Ab initio analysis of some Ge-based 2D nanomaterials
Ali Ghojavand, S. Javad Hashemifar, Mahdi Tarighi Ahmadpour, Alexander V. Shapeev, Amir Alhaji, Qaem Hassanzada
AAb initio analysis of some Ge-based 2D nanomaterials
Ali Ghojavand, ∗ S. Javad Hashemifar, † Mahdi Tarighi Ahmadpour, Alexander V. Shapeev, Amir Alhaji, and Qaem Hassanzada Department of Physics, Isfahan University of Technology, Isfahan, 84156-83111, Iran Skolkovo Institute of Science and Technology, Skolkovo Innovation Center,Bolshoy Boulevard 30, bld. 1, Moscow, 121205, Russia Department of Materials Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran (Dated: January 24, 2020)The structural, electronic and dynamical properties of a group of 2D germanium-based com-pounds, including GeC, GeN, GeO, GeSi, GeS, GeSe, and germanene, are investigated by employ-ing first-principles calculations. The most stable structure of each of these systems is identifiedafter considering the most probable configurations and performing accurate phonon calculations.We introduce a new phase of germanene, which we name the tile germanene, which is significantlymore stable than the known hexagonal germanene. We apply the modern modified Becke-Johnson(mBJ) and DFT1/2 schemes to obtain an accurate band structure for our selected 2D materials. Itis seen that GeO and GeC exhibit the highest band gaps of more than 3 eV in this group of materi-als. Moreover, we argue that, in contrast to the semi-metallic nature of hexagonal germanene, thetile germanene is a very good conductor. The band edges of our semiconducting 2D materials areaccurately aligned to the vacuum level to address the potential photocatalytic application of thissystem for water splitting and carbon dioxide reduction. The optical properties, including dielectricfunctions, refractive index, reflectivity, and Loss function of the samples are investigated in theframework of the Bethe-Salpeter approach.
I. INTRODUCTION
In recent years, elemental sheet of germanium, knownas germanene, has been emerging as a strong contenderin the realm of 2D materials. Germanene is a one-atom-thick germanium layer which has a honeycomb structure(D3d point group) and a zero band gap with a Dirac coneat the K point of the Brillouin zone. In 2009, Cahangirovet al, by using first-principles calculations, predicted alow buckled (corrugated) sheet structure for a 2D ger-manene layer. The main hurdle experienced in realiz-ing individual germanene layers is that, unlike graphene,they do not form a van der Waals layered structure intheir natural form. Hence, top-down approaches are notapplicable for synthesis of individual germanene layers.In 2014, the first synthesis of germanene was realized on agold (111) substrate with a growth mechanism similar tothe formation of silicene layers on silver (111) templates. The absence of band gap dramatically hampers directapplications of germanene in semiconductor devices innanoelectronics, photoelectronics, and sensors. Hence,seeking an effective method to open a sizable band gapin germanene is an active field of research. Chemicalfunctionlization with small molecules, introducing struc-tural defects, and alloying with proper elements are threeconventional methods to engineer the band structure of2D materials. Hydrogen functionalized germanene layers(GeH) were successfully synthesized in 2013 with a bandgap of about 1.5 eV and a similar structure to graphane. Padilha and others considered Stone-Wales (SW), sin-gle vacancy, and divacancies defects in germanene andshowed that the SW defect open a band gap in the systemand destroys the Dirac cone, while the single vacancy de-fect preserves the Dirac cone. Introducing structural de- fects such as StoneWales, single vacancy, and divacanciesstrongly affects the band structure and transport prop-erties of the system, compared with the pristine one. Xuet al. predicted that alloying with Se may lead to the for-mation of two different semiconducting configurations ofGeSe monolayers, which exhibit anisotropic absorptionspectra in the visible region. In this work, we employ first-principles calculations tostudy the structural, electronic, and optical propertiesand dynamical stability of a group of Ge-based 2D ma-terials including GeO, GeS, GeSe, GeN, GeC, GeSi, andgermanene. In the next section, we introduce the compu-tational techniques used in this work. Then the stabilityof the systems will be addressed in section III, electronicproperties will be explained in section IV, and opticalproperties will be presented in section V. The summaryof our work will be given in the last section.
II. METHOD
We performed electronic structure calculations and ge-ometry optimization in the framework of density func-tional theory (DFT) and the Perdew, Burke, and Ernz-erhof (PBE) exchangecorrelation functional by using thefull-potential all-electron numeric atom-centered orbitalmethod implemented in the FHI-aims package. In or-der to obtain very accurate binding energies and opti-mized geometries, the structure relaxation tolerance wasset to 0.001 eV/˚A˙In order to verify dynamical stabilityof the structures, phonon calculations were performedby using the density functional perturbation theory andthe plane wave ultrasoft pseudopotential method, imple-mented in the QUANTUM ESPRESSO package. The a r X i v : . [ c ond - m a t . m t r l - s c i ] J a n tile (4 atoms/cell, nn=4) puckered (4 atoms/cell, nn=3) zigzagbuckled (2 atoms/cell, nn=3) (4 atoms/cell, nn=3) ab a bb aa b FIG. 1. Top and side views of the four candidate structuresfor our Ge based 2D materials. The dotted lines encapsu-late the 2D unitcell of the lattices and the blue and red ballsindicate two nonequivalent atomic positions in the systems. reliability of the pseudopotentials was verified by com-paring the obtained binding energies in the QuantumEspresso and FHI-aims packages. The novel DFT1/2and modified Becke-Johnson (mBJ) schemes were appliedto correct the PBE electronic band structure of the sys-tems. The accuracy of the band gaps in these methodsand especially DFT1/2 were comparable with the expen-sive hybrid functional and GW methods, while theirrequired computational effort is comparable to that ofPBE. The DFT1/2 method , which extends the Slatershalf-occupation technique to bulk materials, were appliedby using the Exciting package which employs the fullpotential linear augmented plane wave method to solvethe single particle Kohn-Sham equations. On the otherhand, the modified Becke-Johnson (mBJ) method wasapplied by using the Wien2k package which has a verysimilar technical structure to the Exciting package. Themonolayer structures were simulated in the slab super-cells with a vacuum thickness of about 13˚A to avoid un-realistic effects from periodic boundary conditions.The optical properties of the systems were calculatedby using the Exciting package which takes into accountthe nontrivial effects of electron-hole interaction andsolves the Bethe-Salpeter equation (BSE): ( ε c k − ε v k ) A Scv k + (cid:88) k (cid:48) c (cid:48) v (cid:48) κ cv k c (cid:48) v (cid:48) k (cid:48) A Sc (cid:48) v (cid:48) k (cid:48) = Ω s A Scv k where the term ( ε c k − ε v k ) refers to the difference betweenthe conduction and valence quasiparticle energies at aspecific k-point, κ describes the electron-hole interaction,and Ω s is the excitation energy. After solving for the BSEexcitation states, the Tomm-Dancoff approximation (TDA) is used to compute the imaginary part of the di-electric function ( ε ):Im( ε iM ( ω )) = ε ( ω ) = 16 π e ω (cid:88) S | (cid:126)e. (cid:104) | (cid:126)v | S (cid:105)| δ ( w − Ω s )where (cid:126)e describes the polarization of the incident lightand (cid:126)v is the velocity operator. Then the Kramers-Kronigrelations are employed to find the real part of the di-electric function ( ε ) and subsequently other linear opti-cal properties including refractive index n ( ω ), reflectivity R ( ω ), and electron loss function L ( ω ): n ( ω ) = (cid:32) (cid:112) ε + ε + ε (cid:33) / k ( ω ) = (cid:32) (cid:112) ε + ε − ε (cid:33) / R ( ω ) = ( n − + k ( n + 1) + k L ( ω ) = ε ε + ε III. STABLE STRUCTURES
After a broad literature survey, we realized that thestructures of the most novel 2D materials may be gen-erally categorized in the puckered and buckled configu-rations, presented in Fig. 1, while other structures arerarely investigated in the literature. Hence, we appliedthese 2D structure patterns to our desired materials; Ge,GeO, GeS, GeSe, GeN, GeC, and GeSi. During geometri-cal optimizations, in some cases, we noticed appearanceof two new structures, called zigzag and tile in Fig. 1,which were added to the candidate structures of our 2Dmaterials. In order to find the most stable structure ofeach of the systems, we calculated their binding energy inthe four candidate structures by comparing the optimizedtotal energies with the energy of corresponding isolatedatoms. The obtained binding energies are presented intable I.It is very interesting to see that these materials mostlyadmit the buckled configuration as a metastable struc-ture while their lowest-energy structure is among other fr e qu e n c y ( c m - ) Γ X S Y Γ Ge - tile
M K Γ Ge - hex
X S Y Γ GeSi
X S Y Γ GeS
X S Y Γ GeSe Γ M K Γ GeC
X S Y Γ GeN fr e qu e n c y ( c m - ) X S Y Γ GeO
FIG. 2. Obtained phonon band structure of our selected 2D materials. The green and orange shaded areas show the phononpartial density of states of Ge and its partner atoms in our 2D systems, respectively.TABLE I. Calculated binding energy (eV/atom) and thick-ness (˚A) (in the parenthesis) of the studied 2D materials inthe four candidate structures.buckled puckered zigzag tileGeO − − (2.23) —GeS − − (2.57) — —GeSe − − (2.60) — —GeN − − (1.26) —GeC − (0.00) — — —GeSi − − (1.95)Ge − − (1.97)TABLE II. Relaxed structural parameters a,b and averagenearest neighbor bond lengths d1 and d2 (˚A) of the investi-gated 2D systems, in their most stable structure.GeO GeS GeSe GeN GeC GeSi Ge-tile a b candidates. It is more fascinating in the case of ger-manene, where its well known hexagonal structure (zerobuckled) is considerably less stable than our discoveredtile structure. A more reliable and detailed investigationof this issue requires studying dynamical stability whichwill be presented later. The obtained results indicatethat the binding energy is decreasing with the increaseof the atomic radius. In other words, oxygen, nitrogen,and carbon atoms which are the smallest atoms in oursamples give rise to the highest binding energies for theGeO, GeN, and GeC compounds. On the other hand,Ge which has the largest atomic radius leads to the low-est binding energy for the 2D germanene. This trendindicates a stronger bonding between atoms with smallerradii which is in agreement with the physical intuition.The equilibrium lattice constants and bond lengths ofthe lowest-energy structures of GeO, GeS, GeSe, GeN, . . . -6-4-2024 E n e r gy ( e V ) G e - til e . G e - h e x . G e S i . Partial Density of States of p orbital (states/eV) G e S e . . G e S . G e C . G e N .
60 1 . . G e O FIG. 3. Contribution of p orbitals of Ge (blue, left oriented)and its partner atom (red, right oriented) in the density ofstates of the investigated 2D systems.
GeC, GeSi, and germanene are calculated and presentedin Table II.As it was mentioned before, phonon calculationsshould be done to confirm the stability of the lowest-energy structures. The phonon dispersion curves of allthe materials at their lowest-energy structure were com-puted by using the density functional perturbation the-ory method. The obtained phonon spectra of the samplesalong their high symmetry paths in the reciprocal spaceare presented in Fig. 2. The absence of any negative(imaginary) frequency mode in the spectra demonstratesdynamical stability of all the systems. We observe thatGeC, GeN, and GeO display the widest range of phononmodes among the studied systems, providing further ev-idence for stronger bonding in these systems, comparedwith others. Especially in GeC, a large distance seenbetween the acoustic modes and the two optical modeswhich is an evidence of a strong directional bonding inthis system. In the new phase of germanene, in con-trast to the hexagonal phase, we observe that the opticalmodes are not well separated from the acoustic modes,which may be attributed to the softer bonding in the tilegermanene.For better understanding of the dynamical features ofthe systems, we calculated the phonon partial densityof states (PDOS) of Ge and its partner atoms in ourinvestigated 2D materials. The results are presented asshaded areas in Fig. 2. It is clearly seen that the heavyelements Ge and Se have a larger contribution to theacoustic phonon modes while the light elements exhibitstronger vibrations in the optical modes.Dynamical stability of the new phase of germanenealong with its lower binding energy, compared with thehexagonal germanene, raises the question of why thisphase has not yet been observed in real samples. In or-der to address this question one should note that withinall successful synthesis of germanene, the [111] surface ofgold or platinum has been used as the substrate. Thesesurfaces involve hexagonal arrangement of the substrateatoms which may enhance formation of hexagonal ger-manene on the surface. Hence, atomistic growth of ger-manene atoms on square-symmetry surfaces may enhancethe formation of tile germanene in realistic samples.
IV. ELECTRONIC STRUCTURE
The obtained electronic PDOS of the systems in theirlowest-energy structure, within PBE, are presented inFig. 3. It is seen that the systems with the highestbinding energies (GeO, GeN, and GeC) exhibit the mostbroadened valence bands, in agreement with strong bond-ing in these systems. Moreover, the valence p orbital ofGe in GeO and GeN is effectively evacuated and trans-ferred to the valence p orbital of its partner atom, indi-cating significant ionic bonding in these two monolayers.GeSi and the tow configurations of germanene represent ametallic electronic structure while the other investigated2D materials are semiconductors. The tile configurationof germanene is likely a very good conductor, because ithas a finite density of high mobility p electrons at theFermi level.In order to obtain more accurate electronic struc-tures, as it was mentioned in the Methods, we apply theDFT1/2 and mBJ schemes which have been proven topredict much more accurate band gaps, compared withthe conventional LDA and GGA functionals. The calcu-lated electronic band structures of GeO, GeS, GeSe, GeN,GeC and GeSi within the DFT1/2 and mBJ schemesalong the high-symmetry directions of the Brillouin zoneare presented in Fig. 4. It is seen that these two meth-ods predict quite similar band structures, except for theposition of Fermi levels which are occasionally different.More discussion on the relative positions of Fermi levelsand band edges needs to the alignment of the energy ref-erences which will be presented in the next paragraphs. -9-6-303 E n e r gy ( e V ) GeO Γ X SY -9-6-303 E n e r gy ( e V ) GeS Γ X S Y
GeSe Γ X S Y
GeN Γ X SY
GeC Γ MK GeSi Γ X S Y
Ge-hex Γ MK m B J Ge-tile Γ X S Y Γ D F T / FIG. 4. Electronic band structures of our Ge based 2Dcompounds within the mBJ (top row) and DFT1/2 (bottomrow) methods.
Among the studied systems, GeC exhibit the most pe-culiar band structure with very high valence band dis-persion, indicating a very stiff directional bonding in thissystem. This observation is consistent with the large dis-tance observed between the acoustic and optical phononmodes of this system (Fig. 2). We observe that the Diraccone of hexagonal germanene is slightly shifted withinmBJ, indicating less accuracy of the mBJ Fermi level,compared with DFT1/2. GeSi and tile germanene ex-hibit similar band structures, because both systems sta-bilize in the tile structure and have similar atomic va-lence shell. The same similarity is visible between GeSand GeSe band structures. The calculated values forthe energy gaps are summarized in Table III. In thistable the reported band gap within the hybrid HSE06functionals are also given for comparison. We ob-serve that the predicted band gaps within DFT1/2 arecloser to the HSE06 gaps, compared with the mBJ func-tional. Due to the lack of experimental data, we arenot able to compare the accuracy of the DFT1/2 andHSE06 methods, although in the case of bulk semicon-ductors there is some evidences for a higher accuracy ofthe DFT1/2 scheme.
2D materials are generally considered as potential can-didates for photocatalytic applications in various chem-ical reactions. Because in these nanomaterials, the rela-tive surface area is very large, the transport distance forthe photo generated carriers to reach the reaction inter-face is very short, and the band gap is likely enlargeddue to the quantum confinement effect . Therefore, wescreen the band edges of our germanium-based 2D mate-rials to investigate their potential photocatalytic applica-tion in the water splitting and carbon dioxide conversion TABLE III. Calculated band gap (eV) of GeS, GeSe, GeC,GeO, and GeN in their most stable structures within thePBE, DFT1/2 and mBJ methods. The most accurate re-ported band gaps within HSE06 and GW0 scheme are alsogiven in the last column as the best theoretical references.PBE DFT1/2 mBJ othersGeO 2.84 3.22 3.27 3.73 [HSE] GeS 1.79 2.27 2.06 2.32 [HSE] GeSe 1.22 1.68 1.41 1.61 [HSE] GeN 1.05 1.29 1.30 —GeC 2.06 3.50 2.32 3.56 [GW] reactions. The water splitting reactions are:2H O + 4h + → + + O ( − .
67 eV)2H + + 2e − → H ( − .
44 eV)where h + and e − are the photo-generated hole and elec-tron and numbers in the parenthesis are the correspond-ing redox potentials at room temperature and zero pH .A proper photocatalyst for these reactions should havea conduction band bottom (CBB) below the hydrogenevolution potential and a valence band top (VBT) abovethe oxygen evolution potential. In addition to the abovewater splitting reactions, we considered carbon dioxideconversion to methanol, formic acid, and methane as fol-lows:CO + 6H + + 6e − → CH OH + H O ( − .
06 eV)CO + 2H + + 2e − → HCOOH ( − .
83 eV)CO + 4H + + 8e − → CH + H O ( − .
20 eV)As it was mentioned before, the numbers in the parenthe-sis are corresponding reduction potentials at room tem-perature and zero pH . In order to consider other tem-perature and pH values, the redox potentials should beshifted by pH × (K B T × ln 10). The CBB of the propersemiconductor photocatalyst for the above mentionedconversions should be above the corresponding reductionpotentials.The band edges of our 2D systems were determinedwith respect to the vacuum level potential within theDFT1/2 and mBJ schemes and the resulting CBBs,VBTs, and Fermi levels are compared with the abovementioned redox potentials in Fig. 5. The vacuum level isdetermined by averaging the electrostatic potential of theslab supercells in the horizontal planes and then plottingthe averaged potential as a function of vertical positionz. The obtained CBB band edges clearly indicate that allour five 2D semiconductors (GeO, GeS, GeSe, GeN, andGeC) are good photocatalyst candidates for the threeconsidered carbon dioxide reduction reactions and theH + /H water splitting half-reaction. On the other hand,GeO, GeS, and GeC may be good photocatalysts for theH O/O water splitting half-reaction while GeSe needs arather little bias of less than 0.2 eV to photocatalyse thisreaction. til e - G e h e x - G e G e O G e S G e S e G e N G e C G e S i -6.0-5.6-5.2-4.8-4.4-4.0-3.6-3.2-2.8-2.4 E n e r gy ( e V ) O / H OH + / H CO / CH CO / CH OHCO / HCO HCBB-DFT1/2CBB-mBJFER-DFT1/2 FER-mBJVBT-DFT1/2VBT-mBJ
FIG. 5. Valence band top (VBT), conduction band bottom(CBB), and Fermi (FER) levels of the studied 2D systemswith respect to the vacuum level within DFT1/2 and mBJschemes.
V. OPTICAL PROPERTIES
As it was mentioned in the section Method, we inves-tigated the optical properties of our 2D materials in theframework of the Bethe-Salpeter approach. The obtainedoptical properties including the real and imaginary partsof the dielectric function, reflectivity, refraction index,and the energy loss function are presented in Fig. 6. Op-tical properties are calculated for two polarization of theincident light electric field along the x and y directions.Comparing the xx and yy components of optical param-eters indicate that GeC and hexagonal germanene arewell isotropic in the xy plane in whole frequency range,while other 2D systems exhibit clear in-plane anisotropiesin the frequencies below 10 eV, being attributed to theanisotropic crystal structure of these materials. The ob-served in-plane anisotropy is more pronounced in GeO,GeS, and tile germanene. It is interesting to see thatthe new invented structure of germanene exhibit stronganisotropy in all optical parameters in low frequencies,below 2 eV. The strong anisotropy of the refractive indexwill likely lead to birefringence behavior of tile germanenein low frequencies. The results show high reflectivity ofGeS, GeSe, GeSi, and hexagonal germanene in a broadfrequency range. On the other hand, GeS, GeSe, andGeSi exhibit very low refractive index around frequencyof 8 eV, in the UV region.The first peak in the imaginary part of the dielectricfunction is expected to give the optical gap of semicon- Ge-tile -80816 ε ( ω ) xxyy ε ( ω ) R ( ω ) Ge-hexxxyy GeOxxyy GeS GeSe GeNxxyy GeCxxyy GeSixxyy n ( ω ) xxyy xxyy L o ss xxyy Photon energy ω (eV) xxyy FIG. 6. Optical properties of our investigated 2D materials, including the real ( ε ( ω )) and imaginary ( ε ( ω )) parts of thedielectric function, reflectivity (R( ω )), refractive index n( ω ), and energy loss function, calculated for the x (green shaded area)and y (red solid line) polarizations of the incident light. ductors, while the characteristic frequency of metals, cor-responding to the collective excitations of valence elec-trons, known as plasma frequency, is given by those peaksof the loss function which are located around nodes ofthe real part of the dielectric function. The optical gapand plasma frequencies of the systems were determinedand presented in table IV. The corresponding zero en-ergy value of the real part of dielectric function, known asstatic relative permittivity, is also extracted and given inthis table. It is seen that GeSi and hexagonal germaneneshows the highest static relative permittivity among ourstudied systems. The largest optical gap is observed inGeC, while GeN has a small optical gap of about 0.45 eV.Since the calculated optical responses are obtained in thepresence of the attractive electron-hole interaction (ex-citonic effects), well described in the Bethe-Salpeter ap-proach, the distance between the optical gap and the elec-tric gap is introduced as a measure of the exciton bindingenergy of the system. In the framework of electronicstructure theory, the electric gap is determined by themany body based GW scheme and as it was mentionedbefore, the GW gaps are expected to be very close to theobtained gaps within the DFT1/2 scheme (table III).Hence, we calculated the exciton binding energy of our2D systems as the difference between their optical andDFT1/2 gaps and presented the results in table IV. Weobserve a very low exciton binding energy for GeC andGeSe, which may indicate a very low carrier recombina-tion rate in these systems after photo-excitations. This TABLE IV. Obtained static value (at ω = 0) of the real partof dielectric function, optical gap (eV), exciton binding energy∆ X (eV), and plasma frequencies ω p (eV) of the investigated2D systems. ε xx ε yy gap ∆ X ω p GeO 2.42 3.33 2.15 1.07 2.85GeS 4.33 4.56 2.04 0.23 8.32GeSe 7.21 7.62 1.68 0.00 8.56GeN 4.61 4.09 0.45 0.84 1.33, 2.55GeC 2.63 2.63 3.50 0.00 4.05, 8.45GeSi 12.94 8.59 — 2.02, 8.14Ge-tile 2.93 8.70 — 1.48, 4.88Ge-hex 14.84 13.49 — — observation encourages photocatalytic and photovoltaicapplications of 2D GeC and GeSe materials.The last column in table IV shows the predictedplasma frequencies of our systems. Although, plasmafrequency is usually a characteristic feature of metallicsystems, as it is seen in the case of our 2D semiconduc-tors, semiconducting materials may also exhibit this kindof collective excitation of valence electrons. While mostof the investigated 2D systems has a plasma frequency inthe UV region, we observe that plasma oscillations in thevisible region may happen in the semiconducting GeOand GeN and metallic GeSi 2D materials. The resultsshow that, in the IR region, only metallic tile germaneneand semiconducting GeN may exhibit plasmonic excita-tions.
VI. CONCLUSIONS
In summary, we carried out a comprehensive first-principles study to investigate electronic and opticalproperties of eight Ge based 2D materials. The calcu-lated binding energies and phonon spectra indicate thatGeO and GeN monolayers stabilize in a zigzag struc-ture, GeS and GeSe sheets prefer a puckered configu-ration, and GeC occurs in a zero buckled (honeycomb)lattice. In the case of germanene monolayer, we ob-tained a new structure, called as tile germanene, which isabout 0.11 eV/atom more stable than the known hexag-onal structure of germanene. The obtained electronicstructures suggest that tile germanene and GeSi are verygood semiconductors, while GeO, GeN, GeS, GeSe, andGeC display a band gap in their electronic structure.The novel DFT1/2 scheme was applied to obtain reli-able and accurate band gaps. The largest band gap is seen in GeC (3.50 eV) while GeN exhibits the smallestband gap (1.29 eV) among these systems. We used thevacuum level potential in the slab supercell as the en-ergy reference to determine the absolute band edges ofthe semiconducting systems. The resulting valence andconduction band edges suggest potential application ofthe GeO, GeS, and GeC monolayers as photocatalyst forwater splitting. The Bethe-Salpeter approach was usedto compute various optical properties, optical band gap,and plasma frequencies of the samples in the presence ofexcitonic effects. We observed a very low exciton bindingenergy in the GeC and GeSe sheets which further encour-age photocatalytic application of these 2D materials.
VII. ACKNOWLEDGMENTS
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