STRESS -TUNED OPTICAL TRANSITIONS IN LAYERED 1T-MX 2 M H F ZR SN X S S E CRYSTALS A P REPRINT

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STRESS-TUNED OPTICAL TRANSITIONS IN LAYERED 1T-MX2

(M= HF, ZR, SN; X= S, SE) CRYSTALS

A PREPRINT

Miłosz Rybak1, Tomasz Wo´zniak1,∗, Magdalena Birowska2, Filip Dybała1, Alfredo Segura3,

Konrad J. Kapcia4,5, Paweł Scharoch1and Robert Kudrawiec1

Department of Semiconductor Materials Engineering, Faculty of Fundamental Problems of Technol-

ogy Wrocław University of Science and Technology, Wybrze˙

ze Wyspia´

nskiego 27, 50-370 Wrocław,

Poland; milosz.rybak@pwr.edu.pl (M.R.); ﬁlip.dybala@pwr.edu.pl (F.D.); pawel.scharoch@pwr.edu.pl (P.S.);

robert.kudrawiec@pwr.edu.pl (R.K.)

Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura St. 5, 02-093 Warsaw, Poland;

birowska@fuw.edu.pl

Departamento de Física Aplicada-ICMUV, Malta-Consolider Team, Universitat de València, 46100 Burjassot, Spain;

alfredo.segura@uv.es

Institute of Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz University in Pozna´

n, ul.

Uniwersytetu Pozna´

nskiego 2, 61-614 Pozna´

n, Poland; konrad.kapcia@amu.edu.pl

Center for Free-Electron Laser Science CFEL, Deutsches Elektronen-Synchrotron DESY, Notkestr. 85,

22607 Hamburg, Germany

* correspondence: tomasz.wozniak@pwr.edu.pl

ABSTRACT

Optical measurements under externally applied stresses allow us to study the materials’ electronic

structure by comparing the pressure evolution of optical peaks obtained from experiments and

theoretical calculations. We examine the stress-induced changes in electronic structure for the

thermodynamically stable 1T polytype of selected MX

compounds (M=Hf, Zr, Sn; X=S, Se), using

the density functional theory. We demonstrate that considered 1T-MX

materials are semiconducting

with indirect character of the band gap, irrespective to the employed pressure as predicted using

modiﬁed Becke–Johnson potential. We determine energies of direct interband transitions between

bands extrema and in band-nesting regions close to Fermi level. Generally, the studied transitions

are optically active, exhibiting in-plane polarization of light. Finally, we quantify their energy

trends under external hydrostatic, uniaxial, and biaxial stresses by determining the linear pressure

coefﬁcients. Generally, negative pressure coefﬁcients are obtained implying the narrowing of the band

gap. The semiconducting-to-metal transition are predicted under hydrostatic pressure. We discuss

these trends in terms of orbital composition of involved electronic bands. In addition, we demonstrate

that the measured pressure coefﬁcients of HfS

and HfSe

absorption edges are in perfect agreement

with our predictions. Comprehensive and easy-to-interpret tables containing the optical features are

provided to form the basis for assignation of optical peaks in future measurements.

Keywords MX2·DFT ·bulk ·band structure ·pressure coefﬁcients ·transition metal dichalcogenides

arXiv:2210.12074v1 [cond-mat.mtrl-sci] 21 Oct 2022

Stress-Tuned Optical Transitions ... A PREPRINT

1 Introduction

Among the large family of van der Waals (vdW) crystals, transition metal dichalcogenides (TMDs) have attracted a great

deal of interest owing to their unique combination of direct band gap, signiﬁcant spin–orbit coupling and exceptional

electronic and mechanical properties, making them attractive for both fundamental studies and applications [

]. In

particular, their semiconducting nature opens a door to potential optoelectronic, photonic and sensing devices such as

light emitting diodes, microlasers, solar cells, transistors or light detectors [3, 4, 5, 6].

Optoelectronic properties of vdW materials can be tuned by multiple external factors. One of them is an effective strain

engineering. Recent theoretical and experimental reports have demonstrated ﬂexible control over their electronic states

via applying external strains [

]. For instance, applying an uniaxial tensile strain to monolayer of MoS

may result

in direct-to-indirect band gap transition [

], whereas applying a biaxial strain gives rise to a semiconductor-to-metal

phase transition [

]. Meanwhile, the prominent mechanical strength of TMDs [

], compared with conventional 3D

semiconductors, allows to use large strains for band structure engineering. For instance, combined studies by means of

density functional theory (DFT) calculations and atomic force microscopy measurements have reported that the fracture

stress of a freely suspended MoS

[

] approaches the theoretical limit of this quantity for defect-free elastic crystal

(one-ninth its Young’s modulus) [

]. In addition, numerous nondestructive optical techniques, including Raman,

absorption, photoreﬂectance, and photoluminescence experiments, can be readily employed to quantitatively determine

strain-tuned optical properties. In addition, high-pressure measurements are highly desirable for detailed band structure

information as well as give useful benchmark to test DFT calculations. Such techniques also provide a direct way

to probe interlayer interaction in the layered structures. In particular, recent experimental reports have demonstrated

that the energies of various optical transitions in TMDs exhibit signiﬁcant pressure dependence [

which allows for the identiﬁcation of the optical peaks, making them attractive for applications in pressure-sensing

devices [

]. Generally, the unique mechanical ﬂexibility and strength of TMDs make them an ideal platform

for band gap engineering by strain, thus, enabling enhancement of their optical properties.

The chemical formula of hexagonal TMDs is MX

, where M stands for a transition metal element, and X is a

chalcogene element (S, Se or Te). TMDs exhibit several structural polytypes of which two most common are trigonal

prismatic (2H) and octahedral (1T) ones (see Figure 1). The difference between 2H and 1T polytypes can be viewed

in different arrangement of atomic planes sequence within the monolayer. Namely, 2H polytype corresponds to

an ABA arrangement, whereas 1T polytype is characterised by ABC sequence order [

]. Although 2H polytype

of TMDs, based on Mo and W, have been extensively studied, the octahedral 1T MX

compounds containing the

M=Hf, Zr and Sn, X=S, Se elements have been less examined. The latter ones are indirect-gap semiconductors with

band gaps ranging from visible to near-infrared wavelengths [

]. The earlier studies on 1T-MX

compounds have

predicted very high electron mobility and sheet current density in HfS

, superior to MoS

[

], which makes

ultrathin HfS

phototransistors appealing for optoelectronics [

]. Thin SnSe

ﬂakes were shown to exhibit high

photoresponsivity [

]. ZrS

nanosheets were found suitable as anodes for sodium ion batteries [

]. These ﬁndings

motivate further studies of electronic properties of 1T-MX

crystals in 1L and bulk form. Despite some works reporting

pressure evolution of Raman spectra [

], as well as X-ray diffraction and transport measurements [

], optical

measurements under pressure are largely missing for 1T-MX2compounds.

In this work, we systematically investigate the impact of external stress on the basic features of the band structure

of MX

(M=Hf, Zr, Sn; X=S, Se) in the 1T bulk polytype by DFT calculations. For each compound, we identify

the dominant direct electronic transitions in BZ. As the structural anisotropy of in-plane and out-of plane directions

in layered systems may result in different response to the strain, we study the evolution of the band structure upon

applying stress types that are most frequently realized in experiments, i.e., compressive isotropic (hydrostatic), biaxial,

and uniaxial stress. We quantify the energy trend for each transition between ambient and band gap closing pressure by

determining the linear pressure coefﬁcients. In addition, we examine the effect of light polarization for optically active

direct transitions using dipole selection rules. Our predicted pressure coefﬁcients and polarization of transitions can

serve for identiﬁcation of the features in measured optical spectra. Meanwhile, we explain the observed chemical trends

by the orbital composition of electronic bands involved in the transitions. Finally, we compare our calculated results to

the pressure trends of absorption edges positions measured in HfS

and HfSe

crystals, ﬁnding an excellent agreement.

It corroborates that our adopted computational strategy is accurate at the quantitative level.

Stress-Tuned Optical Transitions ... A PREPRINT

Figure 1: Top and side views of the (

) trigonal prismatic (2H) and (

) octahedral (1T) polytypes of MX

. (

) The ﬁrst

Brillouin zone (BZ) with high-symmetry k-points and lines denoted in blue.

2 Methods and Materials

The DFT calculations have been performed in Vienna Ab Initio Simulation Package [

]. The electron-ion interaction

was modeled using projector-augmented-wave technique [

]. In the case of tin (Sn) atom, the 4d

states were included

in valence shell, for hafnium and zircon, additional s states were taken (4s

for Zr, 5s

for Hf). The Perdew–Burke–

Ernzerhof (PBE) [

] exchange-correlation (XC) functional was employed. A plane-wave basis cutoff of 500 eV and a

12 ×12 ×8

Monkhorst-Pack [

] k-point grid for BZ integrations were set. These values assured the convergence of

the lattice constants and the electronic gaps were within precision of 0.001 Å and 0.001 eV, respectively. A Gaussian

smearing of 0.02 eV was used for integration in reciprocal space. It is well known that standard exchange correlation

functionals are insufﬁcient to describe a non-local nature of dispersive forces, crucial to obtain a proper interlayer

distance for layered structures [

]. Thus, the semi-empirical Grimme’s correction with Becke–Johnson damping

(D3-BJ) [

] was employed to properly describe the weak vdW forces. The spin–orbit (SO) interaction was taken into

account.

It is well established that the standard approximations to the XC functional lead to a severe underestimation of the

electronic band gap and the lack of inclusion excitonic effects. In this regard, DFT is inaccurate for identiﬁcation of

optical transitions based on their absolute energy values. This issue can be partly improved by using more advanced

techniques such as hybrid functionals or GW method [

], but their computational costs often make the calculations

unfeasible for systems containing more than few atoms. The modiﬁed Becke–Johnson (mBJ) potential is an alternative

approach to improve the band gaps with relatively low computational cost [

]. Recent report shave shown that

mBJ provides reasonable results for identifying the optical transitions in ReS

and ReSe

bulk crystals [

]. It also

yields pressure coefﬁcients of optical transitions in excellent agreement with experimental values [

]. Therefore,

we employ mBJ potential for band structure calculations, on top of the optimized geometry obtained within the

PBE+D3-BJ+SO approach. The direct interband momentum matrix elements were computed from the wave function

derivatives using density functional perturbation theory [46].

Stress-Tuned Optical Transitions ... A PREPRINT

3 Results

3.1 Theoretical Analysis

We start our research by considering the geometry and electronic structure for the unstrained systems. The optimized

lattice parameters, provided in Table 1, are in perfect agreement with experimental values. Similarly to 2H-TMDs,

lattice constants are mostly governed by chalcogene atoms [

]. As it is expected for heavier atoms, the selenium

(Se) compounds possess larger lattice parameters than sulfur (S) ones. The electronic band structures calculated under

ambient conditions are presented in Figure 2. The band edges are located at the same high symmetry k-points for all

studied systems. Namely, the valence band maximum (VBM) and conduction band minimum (CBM) are located at

and

k-points, respectively. Note that the VBM of SnSe

at ambient pressure is not located exactly at

point, but

between the

and K points (on the

-M path the local maximum is 2 meV lower). Under biaxial stress the VBM shifts

to A point, but under hydrostatic and uniaxial stress the position and shape of VBM remain unchanged. This type of

pressure behavior has already been observed in InSe crystals, where VBM exhibits toroidal shape [

]. The toroidal

shape has consequences in transport and optical properties and would require further investigations, which are beyond

the scope of our work. The calculated fundamental gaps exhibit indirect character with values systematically lower by

30–50

with respect to experimental values (see Table 1). The systems containing Se atoms exhibit reduced size of the

energy gaps in comparison to S-containing systems.

The underestimation of the band gap is related to the geometrical structure—a better agreement is obtained with the use

of experimental lattice constants, as shown in Ref. [

] and discussed in Appendix B for ZrSe

. In our study, we focus,

however, not on the absolute value of the band gap, but rather on the pressure dependence of optical transitions, which

requires a full optimization of geometry. Further, the discrepancies between theoretical and experimental bands gaps

stem from the systematic underestimation of the band gap and the lack of including excitonic effects in our theoretical

approach. On the other hand, the quasi-two-dimensional character of layered crystal leads to exciton binding energies on the

order of tens or hundreds of meV [

], which redshifts the optical energies from their band-to-band values.

Incidentally, it can improve the agreement with experiments, but this is fortuitous result and strongly material-dependent. In

contrast to the absolute energy of transition, variation of its energy with respect to pressure, quantified by a linear pressure

coefficient, demonstrates to be in good agreement with measured value [

]. Additionally, the dependence of the

exciton binding energies upon the pressure can be neglected, whenever the exciton binding energy is much smaller than

the transition energy [

]. Aforementioned suggest that the pressure coefficients obtained using mBJ might provide

reasonable values and enable proper identification of the measured optical peaks on a quantitative level. Therefore, in order

to compare the optical experimental results with our theoretical outcomes, the pressure coefficients are computed.

Table 1: Calculated and measured lattice constants and fundamental band gaps of all the compounds.

System aDFT(Å) cDFT(Å) EDFT

g(eV) aexp(Å) cexp(Å) Eexp

g(eV)

HfS23.59 5.75 1.50 3.63 [56] 5.86 [56] 1.96 [57], 1.80 [58], 1.87 [59]

HfSe23.70 6.08 0.71 3.67 [56] 6.00 [56] 1.13 [57], 1.15 [58]

ZrS23.63 5.72 1.12 3.66 [57] 5.82 [57] 1.68 [57], 1.70[60], 1.78 [61]

ZrSe23.74 6.04 0.33 3.77 [57] 6.14 [57] 1.20 [60], 1.10 [62], 1.18 [63]

SnS23.67 5.80 2.14 3.65 [64] 5.90 [64] 2.88 [65]

SnSe23.84 6.00 1.10 3.82 [64] 6.14 [64] 1.63 [65]

Although structural phase transitions under pressure were reported for our compounds [

], they

are out of the scope of this work and we consider only the 1T phase under hydrostatic pressures up to metallization

limit. In our mBJ-PBE+SO calculations they occur at pressures of: 266 kbar for SnS

, 188 kbar for HfS

, 128 kbar

for ZrS

, 84 kbar for SnSe

, 65 kbar for HfSe

and 26 kbar for ZrSe

. We also apply uniaxial and biaxial stresses,

as depicted on Figure 4, which result from reducing the lattice parameters by up to 8% (see Appendix B). Figure 3

presents the band structures of HfS

and SnSe

under hydrostatic, uniaxial, and biaxial stress, as representatives of

(Hf,Zr)X

and SnX

groups. Note, that the indirect character of the band gaps is preserved, irrespective of the pressure

applied. The band edges positions are located at the same k-points as for unstrained samples, except for SnX

systems

under hydrostatic pressure, where CBM moves from L to M point, and biaxial strains, where VBM moves from

A point. Note that, the application of compressive uniaxial strains result in reduction in the band gaps. Notably, the

impact of hydrostatic or biaxial pressures is non-trivial and more complicated. In particular, for SnX

compounds, the

biaxial stresses initially increase the energy gap and move the VBM from

to A point (see Figure 3b and the Appendix

B for a detailed discussion).

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STRESS-TUNEDOPTICALTRANSITIONSINLAYERED1T-MX2(M=HF,ZR,SN;X=S,SE)CRYSTALSAPREPRINTMioszRybak1,TomaszWo´zniak1;,MagdalenaBirowska2,FilipDybaa1,AlfredoSegura3,KonradJ.Kapcia4;5,PaweScharoch1andRobertKudrawiec11DepartmentofSemiconductorMaterialsEngineering,FacultyofFundamentalProblemsofTechnol-ogyWr...

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