nonzero






FIG. S20: Schematic illustration for the microscopic mechanism of the CPGE that arise from

strong spin-orbit and the Rashba-effect observed in the zinc-blende structure-based quantum well

samples. This figure is adapted from Ref. [14].

in the presence of Ma and Mb symmetries in TaAs. The C4c symmetry further results

in 8 independent elements: Tacac = Tbcbc = T ∗acca = T ∗

bccb, Tcaac = Tcbbc = T ∗caca = T ∗

cbcb,

Tabab = Tbaba = T ∗abba = T ∗

baab, Taacc = Tbbcc, Tccaa = Tccbb, Taaaa = Tbbbb, Taabb = Tbbaa and

Tcccc, out of which only the first 3 can be complex valued to lead to a current due to a

circularly polarized drive.

Our observed data corresponds to Tbcba, i.e., a photocurrent along b, a momentum transfer

along c, and the two electric fields along a and b, respectively. Therefore, we exclude the

photon drag effect.






FIG. S21: Schematic illustration for the microscopic mechanism of the CPGE that arise from the

photon drag effect as observed in the zinc-blende structure-based quantum well samples. This

figure is adapted from Ref. [15].






VI. ADDITIONAL PHOTOCURRENT MEASUREMENTS

In Fig. S22, we show photocurrent data in addition to Fig. 3 in the main text. We see

that the photocurrent along c is zero irrespective of the rotation. This again confirms the

cancellation of photocurrent along c

FIG. S22: For a light along the a direction, no photocurrent is observed along c direction irrespective

of the rotation.

In Fig. S23, we show that photocurrent measurements on another TaAs sample. This

sample is purposely filed down so that the out-of-plane direction is c. This allows us to shine

normal incident light along c. As shown in Fig. S23, we observe zero current along both a

and b directions.






a

b

a

bc

A

a

bc

A

0 90 180 270 360

-40

-20

0

20

40

FIG. S23: When the light drives along the c axis, there is no chirality-dependent current generated

in both a and b directions.






VII. PREVIOUS TRANSPORT AND ARPES MEASUREMENTS ON WEYL

SEMIMETALS

Below, we explain why our measurements for the first time detected the WF chirality.

First we wish to emphasize that demonstrating the existence of two opposite chiralities

is different from detecting the chirality of individual Weyl nodes. The negative magneto-

resistance (MR) data due to the chiral anomaly [17, 18] arises because E · B pumps electrons

from Weyl cones of one chirality to the opposite. Therefore, the observation of a negative

MR due to the chiral anomaly clearly proves that there are two opposite chiralities. However,

this is fundamentally different from detecting the chirality of each individual WF.

FIG. S24: a,b, 4 Weyl nodes related by the Mx and My mirror planes. The two cases in panels

(a,b) have identical band structure. Only the chirality of the WFs is opposite. c,d, We imagine

two Weyl semimetal samples A and B with identical band structures. The only difference is that

samples A and B have opposite WF chirality configuration (I and II). We apply parallel electric

and magnetic fields on both samples ( E · B). We expect to see the same negative MR response.

Therefore, the negative MR response is insensitive to WF chirality.






To better illustrate our point, we consider two samples (A,B) with the same band struc-

ture that has 4 Weyl node related by the Mx and My mirror planes (Fig. S24). The only

difference is that the samples A and B have the opposite WF chirality configurations (I and

II) (Figs.S24a,b). By applying parallel electric and magnetic fields E · B, we expect the same

negative longitudinal MR response on both samples. This can be proven more rigorously

as follows: The negative longitudinal MR originates from an additional conduction channel

due to E · B. The total conductivity is given by

σ = σDrude + σchiral anomaly, (S16)

where σDrude is the normal Drude conductivity, and σchiral anomaly is the additional conduc-

tivity due to the chiral anomaly.

Ref. [19] showed that the chiral anomaly contribution σchiral anomaly can be calculated by

σchiral anomaly =e4τa

(2π44g)B2, (S17)

where τa is a relaxation time associated with the chiral anomaly, and g is the total density of

states for all Weyl cones near the Fermi level. We emphasize the two following aspects: 1. All

terms in the equation are positive. Thus, σchiral anomaly is always positive (σchiral anomaly > 0).

In other words, the chiral anomaly always contributes a positive magneto-conductivity and

therefore a negative MR, independent of the chirality configuration (I or II). 2. Samples A

and B have identical band structure and therefore density of states g. The relaxation time,

τa, also should not depends on the two chirality configurations I and II.

Therefore, we conclude that σchiral anomaly does not depend on the chirality of individual

Weyl nodes.

We now proceed to explain why ARPES measurements are not sensitive to the chirality

of individual WFs. This can be clearly seen from the following fact: Chirality is defined

based on Berry curvature, which is a property of the electron wavefunction, rather than

band structure. Specifically, the Berry curvature Ω is calculated by Ω = × in| |n,

where is |n the wavefunction. A Weyl node’s chirality is positive if the Berry curvature

points away from the node (i.e., a monopole), whereas it is negative if the Berry curvature

points toward the node (i.e., an anti-monopole) (Figs. S25).

For example, in Fig. S25a, we consider two Weyl fermions related by a mirror plane.






Although they have opposite chirality, their dispersion can be identical (Figs. S25b,c). This

fact highlights that chirality is a property of the wavefunction rather than the band structure,

which naturally explains why conventional ARPES measured band structure is not sensitive

to the chirality of each individual Weyl node. Therefore, although the linear dispersion

of bulk Weyl cones has been measured by conventional ARPES [20, 22, 23]. These band

structure measurements cannot discern the chirality of individual WFs.

Similarly, researchers have also measured the surface band structure [20, 21, 23, 24].

Measuring the surface is even more indirect, because any information about the bulk can

only be indirectly inferred via the bulk-surface correspondence. Nevertheless, the essential

problem is the same, i.e., the chirality is a property of the electron wavefunction. It cannot

be obtained by a measurement of the band structure, regardless of the bulk or the surface.

For example, although the two systems in Figs. S26a,b have opposite WF chirality, their

surface Fermi surfaces can be identical. Therefore, the surface Fermi surface measured by

ARPES cannot distinguish the two chirality configurations.

Ref. [25] proposed a way to measure the chirality via pump-probe ARPES, taking ad-

vantage of the same chirality selection rule as we used in our CPGE experiments. As shown

in Fig. S27, a circularly polarized mid-IR pump light only excites one side of the Weyl cone

due to the chirality selection rule. The imbalanced optical excitation can be observed by

measuring the transient spectral weight using time-resolved ARPES. One expects to see ex-

cited electrons above EF only on one side of the Weyl cone depending on the chirality of the

particular Weyl node. Similar to our CPGE, the key is the measurement of the imbalanced

excitation due to the chirality selection rule. Without the circularly polarized mid-IR pump

light, one would just measure the band structure of a Weyl cone, which cannot discern the

chirality of an individual WF.






FIG. S25: WF chirality is defined based on the Berry curvature near the Weyl node being a

monopole or an anti-monopole, which is a property of the electron wavefunction, rather than band

structure.






FIG. S26: For the two possible WF chirality configurations, their Fermi arc surface state Fermi

surface (e.g., connectivity, shape and other properties of the Fermi arcs) can be identical. Thus

surface Fermi arcs cannot be used to discern these two possible WF chirality configurations.






FIG. S27: Experimental setup of using pump-probe ARPES to detect chirality as proposed in Ref.

[25].






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