A topological quantum pump in serpentine-shaped semiconducting narrow channels
Sudhakar Pandey Niccolo' Scopigno Paola Gentile Mario Cuoco Carmine Ortix
cond-mat.mes-hall



Institute for Theoretical Solid State Physics, IFW-Dresden, Helmholtzstr. 20, D-01069 Dresden, Germany

Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena, Utrecht University, Princetonplein 5, 3584 CC Utrecht,  Netherlands

CNR-SPIN, I-84084 Fisciano (Salerno), Italy
Dipartimento di Fisica “E. R. Caianiello”,
Università di Salerno I-84084 Fisciano (Salerno), Italy 

CNR-SPIN, I-84084 Fisciano (Salerno), Italy
Dipartimento di Fisica “E. R. Caianiello”,
Università di Salerno I-84084 Fisciano (Salerno), Italy 

Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena, Utrecht University, Princetonplein 5, 3584 CC Utrecht,  Netherlands
Dipartimento di Fisica “E. R. Caianiello”,
Università di Salerno I-84084 Fisciano (Salerno), Italy 
Institute for Theoretical Solid State Physics, IFW-Dresden, Helmholtzstr. 20, D-01069 Dresden, Germany




We propose and analyze theoretically a one-dimensional solid-state electronic setup that operates as a topological charge pump in the complete absence of superimposed oscillating local voltages. The system consists of a 
semiconducting narrow channel
with strong Rashba spin-orbit interaction patterned in a mesoscale serpentine shape. 
A rotating planar magnetic field serves as the external ac perturbation, and cooperates with the Rashba spin-orbit interaction, which is modulated by the geometric curvature of the 
electronic channel
to realize the topological pumping protocol originally introduced by Thouless in an entirely novel fashion. 
We expect the precise pumping of electric charges in our mesoscopic quantum device to be relevant for quantum metrology purposes. 


73.63.Nm, 73.21.Cd, 03.65.Vf, 73.43.-f 
A topological quantum pump in serpentine-shaped semiconducting narrow channels
    Carmine Ortix
    December 30, 2023
==============================================================================





  
 Introduction – Precisely as in an Archimedean screw, where water is pumped by a rotating spiral tube, in a charge pump periodic perturbations induce a dc current <cit.> without an external bias. 
This phenomenon can be entirely adiabatic, as it occurs for instance in open quantum dots subject to a cyclic deformation of the confining potential <cit.>, 
with the system that always remains in its instantaneous ground state. 
In a topological charge pump <cit.> the zero bias dc current is precisely quantized, and the quantization is topologically protected against external perturbations <cit.>. 
Each pump cycle transports an integer number of electronic charges with the integer uniquely determined by a topological invariant: the Chern number 𝒞 of the quantum system <cit.>. 
The dc current generated by periodic variations of one parameter of the system Hamiltonian amounts indeed to I=e  𝒞 ν where ν is the frequency of the variation <cit.>.
Topological charge pumping
can be 
also
understood as a dynamical analog of the integer quantum Hall effect <cit.>, 
with the charge pumped in each cycle that can be mapped exactly to the quantized Hall conductance of a “dual" two-dimensional electronic system. 
Such analogy is mathematically transparent considering the mapping  
between the Hofstadter model <cit.> – perhaps the most simple Hamiltonian to study quantum Hall physics in a two-dimensional lattice system – and the 
one-dimensional Aubry-André-Harper (AAH) model <cit.>, when the superlattice potential is assumed to rigidly slide in time <cit.>. 



Very recently, the advances in constructing optical lattice structures with precise control over lattice intensity and phases have enabled the realization of dynamically controlled optical superlattices, and consequently of topological quantum pumps in 
ultracold atomic systems <cit.>. 
In one-dimensional electronic systems, instead, the creation of a dynamical superlattice  potential critically relies on the presence and control of superimposed oscillating local voltages <cit.>. This, in turns, severely hampers the possibility to bring topological charge pumping within reach in condensed matter experiments. 

In this Rapid Communication,  we propose and validate theoretically an entirely novel solid-state system in which  topological quantum pumping can be achieved even in the complete absence of superimposed voltage leads. The system consists of a Rashba spin-orbit coupled 
semiconducting narrow channel with a
serpentine shape at the mesoscopic scale [c.f. Fig.<ref>(a)]
: It can be obtained either by processing a semiconducting quantum well lithographically <cit.>, or creating a “zigzag" nanowire network of  crystalline quality <cit.>.
To operate, the device makes use of an auxiliary external planar rotating magnetic field, which serves as the periodic (ac) perturbation driving the charge pumping [c.f. Fig.<ref>(b)] . The concomitant presence of the time-dependent Zeeman interaction and the spin-orbit coupling, which is effectively inhomogeneous due to the geometric curvature of the 
nanostructure <cit.>
renders a sliding superlattice potential acting on the electronic charges, and ultimately yields a quantized dc current. 
We also show that as the strength of the rotating magnetic field is increased, the system undergoes a topological phase transition from a state where an even integer number of electronic charges are transported in each rotation period of the magnetic field, to a state where the rotating magnetic field does not pump any electronic charge.  





  
Theoretical model – We start out by expressing the effective k · p Hamiltonian for a planar one-dimensional 
semiconducting channel with curved geometry
as <cit.> 

    H_ k · p   =    - ħ^22 m^⋆∂^2_s - i α_Rτ_𝒩 (s) ∂_s + i α_R2κ(s) τ_𝒯(s)  
           - ħ^28 m^⋆κ(s)^2.

In the equation above, s is the arclength along the 
channel
measured from an arbitrary reference point,  m^⋆ and α_R  indicate the effective mass of the carriers 
and the Rashba spin-orbit coupling constant respectively, and κ(s) is the local curvature explicitly entering in the so-called quantum geometric potential (QGP) <cit.>. 
We also introduced two local Pauli matrices, comoving with the electrons as they propagate along s, explicitly reading 
τ_𝒩(s) = τ·𝒩̂(s) and τ_𝒯(s) = τ·𝒯̂(s),
where the τ's are the usual Pauli matrices. 
𝒯̂(s) and 𝒩̂(s), i.e. the local tangential and normal directions of the narrow 
channel,
can be expressed in Euclidean space by introducing an angle φ(s), in terms of which 𝒩̂(s) = {sinφ(s), cosφ(s), 0 }
and 𝒯̂(s) = {cosφ(s), -sinφ(s), 0 }. 
Using the Frenet-Serret  equation ∂_s 𝒯̂(s) ≡κ(s) 𝒩̂(s), it then follows that the angle φ(s)= - ∫^s κ(s^') d s^' is entirely  determined 
by the local curvature. 


To proceed further, we use that the undulating geometry of a serpentine-shape can be parametrized in the Monge gauge as y(x) = A cos(2 π x / λ) with A 
the maximum departure of the channel from flatness, and λ the serpentine period. 
In the shallow deformation limit A / λ≪ 1, we can express the arclength of the channel s ≃ x while the geometric curvature is 
κ(s) ≃ - (2 π/λ)^2 A cos(2 π s / λ). This periodicity of the curvature transmutes the QGP in an effective superlattice potential <cit.>, 
and yields at the same time a periodic canting of the spin-orbit field axis <cit.>.
In order to study their concomitant effect on the electronic properties of the quantum 
system,
we next introduce a tight-binding model obtained by discretizing Eq. <ref> on a lattice. It can be written as 

    H   =    -t  ∑_i,σc^†_i,σ c_i+1,σ  - ħ^2/8 m^*∑_i,σκ(s_i)^2 c^†_i,σc_i,σ
        -   i α_R/2∑_i,σ,σ^' c^†_i,σ[ τ_N^σ, σ^'(s_i) + τ_N^σ, σ^'(s_i+1) /2]c_i+1,σ^' + H.c.,

where c^†_i , σ, c_i , σ are operators creating and annihilating, respectively, an electron at the ith site with spin projection σ= ↑, ↓ along the ẑ axis, and t = ħ^2 / (2 m^⋆ a^2) is the nearest-neighbour hopping amplitude. 
The atomic positions can be instead written as s_i / λ = p i / q + ϕ, with p and q coprimes whose ratio is p/q = a / λ (a being the tight-binding lattice constant), and ϕ a phase accounting for nonequivalent displacements of the atoms in one superstructure period. 
The Hamiltonian in Eq. <ref> can be seen as the combination of a conventional, commensurate diagonal AAH model with a spin-dependent off-diagonal AAH model <cit.>. Taken separately, both these models realize butterfly spectra <cit.> and are characterized by insulating states with non-trivial Chern numbers if the phase ϕ is assumed to vary in time. 
We will instead consider a constant ϕ value (taking ϕ≡ 0 for simplicity) and monitor the effect of an external magnetic field, which we account for adding the usual Zeeman term H_Z ≡∑_i,σ,σ^'c^†_i,σ[B⃗·τ⃗]_σ,σ^'c_i,σ^' to the Hamiltonian in Eq. <ref>. 


We first consider the ensuing electronic properties in the regime of negligible Rashba spin-orbit coupling, with preserved SU(2) spin symmetry, 
for the simple case of p/q=1/4.  At filling fraction ν=1/4, the geometric curvature of the 
nanostructure
yields, via the QGP, a metal-insulator transition (MIT) for strong enough external magnetic fields. 
In this regime indeed, the lowest energy bands are completely spin-polarized thereby allowing the spin-independent superlattice potential to open up a gap at the centre of the mini-Brillouin zone (mBZ) <cit.>. As explicitly proved in Fig. <ref>, for A / λ→ 0 the critical value of the magnetic field strength at which the MIT takes place 
is B_c ≃ t, and it continuously flows to smaller values as the geometric curvature is increased. In the B<B_c region instead, two degeneracies between opposite spin states at unpinned points in the mBZ  cannot be removed by the action of the QGP thereby implying the existence of a semimetallic state at one-quarter filling. 




Having established the occurrence of a curvature-driven MIT in the presence of SU(2) spin symmetry, we now investigate the effect of the Rashba spin-orbit coupling. 
We first consider a magnetic field B= B  ẑ, orthogonal to the plane in which the spin-orbit field lies. 
Contrary to a conventional 
straight channel,
the canting of the Rashba field resulting from the geometric curvature 
yields a finite gap 
at ν=1/4 filling independent of the strength of the external magnetic field [c.f. Fig. <ref>(a)].  
This is because the degeneracies between opposite spin states encountered at unpinned momenta of the mBZ in the weak magnetic field regime are removed by the action of the curvature-induced modulated Rashba term. 
And indeed the gap increases monotonously with the Rashba coupling α_R for weak magnetic field strengths [c.f. Fig. <ref>(a)]. 
Exactly the same features  occur by tilting the magnetic field direction toward the x̂ direction <cit.>, 
which corresponds to the direction in which the effective spin-orbit interaction averages to zero in one superlattice period. 
When the magnetic field points exactly in the latter direction, however, the situation changes drastically. As shown in Fig. <ref>(b), 
cranking up the magnetic field strength leads first to an increase of the quarter filling gap, which is subsequently followed by a substantial decrease 
until a critical value where the gap undergoes a closing-reopening transition at the center of the mBZ <cit.>.






  
Topological charge pumping – In the same spirit of Thouless seminal work <cit.>, we now consider a slow, adiabatic ac rotation of the magnetic field in the x̂ - ẑ plane. Since the system always remains 
in its instantaneous ground state, we can interpret the angle θ that the magnetic field direction forms with the ẑ axis, as an additional quasi-momentum. 
With this dimensional extension, our system can be viewed as a two-dimensional insulator which, due to the absence of time-reversal symmetry, belongs to the class A of the Altland-Zirnbauer table <cit.>. 
Moreover, the presence of the gap closing-reopening discussed above suggests the existence of two insulating phases characterized by  different ℤ topological invariants. 
We have verified that our quarter-filled model supports an insulating phase with a non-trivial Chern number 𝒞≡ -2 in the weak magnetic field region, 
while a completely trivial 𝒞≡ 0 insulating phase is encountered in the strong-field regime. 
As shown in Fig.<ref>(a), the energy spectrum of a finite size system with open boundary conditions displays  two chiral edge states 
within the one quarter-filling gap for small enough magnetic field. 
They are related to each other by a π rotation of the magnetic field direction, and carry opposite spin content as can be shown with a symmetry argument <cit.>.

In the opposite strong field regime, instead, the edge states do not connect the valence to the conduction band thereby implying a topologically trivial insulating state  [c.f. Fig. <ref>(b)]. 
We have additionally computed the Berry curvature of the insulating states using the method outlined in Ref. tak05. In the weak magnetic field regime, we find that the Berry curvature displays four peaks [c.f. Fig. <ref>(c)], each of which contributing ≃ -1/2 to the total Chern number. In the strong magnetic field regime, instead, the contributions coming from the  two peaks located at the mBZ center for θ=±π/2  [c.f. Fig. <ref>(d)], are identically cancelled by an almost homogeneous background with Berry curvature opposite in sign. 











The fact that our tight-binding model realises a 𝒞≡ -2 topological charge pump when subject to weak rotating magnetic fields is independent of the superlattice period. Specifically, 
for any superlattice period λ= 2 n a with n integer, and at 1 / ( 2 n) filling, all the discussion above remains unaltered. This, in turn, implies that also a  mesoscale serpentine-shaped quantum wire 
can operate as a topological charge pump. 
Since the conditions for this effect to be observed are k_B   T, h ν < E_g where k_B   T  is the energy scale for temperature T, 
ν the magnetic field frequency,
and E_g is the relevant energy gap, we analyzed the continuum k · p effective theory given by Eq. <ref> taking into account 
the Zeeman term H_Z= g^⋆μ_B  B·τ, where g^⋆ is the effective Landé g factor, 
and μ_B the Bohr magneton. 
Fig. <ref> shows the corresponding topological phase diagrams with a map of the bulk gap when the first two Bloch minibands are fully occupied. 
By considering the serpertine period to be of the order of 
λ≃ 0.5 μm,
the effective mass of, e.g., InAs m^⋆≃ 0.023, and a strength of the Rashba spin-orbit interaction <cit.> up to 
0.2 eV Å,
we find that the size of the bulk gap 
E_g≃ 20 μeV
for A/λ=0.1 and a magnetic field strength 
≃ 60mT
(the Landé g factor g_InAs≃ 15 in bulk crystals <cit.>). 
As a result, a topological charge pumping effect characterised by a quantized dc current can be observed for temperatures 
T < 200mK
and frequencies 
ω < 4 GHz. A rotating magnetic field with this freqency can be in principle obtained by running current pulses in two perpendicular conductors with a π/2 phase shift <cit.>. The resulting dc current I ≃ 1nA can be easily detected with present day experimental capabilities. Therefore,
in this physical regime our solid-state device can be used to measure the fundamental electron charge e. 






  
Conclusions – To wrap up, we have shown that a Rashba spin-orbit coupled 
semiconducting channel possessing a mesoscale serpentine shape, as obtained either by lithographic processing <cit.> or using “zigzag" nanowire networks <cit.>,
can operate as a topological charge pump once subject to a relatively weak magnetic field rotating on a plane. 
Being topological in nature, the electronic pumping is robust against additional perturbations, e.g. tiny tilts of the magnetic field in the direction orthogonal to the plane. 
The existence of this phenomenon stems from the 
periodic canting of the Rashba spin-orbit field, which is the prime physical consequence of the geometric curvature of our 
nanostructure,
and provides a bridge between the Zeeman 
interaction and the quantized charge flow in the wire. 
Adiabatic quantum pumping has been recently discussed in one dimensional semiconducting Rashba systems in the presence of local voltage leads or periodically arranged nanomagnets <cit.>. Our proposed set-up, instead, does not necessitate the usage of external local perturbation 
since it only relies on the successful 
growth of a one-dimensional channel with curved undulating geometry. 
Our findings therefore add a prime actor to the recently uncovered series of unique curvature-induced quantum effects in low-dimensional semiconducting systems <cit.>. 



  
Acknowledgements – We acknowledge the financial support of the Future and Emerging Technologies (FET) programme under FET-Open grant number: 618083 (CNTQC). 
C.O. acknowledges support from the Deutsche Forschungsgemeinschaft (Grant No. OR 404/1-1), and from a VIDI grant (Project 680-47-543) financed by the Netherlands Organization for Scientific Research (NWO).
S. P. thanks Ulrike Nitzsche for technical assistance.
N. S. thanks Guido van Miert for many fruitful discussions.
S. P. and N. S. contributed equally to this work.










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