Honeycomb Dirac materials have twofold-degenerate band structures with inequivalent K and K′ valleys, which are located respectively at the corners of the hexagonal Brillouin zone (BZ).^[1–5] Valley scattering between the K and K′ is suppressed due to the large separation between the two valleys. It is fairly stable for electrons sitting in a certain valley. We believe that this feature of valley would provide a firm foundation for manipulation of the valley degree of freedom and may pave a way for the application of valleytronics in the future.

Analogous to the charge degree of freedom in the traditional microelectronics and the spin degree of freedom in spintronics, the valley degree of freedom can also transmit information, therefore, it triggers a new subject of valleytronics and paves a new way for the future information technology.^[6–12] The premise of valleytronics is getting polarized valleys. However, the K and K′ valleys are normally degenerate. So lifting the degenerated K and K′ valleys is crucial.

Many ways have been proposed to lift the degenerated valleys till now. Strong magnetic fields were reported to be able to generate valley splits.^[9–12] However, the required giant magnetic fields and tiny splits (0.1–0.2 meV/T) limit the application. Another way is the circularly polarized light method, which excites specific valleys.^[13–19] Among the honeycomb Dirac material family, graphene is a particular member and often takes the role to represent the family, due to its simple structure, unique physical properties, and the greatest potential in future applications. However, the circularly polarized light method can not be applied to graphene based systems, because it demands a particular gap that graphene does not have.

For practical application, we have proposed a general idea to obtain polarized valleys in graphene and realize valley injection into graphene through an example of the Ni/graphene heterostructure in our previous work.^[6] However, the linear dispersion of the π states of graphene is destroyed to a certain extent by the strong interaction between the graphene and Ni,^[20] which limits the injection efficiency. In this paper, we will improve our former Ni/graphene heterostructure by intercalating a monolayer Au to preserve the unique structure of the graphene, and by covering a Cu layer to fix the Dirac point around the Fermi level. Finally, we propose a new valley injector based on the Cu/graphene/Au/Ni heterojunction.

The electronic structures and transport properties are calculated by the software package NANODCAL, which combines the density functional theory and the non-equilibrium Green function.^[21,22] We use the local density approximation^[23] for the exchange correlation and the double-zeta polarized atomic orbits as the basis sets for all the atom species in all calculations. The cut-off energy for the real space grid is taken as 3000 eV and a k-point mesh of 24× 14× 1 along the X, Y, and Z directions is used for the k-sampling. The external magnetic fields are responded by the spin directions of the nickel atoms. The spin polarizations of the bulk Ni are calculated self-consistently. Convergences are considered to be achieved when the total energy, Hamiltonian matrix, and density matrix all reach the precision of 10⁻⁵ arb. units.

We start the discussion from the graphene/Au/Ni heterostructure as shown in Fig. 1(a). There are two possible configurations for the Au monolayer. The first one is reported in an early Au intercalation work, where the Au atoms form a two-dimensional hexagonal layer in commensuration with the Ni (111) substrate.^[20] In this configuration, the Au–Au bond length is 2.84 Å, which matches well with the experimental value of 2.87 Å. However, the three Au atoms in a unit cell stand respectively on the A-, B-, and H-sites of the graphene, which preserves the A--B symmetry of the graphene, making the valley resolved property disappear.

Fig. 1.

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Fig. 1. Atomic geometry and band structures of the graphene/Au/Ni (111) heterostructure. (a) Side and top views of the atomic structure of the graphene/Au/Ni (111) heterostructure. The gray, red, and blue balls respectively denote the C, Au, and Ni atoms. All of the Au atoms are on the top sites of a specific sub-lattice. Valley-resolved band structures of the graphene/Au/Ni heterostructure for M ∥ X (b) and M ∥ Z (c). The black, blue, and red dotted lines represent the orbital contributions of the p orbits of the C atoms, the 3d orbits of the Ni atoms, and 5d orbits of Au, respectively. The red rectangles in (b) and (c) are replotted in the right to show the valley splitting.

In the other possible configuration, on one side of the Au monolayer, the Au atoms form a two-dimensional epitaxial layer from the Ni (111) substrate, and on the other side, stand on a certain sub-lattice of graphene, which completely breaks the A–B symmetry, as shown in Fig. 1(a). So we consider only this configuration. The lattice constants of the graphene and the Ni (111) surface are set to be 2.46 Å, which is close to the experiment values and makes the interface almost have no mismatch. The distances between the Au layer and the graphene layer, the Ni layer and the Au layer are taken as 2.5 Å and 2.32 Å, respectively. Figures 1(b) and 1(c) plot the valley-resolved band structure of the graphene/Au/Ni heterostructure. For clarity, the Fermi energy E_F is shifted to zero and the momentum is along the G–K–K′–G direction in all of the band plots in this paper. Strong spin–orbital coupling transfers from the Au 5d orbits to the carbon 2p orbits because of the hybridization between them. As a result, the spin–orbit splitting becomes as large as 100 meV, which coincides with the experiments,^[24–26] and the valley splitting is enhanced and the maximum reaches about 50 meV, which is three times larger than that of the Ni/graphene system without the Au intercalation. Due to the screening effect of the Au atoms, the linear dispersion of the graphene is partially recovered. But the gaps, denote by Δ₁ and Δ₂ respectively, hold on at the K and K′ valleys, because of the A–B lattice symmetry breaking. When M ∥ X (in the graphene plane), Δ₁ = Δ₂ = 200 meV, and the two valleys at the K and K′ points are degenerate, as shown in Fig. 1(b). However, when M ∥ Z (perpendicular to the plane), the valley-splitting occurs with Δ₁ = 200 meV and Δ₂ = 300 meV, as shown in Fig. 1(c). This is because the valley magnetic moment only couples to the magnetic field or the magnetic moment in the perpendicular direction.^[6]

Next, we consider the effect of Au concentration on the valley splitting, since the shapes and positions of the graphene π bands are sensitive to the amount of intercalated Au atoms. At the high coverage of 1 monolayer (ML) (1 ML refers to 1 Au per Ni on the Ni (111) surface), the shapes of the Dirac cones are largely preserved. However, the conical points (defined as the midpoints of the gaps of the graphene π bands in this paper, the same as the definition in Ref. [20]) shift upward by 500 meV, indicating a large charge transfer from the graphene to the substrate, which is detrimental to the valley polarized transport because only the states near the Fermi level contribute to the current. With the reduction of the coverage of the Au atoms from 1 ML to 0.25 ML, the Dirac points gradually shift downward.^[20] But without enough Au atoms to block the hybridization between the graphene and the Ni (111) substrate, the linearity of the graphene π bands is disrupted and the valley splitting disappears at low Au coverages.

From the above discussion, we know that the Au monolayer can keep the shape of the graphene bands and increase the valley splitting. But the Dirac points are about 500 meV higher than the Fermi level. To move the Dirac point down to the Fermi surface, we try to deposit a layer of Cu atoms upon the graphene, because electrons transfer from Cu to graphene at the graphene/Cu contact.^[27] The distance between the Cu layer and the graphene layer is taken as 2.5 Å. When the Cu and Au atoms stand on the same sub-lattice of the graphene (A–A stacking), the valley splitting is only slightly smaller than that when the Cu and Au atoms on the opposite sub-lattices of the graphene (A–B stacking). So we only consider the A–B stacking case. As the black dashed lines show, in Figs. 2(a) and 2(b), the Dirac points shift to the vicinity of the Fermi level. Same as the above discussion, when we turn the magnetization direction of Ni from X to Z, the band structure changes from the valley degenerate state to the valley splitting state. The maximal valley splitting slightly reduces to 40 meV. In this Cu/graphene/Au/Ni structure, the shapes of the graphene π bands are largely preserved and the valley splitting occurs near the Fermi level, which makes it an ideal valley injector to realize valley polarized current.

Fig. 2.

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	Fig. 2. Band structures of the Cu/graphene/Au/Ni heterostructure. (a) When M ∥ X, the two valleys are degenerate. (b) When M ∥ Z, the two valleys split.

To study the valley-related transport properties of the Cu/graphene/Au/Ni heterojunction, we design a system shown in Fig. 3(a). The left lead is the Cu/graphene/Au/Ni heterojunction as a valley injector, and the right lead is an individual graphene. The two leads extend to –∞ and + ∞, respectively. The central scattering region including the heterojunction/graphene contact part has a length of 1.7 nm. The transport direction is along the armchair direction of the graphene (y direction). We compare the (E, k_x)-resolved transmission coefficients for M ∥ X in Fig. 3(b) and M ∥ Z in Fig. 3(d). In the two plots, the transmissions focus on k_x = ±1/3, which are exactly the valley positions. This is because the individual graphene in the right lead only has the states at K and K′ near E_F, thus only the carriers at the same states in the left lead are allowed to transport to the right, leading to the sharp peaks at the valley positions. In addition, when M ∥ X, the transmissions in the two valleys are completely identical. However, when M ∥ Z, they are obviously different, which realizes the valley-polarized injection.

Fig. 3.

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Fig. 3. Structure model of the transport system and the (E, kx)-resolved transmission coefficients. (a) Model of the Cu/graphene/Au/Ni heterostructure. The current direction is presented by a red arrow. (b)(d) The (E, kx)-resolved transmission coefficients for M ∥ X and M ∥ Z, respectively. (c) Band structure of the left lead. The red and the blue dotted lines in (c) and (d) represent the regions where the valley splitting occurs.

Additionally, the valley splittings as large as 40 meV occur around 200 meV and –200 meV, as shown in Fig. 3(c). Correspondingly, the transmission coefficients at the two valleys diverge exactly in the same energy regions. For example, in the red dotted-line-marked area in Fig. 3(d), located at E = 0.2–0.24 eV, the K valley has a larger transmission coefficient than the K′ valley. The non-symmetric transmission between the two valleys can be explained by the band structure of the left lead, as shown in Fig. 3(c), where in the energy range of 0.2–0.24 eV, carriers in the K valley have extra conducting modes, marked by the red dotted line, then contribute the extra transmission.

In Figs. 4(a) and 4(b), we plot the valley-polarized current under finite bias, using the non-equilibrium formula

Fig. 4.

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	Fig. 4. Valley-polarized transport properties for the system shown in Fig. 3(a). (a), (b) The valley-dependent I–V characters when M ∥ X and M ∥ Z, respectively. The inset in (b) shows the valley-injecting efficiency (η) when M ∥ X and M ∥ Z.

Based on the unique property of this heterojunction/graphene contact system that the current is valley degenerate when M ∥ X and valley polarized when M ∥ Z, we can imagine a valleytronic device. The left lead is a Cu/graphene/Au/Ni heterojunction to generate a valley-polarized current, which can be easily controlled by the magnetization direction of Ni. The right lead is a graphene to detect the valley-polarized current by the valley Hall effect that carriers at opposite valleys will acquire opposite anomalous velocities in the directions transverse to the electric field,^[8] which has been realized in graphene and MoS₂ systems.^[9,16] Thus, the valley-polarized current will induce the accumulation of different amounts of carriers with opposite valley polarizations at the opposite transverse edges of the graphene lead, then the finite valley Hall voltage appears, which can be easily detected by a voltmeter.

In the Ni/graphene heterostructure, the valley splitting is small and the π states of the graphene are destroyed because of the strong hybridization with the Ni, which is harmful for the use as a practical valley injector. By intercalating a monolayer Au, we can largely preserve the π states and enhance the valley splitting. However, the short-range graphene–Au interaction leads to charge transfer from the graphene to the Au, making a shift of the conical points with respect to the Fermi level.^[28] To offset the shift, we use a Cu monolayer covering the graphene to compensate electrons. As a result, the Cu/graphene/Au/Ni heterojunction shows perfect valley injecting property.

From the viewpoint of practical application, our device has the following advantages. Firstly, the valley energy splitting is as large as 50 meV, which enables the device to work at room temperature. Next, the graphene π bands are largely preserved, which is very important because many peculiar properties of graphene are originated from the π states. Additionally, the valley-injecting states can be controlled simply by changing the magnetization direction of Ni, and detected easily by the valley Hall effect. Because of the large separation comparable to the size of the BZ in the momentum space,^[29–31] the inter-valley scattering is strongly suppressed and the valley-injecting length is long, which means experimental realization is much possible. This valley injection concept and method could be extended to other 2D materials and might open an opportunity of room-temperature valleytronics.