Eective low energy Hamiltonians and unconventional Landau level spectrum of monolayer C 3N Mohsen Shahbaziand Jamal Davoodi

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Eﬀective low energy Hamiltonians and unconventional Landau level spectrum of

monolayer C3N

Mohsen Shahbazi∗and Jamal Davoodi

Department of Physics, Faculty of Science, University of Zanjan, Zanjan, Iran

Arash Boochani

Department of Physics, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran and

Quantum Technological Research Center (QTRC),

Science and Research Branch, Islamic Azad University, Tehran, Iran

Hadi Khanjani

Department of Physics, University of Tehran, P. O. Box 14395-547, Tehran, Iran

Andor Korm´anyos†

Department of Physics of Complex Systems, E¨otv¨os Lor´and University, Budapest, Hungary

We derive a low-energy eﬀective k·pHamiltonians for monolayer C3N at the Γ and Mpoints

of the Brillouin zone where the band edge in the conduction and valence band can be found. Our

analysis of the electronic band symmetries helps to better understand several results of recent ab-

initio calculations [1, 2] for the optical properties of this material. We also calculate the Landau

level spectrum. We ﬁnd that the Landau level spectrum in the degenerate conduction bands at the

Γ point acquires properties that are reminiscent of the corresponding results in bilayer graphene,

but there are important diﬀerences as well. Moreover, because of the heavy eﬀective mass, n-doped

samples may host interesting electron-electron interaction eﬀects.

I. Introduction

Graphene[3] has received a great deal of attention due

to its unique mechanical, electronic, thermal and opto-

electronic properties [4–6]. However, having a zero band

gap limited the applications of graphene in electronic

nano-devices and motivated the search for atomically

thin two-dimensional (2D) materials which have a ﬁnite

band gap. This lead to the discovery of, e.g., monolayer

transition metal dichalcogenides [7–9], silicene [10, 11],

phosphorene [12, 13], germanene [14]. In recent years,

compounds of carbon-nitrides CxNyhave also become at-

tractive 2D materials [15–17]. For example graphitic car-

bon nitride (g-C3N4), which is a direct band gap semicon-

ductor, has potential applications in photocatalysis and

in solar energy conversion due to its strong optical ab-

sorption at visible frequencies [18, 19]. Another carbon-

nitride compound, two-dimensional crystalline C3N has

also been recently synthesized [20, 21]. C3N is an in-

direct band gap semiconductor with energy gap of 0.39

eV [21]. Moreover, it has shown favorable properties,

such as high mechanical stiﬀness [22] and interesting ex-

citonic eﬀects [1, 2]. In addition, its thermal conductiv-

ity properties have been investigated [22–24] and it has

been predicted that the electronic, optical and thermal

properties of monolayer C3N can be tuned by strain en-

gineering [25, 26].

In this work we will employ the k·p[27, 28] approach

∗mohsenshahbazi1984@yahoo.com

†andor.kormanyos@ttk.elte.hu

FIG. 1. a) Crystal structure of C3N monolayer. Purple and

blue circles refer to carbon and nitrogen atoms, respectively.

Dashed black lines show the unit cell. The orange line show

the hexagonal unit cell, which can be useful to understand

certain optical properties, see Sec. IV.

in order to study the electronic properties of the mono-

layer C3N. We obtain the materials speciﬁc parameters

appearing in the k·pmodel from ﬁtting it to density

functional theory (DFT) band structure calculations. A

similar methodology has been successfully used, e.g., for

monolayers of transition metal dichalcogenides [29, 30].

In particular, since the conduction band (CB) minimum

and valence band (VB) maximum are located at Γ and

Mpoints of the Brillouin zone, respectively, we obtain

k·pHamiltonians valid in the vicinity of these points.

The insight given by the k·pmodel allows us to comment

on certain optical properties as well. Moreover, we will

also study the Landau level spectrum of C3N, which, to

our knowledge, has not been considered before.

This paper is organized as follows. In Sec. II we start

arXiv:2210.03472v1 [cond-mat.mes-hall] 7 Oct 2022

FIG. 2. a) DFT band structure calculations for C3N along

the Γ −K−M−Γ line in the BZ. b) orientation of the BZ

and the high symmetry points Γ, K,M.

with a short recap of the band structure obtained with

the help of the density functional theory calculations. In

Sec. III, eﬀective k·pHamiltonians at Γand Mpoints are

obtained, using symmetry groups and perturbation the-

ory. Certain optical properties of this material are dis-

cussed in Sec. IV. In Sec. V the spectra of Landau levels

for this material are calculated at the Γ and Mpoints.

Finally, our main results are summarized in Sec. VI.

II. Band structure calculations

The band structure of monolayer C3N has been calcu-

lated before at the DFT level of theory [25, 31, 32] and

also using the GW approach[1, 2, 33]. The main eﬀect of

the GW approach is to enhance the band gap and this

does not aﬀect our main conclusions below. To be self-

contained, we repeat the band structure calculations at

the DFT level. The schematics of the crystal lattice of

single-layer C3N is shown in the Fig. 1. The lattice of

C3N possess P6/mmm space group with a planar hexag-

onal lattice and the unit cell contains six carbon and

two nitrogen atoms. We used the Wien2K package[34]

to perform ﬁrst-principles calculations based on density

functional theory (DFT). For the exchange-correlation

potential we used the generalized gradient approxima-

tion (GGA) [35]. The optimized input parameters such

as RKmax, lmax, and k-point were selected to be 8.5, 10,

and 14 ×14 ×3, respectively. The convergence accuracy

of self-consistent calculations for the electron charge up

to 0.0001 was chosen and the forces acting on the atoms

were optimized to 0.1dyn/a.u. The optimized lattice con-

stant is a0= 4.86˚

A, in good agreement with previous

studies [31, 36].

The calculated band structure is shown in Fig. 2. The

conduction band minimum is located at the Γ point,

while the valence band maximum can be found at the M

of the BZ. Thus, at the DFT level C3N is an indirect band

gap semiconductor with a band gap of Ebg = 0.48 eV

which is in good agreement with previous works [21, 37].

We have checked that the magnitude of the spin-orbit

coupling is small at the band-edge points of interest and

therefore in the following we will neglect it. The main

eﬀect of spin-orbit coupling is to lift degeneracies at cer-

tain high-symmetry points and lines, e.g., the four-fold

degeneracy of the conduction band at the Γ point would

be split into two, two-fold degenerate bands.

III. Eﬀective k ·p Hamiltonians

We now introduce the k·pfor the Γ point, where the

band edge of the CB is located, and for the Mpoint,

where the band edge of the VB can be found.

A. Γpoint

The pertinent point group at the Γ point of the BZ is

D6h. We obtained the corresponding irreducible repre-

sentations of the nine bands around the Fermi level at

the Γ point with the help of the Wien2k package. Us-

ing this information one can then set up a nine bands

k·pmodel along the lines of Ref. [30], see Appendix A

for details. Here we only mention that there is no k·p

matrix element between the VB and the degenerate CB,

CB+1 which means that direct optical transitions are not

allowed between these two bands. Since it is usually dif-

ﬁcult to work with a nine-band Hamiltonian, we derived

an eﬀective low-energy Hamiltonian which describes the

two (degenerate) conduction bands and the valence band.

Using the L¨owdin partitioning technique [38, 39] we ﬁnd

that

HΓ

eﬀ =HΓ

0+HΓ

k·p,(1a)

HΓ

0=



εvb 0 0

0εcb 0

0 0 εcb+1



(1b)

HΓ

k·p=



α1q20 0

0 (α2+α3)q2−α3(q+)2

0−α3(q−)2(α2+α3)q2

.(1c)

Here εcb =εcb+1 = 0.386 eV and εvb =−1.50 eV are

band edge energies of the degenerate CB minimum and

VB maximum. The wavenumbers qx,qyare measured

from the Γ point, q±=qx±iqyand q2=q2

x+q2

yand in

α2we took into account the free electron term[29].

Note, that there are no linear-in-qmatrix elements be-

tween the VB and the degenerate CB, CB+1 bands. In

higher order of qthese bands do couple, but this is ne-

glected in the minimal model given in Eq. (1c). The mini-

mal model given in Eq. (1) already captures an important

property of the degenerate CB and CB+1 bands from the

DFT calculations, which is that their eﬀective masses

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EectivelowenergyHamiltoniansandunconventionalLandaulevelspectrumofmonolayerC3NMohsenShahbaziandJamalDavoodiDepartmentofPhysics,FacultyofScience,UniversityofZanjan,Zanjan,IranArashBoochaniDepartmentofPhysics,KermanshahBranch,IslamicAzadUniversity,Kermanshah,IranandQuantumTechnologicalResearchCenter...

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Eective low energy Hamiltonians and unconventional Landau level spectrum of monolayer C 3N Mohsen Shahbaziand Jamal Davoodi

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