# Quantum Pumping with Adiabatically Modulated Barriers in Three-Band Pseudospin-1 Dirac–Weyl Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Formalism

## 3. Results and Discussion

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

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of the Boundary Condition of the Spinor Wavefunction

## Appendix B. Detailed Algebra for Obtaining the Scattering Matrix

## References

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**Figure 1.**(

**a**) schematics of the adiabatic quantum pump. Two time-dependent gate voltages with identical width d and equilibrium strength ${V}_{0}$ are applied to the conductor. Time variation of the two potentials ${V}_{1}$ and and ${V}_{2}$ is shown in panel (

**b**). ${V}_{1}$ and ${V}_{2}$ have a phase difference giving rise to a looped trajectory after one driving period; (

**c**) two-dimensional band structure of the pseudospin-1 Dirac–Weyl fermions with a flat band intersected two Dirac cones at the apexes; (

**d**) conductivity of the pseudospin-1 Dirac–Weyl fermions measured by [54] $\sigma ={\textstyle \frac{{e}^{2}{k}_{F}d}{\pi h}}{\int}_{-\pi /2}^{\pi /2}{\left|t({E}_{F},\theta )\right|}^{2}cos\theta d\theta $ in single-barrier tunneling junction as a function of the Fermi energy for three different values of barrier height ${V}_{0}$. ${k}_{F}={E}_{F}/\hslash {v}_{g}$ is the Fermi wavevector and t is the transmission amplitude defined in Equation (5). It can be seen that higher barrier allowing larger conductivity occurs at the Dirac point ${E}_{F}={V}_{0}$ and around ${E}_{F}={V}_{0}/2$ (see the text).

**Figure 2.**(

**a**–

**c**): angular dependence of the pumped for different Fermi energies with the driving phase difference $\phi $ 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, ${V}_{1\omega}={V}_{2\omega}=0.1$ meV, $d=5$ nm, ${L}_{2}-{L}_{1}=10$ nm, and $\phi =\pi /2$.

**Figure 3.**Contours of the Berry curvature $\mathsf{\Omega}\left(l\right)$ 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**) ${10}^{-7}$; (

**b**) ${10}^{-4}$; (

**c**) ${10}^{-4}$; (

**d**) ${10}^{-5}$; (

**e**) ${10}^{-5}$; (

**f**) ${10}^{-2}$; (

**g**) ${10}^{-2}$; (

**h)**${10}^{-2}$; (

**i**) ${10}^{-2}$; and (

**j**) ${10}^{-5}$, respectively. Other parameters are ${V}_{0}=100$ meV, $d=5$ nm, ${L}_{2}-{L}_{1}=10$ nm, ${E}_{F}=100$ meV, and $\theta =0.5$ 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**).

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**MDPI and ACS Style**

Chen, X.; Zhu, R.
Quantum Pumping with Adiabatically Modulated Barriers in Three-Band Pseudospin-1 Dirac–Weyl Systems. *Entropy* **2019**, *21*, 209.
https://doi.org/10.3390/e21020209

**AMA Style**

Chen X, Zhu R.
Quantum Pumping with Adiabatically Modulated Barriers in Three-Band Pseudospin-1 Dirac–Weyl Systems. *Entropy*. 2019; 21(2):209.
https://doi.org/10.3390/e21020209

**Chicago/Turabian Style**

Chen, Xiaomei, and Rui Zhu.
2019. "Quantum Pumping with Adiabatically Modulated Barriers in Three-Band Pseudospin-1 Dirac–Weyl Systems" *Entropy* 21, no. 2: 209.
https://doi.org/10.3390/e21020209