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ARTICLE OPEN

Room-temperature magnetism and tunable energy gaps in

edge-passivated zigzag graphene quantum dots

Wei Hu

1,2

, Yi Huang

3,4,5

, Xinmin Qin

1

, Lin Lin

2,4

, Erjun Kan

6

, Xingxing Li

1

, Chao Yang

2

and Jinlong Yang

1

Graphene is a nonmagnetic semimetal and cannot be directly used as electronic and spintronic devices. Here, we demonstrate that

zigzag graphene nanoﬂakes (GNFs), also known as graphene quantum dots, can exhibit strong edge magnetism and tunable

energy gaps due to the presence of localized edge states. By using large-scale ﬁrst principle density functional theory calculations

and detailed analysis based on model Hamiltonians, we can show that the zigzag edge states in GNFs (C6n2H

6n

,n=1–25) become

much stronger and more localized as the system size increases. The enhanced edge states induce strong electron–electron

interactions along the edges of GNFs, ultimately resulting in a magnetic conﬁguration transition from nonmagnetic to intra-edge

ferromagnetic and inter-edge antiferromagnetic, when the diameter is larger than 4.5 nm (C

480

H

60

). Our analysis shows that the

inter-edge superexchange interaction of antiferromagnetic states between two nearest-neighbor zigzag edges in GNFs at the

nanoscale (around 10 nm) can be stabilized at room temperature and is much stronger than that exists between two parallel zigzag

edges in graphene nanoribbons, which cannot be stabilized at ultra-low temperature (3 K). Furthermore, such strong and localized

edge states also induce GNFs semiconducting with tunable energy gaps, mainly controlled by adjusting the system size. Our results

show that the quantum conﬁnement effect, inter-edge superexchange (antiferromagnetic), and intra-edge direct exchange

(ferromagnetic) interactions are crucial for the electronic and magnetic properties of zigzag GNFs at the nanoscale.

npj 2D Materials and Applications (2019) 3:17 ; https://doi.org/10.1038/s41699-019-0098-2

INTRODUCTION

Engineering techniques that use ﬁnite size effect to introduce

tunable edge magnetism and energy gap are by far the most

promising ways for enabling graphene

1

to be used in electronics

and spintronics.

2,3

Examples of ﬁnize-sized graphene nanostruc-

tures include one-dimensional (1D) graphene nanoribbons

(GNRs)

4–16

and zero-dimensional (0D) graphene nanoﬂakes (GNFs)

(also known as graphene quantum dots).

17–28

It is well known that

electronic and magnetic properties

29

of GNRs and GNFs depend

strongly on the atomic conﬁguration of their edges, which are of

either the armchair (AC) or zigzag (ZZ) types.

8

Edge magnetism has been predicted theoretically

10,11

and

observed experimentally

15,16

in ZZGNRs. The magnetism in

ZZGNRs results from ferromagnetic (FM) coupling for each zigzag

edge and antiferromagnetic (AFM) coupling between two parallel

zigzag edges of ZZGNRs. The strong FM coupling along each

zigzag edge has been predicted in theory

11

and conﬁrmed in

experiments.

16

However, the AFM coupling between two parallel

zigzag edges in ZZGNRs is weak, which cannot be stabilized even

at low temperature below 10 K

14

and rapidly weakens (~w

−2

)as

the ribbon-width wincreases.

13

Furthermore, the energy gap of

GNRs depend on several factors, such as the edge type (armchair

or zigzag) and the width of the nanoribbon,

8

thus cannot be easily

tuned. Such problem does not exist in GNFs due to the quantum

conﬁnement effect.

30

The ability to control the energy gap has

enabled GNFs to be used in promising applications in

electronics.

20

In addition, triangular ZZGNFs are theoretically

predicted to have strong edge magnetism even in small

systems.

31,32

However, triangular ZZGNFs have large formation

energy

24

and have not been synthesized experimentally. For-

tunately, hexagonal ZZGNFs exhibits signiﬁcantly improved

stability in ambient environment.

24

Recent experiments

33

have

also demonstrated that edge magnetism can be observed in

ZZGNFs when the edges are passivated by certain chemical

groups. However, semi-empirical tight-binding model

34,35

and ﬁrst

principle density functional theory (DFT) calculations

25–28

for

hexagonal ZZGNFs have been performed for small-sized systems

but found no magnetism (NM). Thus the prospect of ﬁnding stable

ﬁnite-sized graphene easily fabricated in experiments with both

strong edge magnetism and tunable energy gap seems dim.

In this letter, we systematically investigate the electronic and

magnetic properties of hexagonal ZZGNFs with the diameters in

the range of 1–12 nm (from C

24

H

12

to C

3750

H

150

). Using ﬁrst-

principles DFT calculations, we ﬁnd that both strong edge

magnetism and tunable energy gap can be realized simulta-

neously in large ZZGNFs stabilized at room temperature. We

demonstrate that spin polarization plays a crucial role as the

diameter of a ZZGNF increases beyond 4.5 nm (C

486

H

54

). A spin-

unpolarized calculation shows that edge states become increas-

ingly more localized as the size of a ZZGNF increases. These edge

states form a half-ﬁlled pseudo-band and is thus unstable. Adding

spin-polarization allows the edge states to spontaneously split into

Received: 8 October 2018 Accepted: 18 March 2019

1

Hefei National Laboratory for Physical Sciences at Microscale, Department of Chemical Physics, and Synergetic Innovation Center of Quantum Information and Quantum Physics,

University of Science and Technology of China, 230026 Hefei, Anhui, China;

2

Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA;

3

Department of Applied Physics, Xi’an Jiaotong University, 710049 Xi’an, Shaanxi, China;

4

Department of Mathematics, University of California, Berkeley, CA 94720, USA;

5

School

of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA and

6

Department of Applied Physics and Institution of Energy and Microstructure, Nanjing

University of Science and Technology, 210094 Nanjing, Jiangsu, China

Correspondence: Lin Lin (linlin@math.berkeley.edu) or Chao Yang (cyang@lbl.gov) or Jinlong Yang (jlyang@ustc.edu.cn)

www.nature.com/npj2dmaterials

Published in partnership with FCT NOVA with the support of E-MRS

spin-polarized occupied and unoccupied states. This separation

results in a magnetic conﬁguration transition from an NM

conﬁguration to a strong inter-edge AFM conﬁguration. It also

opens a tunable band gap that can be easily controlled by

quantum conﬁnement effect. These properties make GNFs better

candidate materials for nanoelectronics than GNRs.

8

We also

conﬁrm that ZZGNFs passivated by different chemical groups all

exhibit similar behavior. Such ﬂexibility may facilitate future

experimental synthesis of such ZZGNFs.

RESULTS AND DISCUSSION

We demonstrate the importance of spin polarization using C

864

H

72

(6 nm) as an example (Fig. 2). From a spin-unpolarized calculation,

we observe strong and localized edge states (Fig. 2e). These edge

states contribute to high electron density along the edges of

C

864

H

72

. Furthermore, these edge states become much stronger

and more localized as the ZZGNF size increases.

28

The presence of

strong edge states makes the ZZGNF metallic at the nanoscale.

35

The projected density of states (PDOS) of carbon edges of C

864

H

72

plotted in Fig. 2c clearly show a considerably high density of states

(DOS) near the Fermi level. This ﬁgure conﬁrms that C

864

H

72

is

predicted to be metallic in a spin-unpolarized calculation. The

metallic nature of the ZZGNF can be attributed to the presence of

strong localized edge states.

28

However, a spin-polarized calculation shows that half-ﬁlled

metallic edge states are not stable, and can spontaneously split

into two types of occupied and unoccupied states as shown in

Fig. 2b, d. As a result, a magnetic conﬁguration transition from a

non-magnetic (NM) conﬁguration to a magnetic conﬁguration that

exhibits intra-edge FM and inter-edge AFM characters can be

observed in Fig. 2f. This transition can be interpreted as the

consequence of Mott-type competition between the kinetic

(hopping) energy and the intra-edge (on-site) electron–electron

interaction energy as the system size increases. Lowering kinetic

energy by increasing the system size tends to produce delocalized

spin states across all edges, while reducing the electron–electron

interaction energy as the system size tends to penalize

simultaneous occupation of the same edge by spin up and spin

down electrons. Both semi-local GGA-PBE and hybrid HSE06

calculations (the details are given in the Supplemental Material)

indicate that for small systems, kinetic energy plays a more

dominant role. This observation agrees with previous theoretical

prediction of the NM conﬁguration for hexagonal ZZGNFs.

25–

28,34,35

Only as the system size increases, the effective

electron–electron interaction energy associated with the edge

states starts to dominate and is ultimately responsible for this

magnetic conﬁguration transition.

Figure 1a shows the variation of relative energy of NM, AFM,

and FM magnetic conﬁgurations in ZZGNFs and ZZGNRs,

respectively, with respect to system size. Our calculations show

that AFM states are much more stable than NM and FM states in

large ZZGNFs, and a magnetic conﬁguration transition occurs as

the diameter of the ZZGNF becomes larger than 4.5 nm

(C

486

H

54

).

31

We believe the FM coupling along each zigzag edge

that belong to the same sublattice, are likely to be induced by

intra-edge direct exchange interactions. The AFM coupling

between two nearest-neighbor edges belonging to different

sublattices are likely to be induced by inter-edge superexchange

interactions facilitated by a carbon–carbon double bond (C=C) at

the corner where two nearest-neighbor edges meet in ZZGNFs at

the nanoscale. The local magnetic moment deﬁned by M

i

=

j<^

ni"gt;−<^

ni#>j, where <^

niσgt;is spin electron density with σ

=↑(spin-up) or ↓(spin-down)), at the carbon atom i. Figure 2b

shows the local magnetic moment of carbon in the the middle of

each zigzag edge in ZZGNFs (with the largest magnetic moment)

increases with the system size, and converges to 0.3μ

B

when the

diameter is larger than than 6 nm (C

864

H

72

). Furthermore, there is

no charge transfer (<^

ni"gt;+<^

ni#gt;≈4) between carbon atoms

Fig. 1 aRelative energy per edge atom (ΔE(AFM-NM) and ΔE(AFM-

FM)) of NM, AFM, and FM coupling between different edges in

ZZGNFs and ZZGNRs and (b) spin electron density <^

niσgt;(σ=↑

(spin-up) or ↓(spin-down)) at the carbon atom iin the middle of

each zigzag edge in AFM ZZGNFs under the variation of the

diameter size (ZZGNFs) or ribbon-width length size (ZZGNRs). The

red and blue regions represent the stable NM (ΔE(AFM–NM) ≈0) and

AFM (ΔE(AFM–NM) < 0) coupling between different edges in

ZZGNFs, respectively. The critical temperature is estimated by the

mean-ﬁeld theory T=ΔE/k

B

, where k

B

is the Boltzmann constant

Fig. 2 Electronic structure of edge states in C

864

H

72

in two different

magnetic conﬁgurations (NM and AFM), including the schematic

illustration of orbital diagram of superexchange interaction of edge

states in the (a) NM and (b) AFM conﬁgurations, projected density of

states (PDOS) of edges in the (c) NM and (d) AFM conﬁgurations, (e)

local density of states (LDOS) of Fermi level (pink isosurfaces) in the

NM conﬁguration and (f) spin density isosurfaces in the AFM

conﬁguration. The red and blue isosuraces in (f) represent the spin-

up and spin-down states, respectively. The red and blue lines in (d)

represent the PDOS contributed by sublattice A (spin-up edges) and

B (spin-down edges) atoms in graphene, respectively. The fermi level

is marked by green dotted lines and set to zero

W. Hu et al.

2

npj 2D Materials and Applications (2019) 17 Published in partnership with FCT NOVA with the support of E-MRS

1234567890():,;

sitting on different edges that belong to the same or different

sublattices in ZZGNFs as the system size increases.

Notice that the intra-edge direct exchange interaction via FM

coupling along each zigzag edge in ZZGNFs is similar to that in

ZZGNRs. However, the inter-edge superexchange interaction via

AFM coupling between two nearest-neighbor edges through a

C=C bond (Fig. 2a) in ZZGNFs can be stabilized at room

temperature (298 K) and is much stronger than that via AFM

coupling between two parallel edges though π-bonds in ZZGNRs

as shown in Fig. 1a, where such AFM spin polarization weakens

rapidly as the ribbon-width increases in ZZGNRs

13

and cannot be

stabilized even at ultra-low temperature (3 K).

14

Our DFT calcula-

tions conﬁrm that the energy difference associated with AFM and

FM coupling between two parallel edges in large-scale 1D ZZGNRs

is negligible compared to that reported in ZZGNFs.

The enhanced stability of spin-polarized ZZGNFs can be

understood by using the Heisenberg model. We consider each

FM edge as one site and enumerate all possible magnetic

conﬁgurations, and the Hamiltonian can be written as

^

H¼

XJi;jMi

!Mj

!(1)

where J

i,j

is the exchange parameter between two sites iand j,Mi

!

and Mj

!are the corresponding spin magnetic moments. There are

four different magnetic states in C

864

H

72

, there of which are AFM,

AFM1, and AFM2 conﬁgurations and one is FM conﬁguration as

shown in Fig. 3. The total energies of magnetic conﬁgurations E

(AFM), E(AFM1), E(AFM2), and E(FM) can be computed by DFT

calculations, and the exchange parameters can be evaluated by

solving the following least-squares-ﬁtting problem

36

EðAFMÞ¼ð6J16J2þ3J3ÞM2þE0

EðAFM1Þ¼ð2J1þ2J2J3ÞM2þE0

EðAFM2Þ¼ðJ1þ2J2þ3J3ÞM2þE0

EðFMÞ¼ð6J16J23J3ÞM2þE0

(2)

where J

1

,J

2

, and J

3

are ortho-edge, meta-edge, and para-edge

exchange interaction parameters, respectively, Mis the spin

magnetic moment at each edge, and E

0

is the nonmagnetic

reference total energy. The solution yields J

1

=−0.038351 eV, J

2

=

0.000954 eV, and J

3

=0.001633 eV for two nearest-neighbor edges

of C

864

H

72

, which are 10 times stronger than the exchange

interaction parameters between two parallel edges in ZZGNFs and

ZZGNRs. Therefore, ZZGNFs at the nanoscale have strong edge

magnetism at room temperature and can be directly used in

nanospintronics, superior to that in ZZGNRs at the nanoscale.

7,13

We perform ab initio molecular dynamics (AIMD) simulations on

ZZGNFs and check the effect of temperature on electronic and

magnetic properties of C

486

H

54

in different AFM and FM

conﬁgurations (the details are given in the Supplemental Material).

We ﬁnd that the AFM conﬁguration of C

486

H

54

can remain stable

at room temperature of T=300 K at least within 1.6 ps.

Furthermore, the FM conﬁguration of C

486

H

54

rapidly transfers

into the AFM conﬁguration within 30.0 fs at room temperature of

T=300 K. Furthermore, after t=1.5 ps, C

486

H

54

is slightly bent,

26

although it still keeps the AFM conﬁguration.

We also check the effects of using different types of atoms (e.g.,

bare and ﬂuorine) to passivate ZZGNFs, and how the shape (non-

hexagonal) of ZZGNFs may alter their electronic and magnetic

properties. We ﬁnd that magnetic conﬁguration transition (from

NM to AFM) and semiconductor characteristics (The energy gaps

of 0.54, 0.34, and 0.41 eV, respectively, for bare C

864

,ﬂuorine-

passivated C

864

F

72

and non-hexagonal C

839

H

71

) of ZZGNFs are

independent of the type of passivating atoms

37

as plotted in

Fig. 4. These properties suggest that it is relatively easy to create a

chemical environment in which the synthesis of large scales

ZZGNFs with tunable edge magnetism and energy gaps can be

easily accommodated. The possibility of rapid synthesis makes

ZZGNFs ideal candidates for electronic and spintronic devices.

38

We remark that magnetic conﬁguration transition and the

associated tunable electronic structures in ZZGNFs, especially

energy gaps, can also be understood in terms of the Hubbard

model.

31

From our ﬁrst principle calculations, we ﬁnd that

choosing the parameters t=2.5 eV and U=2.1 eV in the Hubbard

model can well reproduce the size-dependent energy gaps (the

details are given in the Supplemental Material). In Fig. 5, we plot

how the HOMO-LUMO energy gap E

g

changes with respect to the

size of ZZGNFs and ACGNFs in two different magnetic conﬁgura-

tions (NM and AFM). Our DFT calculations and mean-ﬁeld Hubbard

Fig. 3 Spin density isosurfaces of hydrogen-passivated C

864

H

72

in four different magnetic states, three types of antiferromagnetic ((a) AFM, (b)

AFM1, and (c) AFM2) and one type of ferromagnetic ((d) FM) coupling at the inter edges. The red and blue isosurfaces represent the spin-up

and spin-down states, respectively

Fig. 4 Spin density isosurfaces and total density of states ( TDOS) of

(a) bare (C

864

), (b)ﬂuorine-passivated (C

864

F

72

), and (c) non-

hexagonal (C

839

H

71

) ZZGNFs in the AFM conﬁguration. The red

and blue isosurfaces represent the spin-up and spin-down states,

respectively. The energy differences (E

AFM

−E

NM

) between AFM and

NM conﬁgurations of these ZZGNFs are shown above the ﬁgures.

For hexagonal hydrogen-passivated C

864

H

72

,ΔE(AFM−NM) =

−17.9 meV. The fermi level is marked by green dotted lines and

set to zero

W. Hu et al.

3

Published in partnership with FCT NOVA with the support of E-MRS npj 2D Materials and Applications (2019) 17

model show similar results, i.e., the energy gap E

g

of ZZGNF

decreases as its size increases. In particular, we ﬁnd that the

energy gap of NM ZZGNFs decreases more rapidly with respect to

the system size than that of AFM ZZGNFs, due to the presence of

edge states whose electron density near the edges of ZZGNFs as

shown in Fig. 2e. This observation is consistent with previous

results obtained from tight-binding models

34,35

and DFT calcula-

tions.

27,28

However, AFM semiconducting ZZGNFs show similar

energy gap scaling compared to that of NM ACGNFs at the

nanoscale.

28

Therefore, edge states should have little effect on the

energy gaps of AFM ZZGNFs and the quantum conﬁnement

effect

30

is the only factor to control the energy gaps in ZZGNFs

and ACGNFs (Fig. 5a). In detail, NM ZZGNFs exhibits metallic

characters (E

g

is smaller than the thermal ﬂuctuation (25 meV) at

room temperature) when the diameter is larger than 7 nm

(C

1350

H

90

), but AFM ZZGNFs with the diameter of 12 nm

(C

3750

H

150

) still behaves as a semiconductor with a sizable energy

gap E

g

=0.23 eV, similar to the case of NM ACGNFs.

28

Therefore,

ZZGNFs at the nanoscale can be directly used in nanoelectronics.

In summary, using large-scale ﬁrst principle calculations, we

demonstrate that the electronic and magnetic properties of

hexagonal zigzag that graphene nanoﬂakes (ZZGNFs) can be

signiﬁcantly affected by the system size. We found that the zigzag

edge states in ZZGNFs become much stronger and more localized

as the system size increases. The presence of these edge states

induce strong electron–electron interactions along the edges of

ZZGNFs, resulting in a magnetic conﬁguration transition from

nonmagnetic to intra-edge FM and inter-edge AFM when the

diameter is larger than 4.5 nm. On the other hand, such strong and

localized edge states are also responsible for making ZZGNFs

semiconducting with a tunable energy gap. The energy gap can

be controlled by merely adjusting the system size. Therefore,

ZZGNFs with strong edge magnetism, tunable energy gaps and

room-temperature stability may be promising candidates for

practical electronic and spintronic applications.

METHODS

We use the Kohn–Sham DFT-based electronic structure analysis tools

implemented in the Spanish Initiative for Electronic Simulations with

Thousands of Atoms (SIESTA)

39

software package. We use the generalized

gradient approximation of Perdew, Burke, and Ernzerhof (GGA–PBE)

40

exchange correlation functional with collinear spin polarization, and the

double zeta plus polarization orbital basis set (DZP) to describe the valence

electrons within the framework of a linear combination of numerical

atomic orbitals (LCAO).

41

Because semi-local GGA–PBE calculations are less

reliable in predicting the electronic structures of ZZGNFs, the screened

hybrid HSE06

42

calculations implemented in HONPAS

43–45

(Hefei Order-N

Packages for Ab Initio Simulations based on SIESTA) are also used to

compute the electronic and magnetic properties of ZZGNFs. All atomic

coordinates are fully relaxed using the conjugate gradient (CG) algorithm

until the energy and force convergence criteria of 10

−4

eV and 0.02 eV/Å,

respectively, are reached.

For initial magnetic moment setting of spin-polarized DFT calculations in

ZZGNFs, we set all the carbon atoms with initial magnetic moments of 1μ

B

for the FM conﬁguration and only set the edged carbon atoms with initial

magnetic moments of 1 or −1μ

B

, and then optimize the structures and

magnetic moments of ZZGNFs.

Due to the large number of atoms contained in hexagonal hydrogen-

passivated ZZGNFs (C6n2H

6n

,n=1–25), we use the recently developed

Pole EXpansion and Selected Inversion (PEXSI) method

46–48

to accelerate

the eigenvalue problem in the Kohn–Sham DFT calculations. The PEXSI

technique can efﬁciently utilize the sparsity of the Hamiltonian and overlap

matrices generated in SIESTA and overcome the cubic scaling limit for

solving Kohn–Sham DFT, and scales at most as quadratic scaling even for

metallic systems, such as graphene. Furthermore, the PEXSI method is

highly scalable

49

and can scale up to 100,000 processors on high

performance machines.

We perform AIMD simulations on ZZGNFs to check the effect of

temperature on electronic and magnetic properties of ZZGNFs. The

simulations are performed for about 1.6 ps with a time step of 2.0 fs at

room temperature of T=300 K controlled by a Nose–Hoover thermostat.

50,51

DATA AVAILABILITY

The authors conﬁrm that the data supporting the ﬁndings of this study are available

within the article and its Supplementary Materials.

ACKNOWLEDGEMENTS

This work was performed, in part, under the auspices of the U.S. Department of

Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-

07NA27344. Support for this work was provided through Scientiﬁc Discovery

through Advanced Computing (SciDAC) program funded by U.S. Department of

Energy, Ofﬁce of Science, Advanced Scientiﬁc Computing Research and Basic Energy

Sciences (W.H., L.L., and C.Y.), by the Center for Applied Mathematics for Energy

Research Applications (CAMERA), which is a partnership between Basic Energy

Sciences and Advanced Scientiﬁc Computing Research at the U.S. Department of

Energy (L.L. and C.Y.), and by the Department of Energy under Grant No. DE-

SC0017867 (L.L.). This work is also partially supported by the National Key Research

and Development Program of China (Grant no. 2016YFA0200604) and the National

Natural Science Foundation of China (NSFC) (Grant nos. 21688102, 51522206, and

21803066), and the Strategic Priority Research Program of Chinese Academy of

Sciences (Grant no. XDC01000000), the Research Start-Up Grants (Grant no.

KY2340000094) from University of Science and Technology of China, and the

Chinese Academy of Sciences Pioneer Hundred Talents Program. Y.H. acknowledges

support from the Education Program for Talented Students of Xi’an Jiaotong

University. We thank the National Energy Research Scientiﬁc Computing (NERSC)

center, and the USTCSCC, SC-CAS, Tianjin, and Shanghai Supercomputer Centers for

the computational resources.

AUTHOR CONTRIBUTIONS

W.H., L.L., E.K., C.Y., and J.Y. designed the idea of this manuscript and supported this

project. W.H. performed all the DFT calculations in SIESTA. Y.H. wrote the codes of

Hubbard model. X.Q. performed the hybrid HSE06 calculations in HONPAS. All the

authors helped to write, modify, and analyze this manuscript.

ADDITIONAL INFORMATION

Supplementary Information accompanies the paper on the npj 2D Materials and

Applications website (https://doi.org/10.1038/s41699-019-0098-2).

Competing interests: The authors declare no competing interests.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims

in published maps and institutional afﬁliations.

REFERENCES

1. Novoselov, K. S. et al. Electric ﬁeld effect in atomically thin carbon ﬁlms. Science

306, 666–669 (2004).

2. Geim, A. K. & Novoselov, K. S. The rise of graphene. Nat. Mater. 6, 183–191 (2007).

Fig. 5 Energy gap E

g

(eV) of ZZGNFs and ACGNFs in two different

magnetic conﬁgurations (NM and AFM) as a function of the number

of carbon atoms, computed with two different methods: (a) DFT

calculations and (b) Hubbard model (t=2.5 eV and U=2.1 eV)

W. Hu et al.

4

npj 2D Materials and Applications (2019) 17 Published in partnership with FCT NOVA with the support of E-MRS

3. Neto, A. H. C., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The

electronic properties of graphene. Rev. Mod. Phys. 18, 109 (2009).

4. Han, M. Y., Özyilmaz, B., Zhang, Y. & Kim, P. Energy band-gap engineering of

graphene nanoribbons. Phys. Rev. Lett. 98, 206805 (2007).

5. Li, X., Wang, X., Zhang, L., Lee, S. & Dai, H. Chemically derived, ultrasmooth

graphene nanoribbon semiconductors. Science 319, 1229–1232 (2008).

6. Jia, X. et al. Controlled formation of sharp zigzag and armchair edges in graphitic

nanoribbons. Science 323, 1701–1705 (2009).

7. Son, Y.-W., Cohen, M. L. & Louie, S. G. Half-metallic graphene nanoribbons. Nature

444, 347–349 (2006).

8. Son, Y.-W., Cohen, M. L. & Louie, S. G. Energy gaps in graphene nanoribbons. Phys.

Rev. Lett. 97, 216803 (2006).

9. Yang, L., Park, C.-H., Son, Y.-W., Cohen, M. L. & Louie, S. G. Quasiparticle energies

and band gaps in graphene nanoribbons. Phys. Rev. Lett. 99, 186801 (2007).

10. Kan, E., Li, Z., Yang, J. & Hou, J. G. Half-metallicity in edge-modiﬁed zigzag gra-

phene nanoribbons. J. Am. Chem. Soc. 130, 4224–4225 (2008).

11. Yazyev, O. V. & Katsnelson, M. I. Magnetic correlations at graphene edges: Basis

for novel spintronics devices. Phys. Rev. Lett. 100, 047209 (2008).

12. Long, M.-Q., Tang, L., Wang, D., Wang, L. & Shuai, Z. Theoretical predictions of

size-dependent carrier mobility and polarity in graphene. J. Am. Chem. Soc. 131,

17728–17729 (2009).

13. Jung, J., Pereg-Barnea, T. & MacDonald, A. H. Theory of interedge superexchange

in zigzag edge magnetism. Phys. Rev. Lett. 102, 227205 (2009).

14. Kunstmann, J., Özdoğan, C., Quandt, A. & Fehske, H. Stability of edge states and

edge magnetism in graphene nanoribbons. Phys. Rev. B 83, 045414 (2011).

15. Magda, G. Z. et al. Room-temperature magnetic order on zigzag edges of narrow

graphene nanoribbons. Nature 514, 608–611 (2014).

16. Rufﬁeux, P. et al. On-surface synthesis of graphene nanoribbons with zigzag edge

topology. Nature 531, 489–492 (2016).

17. Ponomarenko, L. A. et al. Chaotic dirac billiard in graphene quantum dots. Science

320, 356–358 (2008).

18. Shang, N. G. et al. Catalyst-free efﬁcient growth, orientation and biosensing

properties of multilayer graphene nanoﬂake ﬁlms with sharp edge planes. Adv.

Funct. Mater. 18, 3506–3514 (2008).

19. de Parga, A. L. V. et al. Periodically rippled graphene: Growth and spatially

resolved electronic structure. Phys. Rev. Lett. 100, 056807 (2008).

20. Ritter, K. A. & Lyding, J. W. The inﬂuence of edge structure on the electronic

properties of graphene quantum dots and nanoribbons. Nat. Mater. 8, 235–242

(2009).

21. Kuc, A., Heine, T. & Seifert, G. Structural and electronic properties of graphene

nanoﬂakes. Phys. Rev. B 81, 085430 (2010).

22. Wimmer, M., Akhmerov, A. R. & Guinea, F. Robustness of edge states in graphene

quantum dots. Phys. Rev. B 82, 045409 (2010).

23. Eda, G. et al. Blue photoluminescence from chemically derived graphene oxide.

Adv. Mater. 22, 505–509 (2010).

24. Lin, P.-C. et al. Nano-sized graphene ﬂakes: Insights from experimental synthesis

and ﬁrst principles calculations. Phys. Chem. Chem. Phys. 19, 6338–6344 (2017).

25. Zhou, Y. et al. Hydrogenated graphene nanoﬂakes: semiconductor to half-metal

transition and remarkable large magnetism. J. Phys. Chem. C 116, 5531–5537

(2012).

26. Wohner, N., Lam, P. & Sattler, K. Energetic stability of graphene nanoﬂakes and

nanocones. Carbon 67, 721 (2014).

27. Singh, S. K., Neek-Amal, M. & Peeters, F. M. Electronic properties of graphene

nano-ﬂakes: energy gap, permanent dipole, termination effect, and raman

spectroscopy. J. Chem. Phys. 140, 074304 (2014).

28. Hu, W., Lin, L., Yang, C. & Yang, J. Electronic structure and aromaticity of large-

scale hexagonal graphene nanoﬂakes. J. Chem. Phys. 141, 214704 (2014).

29. Yazyev, O. V. Emergence of magnetism in graphene materials and nanos-

tructures. Rep. Prog. Phys. 73, 056501 (2010).

30. Raty, J., Galli, G. & van Buuren, T. Quantum conﬁnement and fullerenelike surface

reconstructions in nanodiamonds. Phys. Rev. Lett. 90, 037401 (2003).

31. Fernández-Rossier, J. & Palacios, J. J. Magnetism in graphene nanoislands. Phys.

Rev. Lett. 99, 177204 (2007).

32. Wang, W. L., Meng, S. & Kaxiras, E. Graphene nanoﬂakes with large spin. Nano

Lett. 8, 241–245 (2008).

33. Sun, Y. et al. Magnetism of graphene quantum dots. npj Quantum Mater. 2,5

(2017).

34. Zhang, Z. Z., Chang, K. & Peeters, F. M. Tuning of energy levels and optical

properties of graphene quantum dots. Phys. Rev. B 77, 235411 (2008).

35. Güçlü, A. D., Potasz, P. & Hawrylak, P. Excitonic absorption in gate-controlled

graphene quantum dots. Phys. Rev. B 82, 155445 (2010).

36. Li, X., Wu, X. & Yang, J. Room-temperature half-metallicity in la(mn,zn)aso alloy via

element substitutions. J. Am. Chem. Soc. 136, 5664–5669 (2014).

37. Kabir, M. & Saha-Dasgupta, T. Manipulation of edge magnetism in hexagonal

graphene nanoﬂakes. Phys. Rev. B 90, 035403 (2014).

38. Hawrylak, P., Peeters, F. & Ensslin, K. Carbononics—integrating electronics, pho-

tonics and spintronics with graphene quantum dots. Phys. Status Solidi RRL 10,

11–12 (2016).

39. Soler, J. M. et al. The siesta method for ab initio order-n materials simulation. J.

Phys.: Condens. Matter 14, 2745 (2002).

40. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made

simple. Phys. Rev. Lett. 77, 3865 (1996).

41. Junquera, J., Paz, O., Sánchez-Portal, D. & Artacho, E. Numerical atomic orbitals for

linear-scaling calculations. Phys. Rev. B 64, 235111 (1996).

42. Heyd, J., Scuseria, G. E. & Ernzerhof, M. Erratum: “hybrid functionals based on a

screened coulomb potential”[J. Chem. Phys. 118, 8207 (2003)]. J. Chem. Phys.

124, 219906 (2006).

43. Shang, H., Li, Z. & Yang, J. Implementation of exact exchange with numerical

atomic orbitals. J. Phys. Chem. A 114, 1039–1043 (2010).

44. Shang, H., Li, Z. & Yang, J. Implementation of screened hybrid density functional

for periodic systems with numerical atomic orbitals: basis function ﬁtting and

integral screening. J. Chem. Phys. 135, 034110 (2011).

45. Qin, X., Shang, H., Xiang, H., Li, Z. & Yang, J. Honpas: a linear scaling open-

source solution for large system simulations. Int. J. Quantum Chem. 115,

647–655 (2014).

46. Lin, L., Lu, J., Ying, L., Car, R. & E, W. Fast algorithm for extracting the diagonal of

the inverse matrix with application to the electronic structure analysis of metallic

systems. Commun. Math. Sci. 7, 755–777 (2009).

47. Lin, L., Chen, M., Yang, C. & He, L. Accelerating atomic orbital-based electronic

structure calculation via pole expansion and selected inversion. J. Phys.: Condens.

Matter 25, 295501 (2013).

48. Lin, L., Garca, A., Huhs, G. & Yang, C. SIESTA-PEXSI: massively parallel method for

efﬁcient and accurate ab initio materials simulation without matrix diagonaliza-

tion. J. Phys.: Condens. Matter 26, 305503 (2014).

49. Hu, W., Lin, L. & Yang, C. DGDFT: a massively parallel method for large scale

density functional theory calculations. J. Chem. Phys. 143, 124110 (2015).

50. Nosé, S. A uniﬁed formulation of the constant temperature molecular dynamics

methods. J. Chem. Phys. 81, 511 (1984).

51. Hoover, W. G. Canonical dynamics: equilibrium phase-space distributions. Phys.

Rev. A 31, 1695 (1985).

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