Relativistic splitting of surface states at Si-terminated surfaces of the layered intermetallic compounds $R{T}_{2}{\mathrm{Si}}_{2}$ ($R$=rare earth; $T$=Ir, Rh)

Relativistic splitting of surface states at Si-terminated surfaces of the layered intermetallic compounds $R T_{2} {Si}_{2}$ ( $R$ =rare earth; $T$ =Ir, Rh)

I. A. Nechaev and E. E. Krasovskii

Phys. Rev. B 98, 245415 – Published 17 December 2018

Abstract

We present an effective model for surface states at the Si-terminated (001) surface of ternary rare earth (R) intermetallic compounds ${RT}_{2} {Si}_{2}$ with the transition metal element $T$ = Ir or Rh. The model is based on a fully ab initio derivation of the relativistic $k \cdot p$ Hamiltonian and is thereby capable of accurately reproducing the spin polarization of the spin-orbit split surface states, which is very different from the classical Rashba effect. The reliable treatment of spin in our model enables a predictive analysis of the effect of magnetic exchange interaction of the surface-state electrons with ferromagnetically ordered $4 f$ moments of the subsurface rare-earth layer in the magnetic phases of the ${RT}_{2} {Si}_{2}$ compounds.

Received 2 October 2018

DOI:https://doi.org/10.1103/PhysRevB.98.245415

Physics Subject Headings (PhySH)

Electronic structure Exchange interaction Rashba coupling Spin-orbit coupling Surface states

Intermetallic compounds Transition-metal rare-earth alloys

Ab initio calculations k dot p method

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

I. A. Nechaev¹ and E. E. Krasovskii^2,3,4

¹Centro de Física de Materiales CFM – MPC, Centro Mixto CSIC-UPV/EHU, 20018 Donostia/San Sebastián, Basque Country, Spain
²Departamento de Física de Materiales, Facultad de Ciencias Quíimicas, Universidad del Pais Vasco/Euskal Herriko Unibertsitatea, Apdo. 1072, 20080 Donostia/San Sebastián, Basque Country, Spain
³Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, 20018 Donostia/San Sebastián, Basque Country, Spain
⁴IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Basque Country, Spain

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Vol. 98, Iss. 24 — 15 December 2018

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Figure 1
(a) Upper half of the unit cell of the bulk-truncated symmetric 31-layer slab with the equivalent Si-terminated opposite surfaces and density profiles of the surface states at the $\bar{M}$ point. (b) Upper right quadrant of the surface Brillouin zone showing the $\bar{M} - \bar{X}$ line, the CEC region of k space around $\bar{M}$ , and the color code for the spin $y$ and $x$ projections. Band structure along $\bar{M} - \bar{X}$ for ${YbIr}_{2} {Si}_{2}$ (c) and ${YbRh}_{2} {Si}_{2}$ (d) by the ab initio Hamiltonian $H_{k}^{LDA}$ is shown by red ( $S_{y} < 0$ ) and blue ( $S_{y} > 0$ ) fat bands revealing the spin polarization calculated as in-plane spin projection onto the four outermost layers of the upper half of the slab. Gray lines are the band structure of the same surface in a 19-layer slab geometry (space group $P 4 / n m m$ , No. 129), which simulates the behavior of the surface state bands near the $\bar{M}$ and $\bar{X}$ points in the semi-infinite limit (see also Ref. [14]). Spin-resolved ab initio CECs for the surface states of ${YbIr}_{2} {Si}_{2}$ , (e) and (f), and ${YbRh}_{2} {Si}_{2}$ , (g) and (h), are calculated at $0.05$ , $- 0.2$ , $- 0.1$ , and $- 0.35$ eV, respectively.
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Figure 2
Band structure of the Si-terminated surface of ${YbIr}_{2} {Si}_{2}$ (a) and ${YbRh}_{2} {Si}_{2}$ (c) by the $k \cdot p$ Hamiltonian (3) shown by fat bands for the momentum k within one-fifth of the $\bar{M} - \bar{X}$ interval. The CECs at $- 0.2$ eV for ${YbIr}_{2} {Si}_{2}$ is shown in (b) and at $- 0.1$ eV for ${YbRh}_{2} {Si}_{2}$ in (d). Blue (red) fat bands show $S_{y} > 0$ ( $S_{y} < 0$ ) surface states ${\tilde{Φ}}_{k}^{λ}$ of the model. The contributions form the basis states $Φ_{n ↑ (↓)}^{+}$ with $n = 1$ , 2, and 3 are shown by brown, orange, and green fat bands, respectively. Due to the symmetry, the $S_{x}$ CECs are the $S_{y}$ ones rotated clockwise by $π / 2$ as in Figs. 1 and 1. Gray lines are the LDA bands.
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Figure 3
Model Si-terminated surface band structure of magnetic phases of $R T_{2} {Si}_{2}$ compounds with $T$ = Ir (a) and $T$ = Rh (c). The $4 f$ moments of the subsurface rare-earth ( $R$ ) atomic layer are ferromagnetically ordered along the $y$ axis. The effect of exchange (magnetic) interaction on the bands along $\bar{M} \to \bar{X}$ and corresponding CECs at $- 0.2$ eV (b) and at $- 0.1$ eV (d) is modeled by the inclusion of the trivial Zeeman term $H_{kp, Z_{0}}^{Surf}$ with $J = 40$ meV for $T = Ir$ and $J = 8$ meV for $T = Rh$ . Green solid lines in (a) are the $k \cdot p$ bands obtained with the “indirect” exchange for electrons in the Si surface state (see text).
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Figure 4
Same as in Figs. 3 and 3, but for the exchange-interaction parameter $J = 33$ meV and with green lines showing the effect of the “indirect” exchange for the Si surface state (see text).
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Figure 5
(a) Projection of the outermost three layers of the Si-terminated surface onto the transition-metal atomic plane. (b) True (folded, green) and effective (unfolded, brown) Brillouin zones corresponding to the 2D unit cells highlighted green and brown in (a), respectively. Red arrow marks the treated $\bar{M} - \bar{X}$ direction. (c) Spin-resolved contribution from the sublattices “A” and “B” of the transition-metal subsurface layer as obtained from LDA calculations for ${YbIr}_{2} {Si}_{2}$ and ${YbRh}_{2} {Si}_{2}$ . Blue (red) fat bands correspond to $S_{y} > 0$ ( $S_{y} < 0$ ) 31-layer slab states.
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Physical Review B

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Relativistic splitting of surface states at Si-terminated surfaces of the layered intermetallic compounds $R T_{2} {Si}_{2}$ ( $R$ =rare earth; $T$ =Ir, Rh)

I. A. Nechaev and E. E. Krasovskii

Phys. Rev. B 98, 245415 – Published 17 December 2018

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Physical Review B

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Relativistic splitting of surface states at Si-terminated surfaces of the layered intermetallic compounds RT2Si2 (R=rare earth; T=Ir, Rh)

I. A. Nechaev and E. E. Krasovskii

Phys. Rev. B 98, 245415 – Published 17 December 2018

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Relativistic splitting of surface states at Si-terminated surfaces of the layered intermetallic compounds $R T_{2} {Si}_{2}$ ( $R$ =rare earth; $T$ =Ir, Rh)