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Even – odd layer-dependent magnetotransport of high-mobility Q-valley electrons in transition metal disulfides

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ABSTRACT

In few-layer transition metal dichalcogenides (TMDCs), the conduction bands along the ΓK directions shift downward energetically in the presence of interlayer interactions, forming six Q valleys related by threefold rotational symmetry and time reversal symmetry. In even layers, the extra inversion symmetry requires all states to be Kramers degenerate; whereas in odd layers, the intrinsic inversion asymmetry dictates the Q valleys to be spin-valley coupled. Here we report the transport characterization of prominent Shubnikov-de Hass (SdH) oscillations and the observation of the onset of quantum Hall plateaus for the Q-valley electrons in few-layer TMDCs. Universally in the SdH oscillations, we observe a valley Zeeman effect in all odd-layer TMDC devices and a spin Zeeman effect in all even-layer TMDC devices, which provide a crucial information for understanding the unique properties of multi-valley band structures of few-layer TMDCs.

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Layer-dependent spin-valley coupled Q valleys in TMDCs.(a) Calculated band structure of 3L MoS2. The bottom of conduction band is located at the Q (Q') valleys. At the edge of each Q (Q') valley, the Fermi level only crosses the lowest non-degenerate sub-band, whose spin-up and spin-down sub-bands are lifted by 4.3 meV. (b) Calculated band structure of 6L WS2. The energy bands are spin-degenerate at the edge of each Q (Q') valley. These spin-valley coupled band edges are further illustrated in (c), where the red and blue colours denote the spin-down and spin-up bands, respectively. Q1, Q2 and Q3 have the same spin, and Q1', Q2' and Q3' are their time reversals. (d) Schematic diagrams for the Bloch bands, showing the valley Zeeman effect in odd-layer devices and the spin Zeeman effect in even-layer devices. For odd-layer samples, the sub-band at Fermi level is non-degenerate at B=0; at relatively high magnetic field, the degeneracy between Q and Q' valleys is further lifted by the valley Zeeman effect. It follows that an LL sextet can be lifted into two LL triplets caused by the valley Zeeman effect. For even-layer samples, the sub-band at Fermi level is spin-degenerate at B=0; at relatively high magnetic field, the degeneracy between up and down spins is lifted by the spin Zeeman effect. It follows that an LL duodectets can be lifted into LL sextets caused by the spin Zeeman effect.
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f4: Layer-dependent spin-valley coupled Q valleys in TMDCs.(a) Calculated band structure of 3L MoS2. The bottom of conduction band is located at the Q (Q') valleys. At the edge of each Q (Q') valley, the Fermi level only crosses the lowest non-degenerate sub-band, whose spin-up and spin-down sub-bands are lifted by 4.3 meV. (b) Calculated band structure of 6L WS2. The energy bands are spin-degenerate at the edge of each Q (Q') valley. These spin-valley coupled band edges are further illustrated in (c), where the red and blue colours denote the spin-down and spin-up bands, respectively. Q1, Q2 and Q3 have the same spin, and Q1', Q2' and Q3' are their time reversals. (d) Schematic diagrams for the Bloch bands, showing the valley Zeeman effect in odd-layer devices and the spin Zeeman effect in even-layer devices. For odd-layer samples, the sub-band at Fermi level is non-degenerate at B=0; at relatively high magnetic field, the degeneracy between Q and Q' valleys is further lifted by the valley Zeeman effect. It follows that an LL sextet can be lifted into two LL triplets caused by the valley Zeeman effect. For even-layer samples, the sub-band at Fermi level is spin-degenerate at B=0; at relatively high magnetic field, the degeneracy between up and down spins is lifted by the spin Zeeman effect. It follows that an LL duodectets can be lifted into LL sextets caused by the spin Zeeman effect.

Mentions: Figure 4a,b show the calculated band structures of 3L MoS2 and 6L WS2 (see Supplementary Fig. 7 for the band structures of 3L WS2 and 6L MoS2), in which the minima of the conduction bands are not located at the K/K' points, but rather at the Q/Q' points, that is, between K(K') and Γ points, with quadratic sub-bands. As illustrated in Fig. 4c, 3 Q and 3 Q' valleys exist in the first Brillouin zone of the few-layer TMDCs. The C3 rotational symmetry dictates the threefold Q-valley degeneracy. For even-layer TMDCs, the Q and Q' valleys are further related by both time reversal and spatial inversion symmetries, which require Kramers degeneracy. Consider the low carrier density in our 6L WS2 device, the Fermi energy is ∼2.9 meV above the valley edge and crosses only one spin-degenerate sub-band at each valley (see the inset of Fig. 4b). Thus, in the SdH oscillations, we observe a twelve-fold LL degeneracy at low fields and sixfold LL degeneracy at high fields caused by the spin Zeeman splitting within each valley, that is, between /Q↑> and /Q↓> states (see Fig. 4d). The valley Zeeman effect is absent because of the inversion symmetry.


Even – odd layer-dependent magnetotransport of high-mobility Q-valley electrons in transition metal disulfides
Layer-dependent spin-valley coupled Q valleys in TMDCs.(a) Calculated band structure of 3L MoS2. The bottom of conduction band is located at the Q (Q') valleys. At the edge of each Q (Q') valley, the Fermi level only crosses the lowest non-degenerate sub-band, whose spin-up and spin-down sub-bands are lifted by 4.3 meV. (b) Calculated band structure of 6L WS2. The energy bands are spin-degenerate at the edge of each Q (Q') valley. These spin-valley coupled band edges are further illustrated in (c), where the red and blue colours denote the spin-down and spin-up bands, respectively. Q1, Q2 and Q3 have the same spin, and Q1', Q2' and Q3' are their time reversals. (d) Schematic diagrams for the Bloch bands, showing the valley Zeeman effect in odd-layer devices and the spin Zeeman effect in even-layer devices. For odd-layer samples, the sub-band at Fermi level is non-degenerate at B=0; at relatively high magnetic field, the degeneracy between Q and Q' valleys is further lifted by the valley Zeeman effect. It follows that an LL sextet can be lifted into two LL triplets caused by the valley Zeeman effect. For even-layer samples, the sub-band at Fermi level is spin-degenerate at B=0; at relatively high magnetic field, the degeneracy between up and down spins is lifted by the spin Zeeman effect. It follows that an LL duodectets can be lifted into LL sextets caused by the spin Zeeman effect.
© Copyright Policy - open-access
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC5036047&req=5

f4: Layer-dependent spin-valley coupled Q valleys in TMDCs.(a) Calculated band structure of 3L MoS2. The bottom of conduction band is located at the Q (Q') valleys. At the edge of each Q (Q') valley, the Fermi level only crosses the lowest non-degenerate sub-band, whose spin-up and spin-down sub-bands are lifted by 4.3 meV. (b) Calculated band structure of 6L WS2. The energy bands are spin-degenerate at the edge of each Q (Q') valley. These spin-valley coupled band edges are further illustrated in (c), where the red and blue colours denote the spin-down and spin-up bands, respectively. Q1, Q2 and Q3 have the same spin, and Q1', Q2' and Q3' are their time reversals. (d) Schematic diagrams for the Bloch bands, showing the valley Zeeman effect in odd-layer devices and the spin Zeeman effect in even-layer devices. For odd-layer samples, the sub-band at Fermi level is non-degenerate at B=0; at relatively high magnetic field, the degeneracy between Q and Q' valleys is further lifted by the valley Zeeman effect. It follows that an LL sextet can be lifted into two LL triplets caused by the valley Zeeman effect. For even-layer samples, the sub-band at Fermi level is spin-degenerate at B=0; at relatively high magnetic field, the degeneracy between up and down spins is lifted by the spin Zeeman effect. It follows that an LL duodectets can be lifted into LL sextets caused by the spin Zeeman effect.
Mentions: Figure 4a,b show the calculated band structures of 3L MoS2 and 6L WS2 (see Supplementary Fig. 7 for the band structures of 3L WS2 and 6L MoS2), in which the minima of the conduction bands are not located at the K/K' points, but rather at the Q/Q' points, that is, between K(K') and Γ points, with quadratic sub-bands. As illustrated in Fig. 4c, 3 Q and 3 Q' valleys exist in the first Brillouin zone of the few-layer TMDCs. The C3 rotational symmetry dictates the threefold Q-valley degeneracy. For even-layer TMDCs, the Q and Q' valleys are further related by both time reversal and spatial inversion symmetries, which require Kramers degeneracy. Consider the low carrier density in our 6L WS2 device, the Fermi energy is ∼2.9 meV above the valley edge and crosses only one spin-degenerate sub-band at each valley (see the inset of Fig. 4b). Thus, in the SdH oscillations, we observe a twelve-fold LL degeneracy at low fields and sixfold LL degeneracy at high fields caused by the spin Zeeman splitting within each valley, that is, between /Q↑> and /Q↓> states (see Fig. 4d). The valley Zeeman effect is absent because of the inversion symmetry.

View Article: PubMed Central - PubMed

ABSTRACT

In few-layer transition metal dichalcogenides (TMDCs), the conduction bands along the ΓK directions shift downward energetically in the presence of interlayer interactions, forming six Q valleys related by threefold rotational symmetry and time reversal symmetry. In even layers, the extra inversion symmetry requires all states to be Kramers degenerate; whereas in odd layers, the intrinsic inversion asymmetry dictates the Q valleys to be spin-valley coupled. Here we report the transport characterization of prominent Shubnikov-de Hass (SdH) oscillations and the observation of the onset of quantum Hall plateaus for the Q-valley electrons in few-layer TMDCs. Universally in the SdH oscillations, we observe a valley Zeeman effect in all odd-layer TMDC devices and a spin Zeeman effect in all even-layer TMDC devices, which provide a crucial information for understanding the unique properties of multi-valley band structures of few-layer TMDCs.

No MeSH data available.


Related in: MedlinePlus