Persistent spin helix on a diamond surface

Semiconductor surfaces are unstable owing to their dangling bond, resulting in chemical reactivity. To overcome this, surface termination is required. Previously, hydrogen-terminated silicon surfaces were considered for termination semiconductor surfaces by hydrogen, where a silicon substrate with a surface that has been chemically passivated has hydrogen atoms bound to it. ¹⁾ Terminating the silicon surface using hydrogen is the first step in upcoming functionalization devices considering that it is submissive to chemical modification. ²⁾ Additionally, the surface termination of semiconductors such as Ge, ³⁾ GaAs and InP, ⁴⁾ as well as diamond surfaces, ⁵⁾ have been studied.

Termination of the diamond surface strongly affects the surface properties of the diamond. Hydrogen- or oxygen-terminated diamond surfaces have been used to build diamond devices. ^6–8) In the case of a hydrogen-terminated diamond, a hydrogenated diamond increases its electrical conductivity. ^9,10) In contrast, an oxygen-terminated diamond produces non-conductive electrical properties, shows positive electron affinity, and has a larger Schottky barrier height in metal/diamond interactions compared to a hydrogenated diamond. ¹¹⁾ Furthermore, the oxygen-terminated diamond (111) surface tends to be more reactive, and the roughness ¹²⁾ remains on its surface. Therefore, the combination of hydrogen and oxygen (OH) termination may solve the oxygenated diamond roughness issues on the surface. ¹³⁾

Surface termination affects the electronic properties of materials. A hydrogenated diamond surface possesses strong Rashba-type spin–orbit coupling (SOC), which is approximately 9.74 ± 0.1 meV ¹⁴⁾ and 4.6–24.5 meV, ¹⁵⁾ which is yielded from the highly asymmetric confinement potential. Therefore, a diamond surface with hydrogen termination offers enormous promise for the investigation and use of spin transport phenomena. ¹⁵⁾ These findings guide us to study the effect of OH termination of a diamond surface on its electronic properties, especially in the SOC property.

Unlike the hydrogen-terminated (H-terminated) diamond case, which possesses Rashba-type SOC, this study demonstrated that the OH-terminated diamond yields persistent spin helix (PSH) SOC using first-principles density functional calculations. The PSH has a one-dimensional orientation of spin textures, which is predicted to induce a strongly enhanced spin relaxation time. ¹⁶⁾ We found that the coupling constant that defines the SOC strength, α_PSH, for an OH-terminated diamond, is larger than the other α_PSH values observed in the zinc-blende n-type quantum well (QW) of GaAs/AlGaAs, ^17,18) which is promising as a material for miniaturized spintronic devices.

First-principles electronic structure calculations were performed based on the density functional theory using the OpenMX code. ¹⁹⁾ We calculated the OH-terminated diamond (111) surface and focused on the top diamond surface of termination, as shown in Fig. 1(a). The OH-terminated diamond surface (111) was conducted using a slab model comprising 9 C–C (carbon and carbon) bonding layers, as shown in Fig. 1(b). The lattice constant of the diamond (111) surface hexagonal supercell a₁₁₁ equal to b₁₁₁ was set to 2.527 Å, and c₁₁₁ was 39.285 Å, which included the vacuum region to desist interactions between neighboring slabs. a₁₁₁ was calculated from ${a}_{111}=a/\sqrt{2}$ , where a is 3.574 Å, the theoretical lattice constant of the diamond. ²⁰⁾

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We applied the generalized gradient approximation ²¹⁾ as the exchange-correlation functional, with a k-point grid of 20 × 20 × 1, to discretize the first Brillouin zone in the k-space, as shown in Fig. 1(c), and a cutoff energy of 500 Rydberg for the numerical integration as well as the solution of Poisson's equation. ²²⁾ Furthermore, we applied the effective screening medium approach ²³⁾ to eliminate the dipole–dipole interaction between slab models. The bottom side, hydrogen and four carbon atoms were fixed during the geometrical optimization to maintain the bulk structure, until the total force working on each atom was less than 0.005 eV/Å. The linear combination of multiple pseudo-atomic orbitals ^24)–26) was used to expand the wave functions for each atom, defined as C6.0-s3p2d2, H7.0-s2p2d1 and O7.0-s3p2d2. For example, C6.0-s3p2d2 means the cutoff radius of the carbon atom is 6.0 Bohr, and three primitive orbitals for s and two primitive orbitals for the p and d components are used. We used the norm-conserving pseudopotentials including the 2s and 2p electrons for C and O and the 1s electron for H. Through the fully relativistic total-angular-momentum-dependent pseudopotentials ²⁷⁾ the SOC was incorporated in this computation, and the spin textures in k-space were derived using the k-space spin density matrix of the spinor wave function. ^28,29)

In our OH-terminated diamond (111) surface system, as shown in Fig. 1(b), the bond between the carbon atom and oxygen was 1.425 Å, which was slightly larger than that in the previous work (1.39 Å). ³⁰⁾ The bond between the oxygen and hydrogen atoms was 0.99 Å, which was slightly smaller than that in the previous study (1.01 Å). ³⁰⁾ The bonds between the carbon atoms were between 1.526 and 1.550 Å, which was consistent with the previous theoretical work (between 1.51 and 1.55 Å). ³⁰⁾ On the bottom side, the hydrogenated one, the calculated distance between the hydrogen and carbon atoms was 1.11 Å, which was consistent with that in the previous theoretical study work, 1.11 Å. ³⁰⁾

We calculated the energy band dispersion, as shown in Fig. 2(a). The purple and black lines indicate the energy bands of the diamond surface (111) with OH termination and using a supercell, respectively. The energy gap for these systems was 2.94 eV at the Γ point, direct band gap, for the OH-terminated diamond (111) surface system, and 4.4 eV for the bulk diamond (111) system. The OH termination yielded a narrower band gap owing to several bands within the bulk band gap. The occupied surface state was identified by the 2p orbital of an oxygen atom, based on the partial density of states, as shown in Fig. 2(b).

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To understand the SOC in the OH-terminated diamond (111) surface, we first considered the Hamiltonian that describes the Rashba effect ³¹⁾ $H=\tfrac{{{\hslash }}^{2}}{2m}{k}^{2}+{H}_{R}$ in the 2DEG, where ℏ is the Planck constant, and m is the effective mass of electrons. The Rashba Hamiltonian is H_R = α_R(k_x σ_y − k_y σ_x ), where α_R satisfies α_R = 2E_R /k_R , k_x and k_y are wave vector components in the x and y directions, and σ_x and σ_y are the Pauli matrix vectors. The Rashba parameter α_R suggested by a theoretical work ³²⁾ can be written as ${\alpha }_{R}=\tfrac{{{\hslash }}^{2}}{4{m}^{2}{c}^{2}}\int {dz}| \phi (z){| }^{2}{\partial }_{z}V(z)$, where c is the velocity of light, ϕ(z) is the surface wave function, and V(z) the crystal potential averaged for the xy plane. The light elements can produce strong SOC with a strong electric field and charge asymmetry as presented in the previous theoretical works on graphene ³³⁾ and SrTiO₃(001). ³⁴⁾

In the OH-terminated diamond (111) surface system, the OH termination induced the in-plane electric field, resulting in electric polarization along the y-direction. There is a mirror plane at the yz plane. ³⁵⁾ To comprehend the origin of the spin texture, we considered the SOC of surface states based on the system's symmetry. Applying mirror reflection symmetry operation to this material resulting in transformation from (x, y, z) to (−x, y, z), the Pauli matrices σ_x , σ_y , σ_z transforms to σ_x , −σ_y , −σ_z . We should always remember that the symmetry operation should not alter the Hamiltonian (see supplementary data, Sect. 2). By keeping R⁻¹ HR = H, the SOC Hamiltonian can be given as H_SOC = α₁ k_x σ_z + α₂ k_x σ_y + α₃ k_y σ_x , where k_x and k_y are wave vectors in the x and y directions, respectively, σ_x , σ_y and σ_z are Pauli matrices, and α₁, α₂, α₃ are coupling constants that define the SOC strength. Here, α₁ is characterized by the in-plane electric field E_y , whereas α₂ and α₃ relate to the out-of-plane electric field E_z that is owing to the surface effect. The expression of H_SOC is the same as in the case of a ZnO(10 $\overline{1}$ 0) surface. ³⁵⁾

The SOC effect on the surface state of the OH-terminated diamond (111) surface is shown in Fig. 2(c). For the Γ–M direction, k_x ≈ 0, and H_SOC = α₃ k_y σ_x . Because the calculated bands are degenerated on the Γ–M line, α₃ ≈ 0. For the Γ–K' direction, we found that the bands were splitting, as shown in Fig. 2(c). Along this direction, k_y = 0, and hence, only the first and second terms remain in the spin–orbit Hamiltonian, α₁ k_x σ_z + α₂ k_x σ_y . Consequently, the expected spin value of the spin component mostly comprised 〈S_y 〉 and 〈S_z 〉 as shown in Fig. 3(b). The SOC strength parameter ${\alpha }_{{\rm{\Gamma }}-K^{\prime} }$ can be evaluated in the same way as for the Rashba system: ${\alpha }_{{\rm{\Gamma }}-K^{\prime} }=2{\rm{\Delta }}{E}_{{\rm{\Gamma }}-K^{\prime} }/{\rm{\Delta }}{k}_{{\rm{\Gamma }}-K^{\prime} }$. Through the band dispersion method, as shown in Fig. 2(c), we get ${\alpha }_{{\rm{\Gamma }}-K^{\prime} }$ is 14.2 meV·Å. This is larger than other PSH coefficients observed in GaAs/AlGaAs, which were 3.5 to 4.9 meV·Å ¹⁷⁾ and 2.77 meV·Å. ¹⁸⁾ In addition, the α_PSH of the OH-terminated diamond is significantly larger than the Rashba parameter α_R of the H-terminated diamond. The calculated results of the H-terminated diamond are available in Sect. 3 of the supplementary material.

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We calculated the spin texture projected to the k_x –k_y plane and analyzed the spin texture as shown in Fig. 3(a) in the k-space at two meV below the degenerate point. We found the unidirectional spin texture in the k-space at the band around the VBM at the Γ point. Contrary directions can be seen in the spin textures from the inner and outer Fermi arcs, as shown in Fig. 3(a). Except for the Fermi arc surrounding the degenerate points at θ = π/2 and θ = –π/2, the spin components 〈S_x 〉 and 〈S_z 〉 were nearly constant. Performing the global spin rotation U, ¹⁶⁾ when α₁ is equal to α₂, we could perform the global spin rotation π/4 around the x-axis ³⁶⁾ to the present spin Hamiltonian using $U=\left(\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right)\cos (\pi /8)+i{\sigma }_{x}\sin (\pi /8)$, where σ_x is the Pauli matrix. Therefore, ${U}^{\dagger }\tfrac{1}{\sqrt{2}}({\sigma }_{y}+{\sigma }_{z})U={\sigma }_{z}.$ This shows that OH-terminated diamond (111) surfaces correspond to the spin SU(2) symmetry. The spin textures of the surface states at the VBM are shown in Fig. 3(a). The spin textures exhibited a quasi-one-dimensional orientation in the in-plane y-direction, as shown in Fig. 3(a), despite possessing an out-of-plane component, as shown in Fig. 3(b), thereby indicating the formation of PSH. ¹⁶⁾ The PSH state, having unidirectionally oriented spin components, yielded a long spin lifetime. ¹⁶⁾ Yoshida et al. experimentally observed the OH-terminated diamond (111) surfaces by Fourier transform infrared spectroscopy. ¹³⁾

Because H_SOC is mostly affected by the electric field, the origin of the spin textures was further clarified by evaluating the electric polarization. As shown in Fig. 3(c), given that the spin-split surface state is strongly localized in the first two bilayers, a strong electric polarization is expected to occur in the C–C–OH bilayer, as shown in Fig. 3(d). Here, the first bilayer comprising H and O atoms has strong polarity. Moreover, the expected values of spin are dominated by the oxygen atom compared to the hydrogen atom. Furthermore, the oxygen atoms dominated the occupied surface state, as shown in the partial density of the state in Fig. 2(b).

In summary, we performed first-principles density functional calculations for the OH-terminated diamond (111) surface and demonstrated that the spin textures have a quasi-one-dimensional orientation. Furthermore, we found that the persistent spin helix occurred owing to the OH termination on the surface of the diamond (111). The coefficient PSH α_PSH of the OH-terminated diamond in the Γ–K' direction of 14.2 meV·Å was larger than that observed in the zinc-blende n-type QW structure of GaAs/AlGaAs, of 3.5 to 4.9 meV·Å ¹⁷⁾ and 2.77 meV·Å. ¹⁸⁾ These findings suggest that OH-terminated diamonds with in-plane electric polarization and mirror symmetry can be used to create a persistent spin helix state, which is a promising technique for spintronic devices.

Acknowledgments