Topological flat bands without magic angles in massive twisted bilayer graphenes (Srivani. et.al., Phys. Rev. B 101, 125411(2020))

1. Topological flat bands without magic angles in massive
twisted bilayer graphenes
Srivani Javvaji, Jin-Hua Sun, and Jeil Jung
Quantum Materials Theory Group
Department of Physics, University of Seoul, Seoul 02504, Republic of Korea.
Session B51: Graphene: Electronic Structure and
Interactions : Moire, Correlations, and Topology
[Srivani. et.al., Phys. Rev. B 101, 125411(2020)]

2. Motivation
Motivation:
Magic angle twisted bilayer graphene and trilayer graphene boron nitride moire superlattices
have shown Mott insulating and signatures of supercondictivity
Oliver E. Buckley Condensed
Matter Physics Prize 2020
Pablo Jarillo-Herrero
Magic angle twisted bilayer graphene show Mott insulating phases and signatures of
superconductivity
Search for the Flat band systems without the magic angle
Bilayers of Transition Metal Dichalcogenides (TMDC)
Phys. Rev. Lett. 122, 086402;
Twisted bilayer MoS2
arXiv:1908.10399;arXiv:1910.12147;
Massive Dirac model
Twisted bilayer WSe2
G/G

3. Motivation
Moiré BZ Moiré pattern
HtMBG =
[
ht
θ/2(k) T(r)
T†
(r) hb
−θ/2(k)]
Interlayer coupling term:
T0
=
[
ω1 ω2
ω2 ω3]
, T±
=
[
ω1 ω2e∓i2π/3
ω2e±i2π/3
ω3 ]
Continuum model Hamiltonian perturbed by
stacking-dependent interlayer tunneling
for twisted BLG (tBLG) :
Monolayer Hamiltonian,
2(b)
˜K′
(a)Realspacemoirésuperlattice
2
LM
Energy(eV)
−0.2
−0.1
0
0.1
0.2
(a)(b)
K′
+
θ
2
−
θ
2
˜ΓΓ
˜K
˜K′
˜M˜Γ
˜K′
˜K
˜M
˜K
˜K′
˜Γ
˜Γ
K
ΔK
lM
Moiré bands theory (Bistritzer-MacDonald model)
where,
Bistritzer et al, pnas.1108174108
lM ∼ a/θ
• We include additional diagonal and off-diagonal moire pattern terms to the Bistritzer-MacDonald model
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)

4. Motivation
Continuum model Hamiltonian perturbed by
stacking-dependent interlayer tunneling
for twisted BLG :
Band structures of tBLG
Band width
HtMBG =
[
ht
θ/2(k) T(r)
T†
(r) hb
−θ/2(k)]
Moiré bands theory
Yuan Cao et al,
nature26154
• When the two layers are coupled in tBLG, the electronic structure exhibits, flat bands at certain twist angles
called magic angles.

5. Motivation
Continuum model Hamiltonian perturbed by
stacking-dependent interlayer tunneling
for twisted BLG :
Pressure(GPa)
(eV)<latexitsha1_base64="qUg8iXI01zUDf6doeaeQGgCgJ1c=">AAAB7XicbVDLSgNBEOyNrxhfUY9eFoPgKeyKoMegF48RzAOSJcxOepMx81hmZoUQ8g9ePCji1f/x5t84SfagiQUNRVU33V1xypmxQfDtFdbWNza3itulnd29/YPy4VHTqExTbFDFlW7HxCBnEhuWWY7tVCMRMcdWPLqd+a0n1IYp+WDHKUaCDCRLGCXWSc2uEjggvXIlqAZz+KskzEkFctR75a9uX9FMoLSUE2M6YZDaaEK0ZZTjtNTNDKaEjsgAO45KItBEk/m1U//MKX0/UdqVtP5c/T0xIcKYsYhdpyB2aJa9mfif18lsch1NmEwzi5IuFiUZ963yZ6/7faaRWj52hFDN3K0+HRJNqHUBlVwI4fLLq6R5UQ2Danh/Wand5HEU4QRO4RxCuIIa3EEdGkDhEZ7hFd485b14797HorXg5TPH8Afe5w+QMY8b</latexit><latexitsha1_base64="qUg8iXI01zUDf6doeaeQGgCgJ1c=">AAAB7XicbVDLSgNBEOyNrxhfUY9eFoPgKeyKoMegF48RzAOSJcxOepMx81hmZoUQ8g9ePCji1f/x5t84SfagiQUNRVU33V1xypmxQfDtFdbWNza3itulnd29/YPy4VHTqExTbFDFlW7HxCBnEhuWWY7tVCMRMcdWPLqd+a0n1IYp+WDHKUaCDCRLGCXWSc2uEjggvXIlqAZz+KskzEkFctR75a9uX9FMoLSUE2M6YZDaaEK0ZZTjtNTNDKaEjsgAO45KItBEk/m1U//MKX0/UdqVtP5c/T0xIcKYsYhdpyB2aJa9mfif18lsch1NmEwzi5IuFiUZ963yZ6/7faaRWj52hFDN3K0+HRJNqHUBlVwI4fLLq6R5UQ2Danh/Wand5HEU4QRO4RxCuIIa3EEdGkDhEZ7hFd485b14797HorXg5TPH8Afe5w+QMY8b</latexit><latexitsha1_base64="qUg8iXI01zUDf6doeaeQGgCgJ1c=">AAAB7XicbVDLSgNBEOyNrxhfUY9eFoPgKeyKoMegF48RzAOSJcxOepMx81hmZoUQ8g9ePCji1f/x5t84SfagiQUNRVU33V1xypmxQfDtFdbWNza3itulnd29/YPy4VHTqExTbFDFlW7HxCBnEhuWWY7tVCMRMcdWPLqd+a0n1IYp+WDHKUaCDCRLGCXWSc2uEjggvXIlqAZz+KskzEkFctR75a9uX9FMoLSUE2M6YZDaaEK0ZZTjtNTNDKaEjsgAO45KItBEk/m1U//MKX0/UdqVtP5c/T0xIcKYsYhdpyB2aJa9mfif18lsch1NmEwzi5IuFiUZ963yZ6/7faaRWj52hFDN3K0+HRJNqHUBlVwI4fLLq6R5UQ2Danh/Wand5HEU4QRO4RxCuIIa3EEdGkDhEZ7hFd485b14797HorXg5TPH8Afe5w+QMY8b</latexit><latexitsha1_base64="qUg8iXI01zUDf6doeaeQGgCgJ1c=">AAAB7XicbVDLSgNBEOyNrxhfUY9eFoPgKeyKoMegF48RzAOSJcxOepMx81hmZoUQ8g9ePCji1f/x5t84SfagiQUNRVU33V1xypmxQfDtFdbWNza3itulnd29/YPy4VHTqExTbFDFlW7HxCBnEhuWWY7tVCMRMcdWPLqd+a0n1IYp+WDHKUaCDCRLGCXWSc2uEjggvXIlqAZz+KskzEkFctR75a9uX9FMoLSUE2M6YZDaaEK0ZZTjtNTNDKaEjsgAO45KItBEk/m1U//MKX0/UdqVtP5c/T0xIcKYsYhdpyB2aJa9mfif18lsch1NmEwzi5IuFiUZ963yZ6/7faaRWj52hFDN3K0+HRJNqHUBlVwI4fLLq6R5UQ2Danh/Wand5HEU4QRO4RxCuIIa3EEdGkDhEZ7hFd485b14797HorXg5TPH8Afe5w+QMY8b</latexit>
(deg)
FIG. 1. (Color online) Colormap of the flatband bandwidth as a
function of twist angle q and interlayer tunneling strength w. We
identify numerically straight lines in the parameter space given by
Eq. (??), where we represent in orange, green and blue the regions in
the phase diagram corresponding to the first, second and third magic
Bheema. et.al.,
Electr.Struc (2019)
HtMBG =
[
ht
θ/2(k) T(r)
T†
(r) hb
−θ/2(k)]
Band gap Δ = 0Moiré bands theory
Band width
Yuan Cao et al,
nature26154
• The blue, green & orange lines in the colormap of ( , ) space represents the the regions corresponding to
first,second and third magic angles at which bandwidth minima are achieved.
θ ω

6. Motivation
Continuum model Hamiltonian perturbed by
stacking-dependent interlayer tunneling
for twisted BLG : h(θ) = − vk
[
ei(θk−θ)
e−i(θk−θ) ]
Δ
−Δ
HtMBG =
[
ht
θ/2(k) T(r)
T†
(r) hb
−θ/2(k)]
Moiré bands theory
In massive tBLG (MTBG):
Band width
Yuan Cao et al,
nature26154

7. Motivation
Continuum model Hamiltonian perturbed by
stacking-dependent interlayer tunneling
for twisted BLG :
HtMBG =
[
ht
θ/2(k) T(r)
T†
(r) hb
−θ/2(k)]
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)
Moiré bands theory
Band width
h(θ) = − vk
[
ei(θk−θ)
e−i(θk−θ) ]
Δ
−Δ
In massive tBLG (MTBG):
Yuan Cao et al,
nature26154
• Traces of magic angle bandwidth dips of gapless tBLG gradually disappear as the intra-layer gap becomes large.
• They are almost inexistent already for the gaps of the order of 0.2 eV∼

8. υF −Fermi velocity
2Δ −Intralayer gap
Kθ −Shift between ˜K & ˜K′
ωi −Sublattice resolved interlayer couplings with i = 1,2,3
δs −Secondary gap (EC2
− EC1
)
δp −Primary gap (EC1
− EV1
)
δM
−Gap (EC2
− EC1
) at ˜M
W −Band width
Terminology :
2Δ = 0.1 eV, ωi = 0(b)
Kθ
∝ υF
˜Γ ˜K ˜K′ ˜Γ˜K′
2
zone
super lattice
˜Γ ˜K′
2Δ = 0.1 eV, ωi ≠ 0
δs
W
δp
δM
˜Γ ˜K ˜K′ ˜Γ˜K′
Energy(eV)Energy(eV)
− 0.2
− 0.1
0
0.1
0.2
− 0.2
− 0.1
0
0.1
0.2
(b) (c)
˜K1
˜K2
˜K3
˜K4
˜K1
˜K2
˜K3
˜K4
˜K1
˜K2
˜K3
˜K4
Δ = 0, ωi = 0 Δ ≠ 0, ωi = 0 Δ ≠ 0, ωi ≠ 0
2Δ = 0 . 1 eV, ωi = 0
˜Γ ˜K ˜K′ ˜Γ˜K′
Energy(eV)
Kθ
∝ υF
2Δ
0.2
0.1
0
-0.1
-0.2
Energy(eV)
2Δ = 0 . 1 eV, ωi ≠ 0
δs
W
δp
δM
˜Γ ˜K ˜K′ ˜Γ˜K′
0.2
0.1
0
-0.1
-0.2
V1
C1
C2
V2
Evolution of flat bands in massive tBG
Schematic showing
the flat band evolution
in MTBG
Actual band structure
of MTBG at θ = 1∘
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)

9. ˜K ˜K′ ˜Γ ˜Γ ˜K
0.2
0.1
0
-0.1
-0.2
0.2
0.1
0
-0.1
-0.2
0.3
0.2
0.1
-0.1
-0.2
-0.3
0.4
0.3
-0.3
-0.4
1.4
1.3
1.2
-1.2
-1.3
-1.4
C1
C2
C3
V1
V2
V3
2Δ = 0 eV 2Δ = 0 . 1 eV 2Δ = 0 . 5 eV 2Δ = 1 . 0 eV 2Δ = 3 . 0 eV
Energy(eV)
˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K
Evolution of flat bands in massive tBG
Band structure of
MTBG at θ = 1∘
˜Γ
˜K′
˜K
˜M
˜K
˜K′
˜Γ
˜Γ
Moiré BZ
ωi = 0 . 098 eV
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)
• The evolution of electronic structures in MTBG at as a function of bandgap ranging from
shown here
• We observe the enhancement of the flatness in the lower bands along with the enhancement of secondary
isolation gap and primary gap
θ = 1∘
2Δ = 0 to 3 eV

10. 0 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5
θ (deg) θ (deg) θ (deg) θ (deg) θ (deg)
0.2
0.15
0.1
0.05
0
0.25
ω(eV)
W (eV)
W ∝
1
Δ
(at constant θ)W ∝ θ (at constant Δ) &
Evolution of flat bands in massive tBG
˜K ˜K′ ˜Γ ˜Γ ˜K
0.2
0.1
0
-0.1
-0.2
0.2
0.1
0
-0.1
-0.2
0.3
0.2
0.1
-0.1
-0.2
-0.3
0.4
0.3
-0.3
-0.4
1.4
1.3
1.2
-1.2
-1.3
-1.4
C1
C2
C3
V1
V2
V3
2Δ = 0 eV 2Δ = 0 . 1 eV 2Δ = 0 . 5 eV 2Δ = 1 . 0 eV 2Δ = 3 . 0 eV
Energy(eV)
˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K
˜Γ
˜K′
˜K
˜M
˜K
˜K′
˜Γ
˜Γ
Moiré BZ
Bandwidth
Band structure of
MTBG at θ = 1∘
ωi = 0 . 098 eV
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)
• This colormap shows the bandwidth in the parameter space of ( , ).
• As we increase the bandgap, the magic angle line regions starting to expand to become a continuous region of
bandwidth minima
θ ω

11. 0.2
0.15
0.1
0.05
0
0.25
ω(eV)
0 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5
θ (deg) θ (deg) θ (deg) θ (deg) θ (deg)
δS (eV)
Evolution of flat bands in massive tBG
˜K ˜K′ ˜Γ ˜Γ ˜K
0.2
0.1
0
-0.1
-0.2
0.2
0.1
0
-0.1
-0.2
0.3
0.2
0.1
-0.1
-0.2
-0.3
0.4
0.3
-0.3
-0.4
1.4
1.3
1.2
-1.2
-1.3
-1.4
C1
C2
C3
V1
V2
V3
2Δ = 0 eV 2Δ = 0 . 1 eV 2Δ = 0 . 5 eV 2Δ = 1 . 0 eV 2Δ = 3 . 0 eV
Energy(eV)
˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K
˜Γ
˜K′
˜K
˜M
˜K
˜K′
˜Γ
˜Γ
Moiré BZ
0.2
0.15
0.1
0.05
0
0.25
ω(eV)
W (eV)Bandwidth
Secondary
gap
Band structure of
MTBG at θ = 1∘
ωi = 0 . 098 eV
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)
• This colormap shows the lower energy band isolation in the parameter space of ( , ), we expect to find isolated
flat bands are more susceptible to coulomb interactions
θ ω

12. Study of massive twisted Dirac materials
h(θ) = − vk
[
ei(θk−θ)
e−i(θk−θ) ]
Δ
−Δ
Massive Dirac Hamiltonian :
where,
Band structures of TMDC, BN and SiC monolayers at ‘Dirac’ point
can be described by Massive Dirac Hamiltonian
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)
Bandwidth(eV)
0.2
0 108642
(deg)θ
0.15
0.1
0.05
0
0.0
0.1
0.2
0.3
60o)
0o)
• Traces of magic angle bandwidth dips of gapless tBLG gradually disappear as the intra-layer gap becomes large.
• They are almost inexistent already for the gaps of the order of 0.2 eV∼

13. Study of massive twisted Dirac materials
h(θ) = − vk
[
ei(θk−θ)
e−i(θk−θ) ]
Δ
−Δ
Massive Dirac Hamiltonian :
Band structures of TMDC, BN and SiC monolayers at ‘Dirac’ point
can be described by Massive Dirac Hamiltonian
* Di Xiao et al, PRL
108, 196802 (2012)
*
*
*
* * Fencheng Wu et al,
arXiv:1807.03311v1
DFT
(Wannier
Functions)
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)
Bandwidth(eV)
0.2
0 108642
(deg)θ
0.15
0.1
0.05
0
0.0
0.1
0.2
0.3
60o)
0o)

14. Study of massive twisted Dirac materials
Energy(eV)
0.4
0.0
-0.2
-0.82
0.84
0.86
-0.84
-0.86
0.2
-0.4
2.1
2.0
1.9
1.8
-2
-2.1
2.1
2
1.9
-1.8
-1.9
-2
1.14
1.1
1.06
1.02
-0.96
-1
-1.04
˜G/ ˜G TMDC/TMDC BN/BN(0∘
) BN/BN(60∘
) SiC/SiC
Band structure of
MTBGs at θ = 1∘
˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K
2Δ = 0 . 1 eV
Large bandgap systems
2Δ ∼ 1 . 6 eV 2Δ ∼ 4 . 5 eV 2Δ ∼ 2 . 3 eV
Small bandgap system
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)
• The evolution of electronic structures in different real systems represented by MTBG Hamiltonian at
• We observe the enhancement of the flatness in the lower bands along with the enhancement of secondary
isolation gap and primary gap
θ = 1∘

15. Study of massive twisted Dirac materials
1
2
5
3
4
(arb . u.)
× 104
Energy(eV)
0.4
0.0
-0.2
-0.82
0.84
0.86
-0.84
-0.86
0.2
-0.4
2.1
2.0
1.9
1.8
-2
-2.1
2.1
2
1.9
-1.8
-1.9
-2
1.14
1.1
1.06
1.02
-0.96
-1
-1.04
˜G/ ˜G TMDC/TMDC BN/BN(0∘
) BN/BN(60∘
) SiC/SiC
Band structure of
MTBGs at θ = 1∘
˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K ˜K ˜K′ ˜Γ ˜Γ ˜K
At Valence band van Hove singularity
Ry(nm)
Rx (nm)
0
5
10
-5
-10
0 10-10
Rx (nm)
5
15
25
× 102
1000
1500
2000
(arb . u.)(arb . u.)
0 10-10
0
5
10
-5
-10
Rx (nm)
1
3
9
5
7
(arb . u.)
× 104
0
5
10
-5
-10
0 10-10
Rx (nm)
2
4
8
12
(arb . u.)
× 104
0 10-10
Rx (nm)
0
5
10
-5
-10
0 15-15
0
5
10
-5
-10
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
BA
BA
BA
AA-stacking
AB-stacking
BA-stacking
aM
Rx
Ry
AB-stacking AA-stacking BA-stacking
A B
A′ B′
A B
A′ B′
0∘ alignment: 60∘ alignment:
A B
A′ B′
AB-stacking AA-stacking BA-stacking
A B
B′ A′
A B
B′ A′
A B
B′ A′
Local Density of
States θ = 1∘
2Δ = 0 . 1 eV
Large bandgap systems
2Δ ∼ 1 . 6 eV 2Δ ∼ 4 . 5 eV 2Δ ∼ 2 . 3 eV
Small bandgap system
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)
• The stacking resolved real space representation of the local density of states at the valence band van Hove
singularity (VHS)
• In the all real systems shown above, AA-stacking region gives rise to VHS except in BN/BN(60o)

16. Existence of strongly correlated states
ω(eV)
δs (eV)
−0.05 0.050
0 2.521 1.50.5
W (eV)
0.05
0.1
0.15
0.2
0.25
0 2.521 1.50.5
0.1 0.2 0.3 0.4
δp (eV)
0 2.521 1.50.5
−0.05 0.050
Ueff /W
5 100
0 2.521 1.50.5
˜G/ ˜G
θ (deg)θ (deg) θ (deg) θ (deg)
The ratio of effective Coulomb potential to Bandwidth Ueff /W
Debye length,
Moiré length, lM ∼ a/θ
Screened Coulomb
potential :
Small bandgap system
2Δ = 0 . 1 eV
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)
• The phase diagram colormaps of the bandwidth W, secondary gap δs, primary gap δp, the Ueff/W ratio in
• The threshold value of the Ueff/W = 1 ratio from where Coulomb interactions are expected to be significant is
indicated with a blue line.
˜G/ ˜G

17. Existence of strongly correlated states
ω(eV)
δs (eV)
−0.05 0.050
0 2.521 1.50.5
W (eV)
0.05
0.1
0.15
0.2
0.25
0 2.521 1.50.5
0.1 0.2 0.3 0.4
δp (eV)
0 2.521 1.50.5
−0.05 0.050
Ueff /W
5 100
0 2.521 1.50.5
˜G/ ˜G
ω(eV)
500 10000
BN/BN(0∘
)
θ (deg)
0 54321
500 10000
TMDC/TMDC
θ (deg)
0 54321
0.05
0.1
0.15
0.2
0.25
500 10000
BN/BN(60∘
)
0 54321
θ (deg)
500 10000
SiC/SiC
0 54321
θ (deg)
The ratio of effective Coulomb potential to Bandwidth Ueff /W
Debye length,
Moiré length, lM ∼ a/θ
Screened Coulomb
potential :
Small bandgap system
2Δ = 0 . 1 eV
Large bandgap systems
1 < 2Δ ≤ 4 . 5 eV
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)
• In all the real systems we studied, Ueff/W diagram shows that Coulomb interactions are expected

18. Analytical expressions for energy at , & -points˜K ˜Γ ˜M
Energy(eV)
δs
W
δp
δM
˜Γ ˜K ˜K′ ˜Γ˜K′
0.2
0.1
0
-0.1
-0.2
V1
C1
C2
V2
2Δ = 0 . 1 eV, ωi ≠ 0
1
2
3
4
5
64′
Kθ
T+
T−
T0
˜K
˜K′
˜Γ
Bistritzer-MacDonald model
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)

19. Analytical expressions for energy at , & -points˜K ˜Γ ˜M
Energy(eV)
δs
W
δp
δM
˜Γ ˜K ˜K′ ˜Γ˜K′
0.2
0.1
0
-0.1
-0.2
V1
C1
C2
V2
2Δ = 0 . 1 eV, ωi ≠ 0
• from 4x4 -Hamiltonian model (green oval)E( ˜M)
Bistritzer-MacDonald model
Kθ
1
2
3
4
5
64′
Kθ
T+
T−
T0
˜K
˜K′
˜Γ
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)

20. Analytical expressions for energy at , & -points˜K ˜Γ ˜M
Energy(eV)
δs
W
δp
δM
˜Γ ˜K ˜K′ ˜Γ˜K′
0.2
0.1
0
-0.1
-0.2
V1
C1
C2
V2
2Δ = 0 . 1 eV, ωi ≠ 0
• from 4x4 -Hamiltonian model (green oval)E( ˜M)
Bistritzer-MacDonald model
Kθ
1
2
3
4
5
64′
Kθ
T+
T−
T0
˜K
˜K′
˜Γ
• from 8x8 -Hamiltonian model (brown circle)E( ˜K)
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)

21. Analytical expressions for energy at , & -points˜K ˜Γ ˜M
Energy(eV)
δs
W
δp
δM
˜Γ ˜K ˜K′ ˜Γ˜K′
0.2
0.1
0
-0.1
-0.2
V1
C1
C2
V2
2Δ = 0 . 1 eV, ωi ≠ 0
• from 4x4 -Hamiltonian model (green oval)E( ˜M)
Bistritzer-MacDonald model
Kθ
1
2
3
4
5
64′
Kθ
T+
T−
T0
˜K
˜K′
˜Γ
• from 8x8 -Hamiltonian model (brown circle)E( ˜K)
• from 12x12 -Hamiltonian model (blue square)E(˜Γ)
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)

22. Analytical expressions for energy at , & -points˜K ˜Γ ˜M
Energy(eV)
2Δ = 0 . 1 eV, ωi ≠ 0
δs
W
δp
δM
˜Γ ˜K ˜K′ ˜Γ˜K′
0.2
0.1
0
-0.1
-0.2
V1
C1
C2
V2
The relationship between system parameters:
W = 0 . 03 eV
Kθ
1
2
3
4
5
64′
Kθ
T+
T−
T0
˜K
˜K′
˜Γ
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)
• The well agreement between proposed analytical solutions with Numerical method is shown for different vF & ω

23. Chern numbers
Chern phase diagram of massive tBG
θ = 1∘
Valence
flat Band
Conduction
flat Band
ω1 = ω3 = 0 . 85ω
AA-stacking
A B
A′ B′
2Δt
2Δb
ωi
• The non-trivial Chern flat band emerges when the terms are different from each
other ( )
ωi
ωi ≠ ω
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)

24. Chern numbers
Chern phase diagram of massive tBG
θ = 1∘
θ = 3∘
Valence
flat Band
Conduction
flat Band
ω1 = ω3 = 0 . 85ω
AA-stacking
A B
A′ B′
2Δt
2Δb
ωi
• The non-trivial Chern flat band emerges when the terms are different from each
other ( )
ωi
ωi ≠ ω
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)

25. Circular dichroism in massive tBLG
A B
A′ B′
0∘alignment:
60∘ alignment:
A B
B′ A′
AA-stacking
Interlayer potential
difference Vg :
2Δ = 0 . 1 eV, θ = 3∘
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)
• Schematic illustration of MTBGs for zero and
finite interlayer potential differences(Vg) near 0o &
60o rotational alignments , giving rise to same and
opposite mass signs at the mini valleys
• When Vg = 0, we expect macro-valley K & K’
contrasting circular dichroism near 0o alignment &
suppressed circular dichroism at 60o
˜K & ˜K′

26. Circular dichroism in massive tBLG
Interlayer potential
difference Vg :
2Δ = 0 . 1 eV, θ = 3∘
Srivani.et.al.Phys. Rev. B 101, 125411 (2020)
A B
A′ B′
0∘alignment:
60∘ alignment:
A B
B′ A′
AA-stacking
• When Vg 0, it is possible to introduce a finite circular dichroism by applying interlayer bias & Fermi level
change that leads to layer & minimally polarization
≠

27. Summary
• A finite gap in the constituent layers of the MTBGs makes the generation of flat bands
simpler than in TBG
• Band flattening happens for a continuous range of small twist angles
• Our numerical calculations complemented by analytical solutions of the E( ), E( ) and
E( )
• Relationship between the system parameters in MTBG :
• The non-trivial Chern flat band emerges when the terms are different from each
other
• Valley contrasting circular dichroism in MTBG is studied at zero & finite interlayer
potential difference
˜Γ ˜K
˜M
ωi
Thank you !