exchange field effects on the electronic properties of monolayer h-BP

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Journal of Physics D: Applied Physics

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Electron-phonon coupling, spin-polarized Zeeman field, and exchange field effects on the electronic properties of monolayer h-BP

To cite this article: Nguyen T Dung et al 2021 J. Phys. D: Appl. Phys. 54 385301

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J. Phys. D: Appl. Phys.54(2021) 385301 (9pp) https://doi.org/10.1088/1361-6463/ac0eb2

Electron-phonon coupling,

spin-polarized Zeeman field, and

exchange field effects on the electronic properties of monolayer h-BP

Nguyen T Dung¹, Ta T Tho²and Le T T Phuong^3,4,∗

1Vinh University, Nghe An Province, Vietnam

2Department of Physics, Faculty of Mechanical Engineering, National University of Civil Engineering, Hanoi, Vietnam

3Department of Physics, University of Education, Hue University, Hue City, Vietnam

4Center for Theoretical and Computational Physics, University of Education, Hue University, Hue City, Vietnam

E-mail:[email protected]

Received 14 March 2021, revised 7 June 2021 Accepted for publication 25 June 2021 Published 8 July 2021

Abstract

In this work, we propose the electron-phonon coupling (EPC), a spin-polarized Zeeman field, and an exchange field to tune the electronic phase of single-layer hexagonal boron

phosphide (h-BP). In doing so, the tight-binding model and Green’s function approach are employed by focusing on the electronic density of states (DOS) quantity. A

semiconductor-to-insulator (from 1.297 to 6.45 eV) and a semiconductor-to- semimetal-to-insulator (from 1.297 to 0 eV and then to 3.9 eV) are achieved by the

exchange/Zeeman field induced to h-BP. Then, based on the DOS behaviors in the presence of EPC, the weak and strong coupling regimes are derived. The analysis of the EPC effect on DOS shows that the scattered modes from the coupling are formed in the valence and conduction bands in both weak and strong coupling regimes. Furthermore, it is found that EPC does not significantly affect the phase of the system. The results presented here are expected to be helpful to control the electronic phase of low-dimensional materials in practical applications in both nanoelectronics and optoelectronics communities.

Keywords: electron-phonon coupling, spin-polarized Zeeman filed, exchange field, electronic phase, monolayer h-BP

(Some figures may appear in colour only in the online journal)

1. Introduction

Silicon semiconductor technology with versatile use and amazing capabilities has done marvels for the advancement of our society, however, the transistors can only be shrunk down to a certain size and not further beyond. For this reason, the miniaturization of circuits seems to have reached a

∗ Author to whom any correspondence should be addressed.

possible halt. So, to fulfill the future computing requirements, there is a pressing need to complement Si technology with new materials. Graphene and related 2D materials offer pro- spects of unprecedented advances in device performance at the atomic limit, which have attracted scientists in various fields because of their fascinating physical features compared to their bulk counterparts. 2D systems have different versions depending on the flatness of structured atoms. The first well- known flat one since 2004 is graphene [1,2] and recently borophene [3, 4] is another interesting flat 2D material,

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J. Phys. D: Appl. Phys.54(2021) 385301 N T Dunget al

however, one would also mention group-IV single-layers [5–9], transition-metal dichalcogenides [10–15] and phosphorene [16–21] as the buckled and puckered ones. There are numerous interesting and valuable applications based on 2D systems for the next-generation electronics and optoelectronics community [22–33].

Very recently, the single-layer hexagonal boron phosphide (h-BP) has been synthesized experimentally as a stable 2D material at high temperatures and atmospheric condi- tions [34–39]. This 2D structure is a semiconductor with a direct band gap of 0.9–1.37 eV [38–41]. The structure of h-BP is similar to graphene; a honeycomb lattice with boron (B) and phosphorus (P) atoms on the sublattices [42,43]. Up until now, many applications have been proposed/found for h-BP.

For instance, nanosized boron phosphide (BP) structures and the BP-based electrodes are found to be applicable for sensit- ized liquid junction photovoltaic solar cells [44]. Also, it has been found that h-BP acts as an anode material for alkali metal- based (e.g. Li, Na, and K) batteries [45].

In general, 2D crystals have shown an excellent response to the external perturbations which in turn modulate the physical properties for different purposes [46–48]. For example, it has been theoretically shown that the bias voltage modu- lates the electronic and optical properties of monolayer ger- manene [49]. Moreover, the electronic properties of a few- layer BP have been theoretically addressed in the presence of atomic structure, stacking order, and bias voltage [50]. Also, the strain effects on the electronic and optical properties of the germanium carbide have been recently investigated [51].

Using the non-equilibrium Green’s function formalism, the electron transport properties of a p–n junction constructed from h-BP have been studied, too [52].

It is well-known that h-BP is from the family of B-V com- pounds; hexagonal boron nitride (h-BN) is an insulator, while hexagonal boron arsenide (h-BAs) and the h-BP are semiconductor. Both h-BAs and h-BP are also found to be dynam- ically stable [53]. This, in turn, implies that the band gap of h-BAs and h-BP is widely tunable. Most of the works published so far on these systems are based on first-principles calculations and to the best of our knowledge, an investigation beyond the DFT is missing in the literature. The band gap of h-BAs and h-BN has been investigated recently using the electric field [41,54]. On the other hand, we would consider other external and internal perturbations rather than strain and electric field stimuli in the present work such as an exchange field induced by the proximity coupling, by applying the Zeeman magnetic field, and by considering the electron-phonon coupling (EPC) effect. Band gap is one of the particular important quantities in introducing the applicability of 2D materials. Lately, an accurate tight-binding (TB) model describing π andπ^∗ electronic bands in h-BP has been provided [42, 43,55]. Thus, we employ the TB model and Green’s function approach to study the effect of the above-mentioned external and internal factors on the band gap of h-BP by focusing on the electronic density of states (DOS).

The setup of the paper is organized as follows. In section2, the TB model of pristine h-BP will be reviewed, then, the dispersion energy of carriers will be discussed. Section3presents

B

K

M k _x k _y

Γ Γ a

₁

a

₂

δ

1

δ

2

δ

3

P B

Figure 1. Top view of monolayer h-BP. The rhombic dashed lines show the unit cell of the structure. The lattice vectors,

nearest-neighbor vectors, and lattice constant are given respectively by{⃗a1,⃗a2},{⃗δ₁,⃗δ₂,⃗δ₃}anda=1.855 Å [41,43]. The first Brillouin zone (FBZ) is also denoted by the filled green hexagonal shape comprising of high-symmetry pointsΓ,K, andM.

the electronic DOS calculations in the absence and presence of an external exchange field and Zeeman splitting field when the EPC is absent. In section4, we turn to the EPC effects on the electronic DOS employing the perturbation theory for Hol- stein phonons. In section5the paper ends with a summary of remarkable results.

2. TB model

In this section, we present the TB model of h-BP. Similar to graphene, the orbital hybridization in h-BP issp²andp_zorbit- als are dominant in the electronic transport of the system given by the following Hamiltonian in real-space

H₀=∑

f,i

ε_0ff^†_if_i−t∑

⟨i,j⟩

(b^†_ip_j+H.c.), (1)

wherefi(f^†_i)annihilates (creates) an electron on the sublattice f={b,p}. The coefficientt' −1.844 eV is the in-plane hopping energy between nearest-neighbors [43] andε_0f is the on- site energy of sublatticef;ε_0b=−3.276 eV andε_0p=−1.979 eV [43]. As shown in figure1, the primitive unit cell vectors of honeycomb lattice are given by

⃗a1=

√3a 2

ˆi+a 2

ˆj, ⃗a2=

√3a 2

ˆi−a 2

ˆj, (2) where a'1.855 Å is the lattice constant [43]. Also,ˆi and ˆj are unit vectors along the x andy direction, respectively.

Moreover, the nearest-neighbor vectors are given by ⃗δ₁= a(1/2,√

3/2), ⃗δ₂=a(1/2,−√

3/2) and ⃗δ₃=a(−1,0) (see figure1).

The Hamiltonian in equation (1) can be rewritten in terms of the Fourier transform of operators as

H0=∑

f

ε_0f∑

k,q

f^†_k+qfk−t∑

k

[ϕ(k)b^†_kpk+H.c.], (3)

where ϕ(k) =1+2 cos(kx/2)e^−ik^y^√³^/² and the momenta k and q belong to the FBZ of h-BP honeycomb lattice, see

2

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Figure 2. The electronic band structure ofpristinemonolayer h-BP.

We setEF=−2.594 eV [41,43].

figure1. Finally, one can rewrite the TB model Hamiltonian in equation (3) as

H0=∑

k

Ψ^†_kH0(k)Ψ_k (4) in which the vector of electron creation operators is defined by Ψ^†_k= (b^†_k,p^†_k,bk,pk).b^†_kandp^†_kare the Fourier transformations ofb^†_i andp^†_i. The nearest-neighbor approximation gives us the following matrix form

H0(k) =

( ε_0b tϕ(k) tϕ^∗(k) ε_0p

)

. (5)

The quasiparticle excitation energies from the above Hamilto- nian lead to the following two-band dispersion energy:

Eν(k) =ε_0b+ε_0p

2 +ν

√(ε_0b−ε_0p 2

)₂

+t²|ϕ(k)|², (6) whereν= +1(−1)refers to the conduction (valence) band.

We in figure2 plot the above dispersion energy subtract- ing the Fermi energyEF=−2.594 eV [41,43] for the electrons dispersing along theΓ−K−M−Γpath of the FBZ of h-BP with coordinatesΓ =(0, 0),K= (2π/3a,2π/3√

3a)and M=(2π/3a, 0). It is obvious that the direct band gap is located at theK point. The pristine band gap can be obtained from equation (6) settingkx=2π/3aandky=2π/3√

3a. Thus, we have E_g⁰=|ε_0b−ε_0p|=1.297 eV, in agreement with references [41,43]. This, in turn, means that the pristine h-BP is a direct semiconductor. There is one point to be mentioned here. Looking at the band structure one notices that the bands become almost flat aroundE(k)− EF' ±2 eV. They are signs of degenerate energy levels, which manifest themselves in the van Hove singularities in the electronic DOS. We will come to this point later. In what follows, we will try to tune this semiconducting phase through proximity coupling, the Zee- man field and EPC.

Here, we would briefly compare the current h-BP with other well-known 2D materials such as MoS2 and phosphorene.

There are many other systems to be compared here, however, we stick to these two representative ones. The band gaps of monolayer MoS₂ and phosphorene are respectively 1.88 and 1.52 eV, while the band gap of our system h-BP is 1.297 eV. To compare h-BP with MoS₂and phosphorene, one immediately should mention that both MoS₂and phosphorene have buck- ling in their geometry structure, leading to direction-dependent properties, which of course is useful for some purposes, however, in current electrical and optical technology, it is highly desirable to be able not only to tune the semiconductor properties such as band gap as a key property that determines the semiconductor electrical and optical properties but also to miniaturize the corresponding devices toward nearly atomic- ally thin dimensions. Thus, flat structures are of interest more than buckled and puckered ones. Monolayer MoS₂and phosphorene as wider band gap semiconductors compared to h- BP can be operated at high voltages and are thermally stable.

It is also notable that high critical electric field strength results in low ON-state resistance hence conduction losses can be reduced. However, they are in general very difficult to grow up to large diameters in size due to issues involved in growth. Also, it is difficult to make them semi-insulating due to issues related to doping and growth. Thus, one looks for a bit narrower band gap and thinner semiconductor like h-BP.

3. Exchange and Zeeman field effects on the electronic DOS

In this section, we intend to immediately turn to the effects of proximity coupling and Zeeman magnetic field on the electronic DOS of monolayer h-BP [40,56]. In the case of the exchange field Mz, the proximity effect of h-BP into a ferromagnetic semiconductor is studied. The uniform exchange potential induced to h-BP from the ferromagnetic system gen- erates a new electronic phase. The exchange field is always in the perpendicular direction to the 2D plane, up or down.

Usually, within the TB framework, all potentials are treated in a unit of hopping integral among the nearest-neighbor lattice sites, i.e. t in our model. Furthermore, depending on the perpendicular direction of the exchange field to the 2D plane, up or down, we have changed the sign of it. Thus, depending on the 2D ferromagnetic semiconductor such as (In, Fe)As [57,58], the exchange field can be different. From these points, it is easy to conclude that there is a wide range forMz

to be applied. Actually, the range considered here for 2D ferromagnetic semiconductors with magnetization along the−z and+zdirection, is not far from reality, and from the current advanced experimental facilities, it is expected to induce such potentials.

The Zeeman term mathematically can also be resembled due to magnetic coupling to a proximate ferromagnetic film or the effects of magnetic doping [59]. So, the same argument as above is valid for the Zeeman field. Thus, we assume that when an external magnetic field is applied, due to the interaction between the magnetic field and the magnetic dipole moment (spin) associated with the orbital angular momentum,

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host spins are modulated and finally the electronic properties of the system are spin-dependent.

To this end, we first describe the corresponding Hamiltoni- ans, respectively, as

Hex= − Mz

∑

i

f^†_iσ_zfi, (7a)

H^σ_z =−h 2

∑

i

b^†_i,σb_i,−σ+h 2

∑

i

p^†_i,σp_i,−σ, (7b) in whichMzcould be positive or negative if respectively the magnetization direction is parallel or antiparallel to the nor- mal vector of h-BP plane, whereashrefers to the Zeeman field strength. It should be noted that we have constructed the Zee- man field characterized by the positive and negative strengths of boron and phosphorus atoms based on the antiferromagnetic phase of honeycomb lattices [60,61].

After the Fourier transformations, the reciprocal-space Hamiltonian of the system considering the exchange and Zee- man spin splitting fields can be read as

H_ex(k) =

(ε_0b− M_z tϕ(k) tϕ^∗(k) ε_0p+M_z

)

, (8a)

H_z(k) =







ε_0b −h/2 tϕ(k) 0

−h/2 ε_0b 0 tϕ(k) tϕ^∗(k) 0 ε_0p +h/2 0 tϕ^∗(k) +h/2 ε_0p





, (8b)

which respectively lead to the following dispersions Eν^ex(k) =ε_0b+ε_0p

2 +ν

√(|E_g⁰| −2M_z 2

)2

+t²|ϕ(k)|², (9a)

Eν^z^,σ(k) =ε_0b+ε_0p

2 +ν

√(σ|E_g⁰|+h 2

)2

+t²|ϕ(k)|². (9b) Let us find the electronic correlation elements to build Green’s functions components in the Matsubara formalism [62]. The Green’s function matrix components and their Fourier transformations are expressed by

Gαβ(k, τ) = − hTτfk,α(τ)f^†_k,β(0)i, (10a)

Gαβ(k,iω_n) = ˆ ₁/kBT

0

e^iωⁿ^τGαβ(k, τ)dτ , (10b) where α and β refer to the sublattice atoms. Also, τ is the imaginary time,Tτ is the time ordering operator, ω_n= (2n+1)πkBTis the Fermionic Matsubara’s frequency, andT is the absolute temperature of the system [62]. Having all components, the Green’s function matrix of the system including equation (7) can be constructed.

Considering the imaginary part of the Green’s function components, the electronic DOS can be calculated [7,56,62–64]. Therefore, by summing over all quantum num- bers which label the Hamiltonian, the normalized DOS of the system per unit cell would be expressed as

D(E) =−1 2π

∑

k,α=b,p

ImGαα(k,iω_n7→ E+io⁺), (11)

where o⁺=1 meV is a phenomenological parameter in numerical calculations. We set unitskB=ℏ=1 in our numerical results.

To a better understanding of the electronic DOS behaviors with external factors, a few inputs are needed from the dispersion energy relations given by equation (9). We found that the band gap is located at the K point of FBZ. Using K= (2π/3a,2π/3√

3a)coordinates, we respectively find for Mz- andh-induced gap

Eg=|E_g⁰−2Mz|, (12a)

E_g^σ=|σE_g⁰+h|. (12b)

It is quite clear from equation (12a) that forMz=−E_g⁰/2,

−E_g⁰ and−2E_g⁰, one respectively finds Eg=2 E_g⁰, 3 E_g⁰ and 5 E_g⁰, while if we switch the exchange field direction to Mz= +E_g⁰/2,+E_g⁰and+2E_g⁰, the band gaps alter to respect- ivelyEg=0,E_g⁰and 3E_g⁰. This, in turn, means that the band gap increases with all the exchange fields along the −zdirection, while for the ones along the +z direction the band gap decreases for the strengths below the pristine band gap, becomes the same as the pristine band gap as soon as the exchange field gets the same order of magnitude as E_g⁰ and finally increases for the strengths larger than the pristine band gap. On the other hand, we clearly see that the band gap of the system is the same forMz=−E_g⁰and 2E_g⁰and most probably it is easier to consider the one with small strength along the−z direction if one aims at increasing the band gap and finally at exploring the insulating phase.

In addition to the above-mentioned analyses, one would quickly point out the electronic phase transitions dur- ing these processes. In the case of Mz<0, system starts the transition from the semiconducting phase to the insulating one, whereas a semiconductor-to-semimetal and semiconductor-to-insulator phase transitions emerge respectively for 0<Mz<E_g⁰ and{Mz>0 andMz>E_g⁰}. These propose useful information to the experimentalists for different purposes. By these, we mean that proximity coupling to a ferromagnetic material is a proper way here to tune the electronic phase of the system with all possibilities.

After this knowledge, it is time to plot the electronic DOS for collecting more information. One of the most important information is focusing on the van Hove singularities mentioned at the end of section2referring to the degeneracy levels of electronic bands. Nicely, in figure3we first confirm all the

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Figure 3. Electronic DOS of h-BP including the exchange field Mz. The Zeeman field is off here,h=0.

above discussions on the band gap characterized by the zero DOS around the zero energy. In both panels (a) and (b), the black solid lines belong to the pristine semiconducting phase of the system; h-BP with the band gap around 1.297 eV. How- ever, two sharp peaks located at±2 eV can also be observed.

These peaks are exactly the van Hove singularities expected before highlighting the degeneracy level of bands. Degeneracy states are of course present around±0.5 and±5.5 eV, too, resulting in the step-like curves. When the electronic phase of the system is changed with±Mz, the degeneracy level of these states is also simply altered.

ForMz<0 in figure3(a), we reported the semiconductor- to-insulator phase transition. Following this transition, the height and position of peaks are increased as well. This implies that the number of states locating at new band energies is increased, so the local density of the system is denser than before. One may quickly ask that why the position of peaks is shifted? The physical reason behind this shift can be understood from the fact that we neither add any states to the system nor remove any states from the system through the exchange fieldMz. Thus, the area under the electronic DOS curve should be preserved at the end. For this reason, when some states are removing around the zero energy with Mz

to increase the band gap, they locate at other energies. By this, the sum rule belonging to the electronic DOS will be

Figure 4. The electronic band gap changes with the Zeeman spin splitting fieldhin spin-polarized monolayer h-BP.

preserved. It should be noted that we are allowed to argue that the height of peaks is linearly increased withM_z<0 because the band gap is linearly changed with the exchange field from equation (12a).

As for the case ofM_z>0 in figure3(b), the same arguments are valid for the height of van Hove singularities and the position of peaks. For this reason, we do not intend to repeat them. However, it is necessary to mention that the position of peaks are not significantly affected by 0<M_z≤ E_g⁰, while it is shifted significantly withM_z>E_g⁰.

Let us turn to the Zeeman field effect on the band gap of the system through equation (12b). As mentioned before, we have considered the Zeeman field such that the system will be spin- polarized. For this reason, we should see the spin-polarized band gap changes withhas well. We again consider the general Zeeman strengthsh=±E_g⁰/2,±E_g⁰and±2E_g⁰. As can be seen from equation (12b), negative and positive signs ofhdo not behave similarly. The spin-dependent band gaps are obtained respectively asE_g^↑={3E_g⁰/2,E_g⁰/2},{2E_g⁰,0}and{3E_g⁰,E_g⁰}, while for the spin-down one finds E_g^↓={E_g⁰/2,3E_g⁰/2}, {0,2E_g⁰} and {E_g⁰,3E_g⁰}. The band gaps are spin-polarized.

To confirm this, we have plotted the normalized band gap to the pristine band gap as a function of the Zeeman field h in figure4. Like before, we would report the possible phase transitions. From the band gap behavior, it is evident that for −2<h^↑/E_g⁰<0 and 0<h^↓/E_g⁰<2, a semiconductor- to-semimetal-to-semiconductor transition happens, while for h^↑/E_g⁰>0 andh^↓/E_g⁰<0, the system intends to take the insulating phase. It can be understood that the same arguments can be applied to justify the height and position of the van Hove singularities in the electronic DOS. For this reason, we do not show any DOS diagram here to avoid repetition.

4. Self-consistent perturbation theory: EPC effects on the electronic DOS

In this section, we consider the effect of an internal important factor, the EPC effect, on the electronic DOS of h-BP. The interaction of electrons with optical phonons occurs in two ways: (i) due to the atomic in-plane displacements referred

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to as the longitudinal optical and transverse optical phonons which are degenerate at Einstein zero mode, (ii) through the lattice displacements due to the ripple structures, out-of- plane vibrations, which are symmetric concerning their close atoms. However, we comment that optical modes are only considered here within the Holstein model [65–67] which are essentially local [68–70]. In doing so, one needs the phonon and electron-phonon (e-ph) Hamiltonians, respectively, given by

H_ph=ω₀∑

i

c^†_ic_i, (13a)

H_e−ph=∑

f,i

g_ff^†_if_i(c^†_i +c_i), (13b)

where ω₀ and gf are the phonon frequency and EPC constant, respectively. In gapless structures like graphene, the EPC strength of sublattices are the same, while in h-BP as a gapped structure, the inversion symmetry is broken and one requires the relationgp=gb

√Mb/Mp in whichMb(p)are the atomic masses of boron and phosphorus atoms [71–73]. On the other hand,ci(c^†_i)denotes the annihilation (creation) operator for local modes of optical phonons. Applying the Fourier transformation, we have

H_ph=ω₀∑

q

c^†_qc_q, (14a)

He−ph=∑

f

gf

∑

k,q

f^†_k+qfk(c^†_−q+cq). (14b) Now we should turn to the self-energies of electrons on the honeycomb lattice due to EPC. Migdal theorem [62]

gives the lowest order perturbation theory to find the self-energy diagram. Noninteracting phononic propagator is introduced by

D⁽⁰⁾(p,ip_m) = 2ω₀

(ip_m)²−ω²₀, (15) where p_m=2πmk_BT implies the bosonic Matsubara frequency. Using the Feynman rules [62], EPC contributions in the second-order perturbation theory are given by

Σ_αα(k,iω_n) =−g²γ

β

∑

p,m

D⁽⁰⁾(p,ip_m)Gαα(k−p,iΩ_nm), (16) where β=1/kBT and Ω_nm=ω_n−pm. Interacting Green’s functions are related to the spectral function based on the Leh- man representation [62] as

Gαβ(k,iω_n) = ˆ +∞

−∞

dω 2π

Aαβ(k, ω)

iω_n−ω , (17a) Aαβ(k, ω) =−2ImGαβ(k,iω_n7→ω+io⁺), (17b)

Figure 5. Electronic DOS of h-BP in the presence of EPC. We set Mz=0 andh=0 in this plot. The thermal energy and phonon frequency are respectively fixed atkBT=0.01 eV (temperature around 115 (K) andω₀=0.15 eV.

where A(k, ω) is the electronic spectral function matrix.

Finally, the disruption expansion for the interacting propagator in the Matsubara’s notation is given by

G(k,iω_n) = G⁰(k,iω_n)

1−G⁰(k,iω_n)Σ(k,iω_n). (18) Substituting the above equation into the equation (11), the electronic DOS in the presence of EPC can be obtained.

In what follows, we numerically explore the EPCgeffects on the electronic phase as well as on the degenerate states of the system. The phonon frequency ω₀=0.15 eV is fixed throughout the current section because it is well-established that the out-of-plane optical modes of 2D crystals have an energy ofω₀≈0.1–0.2 eV [74,75]. We would stress that the temperature effects are not addressed here because the Fermi temperature is large,TF≈5×10⁴K, while the usual and reli- able temperatures in such studies are much lower thanTF. For this reason, we believe that the degenerated occupied states (the van Hove singularities), as well as the band gap of the system, do not change with temperature within this framework.

We start with the individual role of EPC on the initial gapped phase of the system. So, we have fixed Mz=0 and h=0 in figure5. Before delving into the details of the EPC effects, we first provide some information on the e-ph process physically. The lowest-order process involving the EPC is the scattering of a single electron by a simultaneous creation or annihilation of a single phonon, as shown in equation (14b), which the probability for the scattering process is given byg constant. It is well-known that the electronic states are significantly influenced by lattice vibrations mostly in the close vicinity of the Fermi energy. More importantly, it affects the lifetime of excited electrons, which becomes important at high temperatures. Although both the in-planeσand out-of-planeπ orbitals are affected with the EPC, we will focus on theπones since the largest contribution to the final band gap of pristine h-BP is dedicated to the out-of-plane orbitals.

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Figure5illustrates the effect of different values ofgon the observation of the van Hove singularities and the band gap of the system. Near the zero energy (see the inset panel) the DOS is very small and it seems that the band gap is slightly decreased withg because some states are occupied, characterized by non-zero DOS. For this reason, we would report that EPC does not change the electronic semiconducting phase of the system and mini gaps take place anyway, in agreement with references [76,77]. The EPC is usually stronger in theσ bands than in theπbands and from this point, the effect of EPC is usually weak in the electronic features of low-dimensional materials in whichπbands are responsible for the electronic transport of the system. However, there are interesting behaviors when getting away from zero energy. For the states in the valence band, weak EPCg=0.1 eV does not show a significant change in the electronic DOS and the only change is related to the height of van Hove singularity at−2 eV, origin- ating from the significant changes in the conduction band.

To gain insight into the origin of the new singularities in the conduction band withgwe look at equation (18) in which the q-resolved matrix elements for the self-energy are connected to the scattering process betweenk andpmodes. This process manifests itself in the denominator of Green’s function Gαα(k−p,iΩ_nm)in equation (16) which are tightly connected to the Hamiltonian of both electrons with momentumkand phonons with modesp. This, in turn, leads to the singularities in equation (16) and eventually in the degenerated states in the electronic DOS. In other words, more fermionic momenta k_i emerge due to the scattering, and the creation probability of states with the same energies increases. Note thatg does not affect these singularities significantly and the only change would be induced to the position of the energies corresponding to these singularities. By this, we mean that in both weak and strong regimes of EPC, one expects the same singularities in the electronic DOS.

Let us set a criterion to distinguish between weak and strong regimes of EPC. Here the only way to do so is based on the deviations from the pristine behaviors of the DOS. In the case ofg=0.1 eV, the valence band behaves similarly as the pristine case and we would argue that we are still in the weak regime of EPC, while forg>0.1 eV, the deviations are much stronger and one would refer them to the strong EPC regime.

In both regimes, the electron–hole symmetry is broken through EPC process, which would be important to the experimentalists to achieve different information for various purposes.

In the strong regime, due to the above-mentioned scattering process between electrons and phonons, the new momenta are also induced to the electrons in the valence band, resulting in two singularities instead of one. The physics behind this in the strong EPC regimes backs to the strong bonding of electrons in the valence band to react to the phonons, while in the weak regime, bonding between electrons and phonons is not strong enough and the scattering process is also not strong enough in the valence band. The origin of singularity splitting in the valence band whengis strong can again be understood from the singularities in the self-energy stemming from the Green’s functions.

As the last point in this section, we would focus on another special case of electronic phase tuning of h-BP. This time we turn on one of the exchange fields M_z or h alongside with the EPCg. So, the combined effect of these external factors would lead to another behavior and distribution of electronic waves and eventually to new electronic DOS responses. We are interested in the gapless phase of the system which could be induced to the system through M_z= +E_g⁰/2 or h^↑⁽^↓⁾=

−(+)E_g⁰, see figures3and4. We found that these potentials show gapless states in h-BP. In this case, we would stress that the electronic phase of the system will not be altered with g because of the previously discussed physical reasons, and the same singularities with the same positions regardless of their height will be formed in the system.

5. Conclusions

In conclusion, we have investigated the role and strength of the induced exchange field, spin-polarized Zeeman field, and the EPC in theπband of h-BP and provide relevant and com- plementary discussions to explore the new electronic phases of the system. The origin of the observed phase transitions is also detailed. In doing so, we have employed the TB model and Green’s function technique to focus on the pristine and perturbed electronic DOS of h-BP.

Although theπband does not contribute to the EPC to tune the electronic phase of h-BP, its contribution to the exchange and the spin-polarized Zeeman fields are significant. Inter- estingly, it is found that a semiconductor-to-insulator and a semiconductor-to-semimetal-to-insulator phase transitions occur respectively when the exchange field induced to h-BP stems from the proximity coupling to a ferromagnetic semiconductor. Moreover, depending on the spin polarization of host electrons, the semiconductor-to-semimetal-to-insulator phase transitions emerge. Additionally, the degeneracy of energy bands in the presence of the above-mentioned external factors are physically addressed for which the scattering process in the case of the EPC effect leads to the new modes in both valence and conduction bands associated with the new degenerate states. The weak and strong EPC regimes are also distinguished through the DOS responses in the valence and conduction bands. All these physical insights are useful for different purposes in logic devices based on such low- dimensional materials.

Data availability statement

The data that support the findings of this study are available upon reasonable request from the authors.

Acknowledgments

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 103.01-2019.389.

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ORCID iDs

Nguyen T Dunghttps://orcid.org/0000-0001-7080-9700 Le T T Phuong https://orcid.org/0000-0001-8340-8738

References

[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V and Firsov A A 2005Nature438197

[2] Castro Neto A H, Guinea F, Peres N M R, Novoselov K S and Geim A K 2009Rev. Mod. Phys.81109

[3] Mannix A Jet al2015Science3501513 [4] Feng Bet al2016Nat. Chem.8563

[5] Kara A, Enriquez H, Seitsonen A P, Lew Yan Voon L, Vizzini S, Aufray B and Oughaddou H 2012Surf. Sci. Rep.

671

[6] Houssa M, Dimoulas A and Molle A 2015J. Phys.: Condens.

Matter.27253002

[7] Hoi B D, Yarmohammadi M and Kazzaz H A 2017J. Magn.

Magn. Mater.439203

[8] Deng Jet al2018Nat. Mater.171081 [9] Liao Met al2018Nat. Phys.14344

[10] Manzeli S, Ovchinnikov D, Pasquier D, Yazyev O V and Kis A 2017Nat. Rev. Mater.217033

[11] Zhao X N, Xu W, Xiao Y M, Liu J, Van Duppen B and Peeters F M 2020Phys. Rev.B101245412

[12] Yarmohammadi M 2017aJ. Magn. Magn. Mater.426621 [13] Yarmohammadi M 2017bSolid State Commun.25084 [14] Yan X, Chen Q, Li L, Guo H, Peng J and Peeters F 2020Nano

Energy75104953

[15] Dong H M, Tao Z H, Li L L, Huang F, Xu W and Peeters F M 2020Appl. Phys. Lett.116203108

[16] Li L L, Partoens B, Xu W and Peeters F M 20182D Mater.

6015032

[17] Li L L and Peeters F M 2018Phys. Rev.B97075414 [18] Bui H and Yarmohammadi M 2018Solid State Commun.

28039

[19] Shah N A, Li L L, Mosallanejad V, Peeters F M and Guo G-P 2019Nanotechnology30455705

[20] Liu H, Neal A T, Zhu Z, Luo Z, Xu X, Tom´anek D and Ye P D 2014ACS Nano84033

[21] Zhu Z and Tom´anek D 2014Phys. Rev. Lett.112176802 [22] Chaves Aet al20202D Mater. Appl.429

[23] Zhao Z-Y and Liu Q-L 2018Catal. Sci. Technol.81867 [24] Dileep K, Sahu R, Sarkar S, Peter S C and Datta R 2016J.

Appl. Phys.119114309

[25] Sengupta A, Audiffred M, Heine T and Niehaus T A 2016 J. Phys.: Condens. Matter.28075001

[26] Cai Y, Zhang G and Zhang Y-W 2014Sci. Rep.46677 [27] Li L L and Peeters F M 2019Appl. Phys. Lett.114243102 [28] Yarmohammadi M 2018Phys. Rev.B98155424

[29] Li L L, Bacaksiz C, Nakhaee M, Pentcheva R, Peeters F M and Yagmurcukardes M 2020Phys. Rev.B101134102 [30] Modarresi M, Mogulkoc A, Mogulkoc Y and Rudenko A 2019

Phys. Rev. Appl.11064015

[31] Xiong W, Xia C, Du J, Wang T, Peng Y, Wei Z and Li J 2017 Nanotechnology28195702

[32] Shi J, Ou Y, Migliorato M A, Wang H, Li H, Zhang Y, Gu Y and Zou M 2019Comput. Mater. Sci.160301

[33] Hoi B D and Yarmohammadi M 2018EPL (Eur. Lett.) 12217005

[34] Shohno K, Takigawa M and Nakada T 1974J. Cryst. Growth 24–25193

[35] Odawara M, Udagawa T and Shimaoka G 2005Japan. J. Appl.

Phys.44681

[36] Kumashiro Y, Nakamura K, Enomoto T and Tanaka M 2011 J. Mater. Sci., Mater. Electron.22966

[37] Yu J and Guo W 2015Appl. Phys. Lett.106043107 [38] S¸ahin H, Cahangirov S, Topsakal M, Bekaroglu E, Akturk E,

Senger R T and Ciraci S 2009Phys. Rev.B80155453 [39] Wang S-f and Wu X-j 2015Chin. J. Chem. Phys.28588 [40] Chegel R and Behzad S 2020aSynth. Met.266116476 [41] Yarmohammadi M and Mirabbaszadeh K 2020aJ. Appl. Phys.

128215703

[42] Wang Y, Huang C, Li D, Li P, Yu J, Zhang Y and Xu J 2019a J. Phys.: Condens. Matter.31285501

[43] Wang Y, Huang C, Li D, Huang F, Zhang X, Huang K and Xu J 2019bJ. Phys.: Condens. Matter.31465502 [44] Schroten E, Goossens A and Schoonman J 2019J.

Electrochem. Soc.1462045

[45] Jiang H R, Shyy W, Liu M, Wei L, Wu M C and Zhao T S 2017J. Mater. Chem.A5672

[46] Rold´an R, Chirolli L, Prada E, Silva-Guillén J A, San-Jose P and Guinea F 2017Chem. Soc. Rev.464387

[47] Avouris P, Heinz T and Low T 20172D Materials: Properties and Devices(Cambridge: Cambridge University Press) [48] Yarmohammadi M and Zareyan M 2016Chin. Phys.B

25068105

[49] Chegel R and Behzad S 2020bSci. Rep.10704

[50] Chen X, Tan C, Yang Q, Meng R, Liang Q, Jiang J, Sun X, Yang D Q and Ren T 2016Phys. Chem. Chem. Phys.

1816229

[51] Behzad S and Chegel R 2020J. Electron Spectrosc. Relat.

Phenom.242146969

[52] Çakr D, Kecik D, Sahin H, Durgun E and Peeters F M 2015 Phys. Chem. Chem. Phys.1713013

[53] Ullah S, Denis P A and Sato F 2018ACS Omega316416 [54] Brems M R and Willatzen M 2019New J. Phys.21093030 [55] Heyd J, Scuseria G E and Ernzerhof M 2003J. Chem. Phys.

1188207

[56] Yarmohammadi M and Mirabbaszadeh K 2020bJ. Phys. D:

Appl. Phys.54045105

[57] Nakamura T, Anh L D, Hashimoto Y, Iwasaki Y, Ohya S, Tanaka M and Katsumoto S 2018J. Phys.: Conf. Series 969012036

[58] Nakamura T, Anh L D, Hashimoto Y, Ohya S, Tanaka M and Katsumoto S 2019Phys. Rev. Lett.122107001

[59] Henk J, Flieger M, Maznichenko I V, Mertig I, Ernst A, Eremeev S V and Chulkov E V 2012Phys. Rev. Lett.

109076801

[60] Gruji´c M, Tadi´c M and Peeters F M 2013Phys. Rev.B 87085434

[61] Wu Yet al2020Nat. Electron.3604

[62] Mahan G 2000 Physics of solids and liquidsMany-Particle Physics(US: Springer)

[63] Kittel C 2004Introduction to Solid State Physics(New York:

Wiley)

[64] Yarmohammadi M 2016aPhys. Lett.A3804062

[65] Goodvin G L, Berciu M and Sawatzky G A 2006Phys. Rev.B 74245104

[66] Holstein T 1959aAnn. Phys., NY8325 [67] Holstein T 1959bAnn. Phys., NY8343

[68] Piscanec S, Lazzeri M, Mauri F, Ferrari A C and Robertson J 2004Phys. Rev. Lett.93185503

[69] Piscanec S, Lazzeri M, Robertson J, Ferrari A C and Mauri F 2007Phys. Rev.B75035427

[70] Pisana S, Lazzeri M, Casiraghi C, Novoselov K S, Geim A K, Ferrari A C and Mauri F 2007Nat. Mater.6198

[71] Yarmohammadi M 2016bAIP Adv.6085008

[72] Yarmohammadi M 2017cSolid State Commun.25357

8

(10)

[73] Yarmohammadi M 2017dJ. Electron. Mater.46747 [74] Stauber T and Peres N M R 2008J. Phys.: Condens. Matter.

20055002

[75] Wirtz L and Rubio A 2004Solid State Commun.131141

[76] Mazzola F, Frederiksen T, Balasubramanian T, Hofmann P, Hellsing B and Wells J W 2017Phys. Rev.B95075430 [77] Ulstrup S, Bianchi M, Hatch R, Guan D, Baraldi A, Alfè D,

Hornekær L and Hofmann P 2012Phys. Rev.B86161402

exchange field effects on the electronic properties of monolayer h-BP

Journal of Physics D: Applied Physics

Electron-phonon coupling, spin-polarized Zeeman field, and exchange field effects on the electronic properties of monolayer h-BP

Electron-phonon coupling,

spin-polarized Zeeman field, and

exchange field effects on the electronic properties of monolayer h-BP

K

M k x k y

Γ Γ a

a

δ

δ

δ

P B

M k _x k _y