High-Efficiency Cooper-Pair Splitter in Quantum Anomalous Hall Insulator Proximity-Coupled with Superconductor

The quantum entanglement between two qubits is crucial for applications in the quantum communication. After the entanglement of photons was experimentally realized, much effort has been taken to exploit the entangled electrons in solid-state systems. Here, we propose a Cooper-pair splitter, which can generate spatially-separated but entangled electrons, in a quantum anomalous Hall insulator proximity-coupled with a superconductor. After coupling with a superconductor, the chiral edge states of the quantum anomalous Hall insulator can still survive, making the backscattering impossible. Thus, the local Andreev reflection becomes vanishing, while the crossed Andreev reflection becomes dominant in the scattering process. This indicates that our device can serve as an extremely high-efficiency Cooper-pair splitter. Furthermore, because of the chiral characteristic, our Cooper-pair splitter is robust against disorders and can work in a wide range of system parameters. Particularly, it can still function even if the system length exceeds the superconducting coherence length.


Results
System Model. In this article, we study the quantum tunnelling, Andreev reflection and crossed Andreev reflection in a two-terminal finite-sized QAHI system, with the central region being covered by a superconductor (See Fig. 1). The total Hamiltonian of the hybrid system can be written as H T = H QAHI + H SC + H C , where H QAHI , H SC , and H C correspond respectively to the Hamiltonians of the QAHI, superconductor, and their coupling. As a concrete example, the QAHI is modelled by using a monolayer graphene including the Rashba spin-orbit coupling and an exchange field, and its tight-binding Hamiltonian can be written as 36 : where t measures the nearest-neighbor hopping amplitude that is set as the unit of energy, and α † c i (c iα ) is the electron creation (annihilation) operator at site i with spin α (i.e., ↑ or ↓ ). The second term corresponds to the exchange field with a strength of λ , and σ are the spin-Pauli matrices. The third term describes the external Rashba spin-orbit coupling with a coupling strength of t R , arising from the mirror symmetry breaking, e.g., by applying a vertical electric field 19 . Here, d ij is a unit vector pointing from site j to site i. In the last term, the static Anderson-type disorder is added to ε i with a uniform distribution (1) directly tunnelling through the QAHI as an electron, and (2) proximity-flowing into the SC to form a spatially separated Cooper-pair and ejecting a hole to the right terminal. Red solid and empty circles are used to denote the electrons and holes, respectively.
in the interval of [− W/2, W/2], where W characterizes the strength of the disorder. In addition, the Hamiltonians of the superconductor and its coupling with the QAHI can be respectively expressed as: where  k corresponds to the on-site energy in the momentum space, k = (k x , k y ) is the wave vector, Δ S is the superconducting pair-potential measuring the superconductor gap, t C is the hopping amplitude between the superconductor and the QAHI, and ( ) = ∑ Physical picture of the absence of the normal and Andreev reflections. In the absence of the superconductor, gapless edge modes of the QAHI appear inside the bulk band gap Δ QAHI of the graphene nanoribbon (See Fig. 2a). For a given Fermi-level lying inside the bulk band gap, e.g., the crossing points by the dashed line displayed in Fig. 2a, there correspond four different electron states labelled as "A", "B", "C", and "D". In Fig. 2b, we plot the wavefunction distributions of these states across the width. It can be clearly seen that the wavefunction of each state is mainly localized at the ribbon boundary, i.e., states "A" Wave-function distributions ψ 2 across the width for the eight states labelled in (c). States "A", "C", "E", and "G" are localized at the top boundary, while states "B", "D", "F", and "H" are localized at the low boundary. and "C" are localized at the top boundary, while states "B" and "D" are localized at the bottom boundary, which confirms that they are indeed the edge states. From the dispersion relation shown in Fig. 2a, one can also find that states "A" and "C" propagate along the same direction, which is opposite from that of states "B" and "D". These together indicate the unidirectionally or chirally propagating property of the edge states, which is distinct from the helical edge states of the Z 2 topological insulators. For clarity, the chiral edge states are visually displayed in Fig. 1a with the blue arrows signifying the propagating directions.
We now study how the chiral edge states are affected when the QAHI is covered by a superconductor. Usually, when a conductor or the helical edge states of TIs are covered by a superconductor, a gap can open at the Fermi surface due to the proximity effect from the coupling with the superconductor. The reason behind is that, for the normal conductors and the Z 2 TIs, the dispersion relation of opposite spin states is usually an even function of momentum k due to the time-reversal symmetry, e.g.   = ↑ − ↓ k k . In such cases, a superconducting pair-potential Δ S can open a band gap and the energy spectra become F . However, it is rather different for the situation of the chiral edge states of the QAHI. Fig. 2c shows the spectral function A(E, k x ) of the hybrid system of QAHI covered by a superconductor, where π x r x with G r (E, k x ) being the retarded Green's function [See METHODS for the calculation details]. One can see that at any fixed energy E, the spectral function has a finite value (denoted in red), which reflects that no band gap opens at the Fermi surface in the hybrid system. The underlying reason can be attributed to that the dispersion relation  σ . This new energy dispersion indicates that the chiral edge states still keep gapless, and thus no band gap opens. By analyzing the wavefunction distributions ψ 2 across the width of the QAHI ribbon covered with a superconductor as displayed in Fig. 2d, one can find that all electron states inside the superconductor gap are localized at the system boundaries. For example, the states "A", "C", "E", and "G" propagating from right to left are localized at the top boundary whereas the states "B", "D", "F", and "H" propagating from left to right are localized at the low boundary, which means that the edge states exhibit the chiral propagating characteristic. Therefore, the backscattering is completely forbidden, because that the chiral edge modes are topologically protected by the spatial separation.
We now turn to analyze the scattering processes when an electron with fixed spin (e.g., up spin) incoming from the left terminal flows into the central region of the hybrid system. In general, there are four scattering processes as displayed in Fig. 1a: 1) the direct reflection to the left terminal as a spin-up electron; 2) the quantum tunnelling to the right terminal as a spin-up electron; 3) the Andreev reflection to the left terminal as a spin-down hole; and 4) the crossed Andreev reflection to the right terminal as a spin-down hole. To be specific, in Fig. 1b, we present a schematic illustration of how the Andreev reflection and the crossed Andreev reflection occur in the central hybrid region. Note that the incoming electron from the left terminal propagates along the bottom boundary, while for the direct reflection and the Andreev reflection, the outgoing electron and hole propagate along the top boundary (see Fig. 1a). Furthermore, there exists a bulk band gap no matter whether the QAHI is covered or not by a superconductor, thus the scattering between the top and bottom boundaries are almost impossible for a wide enough ribbon. Therefore it is reasonable to expect that the direct reflection and Andreev reflection will be completely suppressed, while the quantum tunneling and the crossed Andreev reflection will dominate the whole scattering processes. In other words, the two electrons of a Cooper-pair go respectively to the left terminal and right terminal as described in Fig. 1b, which leads to a high efficiency of the Cooper-pair splitting.
Numerical results and discussions. In this Section, we provide a detailed numerical calculation to support our above expectation. Hereinbelow, the parameters for the QAHI are chosen to be λ = 0.18 t and t R = 0.20 t, and the pair potential of the superconductor is set to be Δ S = 0.05 t. It is noteworthy that the size of the QAHI bulk band gap Δ QAHI ≈ 0.26 t is much larger than the superconducting gap 2Δ S . Figure 3 plots the transmission coefficients of the quantum tunneling T QT , Andreev reflection T AR and crossed Andreev reflection T CAR as functions of the Fermi level E F , where the width of the ribbon is fixed at N = 80 that is wide enough to avoid the finite-size effect and the lengths are respectively chosen to be L = 11, 21, and 31 (see METHODS for the calculation details). One can find that, as expected, in all the three different cases the Andreev reflection is completely suppressed to be T AR ≈ 0 regardless of the length L, as long as the Fermi-level lies inside the superconducting gap < ∆ E F S . Because of the absence of both the direct reflection and the Andreev reflection, the electrons incoming from the left terminal propagate into the right terminal in forms of the quantum tunneling and the crossed Andreev reflection, leading to T QT + T CAR ≈ 2 as long as < ∆ E F S . Most importantly, the vanishing of the Andreev reflection also results in an extremely high Cooper-pair splitting efficiency η → 100% independent of the system length L as displayed in Fig. 3, which is defined as η = T CAR /(T AR + T CAR ). This strongly suggests that our proposed system can function as a high-efficiency Cooper-pair splitter. In addition, for the electrons incoming from two opposite spin edge states, they have the same crossed Andreev reflection. This means that the spin-up (spin-down) electron has the same probability to go to left or right terminal. And if the spin-up electron goes to the left terminal, then the spin-down one has to go to the right terminal. So the two spatially-separated electrons from a Cooper-pair still keep the spin and momentum entangled. Another observation in Fig. 3 is that the transmission coefficient of the quantum tunnelling T QT is always comparable with that of the crossed Andreev reflection T CAR , exhibiting a universal characteristic for different system sizes whenever the Fermi energy satisfies < ∆ E F S . To eliminate the influence of the quantum tunnelling for practical applications, one can simply apply the same potential in both the left and right terminals. Then, the crossed Andreev reflection can be utilized to design high-efficiency Cooper-pair splitters by mediating the potential of the central region.
Next, we move to the size dependence of these transmission coefficients at fixed Fermi-levels. Figure 4a,c display the transmission coefficients of the quantum tunnelling, Andreev reflection, and crossed Andreev reflection as functions of the system length L at a fixed system width N. One can find that the Andreev reflection is also vanishing with T AR = 0 for all values of L, while the transmission coefficients T QT and T CAR oscillate as functions of the system length L with the oscillation period being dependent on the Fermi-level E F . The reason behind this observation is that the covered superconductor functions as an applied external potential, and the resulting quantum tunnelling and crossed Andreev reflection can reach a resonance at certain system lengths.
It is noteworthy that the difficulty in realizing Cooper-pair splitter is that the crossed Andreev reflection is intimately affected by the distance between the two normal terminals. For all previous proposed Cooper-pair splitter, the crossed Andreev reflection quickly decreases in parallel to increasing the system length and finally vanishes when it exceeds the superconducting coherence length. Counterintuitively, in our considered system the crossed Andreev reflection can still survive and keep a large value even for relatively long system lengths. And the obtained coefficient of the crossed Andreev reflection is larger than 0.1 for any system lengths L (the distance between the two normal terminals). This is perfectly logical and reasonable, because the chiral edge states exist in the QAHI no matter whether it is covered or not by the superconductor, making the scattering from one boundary to other one almost impossible (except for very narrow ribbons). Therefore, our proposed high-efficiency Cooper-pair splitter is able to function in a long-range junction, even if it farther exceeds the superconducting coherence length. In addition, by choosing proper system lengths or externally adjusting Fermi-levels, our proposed setup can not only reach a high Cooper-pair splitting efficiency ~100%, but also provide a strong signal of the crossed Andreev reflection with T CAR ≈ 2, which is much larger than those reported in previous works. Figure 4b,d show the transmission coefficients of the quantum tunneling, Andreev reflection and crossed Andreev reflection as functions of the sample width N at fixed system length L = 21. One can see that only for small width N there exists very weak Andreev reflection, because, in this case, both At this point, we have proposed a scheme for realizing a high-efficiency Cooper-pair splitter in a hybrid system of the QAHI proximity-coupled with a superconductor. It is clear that external disorders are inevitable in practical devices. Therefore, the question naturally arises as to whether the quantum tunnelling and the crossed Andreev reflection are still robust in real systems? To address this issue, in Fig. 5 we plot the averaged transmission coefficients by collecting over 200 samples in the presence of on-site Anderson disorders in the central scattering region. One can see that these transmission coefficients are nearly unaffected when the relatively strong disorders of W/Δ S = 1, 2 and 5 are introduced. Even for a much stronger disorder strength of W/Δ S = 10, the Andreev reflection is only slightly enhanced. This is because the electrons can be weakly scattered to the opposite edges in the presence of rather strong disorders. Therefore, even in such a case, the quantum tunneling and the crossed Andreev reflection still dominate the whole scattering process. All these observations demonstrate that our proposed high-efficiency Cooper-pair splitter is much robust against external disorders, indicating its experimental feasibility. It should also be noted that although we have used a graphene-based QAHI as a specific example in this article, our findings can be applied to any other systems that can realize the quantum anomalous Hall effect.

Methods
In our numerical calculation, we have mainly employed the non-equilibrium Green's function technique 50 and the recursive transfer matrices method to compute various transmission coefficients in a two-terminal mesoscopic system. For example, the transmission coefficients for the quantum tunneling, the Andreev reflection, and crossed Andreev reflection are expressed as 51,52 : ee ee r ee ee a QT L R  where "e/h" is an abbreviated expression of "electron/hole", and "L/R" indicates "left/right".
1 is the retarded Green's function, where H stands for the Hamiltonian of the central region in the Nambu space, Σ / , r a L R are the self-energies of the left/right terminals and can be numerically obtained 53 , and Σ = ⁎ t g t S r r C S C is the self-energy of the superconductor terminals with g r S representing the surface Green's function of the semi-infinite superconductor terminal that equals to the bulk Green's function for the conventional s-wave superconductor 54 is the line-width function coupling the left/right semi-infinite terminal with the central scattering region. And the line-width constant of the superconductor terminal is set to be g s = 2Δ S for simplicity.
In the calculation of the spectral function in Fig. 2c, we consider an infinite long QAHI ribbon covered by the superconductor. Then the momentum k x is a good quantum number, and