Supporting Information Valence Band Splitting on Multilayer MoS2 ...
Supporting Information
Valence Band Splitting on Multilayer MoS2: Mixing of Spin-Orbit Coupling and Interlayer Coupling Xiaofeng Fana,*, David J. Singhb and Weitao Zhenga,† a. College of Materials Science and Engineering, Jilin University, Changchun 130012, China b. Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211-7010, USA
*, † Correspondence and requests for materials should be addressed, E-mail: [email protected] (X. F. Fan); [email protected] (W.T. Zheng)
Fig. S1
Fig. S1 Band structures of single-layer MoS2 calculated without spin-orbit coupling (a) and with spin-orbit coupling.
Fig. S2
1
Fig. S2 The isosurface of band-decomposed charge density of state at valance band maximum of K of single-layer MoS2 without the consideration of spin-orbit coupling (a) and that of spin-up state (b) and spin-down state (c) at valance band maximum of K of single-layer MoS2 calculated with spin-orbit coupling.
Fig. S3
Fig. S3 The isosurface of band-decomposed charge density of two states (a and b) at valance band maximum of K of bulk MoS2 without the consideration of spin-orbit coupling. The energies of both states are splitted by layer’s coupling, therefore the change distributions of bonding-like state (b) and anti-bonding-like state (a) are similar since the coupling of both states from first-layer and second layer is weak.
Fig. S4
Fig. S4 The isosurface of band-decomposed charge density of fourth states at valance band maximum of K point of bulk MoS2 calculated with spin-orbit coupling including the states ~|1↑〉 (a) and~|2↓〉 (b) from upper main band and states ~|1↓〉 (c) and ~|2↑〉 (d) from lower main band.
2
Fig. S5
Fig. S5 The isosurface of band-decomposed charge density of six states at valance band maximum of K point of triple-layer MoS2 including the states from upper main band (~|1↑〉~|3↑〉-1, ~|2↓〉, ~|1↑〉~|3↑〉-2) and that from lower main band ( ~|1↓〉~|3 ↓〉-1, ~|1↓〉~|3 ↓〉-2 and ~|2↑〉). Note that the states (~|1↑〉~|3↑〉-1 and (~|1↑〉~|3↑〉-2 is named as (~|1↑〉 and ~|3↑〉 and the states ~|1↓〉~|3 ↓〉-1 and ~|1↓〉~|3 ↓〉-2 is named as ~|1↓〉 and ~|3 ↓〉 in the text, respectively. The both main bands are defined by the intra-layer spin-orbit coupling.
Model Hamiltonian without spin-orbit coupling Without the consideration of spin-orbit coupling, the model Hamiltonian for double-layer
MoS2 can be expressed as,
E (k ) M 1 ( k ) H (k ) = ∗ . M 1 (k ) E (k )
(S1)
For triple-layer MoS2, the model Hamiltonian can be expressed as,
E (k ) M 1 (k ) M 2 (k ) H (k ) = M 1∗ (k ) E (k ) M 1 (k ) . M ∗ ( k ) M ∗ (k ) E ( k ) 1 2
(S2)
The model Hamiltonian is used to analyze the splitting of band splitting near band gap and understand the layer’s coupling. M1(k) and M2(k) represent the coupling parameters of nearest-neighbor and second near-neighbor layer-coupling with same k point and same
3
energy level, respectively. As shown in Table S1, it is obvious that the values of band splitting from model Hamiltonian are consistent with that from DFT without considering spin-orbit coupling. Table S1. Band splitting values of the top of valance bands at Γ and K points (VB-Γ and VB-K) and the bottom of conduction bands at Λ points (CB-Λ) for multi-layer MoS2 with 2L and 3L, calculated by DFT and fitted by theoretical model Hamiltonian (TM). Note that ∆1 and ∆2 represent the splitting values between the nearest-neighbor energy levels and ∆ is for total splitting value.
Num. of layers 2L ∆ M1 Bulk ∆ M1 3L ∆1 ∆2 ∆ M1 M2 M2/M1
VB-Γ(eV) DFT TM 0.6180
0.2930 0.5020 0.7952
CB-Λ(eV) DFT TM
0.6180
0.352 0.3520
0.3090
0.1760
0.2926 0.5026 0.7952 0.2800 0.0700 0.2500
0.2410 0.2250 0.4660
0.2409 0.2247 0.4656 0.1646 0.0058 0.0352
VB-K(10-3eV) DFT TM 73.8
73.8 36.9
146.0
146.0 73.0
49.0 55.0 104.0
49.3 55.3 104.7 37.0 2.0
0.0541
Model Hamiltonian with spin-orbit coupling With the consideration of spin-orbit coupling, the model Hamiltonian for double-layer
MoS2 can be expressed as,
M 1 (k ) E (k ) H (k ) = ∗ , M 1 (k ) E (k )-E 0 (k )
(S3)
where E0(k) is the value of band splitting due to the contribution of spin-orbit coupling.
In Table S2, it is obvious that the values of band splitting from model Hamiltonian are consistent with that from DFT. From the results of model Hamiltonian, the band splitting due to spin-orbit coupling in double-layer MoS2 is same with that in single-layer MoS2. The parameter M1 of layer-coupling is also similar to that without spin-orbit coupling in Table S1. 4
Table S2. Band splitting values of the top of VB-Γ, CB-Λ and VB-K for 1L and 2L MoS2, calculated by DFT and fitted by theoretical model Hamiltonian (TM).
Num. of layers 1L ∆ 2L ∆ M1 E0 Bulk ∆ M1 E0
VB-Γ(eV) DFT TM 0 0.6180
CB-Λ(eV) DFT TM 0.079
0.6180 0.3090 0
0.3590
VB-K(10-3 eV) DFT TM 149
0.3590
166.0
0.1752 0.079
166.0 36.6 149.0
208.0
208.6 73.0 149.0
It is proposed that this kind of model Hamiltonian can be expanded to analyze the
splitting of valance bands at K point in triple-layer MoS2. As shown in Fig. S5, the bands are separated into two channels including spin-up and spin-down. For each channel, the model Hamiltonian can be used to analyze the bands splitting due to the interlayer coupling. For spin-up channel, the model Hamiltonian is expressed as,
E (k ) − E0 H (k ) = M 1∗ (k ) M ∗ (k ) 2
M 2 (k ) E (k ) M 1 (k ) , M 1∗ (k ) E (k ) − E0 M 1 (k )
and the model Hamiltonian of spin-down channel is expressed as,
M 1 (k ) E (k ) ∗ H (k ) = M 1 (k ) E (k ) − E0 M ∗ (k ) M 1∗ (k ) 2
M 2 (k ) M 1 (k ) . E (k )
The results are shown in Table S3. It is obvious that the values of band splitting from model
Hamiltonian are consistent with that from DFT for the triple-layer MoS2 under 0 Gap and 15 GPa. After the combination between spin-up and spin-down channel, the splittings of nearest-neighbor energy levels are expressed as, ∆E1= E1d -E1u, ∆E2= E2u -E1d, ∆E3= E2d -E2u , ∆E4= E3d -E2d, and ∆E4= E3u –E3d. In table S4, it can be found that these values 5
including that under 0 Gpa and 15 GPa from model Hamiltonian are also consistent with that from DFT. Fig. S6
Fig. S6 Schematic of the band splitting of valence band maximum at K point (VB-K) due to the spin-orbit coupling in the each layer with inter-layer coupling (LC) between layers for triple-layer MoS2. Note that ∆1u and ∆2u present the splitting between E1u and E2u and that between E2u and E3u for spin-up channel and ∆1d and ∆2d present the splitting between E1d and E2d and that between E2d and E3d for spin-down channel. Table S3. Band splitting values of the VB-K for triple-layer (3L) MoS2, calculated by DFT and fitted by theoretical model Hamiltonian (TM). Note that “u” and “p” mean spin-up and spin-down, respectively.
Num. of layers 3L, 0GPa ∆1u, ∆1d ∆2u, ∆2d ∆u, ∆d M1 M2 E1u, E1d E2u, E2d E3u, E3d 3L, 15 GPa ∆1u, ∆1d ∆2u, ∆2d ∆u, ∆d M1 M2 E1u, E1d E2u, E2d E3u, E3d
VB-K spin-up(10-3 eV) DFT TM 19.1 19.1 173.3 173.5 192.5
192.6 43.9 1.8 20.89 1.8
VB-K spin-down(10-3 eV) DFT TM 156.6 156.8 13.9 13.8 170.4
-171.7 138.6 259.4 397.9
138.7 259.3 398.0 129.0 -9.5 129.2 -9.5 -268.8
170.6 28.1 2.0 9.82 -147.0 -160.8
247.1 100.7 347.9
247.2 100.6 347.8 111.0 0.8 99.0 -148.2 -248.8
6
Table S4. Band splitting values of nearest-neighbor energy levels for the VB-K of triple-layer (3L) MoS2 after the combination between spin-up and spin-down channel, calculated by DFT and fitted by theoretical model Hamiltonian (TM) from Table S3.
3L ∆E1 ∆E2 ∆E3 ∆E4 ∆E5
0GPa (10-3 eV) DFT TM 11.3 11.1 8.0 7.8
15GPa(10-3 eV) DFT TM 27.2 30.2 111.3 108.5
148.7 13.9 10.7
135.8 100.7 22.8
148.8 13.8 10.9
138.7 100.6 20.0
Table S5 Lattice constants a and c of bulk and double-layer MoS2 calculated by PBE-D2 and vdW-DF, compared with the experimental result. a (Å)
c /2 (Å)
Bulk
3.191
6.187
Double-layer
3.191
Lattice constant of MoS2 PBE-D2
6.198 t
3.235 (3.23)
6.443 (6.3)t
vdW-DF
Bulk Double-layer
3.235
6.465
EXP
Bulk
3.160
6.197
Ref. t [Phys. Rev. Lett. 2003, 91, 126402]
Fig. S7
Fig. S7 Band structures of bulk MoS2 calculated without spin-orbit coupling (a) and with spin-orbit coupling.
7
Valence Band Splitting on Multilayer MoS2: Mixing of Spin-Orbit Coupling and Interlayer Coupling Xiaofeng Fana,*, David J. Singhb and Weitao Zhenga,† a. College of Materials Science and Engineering, Jilin University, Changchun 130012, China b. Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211-7010, USA
*, † Correspondence and requests for materials should be addressed, E-mail: [email protected] (X. F. Fan); [email protected] (W.T. Zheng)
Fig. S1
Fig. S1 Band structures of single-layer MoS2 calculated without spin-orbit coupling (a) and with spin-orbit coupling.
Fig. S2
1
Fig. S2 The isosurface of band-decomposed charge density of state at valance band maximum of K of single-layer MoS2 without the consideration of spin-orbit coupling (a) and that of spin-up state (b) and spin-down state (c) at valance band maximum of K of single-layer MoS2 calculated with spin-orbit coupling.
Fig. S3
Fig. S3 The isosurface of band-decomposed charge density of two states (a and b) at valance band maximum of K of bulk MoS2 without the consideration of spin-orbit coupling. The energies of both states are splitted by layer’s coupling, therefore the change distributions of bonding-like state (b) and anti-bonding-like state (a) are similar since the coupling of both states from first-layer and second layer is weak.
Fig. S4
Fig. S4 The isosurface of band-decomposed charge density of fourth states at valance band maximum of K point of bulk MoS2 calculated with spin-orbit coupling including the states ~|1↑〉 (a) and~|2↓〉 (b) from upper main band and states ~|1↓〉 (c) and ~|2↑〉 (d) from lower main band.
2
Fig. S5
Fig. S5 The isosurface of band-decomposed charge density of six states at valance band maximum of K point of triple-layer MoS2 including the states from upper main band (~|1↑〉~|3↑〉-1, ~|2↓〉, ~|1↑〉~|3↑〉-2) and that from lower main band ( ~|1↓〉~|3 ↓〉-1, ~|1↓〉~|3 ↓〉-2 and ~|2↑〉). Note that the states (~|1↑〉~|3↑〉-1 and (~|1↑〉~|3↑〉-2 is named as (~|1↑〉 and ~|3↑〉 and the states ~|1↓〉~|3 ↓〉-1 and ~|1↓〉~|3 ↓〉-2 is named as ~|1↓〉 and ~|3 ↓〉 in the text, respectively. The both main bands are defined by the intra-layer spin-orbit coupling.
Model Hamiltonian without spin-orbit coupling Without the consideration of spin-orbit coupling, the model Hamiltonian for double-layer
MoS2 can be expressed as,
E (k ) M 1 ( k ) H (k ) = ∗ . M 1 (k ) E (k )
(S1)
For triple-layer MoS2, the model Hamiltonian can be expressed as,
E (k ) M 1 (k ) M 2 (k ) H (k ) = M 1∗ (k ) E (k ) M 1 (k ) . M ∗ ( k ) M ∗ (k ) E ( k ) 1 2
(S2)
The model Hamiltonian is used to analyze the splitting of band splitting near band gap and understand the layer’s coupling. M1(k) and M2(k) represent the coupling parameters of nearest-neighbor and second near-neighbor layer-coupling with same k point and same
3
energy level, respectively. As shown in Table S1, it is obvious that the values of band splitting from model Hamiltonian are consistent with that from DFT without considering spin-orbit coupling. Table S1. Band splitting values of the top of valance bands at Γ and K points (VB-Γ and VB-K) and the bottom of conduction bands at Λ points (CB-Λ) for multi-layer MoS2 with 2L and 3L, calculated by DFT and fitted by theoretical model Hamiltonian (TM). Note that ∆1 and ∆2 represent the splitting values between the nearest-neighbor energy levels and ∆ is for total splitting value.
Num. of layers 2L ∆ M1 Bulk ∆ M1 3L ∆1 ∆2 ∆ M1 M2 M2/M1
VB-Γ(eV) DFT TM 0.6180
0.2930 0.5020 0.7952
CB-Λ(eV) DFT TM
0.6180
0.352 0.3520
0.3090
0.1760
0.2926 0.5026 0.7952 0.2800 0.0700 0.2500
0.2410 0.2250 0.4660
0.2409 0.2247 0.4656 0.1646 0.0058 0.0352
VB-K(10-3eV) DFT TM 73.8
73.8 36.9
146.0
146.0 73.0
49.0 55.0 104.0
49.3 55.3 104.7 37.0 2.0
0.0541
Model Hamiltonian with spin-orbit coupling With the consideration of spin-orbit coupling, the model Hamiltonian for double-layer
MoS2 can be expressed as,
M 1 (k ) E (k ) H (k ) = ∗ , M 1 (k ) E (k )-E 0 (k )
(S3)
where E0(k) is the value of band splitting due to the contribution of spin-orbit coupling.
In Table S2, it is obvious that the values of band splitting from model Hamiltonian are consistent with that from DFT. From the results of model Hamiltonian, the band splitting due to spin-orbit coupling in double-layer MoS2 is same with that in single-layer MoS2. The parameter M1 of layer-coupling is also similar to that without spin-orbit coupling in Table S1. 4
Table S2. Band splitting values of the top of VB-Γ, CB-Λ and VB-K for 1L and 2L MoS2, calculated by DFT and fitted by theoretical model Hamiltonian (TM).
Num. of layers 1L ∆ 2L ∆ M1 E0 Bulk ∆ M1 E0
VB-Γ(eV) DFT TM 0 0.6180
CB-Λ(eV) DFT TM 0.079
0.6180 0.3090 0
0.3590
VB-K(10-3 eV) DFT TM 149
0.3590
166.0
0.1752 0.079
166.0 36.6 149.0
208.0
208.6 73.0 149.0
It is proposed that this kind of model Hamiltonian can be expanded to analyze the
splitting of valance bands at K point in triple-layer MoS2. As shown in Fig. S5, the bands are separated into two channels including spin-up and spin-down. For each channel, the model Hamiltonian can be used to analyze the bands splitting due to the interlayer coupling. For spin-up channel, the model Hamiltonian is expressed as,
E (k ) − E0 H (k ) = M 1∗ (k ) M ∗ (k ) 2
M 2 (k ) E (k ) M 1 (k ) , M 1∗ (k ) E (k ) − E0 M 1 (k )
and the model Hamiltonian of spin-down channel is expressed as,
M 1 (k ) E (k ) ∗ H (k ) = M 1 (k ) E (k ) − E0 M ∗ (k ) M 1∗ (k ) 2
M 2 (k ) M 1 (k ) . E (k )
The results are shown in Table S3. It is obvious that the values of band splitting from model
Hamiltonian are consistent with that from DFT for the triple-layer MoS2 under 0 Gap and 15 GPa. After the combination between spin-up and spin-down channel, the splittings of nearest-neighbor energy levels are expressed as, ∆E1= E1d -E1u, ∆E2= E2u -E1d, ∆E3= E2d -E2u , ∆E4= E3d -E2d, and ∆E4= E3u –E3d. In table S4, it can be found that these values 5
including that under 0 Gpa and 15 GPa from model Hamiltonian are also consistent with that from DFT. Fig. S6
Fig. S6 Schematic of the band splitting of valence band maximum at K point (VB-K) due to the spin-orbit coupling in the each layer with inter-layer coupling (LC) between layers for triple-layer MoS2. Note that ∆1u and ∆2u present the splitting between E1u and E2u and that between E2u and E3u for spin-up channel and ∆1d and ∆2d present the splitting between E1d and E2d and that between E2d and E3d for spin-down channel. Table S3. Band splitting values of the VB-K for triple-layer (3L) MoS2, calculated by DFT and fitted by theoretical model Hamiltonian (TM). Note that “u” and “p” mean spin-up and spin-down, respectively.
Num. of layers 3L, 0GPa ∆1u, ∆1d ∆2u, ∆2d ∆u, ∆d M1 M2 E1u, E1d E2u, E2d E3u, E3d 3L, 15 GPa ∆1u, ∆1d ∆2u, ∆2d ∆u, ∆d M1 M2 E1u, E1d E2u, E2d E3u, E3d
VB-K spin-up(10-3 eV) DFT TM 19.1 19.1 173.3 173.5 192.5
192.6 43.9 1.8 20.89 1.8
VB-K spin-down(10-3 eV) DFT TM 156.6 156.8 13.9 13.8 170.4
-171.7 138.6 259.4 397.9
138.7 259.3 398.0 129.0 -9.5 129.2 -9.5 -268.8
170.6 28.1 2.0 9.82 -147.0 -160.8
247.1 100.7 347.9
247.2 100.6 347.8 111.0 0.8 99.0 -148.2 -248.8
6
Table S4. Band splitting values of nearest-neighbor energy levels for the VB-K of triple-layer (3L) MoS2 after the combination between spin-up and spin-down channel, calculated by DFT and fitted by theoretical model Hamiltonian (TM) from Table S3.
3L ∆E1 ∆E2 ∆E3 ∆E4 ∆E5
0GPa (10-3 eV) DFT TM 11.3 11.1 8.0 7.8
15GPa(10-3 eV) DFT TM 27.2 30.2 111.3 108.5
148.7 13.9 10.7
135.8 100.7 22.8
148.8 13.8 10.9
138.7 100.6 20.0
Table S5 Lattice constants a and c of bulk and double-layer MoS2 calculated by PBE-D2 and vdW-DF, compared with the experimental result. a (Å)
c /2 (Å)
Bulk
3.191
6.187
Double-layer
3.191
Lattice constant of MoS2 PBE-D2
6.198 t
3.235 (3.23)
6.443 (6.3)t
vdW-DF
Bulk Double-layer
3.235
6.465
EXP
Bulk
3.160
6.197
Ref. t [Phys. Rev. Lett. 2003, 91, 126402]
Fig. S7
Fig. S7 Band structures of bulk MoS2 calculated without spin-orbit coupling (a) and with spin-orbit coupling.
7
