Effect of sublattice asymmetry and spin-orbit interaction on out-of-plane spin polarization of photoelectrons

We study theoretically the effect of spin-orbit coupling and sublattice asymmetry in graphene on the spin polarization of photoelectrons. We show that sublattice asymmetry in graphene not only opens a gap in the band structure, but in the case of finite spin-orbit interaction it also gives rise to an out-of-plane spin polarization of electrons close to the Dirac point of the Brillouin zone. This can be detected by measuring the spin polarization of photoelectrons, and therefore spin-resolved photoemission spectroscopy can reveal the presence of a band gap even if it is too small to be observed directly by angle-resolved photoemission spectroscopy because of the finite resolution of measurements or because the sample is $p$-doped. We present analytical and numerical calculations on the energy and linewidth dependence of photoelectron intensity distribution and spin polarization.

Figure 1

(a) Schematic of the hexagonal Brillouin zone of graphene with the $\mathbf{\Gamma }$ point and vectors $\mathbf{K}$, ${\mathbf{K}}^{\prime }$ pointing to the corresponding corners of the Brillouin zone. The coordinate system that we use in the momentum space is also shown. (b) Schematic of the energy bands near the $K$ point of the BZ, as obtained from Eq. (3) for $\Delta =0$ (solid) and $\Delta \ne 0$ (dashed). The energy splitting between the spin-split bands is $3{\lambda }_R$ for $\Delta =0$. If $\Delta \ne 0$, a band gap of $\Delta$ opens at $\mathbf{k}=0$. The ($\mu ,\nu$) indices corresponding to a given band are also indicated. (c) Solid (dashed) line: expectation value of the $z$ component of the spin as a function of $k=\left|\mathbf{k}\right|$ (in units of the carbon-carbon bond length $a_0$) in the upper valance band ($\mu =-1$, $\nu =1$) calculated from Eq. (7) [Eq. (8)]. The interplay of sublattice asymmetry and RSOI leads to a finite $\langle S_z\rangle$. The width of the peak, as defined in the figure, is independent of the asymmetry parameter $\Delta$ [see text below Eq. (8)].

Figure 2

(a) Intensity distribution; (b) and (c) spin polarization of photoelectrons emanating from the upper valance band ($\mu =-1$, $\nu =1$) for momenta close to the $K$ point. In (b), the vector plot of the in-plane component of the spin polarization is shown. In (c), the density plot of the $z$ component of spin indicates the regions in the BZ where the out-of-plane polarization is finite. We used $\Delta =40\hspace{0.5em}\mathrm{meV}$ and ${\lambda }_R=66\hspace{0.5em}\mathrm{meV}$ in these calculations.

Figure 3

Constant energy (S)ARPES calculations close to the Dirac point. (a) and (b)Schematic band structure at the $K$ point of the BZ for zero and finite sublattice asymmetry, respectively. Dashed lines indicate a constant energy cross section at $E=-37\hspace{0.5em}\mathrm{meV}$, where the photoelectron intensity maps in (c) and (d) are obtained. (c) and (d) Constant energy intensity maps for $\Delta =0$ and $40\hspace{0.5em}\mathrm{meV}$, respectively. The dashed lines indicate the direction in $\mathbf{k}$ space along which the spin-polarization curves in (e) and (f) are calculated. (e) and (f) The $x$ (dashed), $y$ (dashed-dotted), and $z$ (solid) component of the photoelectron spin polarization for cross sections shown in (c) and (d), respectively (left axis). The dotted lines indicate the intensity profile along the same cross section (right axis). Subfigures in the left (right) column correspond to sublattice anisotropy parameter $\Delta =0$ ($\Delta =40\hspace{0.5em}\mathrm{meV}$). Other parameters of the figure are $\lambda =66\hspace{0.5em}\mathrm{meV}$ and $\Gamma =12.5\hspace{0.5em}\mathrm{\hspace{0.5em}}\mathrm{meV}$.

Figure 4

Constant energy (S)ARPES calculations for low energies and small broadening $\Gamma$. (a) and (b) Schematic band structure at the $K$ point of the BZ for zero and finite sublattice asymmetry, respectively. Dashed lines indicate a constant energy cross section at $E=-110\hspace{0.5em}\mathrm{meV}$ where the photoelectron intensity maps in (c) and (d) are obtained. (c) and (d) Constant energy intensity maps for $\Delta =0$ and $40\hspace{0.5em}\mathrm{meV}$, respectively. The dashed line at $k_y=-0.007\hspace{0.5em}\AA {}^{-1}$ ($k_y=-0.014\mathrm{\hspace{0.5em}}\mathrm{\hspace{0.5em}}\AA {}^{-1}$) indicates the direction in $\mathbf{k}$ space along which the spin-polarization curves in (e) and (f) [(g) and (h)] are calculated. (e) and (g) [(f) and (h)] The $x$ (dashed), $y$ (dashed-dotted), and $z$ (solid) component of the photoelectron spin polarization for the two cross sections shown in (c) [(d)] (left axis). The dotted lines indicate the intensity profile along the same cross section (right axis). Subfigures in the left (right) column correspond to sublattice anisotropy parameter $\Delta =0$ ($\Delta =40\hspace{0.5em}\mathrm{meV}$). Other parameters of the figure are $\lambda =66\hspace{0.5em}\mathrm{meV}$ and $\Gamma =16.7\hspace{0.5em}\mathrm{meV}$.

Figure 5

The effects of the broadening parameter $\Gamma$ on the (S)ARPES spectra. As in Fig. 4, all calculations are for $E=-110\hspace{0.5em}\mathrm{meV}$. (a) and (b) The same as in Fig. 4. (c) and (d) Constant energy intensity maps for $\Delta =0$ and $40\hspace{0.5em}\mathrm{meV}$, respectively. The dashed line at $k_y=-0.007\hspace{0.5em}\AA {}^{-1}$ ($k_y=-0.014\mathrm{\hspace{0.5em}}\mathrm{\hspace{0.5em}}\AA {}^{-1}$) indicates the direction in $\mathbf{k}$ space along which the spin polarization curves in (e) and (f) [(g) and (h)] are calculated. (e) and (g) [(f) and (g)] The $x$ (dashed), $y$ (dashed-dotted), and $z$ (solid) component of the photoelectron spin polarization for the two cross sections shown in (c) [(d)] (left axis). The dotted lines indicate the intensity profile along the same cross section (right axis). Subfigures in the left (right) column correspond to sublattice anisotropy parameter $\Delta =0$ ($\Delta =40\hspace{0.5em}\mathrm{meV}$). Other physical parameters of the figure are $\lambda =66\hspace{0.5em}\mathrm{meV}$ and $\Gamma =83.5\hspace{0.5em}\mathrm{meV}$.

Figure 6

Constant energy (S)ARPES calculations far from the Dirac point. (a) Schematic band structure at the $K$ point. The dashed line indicates a constant energy cross section at $E=-660\hspace{0.5em}\mathrm{meV}$ where the photoelectron intensity maps in (b) and (c) are obtained. (b) and (c) Constant energy intensity maps for $\Gamma =50$ and 134 meV, respectively. The dashed lines at $k_y=0.035\hspace{0.5em}\AA {}^{-1}$ ($k_y=-0.035\hspace{0.5em}\AA {}^{-1}$) indicate the direction in $\mathbf{k}$ space along which the spin polarization curves in (d) and (e) [(f) and (g)] are calculated. (d) and (f) [(e) and (g)] The $x$ (dashed), $y$ (dashed-dotted), and $z$ (solid) component of the photoelectron spin polarization for the two cross sections shown in (b) [(c)] (left axis). The dotted lines indicate the intensity profile along the same cross section (right axis).