Electronic and optical properties of atomically thin InSe

National Graphene Institute

Vladimir Falko

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layered substances with covalent bonding within the layers and van der Waals coupling between the layers

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Zoo of 2D Materials

layered substances with covalent bonding within the layers and van der Waals coupling between the layers

Electronic and optical properties of atomically thin InSe

band and gaps for mono- and few-layer crystals optical transitions k•p theory for InSe paper

KK

M

side view top view Brillouin zone

InIn

Se

Se

Monolayer (In2Se2) N-layer In2NSe2N

z→‐z mirror symmetry

broken z→‐z mirror symmetry

Zolyomi, Drummond, Fal’koPRB 89, 205416 (2014)

Monolayers In2S2, Ga2X2 have qualitatively very similar properties, but multilayer films

have different lattice, bands/gaps and selection properties for optical transitions)

Zolyomi, Drummond, Fal’koPRB 87, 195403 (2013)

DFT bands for monolayer M2X2

Zolyomi, Drummond, Fal’ko - PRB 87, 195403 (2013); PRB 89, 205416 (2014)

mc = 0.2me

why InSe

Relatively light electrons in conduction

band: good for high mobility

almost flat edge of the valence band:

opportunity to get strong correlations in hole-doped material

most stable in ambient conditions bulk γ-InSe

Formulate tight binding model with all s and p orbitals and all nearest neighbour hoppings (MX, MM, XX) for monolayer (In2Se2) and bulk

Fit TB parameters to reproduce DFT bands in monolayer and bulk, after implementing a scissor correction to the band gap determined by comparing experimental and DFT gap values for bulk γ-InSe (1.45eV at low T and 1.25eV at room T)

Compute spectra of N-layer InSe (In2NSe2N) and matrix elements for z- and in-plane polarised optical transition, to compare with the experiment

Formulate k.p theory for aligned and misaligned (‘paper’) InSe layers

p,sp,s

p,s

p,s

Tight-binding model and DFT In2Se2 DFT-parametrised tight-binding model for In2NSe2N

Magorrian, Zólyomi, Fal'ko ‐ PRB 94, 245431 (2016)

Tight-binding model and DFT In2Se2 DFT-parametrised tight-binding model for In2NSe2N

z→‐z mirror symmetry

antisymmetric (odd)

symmetric (even)B

VB-CB transition ‘A’ across the gap is active in z-polarisation;

a higher-energy transition ‘B’ to a deeper double degenerate at the Г-point valence

band is active in the x-y polarisation

Parametrised TB: Δ=3.1eV(low T scissor correction)

A

k•p theory for monolayer InSe

zzsozz dsAdE )(

zzsozz dsAdE )(

A

ecme

A

ecme

Monolayer (In2Se2)

z→‐z mirror 

symmetry

Electronic bands in In2NSe2N

Bilayer (In4Se4)

broken mirror

symmetry

Magorrian, Zólyomi, Fal'ko ‐ PRB 94, 245431 (2016)

Relatively light electrons CB and wide interval of an almost flat VB edge also appear in few-layer InSe

large variation of the band gap as a function

of number of layers.Magorrian, Zólyomi, Fal'ko ‐ PRB 94, 245431 (2016)

Electronic bands in In2NSe2N

room T scissor correction

z ‐z antisymmetricsymmetric

Optical transitions in In2NSe2N

Monolayer In2Se2

in-plane-polarised A

Magorrian, Zólyomi, Fal'ko ‐ PRB 94, 245431 (2016)

transition A (z-polarised and SO-induced in-plane

polasised)

xy-polarised transition B

broken z→-z mirror

symmetry

hBN‐encapsulated In2NSe2N

hBN is sp2 – bonded insulator with a large band gap it is transparent and

takes high voltage drop (useful for electrostatic gating)

hBN and 2DM are glued together by weak van der Waals attraction, hence, they preserve their lattice structures and, hence, retain their basic physical properties.

hBN insulator

hBN provides atomically flat substrate and clean encapsulation environment (low-resistance side contacts are possible)

hBN insulator

active 2DM

hBN‐encapsulated In2NSe2N

hBN and 2DM are glued together by weak van der Waals attraction, hence, they preserve their lattice structures and, hence, retain their basic physical properties.

hBN provides atomically flat substrate and clean encapsulation environment (low-resistance side contacts are possible)

high mobility!

hBN‐encapsulated 2D crystal: QHE in 6L‐InSe

mc(N=3)=0.17memc(N=6)=0.14me

Bandurin, Tyurnina, Yu, Mishchenko, Zólyomi, Morozov, Krishna Kumar, Gorbachev, KudrynskyiPezzini, Kovalyuk, Zeitler, Novoselov, Patanè, Eaves, Grigorieva, Fal’ko, Geim, Cao

DFT/TB with scissor correction

Magnetoluminescence (non‐encapsulated 5L) 

Mudd, Molas, Chen, Zolyomi, Nogajewski, Kudrynskyi, Kovalyuk, Yusa, Makarovsky, Eaves, Potemski, Fal’ko, Patane - Scientific Reports 6, 39619 (2016)

mc=0.14me

What else may be interesting about 2D InSe?

n-type In2Se2time-reversal & mirror (z→-z) symmetry

forbid SO splitting & coupling near Г-point (in contrast to In2NSe2N)

P-doped: Wigner crystal? Mott insulator? ferromagnetic state?Peierls instability? superconductivity?

mc = 0.2me

Can we model electronic properties of InSe ‘paper’?

Laminate of misaligned monolayer in a van der Waals stack.

1 + 1 = 2 ?1 + 2 = 1 + 1 + 1 ?

k.p theory description benefits from the fact that band edges in the layer appear in the vicinity of Г-point, with an almost isotropic 2D dispersions, which facilitates resonant coupling/mixing of the band edges in the consecutive layers.

k•p theory for monolayer InSe near Г-point

B

zzsozz dsAdE )(

zzsozz dsAdE )(

A

ecme

A

ecme

k•p theory for bulk InSe

vv

cc

uHvvuH

vv

cc

uHvvuH

v

c

tttt

''

v

c

tttt

''

top layer bottom

interlayer potentials

interlayer hopping at Γ-point

...cos)(cos)()(

||

22

110

''

'

circlem

circlen

n

Grin

n

Grinbottomtop

ndst

nn

GdtGdtdt

eaeat

displacement of the two layers (AA or AB stacking) can be used in DFT modelling to fix values of t0, t1, t2

similarly for u and v

)(dtmG

truncated as decay fast

with increasing G

‘Г-pont’ k•p theory for misaligned 2-layer InSe

vv

cc

uHvvuH

~~~~

21

21

21

21

v

c

tttt~'~'~~

v

c

tttt~'~'~~

top layer bottom

interlayer potential

interlayer hopping

)(||||~0

2

'

''

' dtaeaeatn

Grin

n

Grinbottomtop

nn

fixed using DFT modelling for aligned (by shifted) layers

vv

cc

uHvvuH

~~~~

21

21

21

21

similarly

)(~0 duu

k•p theory for misaligned multilayer InSe

vv

cc

uHvvuH

~~~~

21

21

21

21

vv

cc

uHvvuH

~~~~

v

c

tttt~'~'~~

v

c

tttt~'~'~~

top layer bottommiddle

vv

cc

uHvvuH

~~~~

21

21

21

21

v

c

tttt~'~'~~

v

c

tttt~'~'~~

moiré superlatticeeffect not included

k•p theory for misaligned multilayer InSe

vv

cc

uHvvuH

~~~~

21

21

21

21

vv

cc

uHvvuH

~~~~

v

c

tttt~'~'~~

v

c

tttt~'~'~~

top bottommiddle

vv

cc

uHvvuH

~~~~

21

21

21

21

vv

cc

uHvvuH

~~~~

v

c

tttt~'~'~~

v

c

tttt~'~'~~

v

c

tttt~'~'~~

v

c

tttt~'~'~~

laminate of 1L-InSe

Two-dimensional InSe strong band gap variation with the number of layers light conduction band mass: potential for high

mobility and quantum circuits potential for strongly correlated states in p-doped

2D InSe band gap variation in InSe paper

S Magorrian (NGI)V Zolyomi (NGI)

X Chen (NGI)N Drummond (Lancaster)

J Lischner (Imperial) A Tyurnina (NGI)D Bandurin (NGI)

R Gorbachev (NGI)K Novoselov (NGI

A Geim (NGI) A Patane (Nottingham)L Eaves (Nottingham)

Z Kovalyuk (IPMS-UAS)U Zeitler (Nijmegen)

M Potemski (Grenoble)

National Graphene Institute