JP6478187B2 - Honeycomb lattice-type material that has a Kekule superlattice structure and exhibits giant effective spin-orbit interaction and topological state - Google Patents

Description

  The present invention relates to a material that develops a topological state, and more particularly, to the transition coefficient of electrons between adjacent lattice points on a honeycomb lattice.

The present invention relates to a honeycomb lattice structure material that induces a giant effective spin-orbit interaction by a superlattice structure (hereinafter referred to as Kekule) and thereby develops a stable topological state.

  With the discovery of the quantum Hall effect (QHE), new progress in physical property research centered on topology has been made (Non-Patent Documents 1 to 11). The topological state is not only interesting from an academic point of view, but is also expected to have a significant impact on applications. This is because the strong surface (or edge) state protected by the bulk topology provides new possibilities for spintronics and quantum computation (Non-Patent Documents 12 to 17).

  However, only a few substances have been confirmed to have a topological state so far, and most of them exhibit topological properties only at very low temperatures. This problem hindered detailed research and manipulation of materials essential for practical applications.

Graphene's unique physical properties are attracting a great deal of attention, and it has been widely studied from the search for basic physical properties to the development of new devices. The characteristic of graphene is the Dirac energy dispersion of carbon π electrons. In other words, the momentum dependence of the electron energy is linear at the Fermi level. This is determined by the C ₃ symmetry honeycomb lattice.

  It was understood that the topological state can be realized using the Dirac dispersion relation of electrons on the honeycomb lattice, which led to the flourishing research of topological materials including topological insulators. In Non-Patent Document 6, Haldane added a so-called staggered flux term accompanying an electron transition between adjacent sites to the electron tight binding model on the honeycomb lattice. As a result, Dirac electrons have mass, the Berry phases at the K and K ′ points in the corner of the Brilliant zone are aligned, and a quantum anomalous Hall effect is produced. Kane and Mele considered the spin degree of freedom of electrons in Non-Patent Document 7 and clarified that the spin-orbit interaction accompanying the electron transition between the next adjacent sites can be aligned with the Berry phase of the K point and the K ′ point. In this case, the Berry phase of the spin upward and downward electrons is opposite, the time reversal symmetry is satisfied, and the quantum spin Hall effect appears.

  However, the spin-orbit interaction of graphene is very small, so it is very difficult to experimentally observe the quantum spin Hall effect. On the other hand, the alternating flux can be realized by irradiation with circularly polarized light, but it is disadvantageous for realizing the device.

  The problem of the present invention is that the introduction of the Kekule superstructure of the transition energy of electrons on the honeycomb lattice in which atoms are arranged in a honeycomb shape makes it possible to stabilize the giant effective spin-orbit interaction and stability compared to the conventionally proposed materials. The object is to provide a material that develops a topological state.

According to one aspect of the present invention, there is a honeycomb lattice type material in which atoms are arranged in a honeycomb shape, the electron transition energy with the nearest site has a Kekule superstructure, and the electron transition energy inside the six-membered ring A honeycomb lattice-type material having a Kekule superlattice, where t ₀ is smaller than the electronic transition energy t ₁ between the nearest six-membered rings and exhibiting giant effective spin-orbit interaction and topological states is provided.
Here, the Kekule superstructure may be introduced by arranging CO molecules on the Cu [111] plane and introducing extra CO molecules into some of the six-membered rings.

According to one aspect of the present invention, there is a honeycomb lattice-type material in which atoms are arranged in a honeycomb shape, the electron transition energy with the nearest site has a Kekule superstructure, and the inside of the six-membered ring of the Kekule superstructure Honeycomb lattice-type material having a Kekule superlattice that exhibits a large effective spin-orbit interaction and a topological state, in which the electron transition energy t ₀ is smaller than the electron transition energy t ₁ superstructure between adjacent six-membered rings Is given.
Here, the Kekule superstructure may be introduced by arranging CO molecules on the Cu [111] plane and introducing extra CO molecules into some of the six-membered rings.

(A) The figure which shows the honeycomb lattice adjusted so that the electronic transition energy (hopping integral) between adjacent sites may have a Kekule superstructure. t ₀ represents the electron transition energy inside the six-membered ring indicated by the black solid line in the hexagon surrounded by the broken line shown in the lower right corner of the figure, and t ₁ represents the electron transition energy between the adjacent six-membered rings. The hexagon surrounded by a broken line has a lattice vector

And the lattice constant is

Is a primitive cell of a triangular lattice. (B) The figure which shows the natural orbital (eigen orbital) of the artificial atom comprised by one 6-membered ring.
The figure which shows the dispersion relation (dispersion relation) of the lattice system which has a hexagonal unit cell. (A) when t ₁ = 0.9t ₀ , (b) when t ₁ = t ₀ and (c) when t ₁ = 1.1t ₀ . In the graphs (a) to (c), in the curves other than the convex curve at the top and the convex curve at the bottom at the top, the dark gray line and the slightly light gray line are The portions corresponding to | p _± > and | d _± >, respectively, in the vicinity of the center of (c) and appearing to be mixed, indicate the combination of the two. Here, the on-site energy was ε ₀ = 0. (A) Both sides of the ribbon-shaped system having 36 six-membered rings and a structure in which the electron transition energy t ₁ = 1.1t ₀ between the six-membered rings has 12 six-membered rings, respectively. It shows a dispersion relation of the composite system across the structure of the electronic transition energy t _{1 =} 0.9t ₀ six-membered rings. The light-colored spread curve group is for the bulk band, and the dispersion corresponding to the red curve is specific to the ribbon system. (B) A state corresponding to a slightly thick curve drawn in two dark colors intersecting at the center in (a) is localized at the two interfaces in real space. Black and white arrows indicate pseudo-spin-up and pseudo-spin-down channels, respectively. (A) A 6-terminal hole bar that measures electrical conduction in a topological state. However, the finely shaded part at the center is the topological area shown in FIG. 3, the gray part slightly lighter than the topological area is not obvious, and the current is on the left side. It is injected from the terminal and taken out from the right terminal. The distribution of current density at the hall bar is shown in shades. (B) Injection electron energy dependence of parallel direction and hole electrical conduction. However, the on-site energy of electrons was set to ε ₀ = 0. Schematic diagram of a molecular graphene system realized by decorating the Cu [111] surface electron system with a triangular lattice of CO molecules: (a) Topological phase when t ₁ > t ₀ , (b) Trivial when t ₁ <t ₀ Corresponds to the phase. Gray circles indicate CO molecules and are arranged by a technique such as STM. A thick bond is shorter than a thin bond, and the Kekule superstructure of electronic transition is realized.

In one form of the present invention, the transition energy of electrons between the nearest lattice points on the honeycomb lattice is strengthened and weakened according to a certain rule, and the Kekule transition superstructure is introduced. It shows that a stable topological state can be realized. Specifically, as shown in FIG. 1 (a), all the sites of the honeycomb lattice are grouped into a six-membered ring consisting of six nearest neighbor sites, and the electron transition energy in the six-membered ring is kept constant. Strengthens the electron transition energy between six-membered rings. By this operation, C ₆ symmetry is realized in the electron system, and accordingly, a pseudo-spin degree of freedom is generated in the electron wave function, and the quantum spin Hall effect becomes possible. In this case, the effective spin orbit interaction corresponding to the pseudo spin is several orders of magnitude larger than the intrinsic spin orbit interaction seen in graphene. Since the effective spin-orbit interaction is very large, the topological state discussed here is stable even at high temperatures.

  The starting point is a tight-binding model of electrons on the honeycomb lattice:

Here, c _i is an electron annihilation operator at site i, and ε ₀ is on-site energy. t ₀ is the energy associated with the electron transition in the six-membered ring, and t ₁ is the electron transition energy between the six-membered rings. The first sum is for all sites, the second sum is for the nearest neighbors in the six-membered ring, and the third sum is for the six-membered rings, and is for the nearest neighbor sites. To simplify the discussion, let's not consider the spin of electrons. In the following, it is considered that t ₁ is adjusted with t ₀ kept constant. Note that if t ₁ and t ₀ are different, the original honeycomb lattice is a triangular lattice with a six-membered ring as a unit. The six-membered ring may conveniently be regarded as one of the artificial atoms, the symmetry of the lattice to make this case the artificial atom is C _6.

  First, consider a single six-membered ring. The Hamiltonian

Given in. Where vector

Is composed of a six-site electron annihilation operator, and n is a six-membered ring number. The Hamiltonian eigenvector of equation (2) is

Given by the respective energy 2,1,1 out of order, -1, -1, which is -2, where the unit is _{t 0.} It is assumed that normalization of eigenstates is tolerated. The state of Equation (3) may be considered as an artificial lattice trajectory, and its shape is shown in FIG. It can be seen that the second to fourth orbits are p-orbit and d-orbit.

By direct diagonalization in the momentum space, the energy dispersion relationship of the system described by equation (1) can be calculated. However, ε ₀ is zero here. FIG. 2A shows the band dispersion relationship calculated when t ₁ = 0.9t ₀ . First, the band has two double degeneracy at the Γ point. This corresponds to the two two-dimensional irreducible expressions of the C ₆ point group.

  When the calculated wave function is projected onto the artificial molecular orbital shown in FIG. 1 (b), it can be seen from the gray scale expression of the dispersion relation shown in FIG. 2 (a) (the original drawing is expressed in color). In addition, the highest energy state in the valence band is d orbital, and the lowest energy state in the conduction electron band is p orbital. This energy sequence is the same as that of an isolated six-membered ring artificial atom, suggesting that the band insulator in this case is continuously connected to the isolated artificial atom.

When t ₁ is increased, the band gap in FIG. When t ₁ = t ₀ , as shown in FIG. 2B, the band gap is closed, the d orbit and the p orbit degenerate at the Γ point, and linear energy dispersion is observed. This doubly degenerated Dirac cone originates from Dirac cones at the K and K ′ points of the honeycomb lattice. At t ₁ = t ₀ , the system returns to the honeycomb lattice and the real cell unit cell is a rhombus composed of two sites. On the other hand, since the unit cell in the real space corresponding to the band structure shown in FIG. 2 is a six-membered ring, the Brilliant zone is convoluted, and the K and K ′ points of the honeycomb lattice become the Γ points in FIG. Become.

When t ₁ is further increased, the band gap at the Γ point opens again. FIG. 2C shows the result when t ₁ = 1.1t ₀ . As can be seen from the density expression of band dispersion in FIG. 2C, the p-orbital component is dominant in the valence band having the highest energy at the Γ point, and the conduction electron band having the lowest energy is the d-orbital. When moving away from the Γ point, the orbital component of the band is the same as in FIG. That is, the band inversion between the p orbit and the d orbit occurs. Since the p-orbit and d-orbital exhibit diametrically opposite parity for the spatial inversion operation, the band inversion can result in a topological state. Regarding this, see also the non-patent document 18 and the BHZ model (non-patent document 8) which are the papers of the present inventor.

  To explain the non-trivial topology, we discuss the pseudo-time reversal symmetry of this system. Pseudo-time reversal symmetric operator

Is the multiple conjugate operator

And operators

Can be synthesized as follows:

However,

Is a Pauli matrix. The important nature is

Corresponds to an operator of p / 2 rotation for the p orbit and p / 4 rotation for the d orbit. For this reason, in the space spanned by p orbit and d orbit

become. This

Can be proved easily. This anti-unitary operator has the same properties as the time-reversal symmetric operator of the electron system considering the spin degree of freedom. In fact, the Kramers pair associated with this pseudo-time-reversal symmetric operator is the opposite state of orbital angular momentum.

Can be easily confirmed.

By considering a six-membered ring artificial atom consisting of the six adjacent sites on the honeycomb lattice, atomic orbitals with angular momentum are created from only the carbon π electrons, and these are created under C ₆ symmetry. It becomes pseudo-spin degree of freedom.

The basis [p ₊ , d ₊ , p _e , d ₋ ] ^T composed of atomic orbits with pseudo-spins in equation (4), k · p model at Γ point written by ^T

Can be used to understand the behavior of FIG. Where H ₊ and H ₋ are respectively

as well as

Given in. Also, δt = t ₁ -t ₀ , a and b are real numbers of about 1,

Is a zero matrix with 2 rows and 2 examples (Non-patent Document 18). In the case of FIG. 2C, since δt> 0 is satisfied, band inversion occurs and a topological state characterized by a Z ₂ topological invariant appears (Non-patent Document 7, Non-patent Document 18, Non-patent Document 22). . On the other hand, in FIG. 2A, δt <0 and the band gap is self-evident.

Importantly, the effective spin orbit interaction is λ _eSOC -δt. For example, in the case of δt = 0.1t ₀ , the effective spin orbital action is intrinsic considering that the intrinsic spin orbit interaction of graphene is λ _SOC ˜0.1 meV and the electron transition energy is t ₀ = 2.7 eV. It can be seen that it is about 3000 times the spin interaction.

  The effective spin-orbit interaction is based on the electron transition energy, whereas the intrinsic spin-orbit interaction is a relativistic effect and generally has a small value. The ability to enable a huge effective spin orbit interaction is the most excellent characteristic of the present invention, whereby the size of the topological band gap corresponds to a temperature of several thousand degrees.

For characterizing better characterized topological _state, the ribbon system becomes t 1 = 1.1t _0, analyzes the case of sandwiching with that of t 1 = 0.9t _0. As shown in FIG. 3, a new dispersion curve (a slightly thick dark curve) having double degeneracy in the band gap appears. Examining the wave functions of the states to which they correspond, it can be seen that the pseudo-spin is upward and downward, respectively, and is localized at the two boundaries of the ribbon system. This is nothing but the countercurrent associated with reverse spin, which is expected for a typical quantum spin Hall effect.

The edges of the ribbons varies C ₆ symmetry to C ₂ symmetry, is broken inversion symmetry between擬時. For this reason, pseudo-spin upward and downward electrons interact with each other, and the edge state dispersion has a small gap at the G point. Quantitatively, it is about 0.01 to ₀ and is not visible on the scale of FIG. This minigap causes backscattering and may hinder the ideal edge transport of the quantum spin Hall effect. To quantitatively estimate this effect, the electrical conductivity of the ribbon system shown in FIG. 3 is calculated. Therefore, using the hole bar structure shown in FIG. 4, the current I is injected from the left electrode, taken out from the right electrode, the parallel voltage V _x and the Hall voltage V _y are measured, and the parallel resistance and the Hall resistance are estimated. Theoretically, a semi-infinite electrode and a wave function in the hole scattering region are connected at the interface (Non-Patent Document 21, Non-Patent Document 22), and the Landauer-Buettiker formula (Non-Patent Document 29) is used. Conductivity of plane waves scattered by six electrodes

Calculate

  As shown in FIG. 4A, the injected current I is divided into two by the pseudospin, and the upward and downward pseudospins flow on the upper and lower edges of the hole bar, respectively. Ideally, as long as the energy of the injected electrons is set within the bulk energy gap,

Is expected (Non-Patent Document 22, Non-Patent Document 23). As in FIG. 4 (b), in among the mini gap ~0.01t _0, electric conduction deviates from the expected value. However, when the energy exceeds the minigap, G _xx and G _xy approach the expected values within several cycles of Fabry-Perot type vibration (Non-patent Document 30) with a maximum amplitude of 0.05 e ² / h. When the Fermi level is set to [0.05 t ₀ , 0.1 t ₀ ], almost perfect quantized electrical conductivity (the above equation) is realized. In this energy region, the energy dispersion in the edge state is almost linear and does not feel the existence of the mini gap.

Attempts have been made to create Dirac energy dispersion with artificial honeycomb lattices in various systems from optical lattices (Non-patent Document 24, Non-patent Document 25) to two-dimensional electron gas (Non-patent Documents 26 to 28). ing. All of these systems are promising platforms for realizing a topological state by adjusting transition energy between lattice points of a honeycomb lattice as described in the present application. In order to proceed with the discussion specifically, here, the Cu [111] surface is limited to creating a topological state by a triangular gate of CO molecules (Non-patent Document 28). If an extra CO molecule is placed in the selected place in the original molecular graphene, the CO atom cluster creates a repulsive potential locally and keeps electrons away (Non-patent Document 27). The bond length is increased, and therefore the electron transition energy is reduced. From the viewpoint of the present invention, it is very important that a large scale Kekule transition pattern has already been experimentally realized in molecular graphene by the above method (Non-patent Document 28). In the present application, CO molecule clusters are arranged according to the rule of the pattern shown in FIG. Obviously, here, compared to the original honeycomb lattice, the electron transition energy in the six-membered ring surrounding the CO molecule cluster is smaller, and that between the six-membered rings is relatively larger. Since t ₁ > t ₀ is satisfied, the topological state is realized in this case according to the theory of the present invention. On the other hand, the Kekule electron transition pattern that has been experimentally realized is shown in FIG. 5B (Non-Patent Document 28), and has a dual relationship with that of FIG. 5A, and t ₁ <t ₀ is satisfied. Therefore, the system becomes a topologically obvious state.

It is possible to estimate the magnitude of the spin-orbit interaction corresponding to the pseudo-spin degree of freedom from the band gap. As can be seen from the band gap in FIG. 2 (c), in this case, the pseudo-spin orbit interaction is approximately λ _eSOC = 0.1t ₀ (eSOC is an effective spin-orbit coupling). ). Considering that t ₀ = 2.7 eV and λ _SOC = 0.1 meV in graphene, the effective pseudo-spin-orbit interaction can be several thousand times the spin-orbit interaction by modulating the electron transition energy by about 10%. Can be realized. The spin-orbit interaction is a relativistic effect and is generally weak, whereas the pseudo-spin-orbit interaction can be very large only by electrical interaction. This is one of the most important features of the present invention, whereby stable topological characteristics can be obtained even at high temperatures.

It is possible to take into account the intrinsic spin degree of freedom of electrons. Without spin-orbit interaction, the two spin channels behave independently of each other and the degree of degeneracy is twice that of the above result. If there is a spin orbit interaction, the degeneracy can be solved. For one distributed, it fell from C ₆ symmetry C ₂ symmetry ribbon edge, minigap appearing at Γ point with the擬時between inversion symmetry loss is closed. On the other hand, for the remaining two states, the minigap increases.

Here, in order to set the ratio between the energy t ₀ associated with the electron transition in the six-membered ring and the electron transition energy t ₁ between the six-membered rings to a desired value, the present invention is not limited to this. However, for example, artificial graphene was created by placing CO molecules on the Cu [111] plane, and an extra CO atom was introduced into some of the six-membered rings as shown in FIG. A technique such as introducing a Kekule superstructure of transition energy can be used (Non-patent Document 28).

  As described above in detail, according to the present invention, a huge effective spin-orbit interaction can be created simply by adjusting the electron transition energy on the honeycomb lattice in which the atoms are arranged in a honeycomb shape so as to have the Kekule superstructure. Thus, a material that exhibits a porologic state even at a much higher temperature than before can be realized. Although such a material is not limited to this, it is expected to be very useful for application to new quantum computation, for example, by coupling with spintronics or superconductivity.

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Claims (2)

A honeycomb lattice-type material in which atoms are arranged in a honeycomb shape,
The electron transition energy with the nearest site has a Kekule superstructure,
The Kekule superstructure has a Kekule superlattice in which the electron transition energy t ₀ inside the six-membered ring of the Kekule superstructure is smaller than the electron transition energy t ₁ superstructure between the nearest neighbor six-membered ring, and the giant effective spin-orbit interaction and topological A honeycomb lattice-type material that develops a state. CO molecules are arranged on the Cu [111] plane,
By introducing an extra CO molecule into some of the six-membered rings, the Kekule superstructure was introduced,
The honeycomb lattice-type material according to claim 1.