**3. Results and Discussion**

Previously, we know that transport properties of the pseudospin-1 Dirac–Weyl fermions differs from free electrons in two ways. One is super Klein tunneling, which gives perfect transmission through a potential barrier for all incident angles while the quasiparticle energy equals one half the barrier height [48]. The other is particle-hole symmetry above and below the Dirac point of a potential barrier, which is a shared property with pseudospin-1/2 Dirac–Weyl fermions on monolayer graphene [56]. It gives that the transmission probability closely above and below the Dirac point is mirror symmetric because hole states with identical dispersion to electrons exist within the potential barrier unlike the potential barrier formed by the energy gap in semiconductor heterostructures. These two properties are demonstrated in the conductivity through a single potential barrier shown in Figure 1d. As a result of super Klein tunneling and because the conductivity also depends on the velocity or Fermi wavevector of the charge carriers, the maximum is parabolically-shaped under the present parameter settings and occurs at the Fermi energy larger than half the barrier height. For higher potential barriers, the maximum can be a sharp Λ-shaped peak appearing at the Fermi energy equal to half the barrier height [57]. Because of the existence of the local maximum peak and the V-shape local minimum in the single-barrier transmission probability and hence in the conductivity, it occurs that under certain conditions higher barrier allows larger quasiparticle transmission. The mechanisms of an adiabatic quantum pump in a mesoscopic system can be illustrated consistently by a classic turnstile picture and by the the Berry phase of the scattering matrix in the parameter space [2,5,6]. The turnstile picture can be illustrated within the framework of the single electron approximation and

coherent tunneling constrained by the Pauli principle. The two oscillating potential barriers work like two "gates" in a real turnstile. Usually, lower potential allows larger transmissivity and thus defines the opening of one gate. When the two potentials oscillate with a phase difference, the two gates open one by one. Constrained by the Pauli principle, only one electron can occupy the inner single-particle state confined in the quantum well formed by the two potential barriers at one time, electrons flow in a direction determined by the driving phase difference. However, in monolayer graphene and in the pseudospin-1 Dirac–Weyl system, Klein tunneling, super Klein tunneling, and particle-hole symmetry at the Dirac point give rise to a reversal of the transmissivity-barrier height relation. As a result, the direction of the DC pumped current is reversed.

Numerical results of the pumped current are shown in Figure 2. It can be seen from Figure 1d that when the value of *EF* is between 70 meV and 100 meV, conductivity through higher potential barriers is larger than that through lower barriers. Angular dependence of the pumped current at Fermi energies selected within this range is shown in Figure 2b. With *ϕ* fixed at *π*/2, potential barrier *V*1 starts lowering first and then it rises and *V*2 starts lowering. Usually (like in a semiconductor heterostructure), higher potential barriers give rise to smaller transmission probability. The process can be interpreted as "gate" *V*1 "opens" first allowing one particle to enter the middle single-particle state from the left reservoir and then it "closes" and "gate" *V*2 "opens" allowing the particle to leave the device and enter the right reservoir. This completes a pump cycle and a DC current is generated. Such is the classical turnstile picture of the pumping mechanism. However, for pseudospin-1 Dirac–Weyl fermions, higher potential barriers give rise to larger transmission probability under certain parameter settings as demonstrated in Figure 1d. In the classical turnstile picture, this means that the definition of "opening" and "closing" of the "gate" is reversed. This is the reason for the negative (direction-reversed) pumped current shown in Figure 2b.

**Figure 2.** (**<sup>a</sup>**–**<sup>c</sup>**): angular dependence of the pumped for different Fermi energies with the driving phase difference *ϕ* fixed; (**d**) angle-averaged pumped current as a function of the Fermi energy. Its inset is the zoom-in close to the Dirac point to show that the large value of the pumped current does not diverge. Other parameters are *V*0 = 100 meV, *V*1*ω* = *V*2*ω* = 0.1 meV, *d* = 5 nm, *L*2 − *L*1 = 10 nm, and *ϕ* = *π*/2.

It can also be seen in Figure 2 that this turnstile picture of quantum pumping works for all parameter settings by comparing with Figure 1d. As a result of particle-hole symmetry above and below the Dirac point of a potential barrier, transmission probability of the pseudospin-1 Dirac–Weyl fermions demonstrate a sharp *V*-shape local minimum at the Dirac point. It should be noted that, at the Dirac point, eigenspinor wavefunction of the Hamiltonian is Ψ0 and the transmission probability is exactly zero. We singled out this point in all of our calculations. Below the Dirac point, higher potential barriers allow larger transmission probability. Above the Dirac point, higher potential barriers allow smaller transmission probability. In addition, the difference is very sharp giving rise to a sharp negative pumped current below the Dirac point and a sharp positive pumped current above the Dirac point as shown in Figure 2d. In vast Fermi energy regime, the pumped DC current flows in the same direction for all incident angles as shown in panels (a), (b), and (c) of Figure 2, giving rise to smooth angle-averaged pumped current shown in Figure 2d. It should also be noted that the sharp current peak close to the Dirac point does not diverge and the current has an exact zero value at the Dirac point by taking into account quasiparticles on the flat band, which is a stationary state while the wavevector in the pump–current direction (*x*-direction in Figure 1a) is imaginary. The finite value of the pump–current peak is shown in the zoom-in inset of Figure 2d.

The previous discussion is based on the classical turnstile mechanism, while the pumped current is evaluated by the Berry phase of the scattering matrix formed from the parameter variation with a looped trajectory (Equation (8)). Such a consistency needs further looking into, which is elucidated in the next section.

### **4. Consistency between the Turnstile Model and the Berry Phase Treatment**

In previous literature, consistency between the turnstile model and the Berry phase treatment is discovered while a clear interpretation is lacking.

Berry curvature <sup>Ω</sup>(*α*) of the scattering matrix **s** is defined by Equation (7) with **s** defined in Equation (5). *t*/*t* and *r*/*r* are the transmission and reflection amplitudes generated by incidence from the left/right reservoir with *t* = *t* and *r* = <sup>−</sup>*r*<sup>∗</sup>*t*/*t*<sup>∗</sup>.

Without losing generosity, we consider a conductor modulated by two oscillating potential barriers *X*1 = *V*1 and *X*2 = *V*2 with the same width and equilibrium height. By defining the modulus and argumen<sup>t</sup> of *t* and *r* as *t* = *ρteiφ<sup>t</sup>* and *r* = *<sup>ρ</sup>reiφ<sup>r</sup>*, we have

$$
\Omega\left(l\right) = \sum\_{i=t,r} \rho\_i \frac{d\rho\_i}{dV\_1} \frac{d\phi\_i}{dV\_2} - \rho\_i \frac{d\phi\_i}{dV\_1} \frac{d\rho\_i}{dV\_2}.\tag{10}
$$

Analytic dependence of *ρi* and *φi* on the parameters *V*1 and *V*2 cannot be explicitly expressed. We show numerical results of the Berry curvature Ω (*l*) and the eight partial derivatives on the right-hand side of Equation (10) in Figure 3. For convenience of discussion, the parameter space in Figure 3a to (i) is divided into four blocks. It can be seen from Figure 3a that Ω (*l*) is negative in block II, positive in block III, and nearly zero in blocks I and IV. For the term *ρt dρt dV*1 *dφt dV*2 − *ρt dφt dV*1 *dρt dV*2 in Equation (10), *ρt* > 0, *dφt dV*2 ≈ *dφt dV*1 is negative throughout the four blocks and *dρt dV*1 ≈ *dρt dV*2 is positive in block II and negative in block III (see Figure 3b,g). As a result, this term approximates zero in blocks II and III. For the term *ρr dρr dV*1 *dφr dV*2 − *ρr dφr dV*1 *dρr dV*2 in Equation (10), *ρr* > 0, *dφr dV*2 − *dφr dV*1 is positive throughout the four blocks (see Figure 3h,i), *dρr dV*1 ≈ *dρr dV*2 is positive in block III and negative in block II. It can also be seen from Figure 3 that in blocks I and IV the values of the two terms cancel out each other giving rise to nearly zero Ω (*l*). Therefore, the combined result of the two terms is that Ω (*l*) > 0 when *dρr dV*1 > 0 and Ω (*l*) < 0 when *dρr dV*1 < 0. This means that the Berry phase is positive and hence the pumped current is positive when higher potential barrier allows larger reflection probability in block III and that the Berry phase is negative and hence the pump–current direction is reversed when higher potential barrier allows smaller reflection probability in block II. Because *ρ*2*r* + *ρ*2*t* = 1, larger reflection probability means smaller transmission probability, and consistency between the Berry phase picture and the classical turnstile model is numerically proved in the pseudospin-1 Dirac–Weyl system.

**Figure 3.** Contours of the Berry curvature Ω (*l*) and the eight derivatives on the right-hand side of Equation (10) in the *V*1-*V*2 parameter space. For all the subfigures, the horizonal and vertical axes are *V*1 and *V*2 in the unit of meV, respectively. The magnitudes of the contours are in the scale of (**a**) <sup>10</sup>−7; (**b**) <sup>10</sup>−4; (**c**) <sup>10</sup>−4; (**d**) <sup>10</sup>−5; (**e**) <sup>10</sup>−5; (**f**) <sup>10</sup>−2; (**g**) <sup>10</sup>−2; (**h)** <sup>10</sup>−2; (**i**) <sup>10</sup>−2; and (**j**) <sup>10</sup>−5, respectively. Other parameters are *V*0 = 100 meV, *d* = 5 nm, *L*2 − *L*1 = 10 nm, *EF* = 100 meV, and *θ* = 0.5 in radians. For convenience of discussion, the parameter space in the nine panels is divided into four blocks: I (−1 < *V*1 < 0 and 0 < *V*2 < 1), II (0 < *V*1 < 1 and 0 < *V*2 < 1), III (−1 < *V*1 < 0 and −1 < *V*2 < 0), and IV (0 < *V*1 < 1 and −1 < *V*2 < 0). The four blocks are illustrated in (**a**).

If we consider normal incidence, consistency between the Berry phase picture and the classic turnstile model becomes straightforward. For normal incidence, derivative of *t*/*r* with respect to *V*2 is equal to derivative of *t*/*r* with respect to *V*1. Hence, we have

$$
\Omega\,\Omega\,(l) = 2\rho\_r \frac{d\rho\_r}{dV\_1} \left(\frac{d\phi\_t}{dV\_1} - \frac{d\phi\_r}{dV\_1}\right).\tag{11}
$$

From Figure 3, we can see that *dφt dV*1 − *dφr dV*1 is positive throughout the parameter space. Therefore, the Berry phase has the same sign with *dρr dV*1 , which demonstrates consistency between the Berry phase picture and the classic turnstile mechanism of the adiabatic quantum pumping.

Besides data shown in Figure 3, which are contours of the Berry curvature Ω (*l*) and the eight derivatives on the right-hand side of Equation (10) in the parameter window of ±1 meV for both *V*1 and *V*2 with *V*0 = 100 meV, *EF* = 100 meV, and *θ* = 0.5 in radians, we have numerically targeted dozens more parameter windows of ±1 meV for *V*1 and *V*2 at other values of *V*0, *EF*, and *θ* and no obtained results violate consistency of the two mechanisms. Although the main focus of the present manuscript is the pseudospin-1 Dirac–Weyl fermions, we numerically confirmed consistency between the two mechanisms in various parameter settings in different systems such as two-dimensional electron gas, graphene, and the pseudospin-1 Dirac–Weyl system by calculating term by term Equation (10). Although we could not provide a general proof, up to now, no numerical evaluation violates such a conclusion.

We observe in this work and previously that the pump–current direction can be reversed in systems with linear bands such as graphene and pseudospin-1 Dirac–Weyl system. Up to now, similar behavior has not been observed in systems with parabolic band dispersion such as in the semiconductor two-dimensional electron gas. We remark that a quantitative argumen<sup>t</sup> of the underlying reason for the dependence of the adiabatic quantum pumping behavior on the band structure as observed numerically is lacking, due to the topological difference of the band structure.
