Artificial Gauge Field and Topological Phase in a Conventional Two-dimensional Electron Gas with Antidot Lattices

Based on the Born-Oppemheimer approximation, we divide the total electron Hamiltonian in a spin-orbit coupled system into the slow orbital motion and the fast interband transition processes. We find that the fast motion induces a gauge field on the slow orbital motion, perpendicular to the electron momentum, inducing a topological phase. From this general designing principle, we present a theory for generating artificial gauge field and topological phase in a conventional two-dimensional electron gas embedded in parabolically graded GaAs/InxGa1−xAs/GaAs quantum wells with antidot lattices. By tuning the etching depth and period of the antidot lattices, the band folding caused by the antidot potential leads to the formation of minibands and band inversions between neighboring subbands. The intersubband spin-orbit interaction opens considerably large nontrivial minigaps and leads to many pairs of helical edge states in these gaps.


Results
General design principle: gauge field from Born-Oppenheimer approximation. First we discuss the emergence of an artificial gauge field in a system of electrons in a 2D system described by a low-energy single-particle Hamiltonian is a gauge potential in the momentum space of the slow orbital motion, due to the interband coupling to the fast spin dynamics 27,28 . For the BHZ Hamiltonian, the gauge potential A n leads to an effective Lorentz force ( ) in the momentum space perpendicular to the electric field E: where s n = ± 1 denotes spin up or down state while t n = ± 1 denotes the conduction or valence band, respectively (n = 1, 2, 3, 4). The Chern number x y n n n BZ 2 is obtained by integrating the field strength F xy,n in the Brillouin zone. The sign change in M would induce a change of the Chern number by 1, which corresponds to the topological phase transition 29 .
For a 2DEG with Rashba and Dresselhaus SOIs, we find ( ) = F k 0 xy , which means that the Chern number vanishes in 2DEG with SOIs. Comparing the Hamiltonian of 2D TIs to that of 2DEGs with SOIs, one can see clearly that 2D TIs posses an additional degree of freedom: the band index τ. In order to generate the gauge field and realize TI phases in a 2DEG, one needs to create minibands and band inversion in 2DEGs. Based on the above designing principle, we will create topological phase in conventional semiconductor 2DEG. This is the first demonstration of the formation of a TI phase in the s-like band systems, i.e., a 2DEG with nanostructured antidot lattice shown schematically in Fig. 1

(a).
Topological phase transition in two-dimensional electron gas: effective model. Nanostructured antidot lattices, consisting of periodically arranged holes that are etched in a 2DEG, form a strongly repulsive egg-carton-like periodic potential in a 2DEG [30][31][32][33][34][35][36][37] . This artificial crystals lead to a wide variety of phenomena, for instance, Weiss oscillation, chaotic dynamics of electrons, the formation of an electronic miniband structure and massless Dirac fermions. At low temperatures, the mean free path of electrons is much longer than the period of antidot lattices ranging from 10 to 100 nanometers. The modulated periodic potential can also be created by electron beam lithography electrodeposition and periodic arrays of metallic nanodots can be realized on semiconductor surfaces. Due to elastic strains producing these dots, a sufficiently strong piezoelectric potential modulation results in miniband effects in the underlying 2DEG 32,33 . Very recently, a honeycomb lattice of coronene molecules was created by using a cryogenic scanning tunneling microscope on a Cu(111) surface to construct artificial graphene-like lattice with the lattice constant approaching 5 nm 37 .
We consider the 2DEG in a GaAs/In x Ga 1−x As/GaAs parabolically graded QW, which was fabricated successfully before [38][39][40] , with a triangular antidot lattice (see Fig. 1(a). Before going to the numerical calculation, we first give a clear physical picture for the emergence of a TI phase in this 2DEG system upon nanostructuring with antidot lattice. The simplest description of the 2DEG system is obtained by reducing the eight-band Kane model to the lowest conduction subbands of the QW (see Methods). This gives the Hamiltonian for the 2DEG with periodic antidot lattice potential V(x, y):  in a symmetric QW are odd. This means that the ISOI can exist in symmetric QWs. Here we neglect Dresselhaus SOI term which is proportional to π ( / ) k d z 2 2 (d is the thickness of the QW) because in our proposal the QW thickness is quite large (300A), therefore the strength of Dresselhaus SOI is quite weak. We would also emphasize that the Dresselhaus SOI only exists in the same subband, which is an intra-subband interaction and will not affect our above analysis and topological nontrivial gap in such system, i.e., the band inversion between two adjacent subbands.
The first and second subbands both form minibands due to the Brillouin zone folding caused by the antidot lattice. The band inversion could occur between the two adjacent minibands of the first subband Topological phase transition in two-dimensional electron gas: numerical calculation. We employ the eight-band Kane k · p model to calculate the subband structure with SOIs in a 40-nm-thick GaAs/In x Ga 1−x As/GaAs parabolically graded QWs 38,39 , as plotted in Fig. 2(a). The energy difference between the minima of the first and second subbands at Γ point is about 90 meV (see Fig. 2(a). In order to calculate the miniband structures caused by an in-plane periodic potential induced by the triangular antidot lattice, we reduce the eight-band model to an effective four-band k · p Hamiltonian by including the lowest 20 electron subbands and 54 highest hole subbands in the QW, to reproduce the energy dispersions of the first and second subbands calculated from the eight-band Kane model (see Fig. 2a). The parameters in the four-band Hamiltonian is given Supplementary Note 2. The minibands from the four-band k · p Hamiltonian are shown in Fig. 2(b,c). These minibands originates from folding the first and second subbands of the QW into the first Brillouin zone of the antidot lattice [ Fig. 1(c)]. By tuning the antidot lattice constant a and the potential height V 0 , i.e., the etching depth of the antidot lattice, many band inversions appear between these minibands, which can be clearly seen in Fig. 2(b,c). The minigaps between these minibands are opened by the ISOI shown in Eq. (4) [see Fig. 2(b,c)].
To demonstrate that these minigaps are topologically nontrivial, we determine the parity of each miniband at the four time-reversal invariant momenta 11 Γ i (i = 0, 1, 2, 3) in the first Brillouin zone shown in Fig. 1(c). For the lowest N spin-degenerate minibands being occupied, the Z 2 invariant is given by is the parity of the 2mth occupied miniband at Γ i . Our calculation gives ν = 1 at all the minigaps, which proves the whole system is in the quantum spin-Hall phase (see Methods).
Next, we demonstrate the emergence of topological edge states upon etching the QW into a Hall bar structure along two different directions (x axis and y axis). As shown in Fig. 3, a pair of topological helical edge states appear inside each nontrivial minigap. For example, we can see topological helical edge states in the lowest two nontrivial minigaps near ~186.5 meV and ~255 meV, respectively. The helical edge state pairs in these minigaps would lead to higher conductance plateaus as the Fermi energy increases by increasing the doping level. The helical edge states do not overlap with the bulk states, making it possible to be detected experimentally.
The lowest nontrivial minigaps is quite small (about 0.5 meV), but the second minigap is larger (about 5 meV). By tuning the period and potential height of the antidot lattice, the nontrivial minigaps can be significantly enhanced [see Fig. 4(a,b)]. For example, the lowest minigaps can be enhanced to 5 meV, which is already comparable with that in HgTe and InAs/GaSb QW systems (~10 meV) 8,9 . The second minigaps can approach 20 meV, which means the TI phase can be realized at liquid nitrigen temperature regime. From Fig. 4(a,b), one can see that the lowest nontrivial minigap is closed as the lattice constant a increases, but the second higher nontrivial minigap survives, i.e., the TI phase can exist even at large lattice constants, e.g., 25 nm. We remark that the randomness of the size and position, i.e., disorder effect, might smear our the nontrivial minigap. However, the previous works [45][46][47] demonstrated that the disorder effect would not cancel topological phase, instead, it will lead to topological Anderson insulator phase where the edge states can exist even for very strong disorder strength, which is much larger than the bandgap. Experimental detection scheme. One way to detect the aforementioned edge states (shown in Fig. 3) is the standard four terminal measurements as demonstrated in previous works 8,9 . In contrast to HgTe and InAs/GaSb quantum well systems, there are many pairs of helical edge states in our system between these inverted minibands, which leads to higher plateaus with increasing the Fermi energy. Another possible way is microwave impedance microscopy which makes spatial-resolved nano-scale images (< 100 nm) of the conductivity and permittivity of a sample 48 . The unoccupied edge states in higher minigaps can be detected using the angle-resolved photonemission technique 49 , which has already been successfully applied to identify occupied and unoccupied surface states in Bi 2 Se 3 and Bi 2 Te x Se 3 [49][50][51] .

Discussion
Our proposal is based on a general analysis about the electron orbital motion in TIs. By using the Born-Oppenheimer approximation, we find that the fast motion will induce a spin-dependent gauge field on slow orbital motion. Based on this general analysis, we demonstrate theoretically the TI phase in a conventional 2DEG embedded in a symmetric parabolically graded GaAs/In x Ga 1−x As/GaAs QW, with antidot lattices created by well-developed etching technique. The key point is to create a ISOI in a symmetric quantum well, in contrast to conventional SOI in asymmetric QWs. This hidden ISOI in  symmetric QWs induces a spin-dependent effective Lorentz force on the electrons, and generates the TI phases in such system. Interestingly, such ISOI exists in conventional semiconductors with a positive bandgap, i.e., normal band structures can generate quite large nontrivial gaps approaching 20 meV. This make it possible to observe the quantum spin Hall effect in liquid nitrigen temperature regime. So far, all members of TI family are narrow bandgap systems containing heavy atoms. Our proposal breaks this constraint, and makes it possible to realize TI phase in conventional semiconductor 2DEG using the well-developed semiconductor fabrication techniques. The presence of the TI phase in parabolically graded QWs with antidot lattice can largely advance the application of this new quantum state in existing electronics and optoelectronics devices. The general designing principle proposed in this work, i.e., the gauge field acting on slow orbital motion induced by interband coupling, paves a new way for generating nontrivial topological phases, such as quantum spin Hall phase and even quantum anomalous Hall phases by doping magnetic ions, in conventional semiconductor 2DEGs, and suggests a promising approach to integrate it in well developed semiconductor electronic devices.

Methods
Effective spin-orbit coupling in a quantum well. For a symmetric quantum well grown along (001) direction (the z axis), effective spin-orbit coupling exists between subbands with opposite parities. This effective spin-orbit coupling comes from interband coupling and can be understand by reducing the 8 × 8 Kane Hamiltonian to a 2 × 2 effective Hamiltonian.
To the first order of k, the 8 × , where , = , n n 0 1 1 2 . The calculated parity eigenvalue of the 2mth (m = 1, 2, 3, 4, 5, 6) occupied energy band at Γ ι are listed: (11) From the above calculation, we can confirm that the Z 2 invariant ν = 1 at the 2mth (m = 2, 5, 6) occupied band where the minigaps open, and the system enters the TI phase and the dissipationless edge states appear.