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Mixed magnetic edge states in graphene quantum dots

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Mixed magnetic edge states in graphene quantum

dots

Junyao Li†a, Xiaofeng Liu†a,b, Lingyun Wana, Xinming Qina,

Wei Hu∗a, and Jinlong Yang∗a

aDepartment of Chemical Physics, Hefei National Laboratory for Physical Sciences at

the Microscale, and Synergetic Innovation Center of Quantum Information and

Quantum Physics, University of Science and Technology of China, Hefei, Anhui

230026, China

bSchool of Electronic Science and Applied Physics, Hefei University of Technology,

Hefei, Anhui 230009, China

†These authors have contributed to this work equally

E-mail: whuustc@ustc.edu.cn (Wei Hu),jlyang@ustc.edu.cn (Jinlong Yang)

September 2021

Abstract. Graphene quantum dots (GQDs) exhibit abundant magnetic edge states

with promising applications in spintronics. Hexagonal zigzag GQDs possess a ground

state with an antiferromagnetic (AFM) inter-edge coupling, followed by a metastable

state with ferromagnetic (FM) inter-edge coupling. By analyzing the Hubbard

model and performing large-scale spin-polarized density functional theory calculations

containing thousands of atoms, we predict a series of new mixed magnetic edge states

of GQDs arising from the size eﬀect, namely mix-n, where nis the number of spin

arrangement parts at each edge, with parallel spin in the same part and anti-parallel

spin between adjacent parts. In particular, we demonstrate that the mix-2 state of bare

GQDs (C6N2) appears when N≥4 and the mix-3 state appears when N≥6, where N

is the number of six-membered-ring at each edge, while the mix-2 and mix-3 magnetic

states appear in the hydrogenated GQDs with N = 13 and N = 15, respectively.

Keywords: Graphene quantum dots, mixed magnetic edge states, Hubbard model,

density functional theory

1. Introduction

Since the successful preparation of isolated single-layered graphene samples [1, 2],

graphene and its derivatives have aroused tremendous interest in theoretical and

experimental research [3, 4]. Graphene, a single layer composed of sp2hybrid carbon

atoms, opens a new era in nanoelectronics due to its outstanding properties containing

remarkably high carrier mobility at room temperature [5], unique optical [6], thermal, [7]

and mechanical properties [8]. In particular, the exploration of the graphene-based

nanostructures involving ﬁnite-sized one-dimensional graphene nanoribbons [9–11] and

zero-dimensional graphene quantum dots (GQDs) has sprung up due to their unique

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Mixed magnetic edge states in graphene quantum dots 2

electronic, magnetic properties and mature fabrication state of the art [9–13] due to the

size eﬀect.

Graphene nanostructures are formed by two triangular interpenetrating sublattices

carbon atoms and are terminated by either zigzag or armchair edges [14], which gives

rise to unconventional electronic and magnetic properties and makes them promising

candidates for spintronics. The carbon atoms at a particular zigzag edge belong to one

of the graphene sublattices, contributing to the localized nonbonding πstates near the

Fermi level. Because of the size eﬀect, these edge states will become much stronger

and more localized as the size of the system increases, which can introduce magnetic

properties in the system, so the engineering techniques based on ﬁnite size eﬀect can

be the most promising way for spintronic applications of graphene and its derivatives.

However, armchair edges contain atoms from diﬀerent (or adjacent) sublattices, which

show no edge states. The edge states of graphene nanostructures with zigzag edges can

induce localized magnetic moment because of the half-ﬁlled ﬂat band at the Fermi level.

The edge magnetism of 1D zigzag graphene nanoribbons (ZZGNRs) [11, 15–17], as well

as rectangular [18], triangular [19, 20], and hexagonal zigzag GQDs [21, 22], has been

studied extensively in theory and experiments.

Among these graphene-based nanostructures, hexagonal zigzag GQDs exhibit

signiﬁcantly improved stability in ambient environment [23]. The molecular formula of

hexagonal zigzag GQDs satisﬁes C6N2, where N denotes the number of six-membered-

ring at each edge. The carbon atoms of hexagonal zigzag GQDs (C6N2) at a particular

edge belong to the same sublattices, while the alternate edge C atoms belong to diﬀerent

sublattices, leading to an antiferromagnetic (AFM) ground state with ferromagnetic

(FM) intra-edge coupling and AFM inter-edge coupling. Moreover, The FM state with

ferromagnetic intra-edge and inter-edge coupling has also been predicted. For example,

Saha-Dasgupta et al. have recently reported a new mixed magnetic state in hexagonal

zigzag GQDs with N= 3 by using a large Coulomb repulsion energy (U≥6.0 eV) and

carrier doping [22].

In this work, by analyzing the Hubbard model, we predict a series of new types

of mixed magnetic states in hexagonal zigzag GQDs with large Ndue to the quantum

size eﬀect. These mixed magnetic states are labeled as mix-n, and here we focus on

the magnetic mix-2 and mix-3 states. We perform large-scale spin-polarized density

functional theory calculations containing thousands of atoms to verify our theoretical

prediction obtained from the Hubbard model. For bare GQDs, while the carbon number

is larger than 54 (N= 3), we ﬁnd that the system transforms from the nonmagnetic

(NM) state to an AFM state with parallel spin arrangement at a particular edge and

anti-parallel spin arrangement between the nearest neighbor edges, which is in good

agreement with the previous results [19, 21, 22]. Furthermore, the mix-2 state appears

when N≥4 and the mix-3 state comes into sight when N≥6. For mix-2, all edges

can be divided into two equal parts with opposite spin orientation, while for mix-3,

the magnetic state of each edge can be cut into three groups and the two side groups

are coupled antiparallel with the middle one. The order of stability of these magnetic

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states for GQDs with the same size is: AFM >FM >mix-2 >mix-3. For GQDs with

edge passivation , the system transforms from nonmagnetic to magnetic when its size

increases to N= 9 (C486 H54), while the mix-2 and mix-3 magnetic states appear when

the sizes increase to N= 13 and N= 15 respectively.

2. Computational Models and Methods

2.1. Hubbard Model

The Hamiltonian of Hubbard model [24] can be expressed as

ˆ

H=−tX

hi,ji,σ

ˆc†

i,σ ˆcj,σ +U

N

X

i=1

ˆni↑ˆni↓(1)

where the ﬁrst term describes the kinetic energy of the system, tis the hopping integral,

and ˆc†

i,σ and ˆci,σ are creation and annihilation operators, respectively. Uis the Coulomb

repulsion energy between two electrons with opposite spins at the same point and ˆni,σ

is the number operator. The second term of the Hamiltonian can be simpliﬁed by

mean-ﬁeld approximation [25], and ˆncan be split into two terms

ˆni↑=hˆni↑i+ ∆ˆni↑(2)

ˆni↓=hˆni↓i+ ∆ˆni↓(3)

where the ﬁrst term of the right-hand side is from the mean-ﬁeld approximation, and

the second one is a small quantity corresponding to the variation of ˆn.

Finally, the Hamiltonian can be divided into three terms according to the direction

of spin, which can greatly reduce the computing time (See Supplementary Materials

(SM) for the detailed derivation process).

ˆ

H=ˆ

H↑+ˆ

H↓−U

N

X

i=1

hˆni↑i hˆni↓i(4)

ˆ

H↑=−tX

hi.ji

ˆc†

i↑ˆcj↑+U

N

X

i=1

hˆni↓iˆni↑(5)

ˆ

H↓=−tX

hi,ji

ˆc†

i↓ˆcj↓+U

N

X

i=1

hˆni↑iˆni↓(6)

Under the mean-ﬁeld approximation, we can iteratively solve the Hamiltonian:

ˆ

H=ˆ

H↑+ˆ

H↓−U

N

X

i=1

hˆni↑i hˆni↓i(7)

The results shown here are obtained by ﬁxing N↑and N↓(Nσ=Piniσ), with N↑+N↓

equal to the number of electrons.

2.2. Density Functional Theory

The ﬁrst-principles density-functional theory (DFT) calculations are carried out by

using the electronic structure analysis tools implemented in the Spanish Initiative for

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Mixed magnetic edge states in graphene quantum dots 4

Electronic Simulations with Thousands of Atoms (SIESTA) [26] software. We perform

the spin-polarized DFT calculations using the generalized gradient approximation

(GGA) of the Perdew-Burke-Ernzerhof (PBE) [27] exchange correlation functional. The

valence electrons are expanded in terms of the double zeta plus polarization orbital

basis set (DZP) within the framework of a linear combination of numerical atomic

orbitals (LCAO) [28]. Because the semi-local GGA-PBE calculations are less reliable

in predicting the electronic and magnetic properties of GQDs, the screened hybrid

HSE06 [29] calculations implemented in the Heifei Order-N Packages for Ab Initio

Simulations (HONPAS) [30, 31] software also used to verify the mixed magnetic states

of GQDs [21]. All atomic coordinates are fully relaxed using the conjugate gradient

(CG) algorithm until the energy and atomic forces are smaller than 10−3eV and 0.04

eV/˚

A, respectively. We set all the carbon atoms with initial magnetic moments of 1.0

µBfor the FM conﬁguration and only set the carbon atoms of zigzag edges with initial

magnetic moments of 1.0 or -1.0 µB, respectively, for the AFM conﬁguration, as shown

in Figure 1. To check the stability of AFM, FM, mix-2, and mix-3 magnetic states in

GQDs, we also perform ab initio molecular dynamics (AIMD) simulations for N= 9

GQDs (see SM). The AIMD simulations are performed by using SIESTA software lasting

for 4 ps with a time step of 1 fs at 400 K controlled by a Nos´e-Hoover thermostat.

3. Results and Discussion

The edge-state magnetism of GQDs has been predicted by various theoretical models,

indicating that the edge magnetism of GQDs is robust. In this work, a mean-ﬁeld

Hubbard model is built to study the magnetic states of GQDs. To get diﬀerent

magnetic states, we consider two initial spin electron distributions for the Hubbard

model, as shown in Figure 1. In Figure 1(a), the spin-up and spin-down electrons align

alternately and the spin orientation of electrons is parallel at a particular edge and anti-

parallel between adjacent edges. The antiferromagnetic conﬁguration can be obtained

by iterating this initial distribution. To obtain the ferromagnetic conﬁguration, the

initial distribution should have the same spin arrangement for all the edges, as shown

in Figure 1(b).

We ﬁrst investigate the magnetic structures of hexagonal zigzag GQDs with various

sizes through the Hubbard model [19].In our previous work, we choose the parameters

t= 2.5 eV and U= 2.1 eV for the Hubbard model [21]. The zigzag edge states

become much more localized as the size increases [32], which results in much stronger

electron-electron interaction along each edge of GQDs. Therefore, the system transforms

from nonmagnetic states to an AFM state with FM intra-edge coupling and AFM

inter-edge coupling when N= 8, as shown in Figure 2(a). The FM state shown in

Figure 2(b) has higher energy compared with the AFM state, and all zigzag-edge C

atoms possess the same spin polarization. These results are in good consistency with

previous works [19, 22].

For the AFM conﬁguration, the total number of spin-up and spin-down electrons

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Figure 1. (a) Initial spin distributions of the GQDs with AFM and (b) FM states for the Hubbard model. The radius

of circle represents the value of |ˆni↑−ˆni↓|, which means the magnitude of magnetic moment. The red color

represents that the spin-up electrons are in majority, while the blue color means that the spin-down electrons are in

majority.

Figure 2. Spin density of GQDs with (a) AFM (N= 8), (b) FM (N= 8), mix-2 (N= 12) and mix-3 (N= 15) states

calculated by the Hubbard model. Red and blue colors denote spin-up and down density, respectively.

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Table 1. Electronic and magnetic properties of C6N2(N= 3, 4, 5, 6, 7, 8, 9, 10) with magnetic conﬁgurations of AFM,

FM, mix-2 and mix-3, including energy gap Egap (eV) of AFM ground state and maximum magnetic moment of carbon

atoms on zigzag edge (µB) computed with spin-polarized DFT calculations. The magnetic moment is absolute value.

C6N2345678910

µAFM 0.98 1.05 1.14 1.19 1.24 1.25 1.27 1.27

µFM 1.00 0.98 1.02 1.03 1.30 1.28 1.29 1.27

µmix−2– 0.95 1.05 1.05 1.16 1.17 1.24 1.25

µmix−3– – – 0.98 1.05 1.29 1.32 1.29

Egap 1.52 1.43 1.18 1.00 0.89 0.79 0.71 0.64

Table 2. Relative energies(∆E=EFM/mix−2/mix−3−EAFM) (eV) of N = 15 GQDs with AFM, FM, mix-2 and mix-3

states carried out by the Hubbard model (t= 2.5 eV and U= 2.1 eV) and DFT calculations, respectively.

∆E EFM −EAFM Emix−2−EAFM Emix−3−EAFM

DFT 0.93 1.32 2.45

Hubbard 0.19 0.36 0.68

are always the same, but one of which will always be in the majority for FM. When

adjusting the initial hˆni↑iand hˆni↓i, we can always obtain these two relatively stable

states. By forcing the number of electrons with diﬀerent spin orientations to be equal

during the iteration, new mixed magnetic states can be obtained, labeled as mix-n,

where nis the number of spin arrangement parts at each edge and spin arranges in

parallel for the same parts and anti-parallel for adjacent parts. For instance, the mix-2

state with AFM inter-edge coupling appears when N= 12 and the mix-3 state shows

up when N= 15. For the mix-2 state, all edges can be grouped into two parts with

opposite spin polarization, as shown in Figure 2(c). In the case of the mix-3 state, the

spin arrangement of each edge can be divided into three parts, as shown in Figure 2(d),

where the spin arrangement of the two side parts is coupled antiferromagnetically with

the middle one and adjacent edges are also antiferromagnetically coupled. Furthermore,

the mix-4/5 state appears when N= 20/34, as shown in Figure S1. The energies of

N=34 GQDs with mix-n (n = 2, 3, 4, 5) states are calculated by the Hubbard model

and their stability follows the order: mix-2 >mix-3 >mix-4 >mix-5 [Figure S2(c)].

This phenomenon originates from much more localized edge state of GQDs with larger

size due to the ﬁnite size eﬀect, and we believe the mix-nstate with high level nwill

come into view as the GQDs’ size increases. In this work, we only discuss the case of

mix-2 and mix-3 states. The energies of N = 15 GQDs with AFM, FM, mix-2, and

mix-3 states are calculated by using the Hubbard model, as shown in Table 2. It’s clear

that the energy of mix-2 is lower than mix-3, and they are all higher than the FM state

and AFM ground state. For GQDs with mix-2 and mix-3 states, adjacent edges can

take an AFM coupling or FM coupling (Figure S3) and the former is preferred. Unless

otherwise speciﬁed, the following discussion is based on the AFM inter-edge coupling.

First of all, we give a graphical explanation for this phenomenon according to

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Lieb’s theorem [33]. NAand NBare the number of sites on sublattices A and B,

respectively. Assuming that the number of electrons is Ne=NA+NB(half ﬁlling),

the ground state has total spin S=1

2|NA−NB|, hence the ground state of that lattice

is antiferromagnetic. For the edge of hexagonal GQDs, the adjacent zigzag-edge C

atoms belong to diﬀerent sublattices, which means the spin polarization of the adjacent

zigzag-edge C atoms is opposite (Figure 2). The quantum dot is an open boundary

system, we can envisage that it is equivalent to a system equipping a barrier potential

with a periodic boundary. For convenience, we can simplify the boundary to a one-

dimensional potential well model, where the basic solution is trigonometric functions

and the energy increases as the number of nodes in the spatial distribution of the wave

function increases. Then, the energy of the mix-3 state exhibits higher energy than the

mix-2 state.

Figure 3. Spin density of GQDs with (a) AFM (N= 3), (b) FM (N= 3), (c) mix-2 (N= 4), and (d) mix-3 (N= 6)

states based on the DFT calculations, respectively. Red and blue colors represent spin up and down. The isosurface

value is set to 0.004 e·˚

A−3.

Here, we also perform ﬁrst-principles spin-polarized DFT calculations to validate

the two new mixed magnetic states predicted from the Hubbard model. After optimizing

the structures and magnetic moments of bare hexagonal zigzag GQDs, we ﬁnd that the

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edges of these pristine GQDs are planar after optimization [Figure S8(a)] and the system

transforms from nonmagnetic to antiferromagnetic when the number of C atoms is larger

than 54 (N= 3) [22]. The mix-2 state appears when N≥4 and the mix-3 state can

be created when N≥6, as shown in Figure 3, which veriﬁes the prediction from the

Hubbard model. The stability of various magnetic states for N = 15 GQDs follows the

order: AFM >FM >mix-2 >mix-3, which is consistent with the results obtained from

the Hubbard model (see Table 2). Furthermore, the mix-4 and mix-5 states are also

found in N = 15 GQDs by conducting DFT calculations and the order of stability of

mixed states is mix-2 >mix-3 >mix-4 >mix-5, as shown in Figure S2. The maximum

local magnetic moments of edge C atoms of GQDs with AFM, FM, mix-2, and mix-3

states, as well as the energy gap (Egap =ELOMO -EHOMO) of GQDs with the AFM

magnetic ground state are listed in Table 1. For all magnetic states, it can be seen

that the maximum magnetic moments of edge C atoms increase with Nand gradually

converges to 1.3 µB, which means these magnetic states can all exist when the size of

GQDs increases. In addition, the energy gap (Egap) of AFM GQDs decreases as its

size increases, which is in good agreement with previous work [21]. Interestingly, in the

rectangle GQDs, a mixed spin conﬁguration is also obtained on the same zigzag-edge

carbon atoms, forming a short spin wave, which is similar to our case [43]. Furthermore,

a mixed magnetic state can be achieved in hexagonal zigzag GQDs with N = 3 through

carrier doping [22], while the carrier doping can be realized experimentally by using

electrolyte gate dielectrics [44], ionic liquid gate dielectrics [45, 46], and so on.

The projected density of states (PDOS) of zigzag-edge spin-up (down) C atom,

C1 (C2), and spin density of GQDs (N= 6) with NM, FM, AFM, mix-2, and mix-3

states are also calculated (Figure 4). For the NM-1 case, there are strongly localized

electronic states around the Fermi level mainly contributed by the pxand pyorbitals of

zigzag-edge C atoms. According to the Stoner criterion [34], these localized states are

not stable and would result in spontaneous polarization. The electronic states near the

Fermi level would become more localized as the GQDs size increases and hence lead to

the transition from an NM conﬁguration to an AFM conﬁguration. In the case of AFM

states, the valence band maximum (VBM) and conduction band minimum (CBM) are

all contributed by the pzorbitals of zigzag-edge atoms, while the magnetic moments

mainly consist of the pxand pyorbitals of C1 and C2 with opposite spin polarization.

The situations of GQDs with other magnetic states are the same as AFM cases, but the

energy gaps of these cases are diﬀerent. The AFM case has the largest gap, while the

FM case has the smallest. The C1 and C2 exhibit the same energy gap, and other C

atoms with spin-up/spin-down polarization on the zigzag edges have similar properties

as C1/C2, as shown in Figure S4.

We also check the magnetic properties of C486 GQDs using the screened hybrid

HSE06 calculations, and the same magnetic conﬁgurations are observed as the results

calculated by the PBE functional (see Figure S5). Usually, these edge states of GQDs

are susceptible to chemical modiﬁcation and these synthetic GQDs are stabilized by

hydrogenation or other functional groups, which have been studied theoretically and

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Mixed magnetic edge states in graphene quantum dots 9

Figure 4. Density of states and magnetic states of GQDs (N= 6) carried out by the DFT calculations. (a, b)

Projected density of states (PDOS) of C1 in NM and FM GQDs, respectively. (c) Spin density of GQDs with FM

states. (d, e) PDOS of C1 (left) and C2 (middle) and spin density (right) of GQDs with AFM state. (f, g) PDOS of C1

and C2 and spin density of GQDs with mix-2 state. (h, i) PDOS of C1 and C2 and spin density of GQDs with mix-3

state. Red and blue colors represent spin-up and spin-down density with a isosurface value of 0.004 e·˚

A−3, respectively.

experimentally [14, 20, 35, 36]. Hence we also investigate the magnetic structure of

GQDs with edges passivated by hydrogen atoms and ﬁnd that the system transforms

from nonmagnetic to magnetic when Nis larger than 9 (C486H54 ), which is consistent

with previous works [19, 21] (see Figure S6), and the mix-2 state appears when N≥

13 and the mix-3 state appears when N≥15. The mixed magnetic states found in

the edge passivated GQDs remain the same as that of the pristine GQDs, which means

these states can exist stably. Furthermore, AIMD simulations are performed under 400

K to check the stability of C486 with diﬀerent magnetic conﬁgurations, including the

FM, AFM, mix-2, and mix-3 states (see Figure S7). In the molecular dynamics process,

the GQDs can maintain their basic framework with slight ﬂuctuations, and no C-C

bond breakage and formation are observed [Figure S8(b)]. Because the energy of GQDs

with AFM, FM, mix-2, and mix-3 states can reach equilibrium within 0.2 ps, we further

calculate the spin distribution of the N = 9 GQDs after 1.5 ps AIMD simulations,

as shown in Figure S9, and observe that the AFM, FM, mix-2, and mix-3 states can

maintain stability after the molecular dynamics process and not be hardly aﬀected by

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Mixed magnetic edge states in graphene quantum dots 10

the structure wrinkle, and the magnetic moments of the edge carbon atoms (taking C6

labeled in Figure S9(d) as an example) remain almost constant as well (Table S1), which

means they are robust against thermal perturbation.

4. Conclusion

In summary, by analyzing the Hubbard model, we predict a series of new types of

mixed magnetic states in hexagonal zigzag GQDs due to the quantum size eﬀect. The

mix-2 state has two parts of spin alignment on each edge and mix-3 has three, both

of which show parallel spin orientation within the same part and antiferromagnetic

coupling between adjacent parts. To verify our theoretical prediction obtained from

the Hubbard model, large-scale spin-polarized DFT calculations containing thousands

of atoms are performed for bare and hydrogenated GQDs. For the bare GQDs, the

system transforms from nonmagnetic to magnetic while the carbon number is larger

than 54 (N= 3), and the mix-2 (3) states can be observed when the sizes increase

to N= 4 (6), respectively. For the hydrogenated GQDs, the system transforms from

nonmagnetic to magnetic when its size increases to N= 9 (C486H54), while the mix-2

and mix-3 magnetic states appear when N≥13 and N≥15, respectively. Because of

their abundant edge magnetic states, hexagonal zigzag GQDs hold broad application

prospects in the ﬁelds of spintronics.

Supporting Information

The Supporting Information is available free of charge.

•The detail derivation process of the Hamiltonian of Hubbard model.

•The mix-4 and mix-5 magnetic states of C2400 and C6936 GQDs obtained from the

Hubbard model.

•Spin distribution of C1350 GQDs with the mix-4 and mix-5 states obtained from

DFT calculations and the energies of N = 34/15 GQDs with mix-2, mix-3,

mix-4, and mix-5 states obtained by the Hubbard model and DFT calculations,

respectively.

•The mix-2 and mix-3 magnetic states with inter-edge FM coupling of C864 and C1350

GQDs obtained from the Hubbard model.

•Density of states of zigzag-edge spin-up/down C atoms (C3, C5, C7/C4, C6, C8)

and AFM magnetic state of GQDs (N= 6).

•The mix-2 and mix-3 magnetic states of C486 GQDs computed with the screened

hybrid HSE06 calculations implemented in the HONPAS software.

•The magnetic conﬁgurations of GQDs with edges passivated by hydrogen atoms.

•AIMD simulations of C486 with the AFM, FM, mix-2, and mix-3 conﬁgurations.

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REFERENCES 11

•The geometry a GQD with N = 9 after structure optimization and AIMD simulation

at 400 K.

•The spin density of GQDs with AFM, FM, mix-2, and mix-3 states after 1.5 ps.

Notes

The authors declare no competing ﬁnancial interest.

Acknowledgements

This work is partly supported by the National Natural Science Foundation of China

(21688102, 21803066, 22003061, 22173093), by the Hefei National Laboratory for

Physical Sciences at the Microscale (KF2020003), by the Chinese Academy of Sciences

Pioneer Hundred Talents Program (KJ2340000031, KJ2340007002), by the National Key

Research and Development Program of China (2016YFA0200604), the Anhui Initiative

in Quantum Information Technologies (AHY090400), the Strategic Priority Research

Program of Chinese Academy of Sciences (XDC01040100), the CAS Project for Young

Scientists in Basic Research (YSBR-005), the Hefei National Laboratory for Physical

Sciences at the Microscale (SK2340002001), the Fundamental Research Funds for the

Central Universities (WK2340000091, WK2060000018), the Research Start-Up Grants

(KY2340000094) and the Academic Leading Talents Training Program (KY2340000103)

from University of Science and Technology of China. The authors thank the Hefei

Advanced Computing Center, the Supercomputing Center of Chinese Academy of

Sciences, the Supercomputing Center of USTC, the National Supercomputing Center in

Wuxi, Tianjin, Shanghai, and Guangzhou for the computational resources.

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