Adiabatic topological pumping in a semiconductor nanowire

In recent years adiabatic topological pumping has attracted considerable attentions as it plays an important role in implementing quantized particle transports Meidan2010 ; Citro2016 and in exploring higher-dimensional topological effects Prodan2015 ; Kraus2016 ; Zilberberg2018 ; Lohse2018 . The concept of topological pumping was first introduced by Thouless who showed that the quantization of charge transport can be realized by a cyclic modulation of a one-dimensional periodic potential adiabatically and the charge pumped per cycle can be determined by the Chern number defined over the two-dimensional Brillouin zone formed in the momentum and time spaces Thouless1983 ; Niu1984 . It has been confirmed that some low-dimensional systems, such as quasicrystals and optical supperlattices, can exhibit the higher-dimensional topological effects, when subjected to a nontrivial modulation, and can thus be used to implement the adiabatic topological pumping   Tangpanitanon2017 ; Ke2020 ; Lin2020 ; Kraus2012 ; Kraus2013 ; Verbin2015 ; Schweizer2016 ; Nakajima206 ; Lohse2015 . Recently, the Thouless pumping of ultracold bosonic/fermionic atoms in an optical superlattice has been experimentally achieved via periodically modulating the superlattice phase Nakajima206 ; Lohse2015 . However, up to now, realizations of the topological pumping are almost limited to the optical systems and, therefore, it is of interest to explore the exotic phenomena of topological pumping in low-dimensional systems made from conventional semiconductors.

Both experimentally and theoretically, the adiabatic quantum pumping of electron charge and spin in semiconductor nanostructures has been one of the hotspot issues Thomas1994 ; Brouwer1998 ; OEA2002 ; Altshuler1999 ; Switkes1999 ; Aleiner1998 ; Hasegawa2019 ; OEA2002a ; Aharony2002 ; Zhou1999 . Especially, the parametric pumping facilitates the adiabatic transfer of noninteracting electrons in unbiased quantum dots Brouwer1998 ; Switkes1999 ; OEA2002 , but the average transferred charge in a cycle is not necessarily quantized under such a parametric pumping scheme. It has been shown that the pumped number is quantized through the adiabatic topological pumping and the physics behind this deeply roots into the nontrivial topology of the periodic wave functions Xiao2010 ; Wang2013 ; Hatsugai2016 . Moreover, the presence of fractional boundary charges in quantum dot arrays has been theoretically demonstrated by manipulating the onsite potentials periodically Park2016 ; Thakurathi2018 . However, a systematic analysis of the topological pumping in a one-dimensional periodic semiconductor nanostructure is still scarce.

In this paper, we propose a scheme for implementing the adiabatic topological pumping in a semiconductor nanowire double-quantum-dot (DQD) chain. We first demonstrate that the topological charge pumping can be achieved by simultaneously modulating the quantum-dot well and barrier potentials of the DQD chain. In the absence of the quantum-dot well potential modulation, we show that the topological spin pumping can be achieved by employing a time-dependent staggered magnetic field and by a nontrivial modulation of the barrier potentials and magnetic field in a cycle. When the DQD chain of a finite length is coupled to the external leads, we show that the quantized charge and spin transport can be calculated by exploiting the scattering matrix formalism under a nontrivial modulation. Especially, we demonstrate the quantized spin pumping in the presence of the Rashba spin-orbit interaction (SOI), but the pumped spin can have the spin quantization axis different from the applied staggered magnetic field direction. We also demonstrate, based on the changing of time-reversal charge polarization in a half cycle, that the periodic DQD chain in the presence of SOI can serve as a dynamic version of a topological insulator under a nontrivial modulation of the barrier potentials and magnetic field.

The rest part of the paper is organized as follows. In Sec. II, a periodic potential to define the DQD chain in a one-diemnsional semiconductor is introduced and the implementation of topological charge pumping is demonstrated. In Sec. III, the topological spin pumping in the semiconductor DQD chain in the presence of a time-dependent staggered magnetic field is analyzed. Section IV devotes to the formalism for calculating the charge and spin pumping through a finite DQD chain under topological modulations. In the presence of the Rashba SOI, the topological spin pumping in the periodic DQD chain is discussed in Sec. V. Finally, we summarize the paper in Sec. VI.

We first consider a periodic semiconductor nanowire DQD system subjected to time-dependent periodic barrier potential $U_{\Lambda}(t,x)$ and quantum-dot well potential $U_{V}(t,x)$, as shown in Fig. 1(a), described by the effective one-dimensional Hamiltonian,

where $p=-i\hbar\frac{\partial}{\partial x}$ represents the momentum operator and $m^{\ast}$ is the effective electron mass. In this system, a unit cell, as indicated by the dashed square in Fig. 1(a), comprises two quantum dots (QDs), labeled as QD1 and QD2. Structurally, $U_{\rm\Lambda}(t,x)$ consists of barrier potentials $V_{1}(t)$ and $V_{2}(t)$ and $U_{\rm V}(t,x)$ comprises two QD well potentials $\pm V_{0}(t)/2$ in a unit cell. Explicitly, for a unit cell of $-L\leq x<L$, the two potentials can be written as

where $2L$ denotes the size of unit cell, $2d$ ($2d\ll 2L$) the width of each potential barrier, $f^{\pm}_{1}(x)=\Theta(x+d)\pm\Theta(x-d)$ and $f^{\pm}_{2}(x)=\Theta(x+d-L)\pm\Theta(x+L-d)$, with $\Theta(x)$ being the Heaviside function, the spatial distribution functions.

In fact, the topological charge pumping can be achieved by a nontrivial modulation of system parameters $V_{0}(t)$ and $\delta V(t)\equiv V_{2}(t)-V_{1}(t)$. To see this, we consider the case that there is only one energy level in each QD and rewrite Hamiltonian Eq. (1) in the second quantization form

with $n$ representing the unit-cell index, $a^{\dagger}_{n}$ ($a_{n}$) and $b^{\dagger}_{n}$ ($b_{n}$) denoting the creation (annihilation) operators of the energy levels in QD1 and QD2 of the $n$th unit cell, $t_{\rm in/ex,0}$ being the intra/inter unit-cell hopping amplitudes, and $\pm\Delta_{0}/2$ the effective on-site energies (i.e., the level energies) in QD1 and QD2. Note that the Hamiltonian in Eq. (3) is derived from Eq. (1) by subtracting a constant potential value and this Hamiltonian is identical to the Rice-Mele model Hamiltonian Rice1982 . In analogy with the Thouless pumping in Refs. Nakajima206, ; Lohse2015, , it is found that the topological charge pumping can be realized when the changing trajectory of the parameter vector $(\Delta_{0},\delta t_{0})$, with $\delta t_{0}=t_{\rm in,0}-t_{\rm ex,0}$, obeys a nonzero winding number in a period. Here, because $\Delta_{0}$ and $\delta t_{0}$ can be regulated by potentials $V_{0}$ and $\delta V$, respectively, we can deduce that the topological charge pumping can be implemented if the changing contour of the modulation vector $\widehat{V}=(V_{0},\delta V)$ possesses a nonzero winding number in a cycle.

For an infinite periodic DQD chain system under a nontrivial modulation with $V_{0}(t)=\varepsilon_{0}\sin(2\pi t/T)$ and $\delta V(t)=\varepsilon_{0}\cos(2\pi t/T)$, a three-dimensional (3D) plot of the energy spectrum for the two lowest-energy Bloch bands is displayed in Fig. 1(b), where $\varepsilon_{0}$ is the energy modulation amplitude, $T$ is the cyclic time of the pumping and $k$ is the wave vector in a unit of $1/(2L)$. It is seen in Fig. 1(b) that the energies of the two lowest-energy Bloch bands vary with time and are separated by a nonzero bandgap throughout the pumping. Figure 2(a) shows the energy spectrum of a finite DQD chain under the same modulation of $V_{0}(t)=\varepsilon_{0}\sin(2\pi t/T)$ and $\delta V(t)=\varepsilon_{0}\cos(2\pi t/T)$. Here it is seen that two gapless bands are present in the Bloch band gap. Figure 2(b) shows the probability density distributions of two gapless band states at a selected energy. Here, it is seen that the two gapless band states are highly localized at the ends of the DQD chain. Thus these gapless band states are of edge states as one commonly observes in a topological insulator. Figure 2(c) shows the energy spectrum of the finite DQD chain under a trivial modulation of $V_{0}(t)=\varepsilon_{0}\sin(2\pi t/T)$, $\delta V(t)=\varepsilon_{\iota}+\varepsilon_{0}\cos(2\pi t/T)$ with the shifting energy $\varepsilon_{\iota}>\varepsilon_{0}$. It is evident that no gapless edge state is presented in this trivial modulation case. Here and hereafter, we assume that the periodic DQD chain is defined in an InAs nanowire with $m^{\ast}=0.023m_{0}$ Winkler2003 ($m_{0}$ is the free electron mass), $2L=120$ nm, $2d=20$ nm, $\varepsilon_{0}=6.0$ meV, and the sum of the two barrier potentials fixed as $V_{2}(t)+V_{1}(t)=36$ meV, and the cyclic time $T$ is large enough to ensure the adiabaticity of pumping.

Based on the bulk-boundary correspondence Hatsugai2016 , the presence of the gapless edge states is correlated to the nontrivial topology of the chain with periodic boundary conditions. By regarding the time as a virtual momentum axis orthogonal to the wire axis, the Berry curvature of a Bloch band in the two-dimensional Brillouin zone can be introduced Thouless1983 ; Wang2013 ,

with $|u(k,t)\rangle$ being the Bloch function below the energy gap, and the nontrivial topology is quantified by a nonzero Chern number Thouless1983 ; Wang2013 ,

For the case of a nontrivial modulation as illustrated in the inset of Fig. 2(a), the calculated distribution of the Berry curvature on the $k-t$ plane is displayed in Fig. 2(d). Numerical evaluation of Eq. (5) performed as in Ref. Fukui2005, verifies the nonzero Chern number of $\mathcal{C}=1$, confirming that the DQD chain under the nontrivial modulation is in the topological nontrivial phase. In addition, based on the theory of topological polarization Smith1993 ; Asboth2016 , $\mathcal{Q}(t)=\frac{1}{2\pi}\int^{t}_{0}\int^{\pi}_{-\pi}F(k,t)dtdk$ can be interpreted as the time-dependent charge polarization (in a unit of $-e$) and the topological charge pumping can be visualized by the time-evolution of $\mathcal{Q}(t)$ in a cycle, just as shown in the inset of Fig. 2(b), with $\mathcal{Q}(T)=1$ under this nontrivial modulation.

In this section, the implementation of topological spin pumping is proposed in the periodic DQD chain with a time-dependent staggered magnetic field $B(t,x)$ applied in a direction perpendicular to the wire axis, say the $z$ direction, as shown in Fig. 3(a). Here, we set the QD well potential unmodulated but keep modulation on the barrier potential difference $\delta V(t)$. The Hamiltonian of the periodic DQD chain system can now be written as

with $g^{\ast}$ being the electron Landé factor, $\mu_{\rm B}$ the Bohr magneton, and $\boldsymbol{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ representing the three Pauli matrices. Specifically, the time-dependent staggered magnetic field [see Fig. 3(b)] within a unit cell of $-L\leq x<L$ takes the form of

with $B(t)$ denoting the strength of the field and $f^{+}_{1}(x)$ and $f^{+}_{2}(x)$ representing the same spatial functions as before. It is important to note that the effect of the vector potential within the Landau gauge, i.e, $\mathbf{A}=(-By,0,0)$, is ignored in Eq. (6), due to the one-dimensional nature of the system, i.e., an extremely strong confinement in the transverse directions of the nanowire.

Using the commutation relation $[H_{\rm Z},\sigma_{z}]=0$, the Hamiltonian of the periodic DQD chain in the second quantization form can be written as

with the two spin-polarized terms given by

Here, the subscripts $\chi=\uparrow,\downarrow$ are spin indexes with the two spin states satisfying $\sigma_{z}|\uparrow\rangle=|\uparrow\rangle$ and $\sigma_{z}|\downarrow\rangle=-|\downarrow\rangle$, $a^{\dagger}_{n,\chi}$ ($a_{n,\chi}$) and $b^{\dagger}_{n,\chi}$ ($b_{n,\chi}$) are the creation (annihilation) operators of the electron spin states locating in the two QDs in the $n$th cell, and $\Delta_{\rm z}(t)=g^{\ast}\mu_{\rm B}B(t)$ is the time-dependent Zeeman splitting energy.

To achieve topological spin pumping in this system, we consider a nontrivial modulation of the parameter vector $\widehat{M}=(\Delta_{\rm z},\delta V)$ in a cycle. For the spin-up term, by mapping $\Delta_{\rm z}$ to $\Delta_{0}$, the Hamiltonian $H_{\rm Z,\uparrow}$ in Eq. (9) is equivalent to the Hamiltonian $H_{0,T}$ in Eq. (3) and, therefore, the spin-up electron is pumped when the changing trajectory of $\widehat{M}$ has a nonzero winding number in a cycle. The analysis can also be applied to the spin-down electron with the parameter vector $\widehat{M}$ replaced by $\widehat{M}^{\ast}=(-\Delta_{\rm z},\delta V)$. Figure 4(a) shows a 3D plot of the energy spectrum for the Bloch bands of the DQD chain under a nontrivial modulation of $\Delta_{\rm z}(t)=\varepsilon_{z}\sin(2\pi t/T)$ and $\delta V(t)=\varepsilon_{0}\cos(2\pi t/T)$. Here, we see again that the energies of the two lowest-energy Bloch bands vary with time and are separated by a nonzero bandgap throughout the pumping. Note that, due to the spin degeneracy, each energy band shown in Fig. 4(a) is twofold degenerate and comprised of two different spin states. Figure 4(b) shows the corresponding energy spectrum of a finite DQD chain under this nontrivial modulation. It is evident that two gapless edge-state Bloch bands indicated by the thick red curves are prresent in the bulk band gap. Because the modulation vectors $\widehat{M}$ and $\widehat{M}^{\ast}$ have different winding directions in a cycle [see the inset of Fig. 4(d)], the spin-up and spin-down electrons are therefore propagated in the opposite directions and results in the effective quantized spin pumping.

To further illustrate this point, the Berry curvatures of the two spin-polarized Bloch bands are introduced $F_{\chi=\uparrow,\downarrow}(k,t)=i\big{(}\langle\partial_{k}u_{\chi}|\partial_{t}u_{\chi}\rangle-\langle\partial_{t}u_{\chi}|\partial_{k}u_{\chi}\rangle\big{)}$, with $|u_{\chi}(k,t)\rangle$ denoting the spin-dependent Bloch functions below the energy gap. Figure 4(c) shows the distributions of the two Berry curvatures on the $k-t$ plane under this nontrivial modulation. Figure 4(d) displays the corresponding time-evolution of the pumped spin polarization $\mathcal{Q}_{\chi}(t)=\frac{1}{2\pi}\int^{t}_{0}\int^{\pi}_{-\pi}F_{\chi}(k,t)dkdt$. The different signs in $\mathcal{Q}_{\uparrow}(t)$ and $\mathcal{Q}_{\downarrow}(t)$ implies the different propagating directions of the two spin polarizations Schweizer2016 and the topological spin pumping (in a unit of $-\hbar$) in a cycle becomes $\frac{1}{2}\big{[}\mathcal{Q}_{\uparrow}(T)-\mathcal{Q}_{\downarrow}(T)\big{]}=1$. In addition, the equality $\mathcal{Q}_{\uparrow}(T)+\mathcal{Q}_{\downarrow}(T)=0$ demonstrates no net charge pumping in the whole process.

In this section, we consider the case of a finite DQD chain coupled to two external leads, as illustrated in Fig. 5(a). The electron energy spectrum of the two semi-infinite leads in the momentum space can be derived as $E=2J_{0}\cos(k)$, with $J_{0}$ denoting the inter-site tunneling coupling strength in the leads and $J^{\prime}$ being the strength of the tunneling coupling of the leads to the finite DQD chain (comprising $N$ cells or $2N$ sites). The eigenstates of the leads are the degenerate plane waves $\exp\left(\pm ikl\right)$ Aharony2002 , with the site index $l\leq 0$ for the left lead and $l\geq 2N+1$ for the right lead. When the finite chain is subjected to an incoming wave from the left side, the scattered solutions of the two leads takes the form of OEA2002a

with $\mathcal{T}(t)$ and $\mathcal{R}(t)$ representing the time-dependent transmission and reflection coefficients, respectively. Practically, the two time-dependent coefficients and the transferred charge or spin in a cycle can be ascertained by the second quantization Hamiltonian of the system based on the scattering matrix method Brouwer1998 ; Zhou1999 .

For a spinless chain, the second quantization Hamiltonian of the finite DQD chain is given by Eq. (3) and the reflection coefficient can be determined by the transfer matrix method (see Appendix). In terms of the reflection coefficient, the transferred charge in a cycle can be expressed as

For a spinful DQD chain, $\boldsymbol{\mathcal{R}}(t)$ corresponds to a $2\times 2$ matrix in spin space and the transferred spin per cycle can be derived as Meidan2010 ; Fu2006

In the absence of spin-orbit interaction, $\boldsymbol{\mathcal{R}}(t)$ can be simplified to a diagonal form and the pumped spin is polarized by the external magnetic field along the $z$ direction. Below, we will demonstrate the quantized charge and spin transports in a short DQD chain under the nontrivial modulation described above.

Figure 5(b) shows the time evolutions of the transferred charge through a finite DQD chain with $N=9$, $J_{0}=0.4$ meV, $J^{\prime}=0.3$ meV, $k\simeq\pm\pi/2$, and under the two different modulations as illustrated in the inset of the figure. Here, the effective charge pumping, i.e., $\Delta q=1$, is observed when the parameter vector $\widehat{V}$ is subjected to a nontrivial modulation [see the red solid line]. The conclusion can also be applied to the spin pumping, but with the modulation vector $\widehat{V}$ replaced by $\widehat{M}=(\Delta_{\rm z},\delta V)$. Figure 5(c) shows the time evolutions of the transferred spin $\Delta s_{z}$ under two different modulations. It is also evident that the transferred spin becomes quantized under a nontrivial modulation (see the purple solid line) and is zero under a trivial modulation with $\Delta_{\rm z}(t)=\varepsilon_{z}\cos(2\pi t/T)$, $\delta V(t)=\varepsilon_{\iota}+\varepsilon_{0}\cos(2\pi t/T)$ and the shifting energy $\varepsilon_{\iota}>\varepsilon_{0}$ (see the blue dashed line).

It is well known that the spin-orbit interaction (SOI) plays an important role in the development of the topological insulator Hasan2010 ; Qi2011 ; Kane2005 ; Bernevig2006 . In this section, we will show that in the presence of the Rashba SOI, the DQD chain can serve as a dynamic version of a topological insulator and a spin pump if the parameter vector $\widehat{M}=(\Delta_{\rm z},\delta V)$ is driven by a nontrivial modulation.

The Hamiltonian of the DQD chain in the presence of the staggered magnetic field and the Rashba SOI can be written as

where $H_{\rm Z}$ is to the Hamiltonian given in Eq. (6) and $\alpha$ represents the strength of the SOI, which is related to the spin-orbit length $x_{\rm so}=\hbar/(m_{\rm e}\alpha)$. For $x_{\rm so}\gg L$, the Hamiltonian of the DQD chain in the second quantization form can be derived as

where $a_{n}=\{a_{n,\Uparrow},a_{n,\Downarrow}\}$ and $b_{n}=\{b_{n,\Uparrow},b_{n,\Downarrow}\}$ represent the spinors constituted of two spin-orbit states in the QDs and the intra/inter-cell tunneling amplitudes can be expressed as

with $\varphi_{\rm in/ex}$ being the functions of $L/x_{\rm so}$. In fact, the phase factors appearing in the right side of Eq. (15) arise purely from the interdot electron tunneling in the presence of the SOI AC1984 ; Shahbazyan1994 ; Liu2021 .

Figure 6(a) shows a 3D plot of the energy spectrum for the four lowest-energy Bloch bands of the DQD chain with $x_{\rm so}=180$ nm Scherubl2016 and with the parameter vector $\widehat{M}$ driven by the nontrivial modulation as illustrated by the solid line in the inset of Fig. 6(d). In contrast to the case without SOI, it indicates that the twofold degeneracy is generally lifted, except in the case of $t=0$, $T/2$, and $T$, for which the staggered magnetic field is zero and the system exhibits the Kramers’ degeneracy. Figure 6(b) shows the energy spectrum of a corresponding finite chain. As expected, there exist two pairs of gapless edge-state Bloch bands crossing the band gap in the finite system under this nontrivial modulation. Figure 6(c) shows the energy spectrum of the finite DQD chain under a trivial modulation as illustrated by the blue dashed line in the inset of Fig. 5(d). Clearly, there is no edge-state bands in the band gap in this case.

In analogy with a topological insulator, the presence of the gapless edge-state band in the band gap is correlated with the changing of time-reversal polarization in a half cycle Fu2006 . To be more specific, this kind of polarization is determined by the difference between the charges on the two Wannier centers in a unit cell constructed from the Bloch functions of the occupied bands Soluyanov2011 ; Yu2011 ,

where $\bar{x}_{1}(t)$ and $\bar{x}_{2}(t)$ represent the time-dependent charges on the Wannier centers in a unit cell with $x_{\nu=1,2}(t)\in[-0.5,0.5]$. The changing of time-reversal polarization in a half cycle can actually be identified as the (double-valued) $\mathbb{Z}_{2}$ invariant and the gapless edge states only appear when the topological invariant is nonzero Fu2006 . The inset of Fig. 6(b) shows the time-evolutions of the charges on the two Wannier centers in a unit cell of the infinite DQD chain under the nontrivial modulation. Indeed, it shows that the time-reversal polarization is increased by one in a half cycle, i.e., $\mathcal{P}(T/2)-\mathcal{P}(0)=1$, and this is in contrast to a trivial modulation for which $\mathcal{P}(T/2)-\mathcal{P}(0)=0$, see the inset of Fig. 6(c). Therefore, the periodic DQD chain can serve as a dynamical version of a topological insulator, with the topological property determined by the changing trajectory of $\widehat{M}$.

Similar to the topological spin pumping stated in Sec. III, there is no net charge pumped in a fully cycle and, because of the spin-orbit interaction, the spin pumped per cycle is no longer with the quantization axis along the $z$ direction. However, the results of the spin pumping depends on whether the changing of the parameter vector $\widehat{M}$ is in a trivial or nontrivial trajectory in a cycle. Figure 6(d) shows the time evolutions of the transferred spin in a finite DQD chain with the spin-orbit length $x_{\rm so}=180$ nm and the vector $\widehat{M}$ driven by the two different modulations as illustrated in the inset of the figure. It is evident that the spin transferred per cycle is nonzero under the nontrivial modulation (see the solid lines). Here, we would like to note that for this particular case the pumped spin has both $z$ and $y$ components. It is also clearly seen that the spin transferred in a cycle is zero under a trivial modulation (see the dashed curves).

In this paper, we propose a scheme to implement adiabatic topological pumping in a semiconductor nanowire DQD chain. The topological property of the pumping is related to the changing trajectory of the modulation parameters. We show that the topological charge pumping can be achieved by periodically modulating the QD well and barrier potentials, simultaneously, and the quantized charge transfer can only be realized if the corresponding changing contour is characterized with a nonzero winding number in a cycle. When the QD well potential is replaced by a time-dependent staggered magnetic field, the topological spin pumping can be achieved by a nontrivial modulation of the barrier potentials and magnetic field. We, in addition, demonstrate that in the presence of the Rashba SOI, the periodic DQD chain can serve as a dynamic version of a topological insulator and as a spin pump under a nontrivial modulation of the barrier potentials and magnetic field. Even though the topological adiabatic pumping studied in this paper is based on an InAs nanowire, it can also be extended to other 1D nanostructures, such as carbon nanotubes and Ge/Si heterostructure nanowires Biercuk2005 ; Hu2007 , with long spin-relaxation times. Our theoretical study presented in this work should boost the exploration of adiabatic topological pumping and higher-dimensional topological phases of matter in one-dimensional semiconductor nanostructures.

ACKNOWLEDGMENTS

This work is supported by the Ministry of Science and Technology of China through the National Key Research and Development Program of China (Grant Nos. 2017YFA0303304 and 2016YFA0300601), the National Natural Science Foundation of China (Grant Nos. 91221202, 91421303, and 11874071), the Beijing Academy of Quantum Information Sciences (No. Y18G22), and the Key-Area Research and Development Program of Guangdong Province (Grant No. 2020B0303060001).

CONFLICT OF INTEREST

The authors have no conflicts to disclose.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.