#
Signatures of Electric Field and Layer Separation Effects on the Spin-Valley Physics of MoSe_{2}/WSe_{2} Heterobilayers: From Energy Bands to Dipolar Excitons

^{*}

## Abstract

**:**

## 1. Introduction

## 2. General Features of MoSe${}_{2}$/WSe${}_{2}$ High-Symmetry Stackings

**k.p**framework [76,86,87,88,89,90], in which additional symmetry-breaking terms can be easily incorporated. Specifically for the K point, the irreps of the energy bands are given in Table 2. As the ${K}_{4}$ and ${K}_{5}$ irreps are complex, the irreps at −K can be found by taking the complex conjugate ${K}_{4}\to {K}_{5}={K}_{4}^{*}$, ${K}_{5}\to {K}_{4}={K}_{5}^{*}$. The top valence band at the $\mathsf{\Gamma}$ point (2-fold degenerate) belongs to the (real) irrep ${\mathsf{\Gamma}}_{4}$ of the ${C}_{3v}$ group, and the lower conduction bands at the Q point both belong to the (real) irrep ${Q}_{2}$ of the ${C}_{1}$ group. To clarify our notation, we identify the irreps by their reciprocal space point, i.e., K${}_{i}$ irreps belong to the K point, ${\mathsf{\Gamma}}_{i}$ irreps for the $\mathsf{\Gamma}$ point, and Q${}_{i}$ irreps for the Q point. We also emphasize that the irreps are obtained using the full wave function calculated within DFT, as implemented in WIEN2k [81]. All the irreps and symmetry groups discussed here follow the character tables of Ref. [91].

## 3. Spin-Valley Physics at the Band Edges

#### 3.1. Pristine Heterostructures

#### 3.2. Electric Field Dependence

#### 3.3. Interlayer Distance Variation

## 4. Low-Energy Dipolar Excitons

#### 4.1. Symmetry-Based Selection Rules

#### 4.2. Effective g-Factors

#### 4.3. Electric Field Dependence

**k.p**formalism, allowing the investigation of spin-dependent physics beyond the parabolic approximation [73,75].

#### 4.4. Interlayer Distance Variation

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Computational Details

## References

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**Figure 1.**(

**a–f**) calculated band structures, with the color-coded layer decomposition of the wave function for (

**a**) R${}_{\mathrm{h}}^{\mathrm{h}}$, (

**b**) R${}_{\mathrm{h}}^{\mathrm{M}}$, (

**c**) R${}_{\mathrm{h}}^{\mathrm{X}}$, (

**d**) H${}_{\mathrm{h}}^{\mathrm{h}}$, (

**e**) H${}_{\mathrm{h}}^{\mathrm{M}}$, and (

**f**) H${}_{\mathrm{h}}^{\mathrm{X}}$ stackings; the insets show the side-view of the crystal structures (solid lines connect the hollow position of the bottom layer to the atomic registry of the top layer); (

**g**) relevant low-energy bands and possible interlayer exciton transitions originating from the top valence bands at the $\mathsf{\Gamma}$ (v${}_{\mathsf{\Gamma}}$) and K points (v${}_{\mathrm{W}}$); the transitions involving the time-reversal partners (−K and −Q points) are not shown, for simplicity; (

**h**) schematic representation of the applied external electric field and the interlayer distance.

**Figure 2.**Spin-resolved band structures for the (

**a**) R${}_{\mathrm{h}}^{\mathrm{h}}$, (

**b**) R${}_{\mathrm{h}}^{\mathrm{M}}$, (

**c**) R${}_{\mathrm{h}}^{\mathrm{X}}$, (

**d**) H${}_{\mathrm{h}}^{\mathrm{h}}$, (

**e**) H${}_{\mathrm{h}}^{\mathrm{M}}$, and (

**f**) H${}_{\mathrm{h}}^{\mathrm{X}}$ stackings.

**Figure 3.**Energy dependence with respect to the electric field of the relevant low-energy bands (see Figure 1g) for all considered stackings: the top row, panels (

**a**–

**f**), indicates the color-coded layer decomposition of the K point bands (same color code as in Figure 1a–f); the bottom row, panels (

**g**–

**l**), indicates the color-coded spin decomposition of the K point bands (same color code as in Figure 2a–f); the insets in panels (

**g**–

**l**) show the energy difference between ${\mathrm{c}}_{\mathrm{Q}+}$ and ${\mathrm{c}}_{\mathrm{Q}-}$ bands, emphasizing an anti-crossing at larger electric fields.

**Figure 4.**Spin degree of freedom, ${S}_{z}$, of the low-energy bands as a function of the applied electric field for the studied stackings R${}_{\mathrm{h}}^{\mathrm{h}}$ (

**a**,

**g**,

**m**), R${}_{\mathrm{h}}^{\mathrm{M}}$ (

**b**,

**h**,

**n**), R${}_{\mathrm{h}}^{\mathrm{X}}$ (

**c**,

**i**,

**o**), H${}_{\mathrm{h}}^{\mathrm{h}}$ (

**d**,

**j**,

**p**), H${}_{\mathrm{h}}^{\mathrm{M}}$ (

**e**,

**k**,

**q**), and H${}_{\mathrm{h}}^{\mathrm{X}}$ (

**f**,

**l**,

**r**).

**Figure 7.**Same as Figure 3 but as a function of the interlayer distance variation.

**Figure 8.**Same as Figure 4 but as a function of the interlayer distance variation.

**Figure 9.**Same as Figure 5 but as a function of the interlayer distance variation.

**Figure 10.**Same as Figure 6 but as a function of the interlayer distance variation.

**Figure 11.**Calculated values of the electric dipole moments for (

**a**) R${}_{\mathrm{h}}^{\mathrm{h}}$, (

**b**) R${}_{\mathrm{h}}^{\mathrm{M}}$, (

**c**) R${}_{\mathrm{h}}^{\mathrm{X}}$, (

**d**) H${}_{\mathrm{h}}^{\mathrm{h}}$, (

**e**) H${}_{\mathrm{h}}^{\mathrm{M}}$, and (

**f**) H${}_{\mathrm{h}}^{\mathrm{X}}$ stackings. Calculated values of the polarizabilities for (

**g**) R${}_{\mathrm{h}}^{\mathrm{h}}$, (

**h**) R${}_{\mathrm{h}}^{\mathrm{M}}$, (

**i**) R${}_{\mathrm{h}}^{\mathrm{X}}$, (

**j**) H${}_{\mathrm{h}}^{\mathrm{h}}$, (

**k**) H${}_{\mathrm{h}}^{\mathrm{M}}$, and (

**l**) H${}_{\mathrm{h}}^{\mathrm{X}}$ stackings. The x-axis indicates the type of dipolar excitons (see Figure 1g. The values originating from ${\mathrm{c}}_{\mathrm{M}/\mathrm{Q}-}$ (${\mathrm{c}}_{\mathrm{M}/\mathrm{Q}+}$) are shown with colored (open) boxes.

**Figure 12.**Absolute value of the dipole matrix element for interlayer transitions at the K point as a function of the electric field for (

**a**) R${}_{\mathrm{h}}^{\mathrm{h}}$, (

**b**) R${}_{\mathrm{h}}^{\mathrm{M}}$, (

**c**) R${}_{\mathrm{h}}^{\mathrm{X}}$, (

**d**) H${}_{\mathrm{h}}^{\mathrm{h}}$, (

**e**) H${}_{\mathrm{h}}^{\mathrm{M}}$, and (

**f**) H${}_{\mathrm{h}}^{\mathrm{X}}$ stackings.

**Figure 13.**Calculated ratio ${\tau}_{\mathrm{rad}}\left({F}_{z}\right)/{\tau}_{\mathrm{rad}}\left(0\right)$ as a function of the electric field for (

**a**) R${}_{\mathrm{h}}^{\mathrm{h}}$, (

**b**) R${}_{\mathrm{h}}^{\mathrm{M}}$, (

**c**) R${}_{\mathrm{h}}^{\mathrm{X}}$, (

**d**) H${}_{\mathrm{h}}^{\mathrm{h}}$, (

**e**) H${}_{\mathrm{h}}^{\mathrm{M}}$, and (

**f**) H${}_{\mathrm{h}}^{\mathrm{X}}$ stackings. We use ${E}_{0}=1.35$ eV for all cases, and the calculated values of $\mu $ given in Figure 11. The contribution of $\alpha $ is neglected, as they nearly vanish for KK dipolar excitons.

**Figure 14.**Dipolar exciton g-factors as a function of the electric field for the R- and H-type systems studied for the cases (

**a**–

**f**) $\left|g\left(c\right)-g\left({\mathrm{v}}_{\mathrm{W}}\right)\right|$, (

**g**–

**l**) $\left|g\left(c\right)+g\left({\mathrm{v}}_{\mathrm{W}}\right)\right|$, (

**m**–

**r**) $\left|g\left(c\right)-g\left({\mathrm{v}}_{\mathsf{\Gamma}}\right)\right|$ and (

**s**–

**v**,

**x**,

**y**) $\left|g\left(c\right)+g\left({\mathrm{v}}_{\mathsf{\Gamma}}\right)\right|$.

**Figure 15.**Same as Figure 12 but as a function o the interlayer distance.

**Figure 16.**Same as Figure 14 but as a function of the interlayer distance.

R${}_{\mathbf{h}}^{\mathbf{h}}$ | R${}_{\mathbf{h}}^{\mathbf{M}}$ | R${}_{\mathbf{h}}^{\mathbf{X}}$ | H${}_{\mathbf{h}}^{\mathbf{h}}$ | H${}_{\mathbf{h}}^{\mathbf{M}}$ | H${}_{\mathbf{h}}^{\mathbf{X}}$ | |
---|---|---|---|---|---|---|

d (Å) | 3.7237 | 3.0803 | 3.0869 | 3.0923 | 3.6885 | 3.1833 |

Mo-W (Å) | 7.0612 | 6.6922 | 6.6985 | 6.7037 | 7.2775 | 6.5208 |

**Table 2.**Irreducible representations at the K point (${C}_{3}$ point group) for the relevant energy bands indicated in Figure 1g for all considered stackings.

R${}_{\mathbf{h}}^{\mathbf{h}}$ | R${}_{\mathbf{h}}^{\mathbf{M}}$ | R${}_{\mathbf{h}}^{\mathbf{X}}$ | H${}_{\mathbf{h}}^{\mathbf{h}}$ | H${}_{\mathbf{h}}^{\mathbf{M}}$ | H${}_{\mathbf{h}}^{\mathbf{X}}$ | |
---|---|---|---|---|---|---|

${\mathrm{c}}_{\mathrm{W}+}$ | ${K}_{5}$ | ${K}_{6}$ | ${K}_{4}$ | ${K}_{4}$ | ${K}_{6}$ | ${K}_{5}$ |

${\mathrm{c}}_{\mathrm{W}-}$ | ${K}_{4}$ | ${K}_{5}$ | ${K}_{6}$ | ${K}_{6}$ | ${K}_{5}$ | ${K}_{4}$ |

${\mathrm{c}}_{\mathrm{M}+}$ | ${K}_{4}$ | ${K}_{4}$ | ${K}_{4}$ | ${K}_{5}$ | ${K}_{5}$ | ${K}_{5}$ |

${\mathrm{c}}_{\mathrm{M}-}$ | ${K}_{5}$ | ${K}_{5}$ | ${K}_{5}$ | ${K}_{4}$ | ${K}_{4}$ | ${K}_{4}$ |

${\mathrm{v}}_{\mathrm{W}}$ | ${K}_{4}$ | ${K}_{5}$ | ${K}_{6}$ | ${K}_{6}$ | ${K}_{5}$ | ${K}_{4}$ |

${\mathrm{v}}_{\mathrm{M}}$ | ${K}_{4}$ | ${K}_{4}$ | ${K}_{4}$ | ${K}_{5}$ | ${K}_{5}$ | ${K}_{5}$ |

${\mathrm{v}}_{+}$ | ${K}_{6}$ | ${K}_{6}$ | ${K}_{5}$ | ${K}_{6}$ | ${K}_{4}$ | ${K}_{6}$ |

${\mathrm{v}}_{-}$ | ${K}_{6}$ | ${K}_{4}$ | ${K}_{6}$ | ${K}_{5}$ | ${K}_{6}$ | ${K}_{6}$ |

MoSe${}_{2}$ | WSe${}_{2}$ | |||
---|---|---|---|---|

${\mathit{S}}_{\mathit{z}}$ | ${\mathit{L}}_{\mathit{z}}$ | ${\mathit{S}}_{\mathit{z}}$ | ${\mathit{L}}_{\mathit{z}}$ | |

${\mathrm{c}}_{+}$ | $-0.99$ | $1.53$ | $0.99$ | $2.98$ |

${\mathrm{c}}_{-}$ | $1.00$ | $1.81$ | $-0.89$ | $1.88$ |

${\mathrm{v}}_{+}$ | $1.00$ | $3.94$ | $1.00$ | $5.02$ |

${\mathrm{v}}_{-}$ | $-0.99$ | $3.67$ | $-0.98$ | $4.07$ |

${\mathrm{v}}_{\mathsf{\Gamma}}$ | $0.96$ | $0.06$ | $0.87$ | $0.19$ |

${\mathrm{c}}_{\mathrm{Q}+}$ | $-1.00$ | $-0.09$ | $-0.99$ | $0.32$ |

${\mathrm{c}}_{\mathrm{Q}-}$ | $1.00$ | $-0.16$ | $0.96$ | $0.05$ |

**Table 4.**Calculated values of ${S}_{z}$ and ${L}_{z}$ for the relevant energy bands in R- and H- type stackings.

R${}_{\mathbf{h}}^{\mathbf{h}}$ | R${}_{\mathbf{h}}^{\mathbf{M}}$ | R${}_{\mathbf{h}}^{\mathbf{X}}$ | H${}_{\mathbf{h}}^{\mathbf{h}}$ | H${}_{\mathbf{h}}^{\mathbf{M}}$ | H${}_{\mathbf{h}}^{\mathbf{X}}$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{S}}_{\mathit{z}}$ | ${\mathit{L}}_{\mathit{z}}$ | ${\mathit{S}}_{\mathit{z}}$ | ${\mathit{L}}_{\mathit{z}}$ | ${\mathit{S}}_{\mathit{z}}$ | ${\mathit{L}}_{\mathit{z}}$ | ${\mathit{S}}_{\mathit{z}}$ | ${\mathit{L}}_{\mathit{z}}$ | ${\mathit{S}}_{\mathit{z}}$ | ${\mathit{L}}_{\mathit{z}}$ | ${\mathit{S}}_{\mathit{z}}$ | ${\mathit{L}}_{\mathit{z}}$ | |

${\mathrm{c}}_{\mathrm{W}+}$ | $0.97$ | $2.98$ | $0.97$ | $3.02$ | $0.97$ | $2.97$ | $0.97$ | $2.92$ | $0.97$ | $2.98$ | $0.97$ | $3.02$ |

${\mathrm{c}}_{\mathrm{W}-}$ | $-0.88$ | $1.88$ | $-0.87$ | $1.93$ | $-0.88$ | $1.88$ | $-0.85$ | $1.86$ | $-0.89$ | $1.89$ | $-0.88$ | $1.90$ |

${\mathrm{c}}_{\mathrm{M}+}$ | $-0.98$ | $1.53$ | $-0.98$ | $1.55$ | $-0.98$ | $1.54$ | $0.98$ | $-1.54$ | $0.98$ | $-1.52$ | $0.98$ | $-1.56$ |

${\mathrm{c}}_{\mathrm{M}-}$ | $1.00$ | $1.81$ | $0.99$ | $1.84$ | $1.00$ | $1.81$ | $-1.00$ | $-1.80$ | $-1.00$ | $-1.80$ | $-1.00$ | $-1.81$ |

${\mathrm{v}}_{\mathrm{W}}$ | $1.00$ | $5.02$ | $1.00$ | $4.99$ | $1.00$ | $4.98$ | $1.00$ | $4.79$ | $1.00$ | $5.01$ | $1.00$ | $5.02$ |

${\mathrm{v}}_{\mathrm{M}}$ | $1.00$ | $3.94$ | $0.99$ | $3.94$ | $1.00$ | $3.91$ | $-1.00$ | $-3.25$ | $-1.00$ | $-3.93$ | $-1.00$ | $-3.95$ |

${\mathrm{v}}_{+}$ | $-0.98$ | $4.04$ | $-0.99$ | $3.63$ | $-0.98$ | $4.01$ | $0.99$ | $-3.45$ | $-0.98$ | $4.08$ | $0.24$ | $-0.67$ |

${\mathrm{v}}_{-}$ | $-0.99$ | $3.70$ | $-0.96$ | $4.02$ | $-0.99$ | $3.62$ | $-0.98$ | $3.41$ | $0.99$ | $-3.66$ | $-0.22$ | $1.06$ |

${\mathrm{v}}_{\mathsf{\Gamma}}$ | $0.92$ | $0.09$ | $0.95$ | $0.04$ | $0.95$ | $0.05$ | $0.95$ | $0.05$ | $0.93$ | $0.08$ | $0.95$ | $0.05$ |

${\mathrm{c}}_{\mathrm{Q}+}$ | $-0.93$ | $0.11$ | $-0.91$ | $0.23$ | $-0.87$ | $0.14$ | $-0.80$ | $0.22$ | $-0.59$ | $0.12$ | $-0.69$ | $0.23$ |

${\mathrm{c}}_{\mathrm{Q}-}$ | $0.95$ | $0.06$ | $0.92$ | $0.19$ | $0.92$ | $0.09$ | $0.79$ | $0.22$ | $0.60$ | $0.17$ | $0.68$ | $0.23$ |

**Table 5.**Symmetry-based selection rules for the direct interlayer excitons at the K point in the ${C}_{3}$ symmetry group.

${\mathbf{v}}_{\mathbf{W}}\left(\mathbf{K}\right)\to {\mathbf{c}}_{\mathbf{M}-}\left(\mathbf{K}\right)$ | ${\mathbf{v}}_{\mathbf{W}}\left(\mathbf{K}\right)\to {\mathbf{c}}_{\mathbf{M}+}\left(\mathbf{K}\right)$ | |
---|---|---|

R${}_{\mathrm{h}}^{\mathrm{h}}$ | ${K}_{4}^{*}\otimes {K}_{5}={K}_{3}\Rightarrow {\sigma}_{+}$ | ${K}_{4}^{*}\otimes {K}_{4}={K}_{1}\Rightarrow z$ |

R${}_{\mathrm{h}}^{\mathrm{M}}$ | ${K}_{5}^{*}\otimes {K}_{5}={K}_{1}\Rightarrow z$ | ${K}_{5}^{*}\otimes {K}_{4}={K}_{2}\Rightarrow {\sigma}_{-}$ |

R${}_{\mathrm{h}}^{\mathrm{X}}$ | ${K}_{6}^{*}\otimes {K}_{5}={K}_{2}\Rightarrow {\sigma}_{-}$ | ${K}_{6}^{*}\otimes {K}_{4}={K}_{3}\Rightarrow {\sigma}_{+}$ |

H${}_{\mathrm{h}}^{\mathrm{h}}$ | ${K}_{6}^{*}\otimes {K}_{4}={K}_{3}\Rightarrow {\sigma}_{+}$ | ${K}_{6}^{*}\otimes {K}_{5}={K}_{2}\Rightarrow {\sigma}_{-}$ |

H${}_{\mathrm{h}}^{\mathrm{M}}$ | ${K}_{5}^{*}\otimes {K}_{4}={K}_{2}\Rightarrow {\sigma}_{-}$ | ${K}_{5}^{*}\otimes {K}_{5}={K}_{1}\Rightarrow z$ |

H${}_{\mathrm{h}}^{\mathrm{X}}$ | ${K}_{4}^{*}\otimes {K}_{4}={K}_{1}\Rightarrow z$ | ${K}_{4}^{*}\otimes {K}_{5}={K}_{3}\Rightarrow {\sigma}_{+}$ |

**Table 6.**Direct product of valence bands at the −K point and conduction bands at the K point in the ${C}_{3}$ symmetry group.

${\mathbf{v}}_{\mathbf{W}}(-\mathbf{K})\to {\mathbf{c}}_{\mathbf{M}-}\left(\mathbf{K}\right)$ | ${\mathbf{v}}_{\mathbf{W}}(-\mathbf{K})\to {\mathbf{c}}_{\mathbf{M}+}\left(\mathbf{K}\right)$ | |
---|---|---|

R${}_{\mathrm{h}}^{\mathrm{h}}$ | ${K}_{5}^{*}\otimes {K}_{5}={K}_{1}$ | ${K}_{5}^{*}\otimes {K}_{4}={K}_{2}$ |

R${}_{\mathrm{h}}^{\mathrm{M}}$ | ${K}_{4}^{*}\otimes {K}_{5}={K}_{3}$ | ${K}_{4}^{*}\otimes {K}_{4}={K}_{1}$ |

R${}_{\mathrm{h}}^{\mathrm{X}}$ | ${K}_{6}^{*}\otimes {K}_{5}={K}_{2}$ | ${K}_{6}^{*}\otimes {K}_{4}={K}_{3}$ |

H${}_{\mathrm{h}}^{\mathrm{h}}$ | ${K}_{6}^{*}\otimes {K}_{4}={K}_{3}$ | ${K}_{6}^{*}\otimes {K}_{5}={K}_{2}$ |

H${}_{\mathrm{h}}^{\mathrm{M}}$ | ${K}_{4}^{*}\otimes {K}_{4}={K}_{1}$ | ${K}_{4}^{*}\otimes {K}_{5}={K}_{3}$ |

H${}_{\mathrm{h}}^{\mathrm{X}}$ | ${K}_{5}^{*}\otimes {K}_{4}={K}_{2}$ | ${K}_{5}^{*}\otimes {K}_{5}={K}_{1}$ |

**Table 7.**Direct product of valence bands at the $\mathsf{\Gamma}$ point and conduction bands at the K point in the ${C}_{3}$ symmetry group.

R${}_{\mathbf{h}}^{\mathbf{h}}$, R${}_{\mathbf{h}}^{\mathbf{M}}$, R${}_{\mathbf{h}}^{\mathbf{X}}$ | H${}_{\mathbf{h}}^{\mathbf{h}}$, H${}_{\mathbf{h}}^{\mathbf{M}}$, H${}_{\mathbf{h}}^{\mathbf{X}}$ | |
---|---|---|

${\mathrm{v}}_{\mathsf{\Gamma}}\left({K}_{4}\right)\to {\mathrm{c}}_{\mathrm{M}-}\left(\mathrm{K}\right)$ | ${K}_{4}^{*}\otimes {K}_{5}={K}_{3}$ | ${K}_{4}^{*}\otimes {K}_{4}={K}_{1}$ |

${\mathrm{v}}_{\mathsf{\Gamma}}\left({K}_{5}\right)\to {\mathrm{c}}_{\mathrm{M}-}\left(\mathrm{K}\right)$ | ${K}_{5}^{*}\otimes {K}_{5}={K}_{1}$ | ${K}_{5}^{*}\otimes {K}_{4}={K}_{2}$ |

${\mathrm{v}}_{\mathsf{\Gamma}}\left({K}_{4}\right)\to {\mathrm{c}}_{\mathrm{M}+}\left(\mathrm{K}\right)$ | ${K}_{4}^{*}\otimes {K}_{4}={K}_{1}$ | ${K}_{4}^{*}\otimes {K}_{5}={K}_{3}$ |

${\mathrm{v}}_{\mathsf{\Gamma}}\left({K}_{5}\right)\to {\mathrm{c}}_{\mathrm{M}+}\left(\mathrm{K}\right)$ | ${K}_{5}^{*}\otimes {K}_{4}={K}_{2}$ | ${K}_{5}^{*}\otimes {K}_{5}={K}_{1}$ |

**Table 8.**Dipolar exciton g-factors involving the top valence band states (at $\mathsf{\Gamma}$, K, and −K points) to conduction bands of Mo at the K point (${\mathrm{c}}_{\mathrm{M}\pm}$) or to the conduction bands at the Q point (${\mathrm{c}}_{\mathrm{Q}\pm}$) at zero electric field and equilibrium interlayer distance. The g-factors with unambiguously determined signs are given in bold.

R${}_{\mathbf{h}}^{\mathbf{h}}$ | R${}_{\mathbf{h}}^{\mathbf{M}}$ | R${}_{\mathbf{h}}^{\mathbf{X}}$ | H${}_{\mathbf{h}}^{\mathbf{h}}$ | H${}_{\mathbf{h}}^{\mathbf{M}}$ | H${}_{\mathbf{h}}^{\mathbf{X}}$ | |
---|---|---|---|---|---|---|

${\mathrm{v}}_{\mathrm{W}}\left(\mathrm{K}\right)\to {\mathrm{c}}_{\mathrm{M}-}\left(\mathrm{K}\right)$ | $-\mathbf{6}.\mathbf{42}$ | $6.34$ | $+\mathbf{6}.\mathbf{34}$ | $-\mathbf{17}.\mathbf{18}$ | $+\mathbf{17}.\mathbf{62}$ | $17.66$ |

${\mathrm{v}}_{\mathrm{W}}\left(\mathrm{K}\right)\to {\mathrm{c}}_{\mathrm{M}+}\left(\mathrm{K}\right)$ | $10.93$ | $+\mathbf{10}.\mathbf{85}$ | $-\mathbf{10}.\mathbf{84}$ | $+\mathbf{12}.\mathbf{70}$ | $13.10$ | $-\mathbf{13}.\mathbf{18}$ |

${\mathrm{v}}_{\mathrm{W}}\left(\mathrm{K}\right)\to {\mathrm{c}}_{\mathrm{Q}-}\left(\mathrm{Q}\right)$ | $10.01$ | $9.78$ | $9.94$ | $9.56$ | $10.49$ | $10.22$ |

${\mathrm{v}}_{\mathrm{W}}\left(\mathrm{K}\right)\to {\mathrm{c}}_{\mathrm{Q}+}\left(\mathrm{Q}\right)$ | $13.67$ | $13.34$ | $13.42$ | $12.74$ | $12.94$ | $12.94$ |

${\mathrm{v}}_{\mathrm{W}}(-\mathrm{K})\to {\mathrm{c}}_{\mathrm{M}-}\left(\mathrm{K}\right)$ | $17.64$ | $17.63$ | $17.57$ | $5.99$ | $6.43$ | $6.41$ |

${\mathrm{v}}_{\mathrm{W}}(-\mathrm{K})\to {\mathrm{c}}_{\mathrm{M}+}\left(\mathrm{K}\right)$ | $13.13$ | $13.12$ | $13.07$ | $10.47$ | $10.94$ | $10.89$ |

${\mathrm{v}}_{\mathrm{W}}(-\mathrm{K})\to {\mathrm{c}}_{\mathrm{Q}-}\left(\mathrm{Q}\right)$ | $14.05$ | $14.19$ | $13.97$ | $13.61$ | $13.55$ | $13.85$ |

${\mathrm{v}}_{\mathrm{W}}(-\mathrm{K})\to {\mathrm{c}}_{\mathrm{Q}+}\left(\mathrm{Q}\right)$ | $10.39$ | $10.63$ | $10.49$ | $10.43$ | $11.10$ | $11.12$ |

${\mathrm{v}}_{{\mathsf{\Gamma}}_{1}}\to {\mathrm{c}}_{\mathrm{M}-}\left(\mathrm{K}\right)$ | $3.58$ | $3.67$ | $3.62$ | $7.58$ | $7.62$ | $7.62$ |

${\mathrm{v}}_{{\mathsf{\Gamma}}_{1}}\to {\mathrm{c}}_{\mathrm{M}+}\left(\mathrm{K}\right)$ | $0.92$ | $0.85$ | $0.87$ | $3.10$ | $3.10$ | $3.14$ |

${\mathrm{v}}_{{\mathsf{\Gamma}}_{1}}\to {\mathrm{c}}_{\mathrm{Q}-}\left(\mathrm{Q}\right)$ | $0.01$ | $0.23$ | $0.03$ | $0.03$ | $0.49$ | $0.18$ |

${\mathrm{v}}_{{\mathsf{\Gamma}}_{1}}\to {\mathrm{c}}_{\mathrm{Q}+}\left(\mathrm{Q}\right)$ | $3.66$ | $3.33$ | $3.46$ | $3.14$ | $2.94$ | $2.90$ |

${\mathrm{v}}_{{\mathsf{\Gamma}}_{2}}\to {\mathrm{c}}_{\mathrm{M}-}\left(\mathrm{K}\right)$ | $7.63$ | $7.63$ | $7.60$ | $3.60$ | $3.58$ | $3.63$ |

${\mathrm{v}}_{{\mathsf{\Gamma}}_{2}}\to {\mathrm{c}}_{\mathrm{M}+}\left(\mathrm{K}\right)$ | $3.13$ | $3.12$ | $3.11$ | $0.88$ | $0.94$ | $0.85$ |

${\mathrm{v}}_{{\mathsf{\Gamma}}_{2}}\to {\mathrm{c}}_{\mathrm{Q}-}\left(\mathrm{Q}\right)$ | $4.04$ | $4.19$ | $4.01$ | $4.01$ | $3.55$ | $3.81$ |

${\mathrm{v}}_{{\mathsf{\Gamma}}_{2}}\to {\mathrm{c}}_{\mathrm{Q}+}\left(\mathrm{Q}\right)$ | $0.39$ | $0.63$ | $0.52$ | $0.83$ | $1.10$ | $1.08$ |

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## Share and Cite

**MDPI and ACS Style**

Faria Junior, P.E.; Fabian, J.
Signatures of Electric Field and Layer Separation Effects on the Spin-Valley Physics of MoSe_{2}/WSe_{2} Heterobilayers: From Energy Bands to Dipolar Excitons. *Nanomaterials* **2023**, *13*, 1187.
https://doi.org/10.3390/nano13071187

**AMA Style**

Faria Junior PE, Fabian J.
Signatures of Electric Field and Layer Separation Effects on the Spin-Valley Physics of MoSe_{2}/WSe_{2} Heterobilayers: From Energy Bands to Dipolar Excitons. *Nanomaterials*. 2023; 13(7):1187.
https://doi.org/10.3390/nano13071187

**Chicago/Turabian Style**

Faria Junior, Paulo E., and Jaroslav Fabian.
2023. "Signatures of Electric Field and Layer Separation Effects on the Spin-Valley Physics of MoSe_{2}/WSe_{2} Heterobilayers: From Energy Bands to Dipolar Excitons" *Nanomaterials* 13, no. 7: 1187.
https://doi.org/10.3390/nano13071187