Topological two-dimensional polymers

Supporting Information

Maximilian A. Springer1,2, Tsai-Jung Liu2, Agnieszka Kuc1, and

Thomas Heine1,2

1Helmholtz-Zentrum Dresden-Rossendorf, Institute of Resource Ecology, Research Site Leipzig,

Permoserstrasse 15, 04138 Leipzig, Germany2TU Dresden, Faculty for Chemistry and Food Chemistry, Bergstrasse 66c, 01069 Dresden,

Germany

Contents

1 Band structures and sketches for 2D nets 1

2 Technical tutorial for tight-binding calculations 30

1

Electronic Supplementary Material (ESI) for Chemical Society Reviews.This journal is © The Royal Society of Chemistry 2020

1 Band structures and sketches for 2D nets

On the following pages, Tables S1-S4 show the tight-binding band structures that were

calculated using scripts similar to the accompanying example script, based on the pythTB

package.[1] The Fermi level was estimated by integrating over a grid of equidistant points

in the first Brillouin zone. The investigated topologies are separated by the manifold of

faces and vertices. First, Table S1 shows the band structures for the face- and vertex-

transitive regular nets hcb, hxl, and sql. In Table S2 and S3, band structures for the

semiregular (one type of vertex but different types of faces) and demiregular nets (ex-

actly two types of vertices) can be found. Additionally, we show the tight-binding band

structures for selected topologies that do not fit into one of these groups in Table S4.

1

Tab

leS

1:S

ket

ches

and

tigh

t-b

indin

gb

and

stru

ctu

res

for

the

regu

lar

net

shcb

,hxl,

andsq

l.

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

+S

OC

hcb

ΓM

KΓ

−5−4−3−2−101234

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

hxl

ΓM

KΓ

−9−8−7−6−5−4−3−2−10123

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

sql

ΓX

MΓ

−6−5−4−3−2−1012345

Energy / |β1|

ΓX

MΓΓ

XM

ΓΓ

XM

Γ

2

Tab

leS

2:S

ket

ches

and

tigh

t-b

indin

gb

and

stru

ctu

res

for

the

sem

ireg

ula

rn

etscem

,fes,

fsz,fxt,

hca

,htb

,kgm

,an

dtts.

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

+S

OC

cem

ΓXM

ΓY

S−8−7−6−5−4−3−2−101234

Energy / |β1|

ΓXM

ΓY

SΓ

XM

ΓY

SΓ

XM

ΓY

S

fes

ΓX

MΓ

−5−4−3−2−101234

Energy / |β1|

ΓX

MΓΓ

XM

ΓΓ

XM

Γ

fsz

ΓM

KΓ

−8−7−6−5−4−3−2−10123

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

3

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

+S

OC

fxt

ΓM

KΓ

−5−4−3−2−101234

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

hca

ΓM

KΓ

−4−3−2−10123

Energy / |β1|Γ

MK

ΓΓ

MK

ΓΓ

MK

Γ

htb

ΓM

KΓ

−6−5−4−3−2−101234

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

kgm

ΓM

KΓ

−7−6−5−4−3−2−10123

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

4

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

+S

OC

tts

ΓX

MΓ

−8−7−6−5−4−3−2−10123

Energy / |β1|

ΓX

MΓΓ

XM

ΓΓ

XM

Γ

5

Tab

leS

3:S

ketc

hes

and

tigh

t-b

ind

ing

ban

dst

ruct

ure

sfo

rth

ed

emir

egu

lar

net

sbew

,cph

,kra

,krb

,krc

,krd

,kre

,krf

,krg

,krh

,krj

,krk

,krl

,krm

,krn

,krq

,krr

,krs

,krt

,an

dusm

.

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

+S

OC

bew

ΓX

MΓ

YS

−6−5−4−3−2−10123

Energy / |β1|

ΓX

MΓ

YSΓ

XM

ΓY

SΓ

XM

ΓY

S

cph

ΓX

MΓ

−6−5−4−3−2−1012

Energy / |β1|

ΓX

MΓΓ

XM

ΓΓ

XM

Γ

kra

ΓM

KΓ

−9−8−7−6−5−4−3−2−10123

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

6

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

+S

OC

krb

ΓM

KΓ

−8−7−6−5−4−3−2−10123

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

krc

ΓXM

ΓY

S−9−8−7−6−5−4−3−2−10123

Energy / |β1|Γ

XM

ΓY

SΓ

XM

ΓY

SΓ

XM

ΓY

S

krd

ΓXM

ΓY

S−8−7−6−5−4−3−2−10123

Energy / |β1|

ΓXM

ΓY

SΓ

XM

ΓY

SΓ

XM

ΓY

S

kre

ΓM

KΓ

−8−7−6−5−4−3−2−101234

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

7

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

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eigh

bou

r1s

t+

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OC

krf

ΓM

KΓ

−7−6−5−4−3−2−10123

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

krg

ΓM

KΓ

−7−6−5−4−3−2−10123

Energy / |β1|Γ

MK

ΓΓ

MK

ΓΓ

MK

Γ

krh

ΓX

MΓ

YS

−7−6−5−4−3−2−10123

Energy / |β1|

ΓX

MΓ

YSΓX

MΓ

YSΓX

MΓ

YS

krj

ΓX

MΓ

YS

−8−7−6−5−4−3−2−10123

Energy / |β1|

ΓX

MΓ

YSΓX

MΓ

YSΓX

MΓ

YS

8

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

+S

OC

krk

ΓM

KΓ

−7−6−5−4−3−2−101234

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

krl

ΓXM

ΓY

S−7−6−5−4−3−2−101234

Energy / |β1|Γ

XM

ΓY

SΓ

XM

ΓY

SΓ

XM

ΓY

S

krm

ΓXM

ΓY

S−7−6−5−4−3−2−101234

Energy / |β1|

ΓXM

ΓY

SΓ

XM

ΓY

SΓ

XM

ΓY

S

krn

ΓM

KΓ

−7−6−5−4−3−2−101234

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

9

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

+S

OC

krq

ΓM

KΓ

−6−5−4−3−2−10123

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

krr

ΓX

MΓ

YS

−6−5−4−3−2−10123

Energy / |β1|Γ

XM

ΓY

SΓ

XM

ΓY

SΓ

XM

ΓY

S

krs

ΓXM

ΓY

S−6−5−4−3−2−10123

Energy / |β1|

ΓXM

ΓY

SΓ

XM

ΓY

SΓ

XM

ΓY

S

krt

ΓM

KΓ

−5−4−3−2−101234

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

10

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

+S

OC

usm

ΓX

MΓ

−8−7−6−5−4−3−2−10123

Energy / |β1|

ΓX

MΓΓ

XM

ΓΓ

XM

Γ

11

Tab

leS

4:S

ket

ches

and

tigh

t-b

indin

gb

and

stru

ctu

res

for

add

itio

nal

net

s.

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

+S

OC

bex

ΓX

MΓ

YS

−5−4−3−2−101234

Energy / |β1|

ΓX

MΓ

YSΓ

XM

ΓY

SΓ

XM

ΓY

S

bey

ΓXM

ΓY

S−5−4−3−2−101234

Energy / |β1|

ΓXM

ΓY

SΓ

XM

ΓY

SΓ

XM

ΓY

S

bhd

ΓX

MΓ

YS

−6−5−4−3−2−1012345

Energy / |β1|

ΓX

MΓ

YSΓ

XM

ΓY

SΓ

XM

ΓY

S

12

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

+S

OC

car

ΓX

MΓ

YS

−4−3−2−101234

Energy / |β1|

ΓX

MΓ

YSΓX

MΓ

YSΓX

MΓ

YS

cem-d

ΓXM

ΓY

S−5−4−3−2−101234

Energy / |β1|Γ

XM

ΓY

SΓ

XM

ΓY

SΓ

XM

ΓY

S

cpa

ΓX

MΓ

−5−4−3−2−101234

Energy / |β1|

ΓX

MΓΓ

XM

ΓΓ

XM

Γ

cpb

ΓX

MΓ

YS

−6−5−4−3−2−10123

Energy / |β1|

ΓX

MΓ

YSΓ

XM

ΓY

SΓ

XM

ΓY

S

13

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

+S

OC

cpc

ΓX

MΓ

−5−4−3−2−10123

Energy / |β1|

ΓX

MΓΓ

XM

ΓΓ

XM

Γ

cpd

ΓXM

ΓY

S−5−4−3−2−101234

Energy / |β1|Γ

XM

ΓY

SΓ

XM

ΓY

SΓ

XM

ΓY

S

cpe

ΓM

KΓ

−6−5−4−3−2−10123

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

cpf

ΓX

MΓ

YS

−6−5−4−3−2−1012

Energy / |β1|

ΓX

MΓ

YSΓX

MΓ

YSΓX

MΓ

YS

14

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

+S

OC

cpg

ΓM

KΓ

−6−5−4−3−2−1012

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

cpj

ΓX

MΓ

−7−6−5−4−3−2−10123

Energy / |β1|Γ

XM

ΓΓ

XM

ΓΓ

XM

Γ

cpk

ΓX

MΓ

YS

−5−4−3−2−10123

Energy / |β1|

ΓX

MΓ

YSΓ

XM

ΓY

SΓ

XM

ΓY

S

cpl

ΓM

KΓ

−6−5−4−3−2−10123

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

15

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

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OC

cpm

ΓM

KΓ

−5−4−3−2−1012

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

cpn

ΓX

MΓ

−5−4−3−2−10123

Energy / |β1|Γ

XM

ΓΓ

XM

ΓΓ

XM

Γ

cpo

ΓX

MΓ

YS

−7−6−5−4−3−2−101234

Energy / |β1|

ΓX

MΓ

YSΓ

XM

ΓY

SΓ

XM

ΓY

S

cpp

ΓX

MΓ

−4−3−2−10123

Energy / |β1|

ΓX

MΓΓ

XM

ΓΓ

XM

Γ

16

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

+S

OC

cpq

ΓX

MΓ

YS

−5−4−3−2−101234

Energy / |β1|

ΓX

MΓ

YSΓ

XM

ΓY

SΓ

XM

ΓY

S

cpr

ΓX

MΓ

YS

−5−4−3−2−10123

Energy / |β1|Γ

XM

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XM

ΓY

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XM

ΓY

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cps

ΓX

MΓ

−5−4−3−2−101234

Energy / |β1|

ΓX

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XM

ΓΓ

XM

Γ

cpt

ΓM

KΓ

−6−5−4−3−2−1012

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

17

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

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OC

cpu

ΓX

MΓ

−6−5−4−3−2−10123

Energy / |β1|

ΓX

MΓΓ

XM

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XM

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cpv

ΓX

MΓ

−5−4−3−2−10123

Energy / |β1|Γ

XM

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XM

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XM

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ΓM

KΓ

−5−4−3−2−101234

Energy / |β1|

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MK

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MK

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cpx

ΓM

KΓ

−6−5−4−3−2−10123

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

18

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

+S

OC

cpy

Γ XM

ΓYS

−6−5−4−3−2−101234

Energy / |β1|

ΓXM

ΓYS

Γ XM

ΓYS

Γ XM

ΓYS

cpz

ΓX

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YS

−6−5−4−3−2−10123

Energy / |β1|Γ

XM

ΓY

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ΓY

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XM

ΓY

S

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ΓM

KΓ

−5−4−3−2−10123

Energy / |β1|

ΓM

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MK

ΓΓ

MK

Γ

cqb

ΓX

MΓ

YS

−6−5−4−3−2−101234

Energy / |β1|

ΓX

MΓ

YSΓ

XM

ΓY

SΓ

XM

ΓY

S

19

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

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eigh

bou

r1s

t+

SO

C1s

t+

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OC

cqc

ΓX

MΓ

YS

−6−5−4−3−2−101234

Energy / |β1|

ΓX

MΓ

YSΓX

MΓ

YSΓX

MΓ

YS

cqd

ΓX

MΓ

YS

−6−5−4−3−2−101234

Energy / |β1|Γ

XM

ΓY

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XM

ΓY

SΓ

XM

ΓY

S

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Γ XM

ΓYS

−6−5−4−3−2−10123

Energy / |β1|

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cqp

ΓM

KΓ

−6−5−4−3−2−10123

Energy / |β1|

ΓM

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MK

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MK

Γ

20

Top

olog

yS

ketc

h1st

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ghb

our

1st

+2n

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t+

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OC

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ΓXM

ΓY

S−6−5−4−3−2−101234

Energy / |β1|

ΓXM

ΓY

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XM

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SΓ

XM

ΓY

S

cru

ΓX

MΓ

YS

−6−5−4−3−2−101234

Energy / |β1|Γ

XM

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XM

ΓY

SΓ

XM

ΓY

S

crz

ΓM

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−7−6−5−4−3−2−10123

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

dha

ΓX

MΓ

−6−5−4−3−2−10123

Energy / |β1|

ΓX

MΓΓ

XM

ΓΓ

XM

Γ

21

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

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r1s

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−7−6−5−4−3−2−10123

Energy / |β1|

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−8−7−6−5−4−3−2−101234

Energy / |β1|Γ

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XM

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fwb

ΓM

KΓ

−6−5−4−3−2−1012

Energy / |β1|

ΓM

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MK

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MK

Γ

fwe

ΓM

KΓ

−7−6−5−4−3−2−101234

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

22

Top

olog

yS

ketc

h1st

Nei

ghb

our

1st

+2n

dN

eigh

bou

r1s

t+

SO

C1s

t+

2nd

+S

OC

hca-a

ΓM

KΓ

−5−4−3−2−10123

Energy / |β1|

ΓM

KΓΓ

MK

ΓΓ

MK

Γ

hna

ΓX

MΓ

−5−4−3−2−10123

Energy / |β1|Γ

XM

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XM

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XM

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hnb

ΓM

KΓ

−6−5−4−3−2−1012

Energy / |β1|

ΓM

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ΓM

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−5−4−3−2−101234

Energy / |β1|

ΓM

KΓΓ

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MK

Γ

23

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29

2 Technical tutorial for tight-binding calculations

In this review, we have used the pythTB package, which can be downloaded from

http://www.physics.rutgers.edu/pythtb/. Here, we explain the elementary steps to per-

form a tight-binding (TB) calculation with the example of the honeycomb (hcb) lattice

(this tutorial is written for python 3.6 and pythTB 1.7.2). For setting up the calcula-

tions, a python script with the following content is needed:

1. Load necessary modules:

from __future__ import print_function

from pythTB import *

import numpy as np

2. Define latticelat and fractional coordinates of sites orb:

lat = [[1, 0], [0.5, np.sqrt(3)/2]]

orb = [[1/3, 1/3], [2/3, 2/3]]

3. Initialise the pythTB class:

my_model = tb_model(dim_k, dim_r, lat, orb, nspin=2)

Here, dim_k and dim_r define the dimensionality in reciprocal and real space,

respectively. For the hcb lattice, dim_k = 2 and dim_r = 2 are correct settings

corresponding to both a two-dimensional real space and a two-dimensional re-

ciprocal space. nspin = 2 enables spin-polarised calculations, while the default

nspin = 1 is the setting for spin-restricted (diamagnetic) systems.

4. Setting on-site energies α for each site as a list with the same ordering as in orb:

my_model.set_onsite([alpha, alpha])

For different chemical potentials, different values for α are possible. Herein, we

use α = 0.

30

5. Setting hopping integrals β:

my_model.set_hop(beta, i, j, [A, B])

Hopping integrals need to be specified one by one. In this function, β is the

hopping integral for interaction between vertex i in the first cell and vertex j

in the cell replicated by the translation vector R[A, B]. Where A and B are in

units of the respective lattice vector, moving from the first cell to the replicated

cell. For first-neighbour hoppings within the first cell, A and B can be set to 0.

In pythTB, symmetric hoppings are set automatically. i -> j + R[A, B] and

i -> j - R[A, B] are set simultaneosly, and cannot be specified twice in the

script. For example, if we have

my_model.set_hop(beta, 1, 0, [0, 1])

then there should be no my_model.set_hop(beta, 0, 1, [0, -1]), since this

is the same translation vector in opposite direction. In addition, individual hop-

ping elements, e.g. β′, can be specified afterwards by setting the mode add:

my_model.set_hop(beta’, 1, 0, [0, 1], mode=’add’)

In this case, the final hopping from site 0 to site 1 in the cell with translation

vector ~R = (0, 1) would be β + β′. In the case of hcb, three hoppings have to be

set:

my_model.set_hop(beta, 0, 1, [0, 0])

my_model.set_hop(beta, 1, 0, [0, 1])

my_model.set_hop(beta, 1, 0, [1, 0])

6. Setting the path in k -space which should be used for the band structure calculation:

(k_vec, k_dist, k_node) = my_model.k_path(path, nk)

With path being a list with the fractional coordinates of high-symmetry points

in the Brillouin zone and nk the total number of calculated eigenvalues for this

path. Returns k_vec, a list of the k -points, for which the eigenvalue problem

should be solved. k_dist is the accumulated distance of each k -point to the first

k -point. k_node is the distance of high-symmetry points to the first k -point.

31

These are, therefore, selected values from k_dist. For hexagonal cells, the setting

path = [[0.0, 0.0], [0.5, 0.5], [2/3, 1/3], [0.0, 0.0]]

will return a band structure along Γ -M -K -Γ . Additionally, the labels for these

high-symmetry points can be given:

label = [r’$\Gamma$’, ’M’, ’K’, r’$\Gamma$’]

7. Start the calculation and return eigenvalues for each k -point in k_vec:

evals = my_model.solve_all(k_vec)

8. Plotting can be done by, e.g., using matplotlib, which requires to load the package

via import matplotlib.pyplot as plt. Band structure plots can be created by

the following:

[plt.plot(k_vec, eigenvalue, color=’black’) for eigenvalue in evals]

plt.xticks(k_nodes, labels)

plt.ylabel(’Energy’)

plt.show()

9. Additional notes:

a) In order to use the formalism given by Kane and Mele[2] to take spin-orbit

coupling (SOC) into account, one needs to use the spin-polarised Hamiltonian

(thus, nspin = 2) and to manually define the Pauli matrix σz:

sigma_z = np.array([0, 0, 0, 1])

tsoc = -1.j*Lambda*sigma_z

With a SOC strength of λ. Note that for the Haldane model[3] is possible

for a spin-restricted calculation:

tsoc = Lambda*np.exp((1.j)*np.pi/2)

Note that if there are two equivalent paths electrons could use to travel from

a site to the second neighbour, the effect of SOC cancels out, e.g., two ver-

32

tices laying diagonally in a square show no spin-orbit contribution. Since the

second neighbour hopping includes both the normal hopping β2 and the SOC

contribution, two lines per second neighbour and the mode add can be used

to assign the hopping and SOC:

my_model.set_hop(beta_2, 0, 0, [1, 0])

my_model.set_hop(tsoc, 0, 0, [1, 0], mode=’add’)

b) Calculation details can be printed by my_model.display().

c) In order to estimate the Fermi level, it is necessary to integrate over the

whole Brillouin zone. In a k -grid of NG points, each grid point contributes

1/NG electrons (2/NG for spin-restricted calculations). The bands are occu-

pied from the lowest to higher bands, until the number of electrons matches

the number of electrons in the cell (see Section 3.1 and the example input file).

d) Obtaining the Chern number and Z2 invariant:

The pythtb.wf_array class calculates the Berry phase and related proper-

ties. After setting up the Hamiltonian, this class can be initialised with

my_array = wf_array(my_model, [nk, nk])

with nk being the number of grid points per axis.

my_array.solve_on_grid([x, y])

solves the model in a regular grid with the origin being at the point [x, y].

The Chern number equals the Berry phase

berry = [my_array.berry_flux([i]) for i in range(len(my_array))]

divided by 2π (with i being the band index). It is worth noting that for

calculating Chern number/Berry phase, a spin-restricted calculation is re-

quired. In order to get the Z2 invariant, the Wannier Charge Centres (WCC)

have to be calculated. An example for this is given on the pythTB website.[4]

33

References

1. http://www.physics.rutgers.edu/pythtb/, accessed 11 February 2020.

2. C. L. Kane and E. J. Mele, Phys. Rev. Lett., 2005, 95, 226801.

3. F. D. M. Haldane, Phys. Rev. Lett., 1988, 61, 2015.

4. http://www.physics.rutgers.edu/pythtb/examples.html#kane-mele-model-using-spinor-

features, accessed 11 February 2020.

34