Remarkably low-energy one-dimensional fault line defects in single-layered phosphorene
aa r X i v : . [ c ond - m a t . m t r l - s c i ] O c t Remarkably low-energy one-dimensional fault line defects in single-layeredphosphorene
Woosun Jang, Kisung Kang, and Aloysius Soon a) Global E Institute and Department of Materials Science and Engineering,Yonsei University, Seoul 120-749, Korea (Dated: 10 October 2018)
Systematic engineering of atomic-scale low-dimensional defects in two-dimensionalnanomaterials is a promising way to modulate the electronic properties of these nano-materials. Defects at interfaces such as grain boundaries and line defects can oftenbe detrimental to technologically important nanodevice operations and thus a funda-mental understanding of how such one-dimensional defects may have an influence onits physio-chemical properties is pivotal to optimizing their device performance. Oflate, two-dimensional phosphorene has attracted much attention due to its high car-rier mobility and good mechanical flexibility. In this study, using density-functionaltheory, we investigate the temperature-dependent energetics and electronic structureof a single-layered phosphorene with various fault line defects. We have generateddifferent line defect models based on a fault method, rather than the conventionalrotation method. This has allowed us to study and identify new low-energy linedefects, and we show how these low-energy line defects could well modulate the elec-tronic band gap energies of single-layered two-dimensional phosphorene – offering arange of metallic to semiconducting properties in these newly proposed low-energyline defects in phosphorene. a) Corresponding author. E-mail: [email protected] raphical Abstract for Table-of-Content: One-dimensional (1D) line defects are ubiquitous in polycrystalline two-dimensional nano-materials. Various structures of phosphorene with 1D fault line defects were designed andstudied using first-principles density-functional theory calculations. We identified new low-energy fault line defects (i.e. even much lower fault defect formation energy when comparedto those found on other two-dimensional (2D) nanomaterials) and showed how these low-energy fault line defects may help to tune the phosphorene-based nanodevices’ performance.2 . INTRODUCTION
In real materials, a perfect crystal is an “ennoblement”, i.e. an idealized state. It isfundamentally important to understand the imperfections in real crystalline materials, andstudying of the science behind these crystal irregularities is critical to functional materialsdesign and engineering.
It is the feasibility of creating geometrically-imperfect materialsthat allows us to “custom-make” new materials with assorted materials functions and prop-erties that modern technological devices desire. In this sense, one of the most importantcharacteristics of a material’s microstructure are these very imperfections that are exploitedfor the engineering design of new materials.
Of late, two-dimensional (2D) nanomaterials have attracted a lot of interest from the op-toelectronics community.
Since the successful isolation of graphene (from its bulk form– graphite) and the 2D metal dichalcogenides (e.g. MoS ), the high carrier mobilitiesof these 2D nanomaterials has sparked great expectations as the most probable candidatesto supersede and replace traditional semiconductors in key nanodevices e.g. the field-effecttransistor (FET). However, after much enthusiasm and anticipation, it has become clear that graphene,which is a semi-metal, does not possess a band gap needed for proper device operationswhile many of these 2D metal dichalcogenides suffer from the existence of deep defect statesin the middle of the band gap.
These defects are often a result of line defects at thegrain boundaries (GBs) which can act as unwanted electron-hole recombination centers thathinders and degrades nanodevice performance. Yet another recently discovered 2D element – phosphorene, the 2D analogue of bulkphosphorous, has been placed in the spotlight for its high carrier mobility and a tunabledirect electronic band gap.
2D phosphorene, which has puckered repeating unit geom-etry (as illustrated in Figs. 1a (top-view) and 1b (side-view)), is also found to have superiormechanical flexibility, as compared to other 2D nanomaterials. To date, much attention has been given to point and substitutional defects in 2D nano-materials. Here, we focus on one-dimensional (1D) line defects where these prevalent (butless studied) defects may be found at the interfaces of such two-dimensional nanomaterials(e.g. grain boundaries and micro-faceted nanocrystals).A detailed discussion of electronically benign point and line defects in phosphorene was3ecently reported by Liu et al. , including a survey of the role of GB orientation on theoverall electronic structure of the defected system. Here they found that, in contrary tothe 2D metal dichalcogenides, these one-dimensional (1D) defects at the GBs do not inducemid-gap defect trap states, and further show that the carrier type and concentration of thisrecently identified 2D phosphorene could be controlled via selective chemical doping.Conventionally in literature, two different methods have been proposed to generate linedefects in these 2D nanosystems. First of all, one may apply the so-called rotationmethod , whereby one rotates part of the 2D nanostructure while keeping the remainingpart fixed, i.e. generating an “angled” line defect. Upon changing the degrees of rotation,different line defect structures can be formed with varying contact angles between the tworotated parts (as in Fig. 1c).Alternately, one might adopt the so-called fault method , following after the recent theo-retical study of line defects in both graphene and hexagonal boron nitride ( h -BN). Here,two different approaches to create line faults in 2D nanomaterials were suggested, namelythe stacking fault method and the growth fault method, respectively. In essence, the stack-ing fault method involves the addition of an extra row of atoms in the middle of the normal
FIG. 1. (Color online) (a) Top- and (b) side-views of the atomic geometry of a single-layeredphosphorene. To aid viewing, atoms at different vertical heights are discriminated by differentshades of color. The grayed area in (a) denotes the p (1 ×
1) Wigner-Seitz cell. To generateline/fault defects, namely (c) the rotation method (upper panel) and (d) the fault method (lowerpanel) can be employed. rotation method and fault method are atomically different, and thus a comparative study of both methods will further ourunderstanding of various one-dimensional line defects in phosphorene. In this work, we usedfirst-principles density-functional theory calculations to discuss the energetics and electronicstructure of these 1D fault line defects, and compare with those reported in Ref. 13 as wellas other 2D nanomaterials like graphene and h -BN. II. COMPUTATIONAL SETUP AND METHODOLOGY
In this work, periodic density-functional theory (DFT) calculations are performed usingthe projector-augmented wave (PAW) method, as implemented in the Vienna
Ab initio
Simulation Package (VASP).
The generalized gradient approximation (GGA) to theexchange-correlation functional due to Perdew, Burke and Ernzerhof (PBE) is used.
A planewave kinetic energy cutoff of 500 eV is applied and the irreducible Brillouin zoneintegrations are performed using a Γ-centered k -point mesh of 4 × × p (3 × × × k -mesh for the p (7 ×
2) supercell (with 104 atoms).To accurately describe the electronic structures, HSE06 hybrid functional calculations areperformed to obtain density-of-states (DOS) and electronic work function with the same k -meshes used in PBE calculations. To minimize the spurious interaction between repeatingphosphorene layers in the out-of-plane direction (i.e. the z -direction), a vacuum region of15 ˚A is applied along the z -direction of supercell. Also, in the effort to minimize the lateralinteractions between each periodic image of the line defects in these phosphorene layers, adefect-defect distance of at least 9 ˚A is ensured. All energies and forces are converged to lessthan 20 meV and 0.02 eV/˚A, respectively. III. ATOMIC STRUCTURES OF ONE-DIMENSIONAL LINE DEFECTSA. One-dimensional line defects by the stacking fault method
Due to the puckered atomic geometry of a single-layered phosphorene, there exists twodistinct ways to add an additional row of atoms in pristine phosphorene, along the so-called5
IG. 2. (Color online) Atomic structures of one-dimensional line defects in phosphorene:(a) and(b) show the top-and side-views of the defect geometries of 4:8p and 4:8c, respectively, while (c)and (d) show that of 4:4s and 4:4t, accordingly. The top- and side-views of the atomic structure of6:5:8:4:8:5 are depicted in (e), whereas that of 5:5:8s and 5:5:8t are shown in (f) and (g), respectively.The line defect region is shaded in orange, while the essential atoms of each line defect is shown ina darker blue to aid viewing. “armchair” direction. On one hand, an additional row of atoms can be added with thesame stacking sequence as found in pristine phosphorene. This results in the 4:8p line defectstructure – a parallel (hence, p) puckered structure with sequential rectangular (hence, 4)and octagonal (hence, 8) ring pair along this fault line, as in Fig. 2a.On the other hand, a single atomic row can be inserted with a reversed “up-and-down”-like structure, owing to its puckered nature. Here, the 4:8c line defect structure is obtained– a crossing (hence, c) puckered structure with a sequential rectangular ring and octagonalring pair along the fault line, as shown in Fig. 2b.For both 4:8p and 4:8c, minimal structural relaxation is observed. In the case of 4:8p, thecalculated distance between repeated atomic rows is slightly increased from 2.21 ˚A to 2.25 ˚A,while for the 4:8c structure, a slight outward relaxation of the rectangular ring towards thevacuum is observed, and the distance between repeated atomic rows is also increased furtherto 2.36 ˚A. 6 . One-dimensional line defects by the growth fault method
Similarly to the stacking fault method, due to its puckered nature, the growth faultmethod presents two different ways of removing a single atomic row from the pristine phos-phorene layer, along the so-called “zigzag” direction. Thus, the two adjacent planes willthen have two distinct ways of attachment across the fault. In particular, the two planescould either be directly connected, or by flipping just one plane by 180 ◦ about the fault linebefore connecting to the other. For the latter case, this means that the uppermost atomiclayer now matches that of the bottom-most layer, and vice versa.From these considerations, two different initial growth fault line defect structures, namely4:4t and 4:4s are formed, respectively. Specifically, in the case of 4:4s, upon removing one ofuppermost atomic row, a single (hence, s) atomic row remains in that layer along the faultline with sequential rhombus rings (hence, 4 as seen in Fig. 2c), while for 4:4t, due to thevertical flipping over of one plane, there now exist three (hence, t) atoms in the same layeralong the fault line with sequential rhombus rings (hence, 4 as in Fig. 2d).In addition to these 4:4-type line defects, Singh et al. proposed two other 4:4-derivedfamilies of line defects for h -BN and graphene, specifically the 6:5:8:4:8:5 and 5:5:8 line de-fect structures. In accordance with Ref. 20, we have also constructed similar line defectstructures for single-layered phosphorene – 6:5:8:4:8:5, 5:5:8s, and 5:5:8t, respectively. Fol-lowing the naming convention for the line defects mentioned above, as shown in Fig. 2e,the 6:5:8:4:8:5 structure has as many n th-sided polygons joining sequentially along the linedefect as listed accordingly. For 5:5:8s (as in Fig. 2f), it shows two slightly folded pentag-onal ring paired with one octagonal ring, and a single outermost top atom exposed alongthe fault line, which dimerizes with the neighboring atom (i.e. forming P-P pairs) aftergeometry relaxation. It is worth noting that this structure relaxes to a rather non-planar2D structure, exhibiting an bent angle of 136 . ◦ . For the 5:5:8t structure in Fig. 2g, it hasthree outermost atoms along the fault line defect, with one almost flat pentagonal ring andone almost 90 ◦ -folded pentagonal ring paired with a similarly folded octagonal ring.Essentially, many of these stacking fault line defects maintain their perfect planar geom-etry (See Figs. 2a and 2b) while those due to growth faults relax to a non-planar geometrywith much surface rumpling and wrinkling (as seen in Figs. 2c to 2g). This off-planar buck-ling behavior has been reported for similar multi-center defects in other 2D nanomaterials7 ABLE I. Line defect formation energy ( E LD ), band gap ( E g ), and work function (Φ) of 1D linedefects (LDs) in various 2D nanomaterials: Phosphorene, h -BN, and graphene. Type of LDs System E LD (eV/˚A) E g (eV) Φ (eV)Pristine Phosphorene a a a h -BN b b a a h -BN b a a h -BN b h -BN b b a h -BN b h -BN b b a This work b Reference 20 (e.g. large-angle grain-boundary defects in graphene and h -BN). IV. ENERGETICS OF ONE-DIMENSIONAL LINE DEFECTS
To study and determine the relative energetic stability of these 1D line defects in single-layered phosphorene, we define the 1D line defect formation energy per unit length of theline defect, E LD using the following equation: E LD = E LDP − N × E SL l , (1)where E LDP , E SL , N , and l are the total energy of the single-layered phosphorene with a linedefect, the total energy of pristine, defect-free single-layered phosphorene, the number of Patoms in the defect system, and the length of the line defect in the defect system, respectively.Since the basal unit is the length of the line defect, we can think of the calculated E LD (in8 .02.52.01.51.00.50.0 D e f ec t a r ea d e n s it y ( x m - ) D e f ec t a r ea d e n s it y ( x m - ) h -BN (a) (b) FIG. 3. (Color online) Simulated defect area density as a function of temperature: (a) Line defectsin phosphorene via the fault method (in blue lines, this work) and the rotation method (in orangelines, Ref. 13), and (b) corresponding fault line defects in graphene (in black lines) and h -BN (in red lines). Note that the scale is about an order of magnitude smaller in (b) eV/˚A) as the additional energy required to form an unit length of line defect for a givenpristine single-layered phosphorene.Now, to collectively address the various line defects in phosphorene, graphene, and h -BNunder typical growth conditions, we relate the defect area density, ρ def as a function oftemperature, T using the simple Arrhenius equation, ρ def = ρ pris e − E D k B T . (2)Here, the atom area density of the defect-free pristine 2D material is represented by ρ pris and is found to be 2 . × m − , 3 . × m − , and 3 . × m − for phosphorene,graphene, and h -BN, respectively. In this case, the defect formation energy, E D is normalizedwith respect to number of atoms per line defect length.As seen in in Tab. I, the formation energies of the fault line defects in phosphoreneconsidered in this work are found to be fairly small in magnitude and are also considerablymuch lower than their counterparts in either h -BN or graphene, as reported in Ref. 20. In9ine with Ref. 13, the energetics of phosphorene grain boundary line defects were calculatedto be between 0.05 to 0.14 eV/˚A, lending to the fact that all these 1D defects in phosphorenemay have fairly similar chemical stability to defect-free phosphorene than the case for other2D nanomaterials. Our calculations suggest that many, if not all, of these low energy linedefects may very well occur in potential polycrystalline phosphorene-based optoelectronicnanodevices. In comparison to other 2D materials like graphene and h -BN (in Fig. 3b), theirline defects tend to occur only at higher temperatures (above 400 K) and the corresponding ρ def are almost an order of magntitude less than that of phosphorene line defects.In fact, uniquely, the 5:5:8s line defect exhibits a remarkably low defect formation energyof 0.01 eV/˚A, i.e. almost close-to-zero value. In Fig. 3a, under typical growth conditionswith T between 300 to 800 K, the 5:5:8s fault line defect structure is predicted to have avery high ρ def as compared to other line defects (including those rotational line defects asreported in Ref. 13). Given its slightly bent geometry with P-P dimer pairs along the faultline, we further investigate the effect of lateral strain on the ultra-low energy line defect5:5:8s by varying the length of the longest edge up to ±
10 %. We find that the averageP-P bond length, the bent angle, and the total energy of this line defect change negligiblyby less than 0.003 ˚A, 10 ◦ , and 1 meV per phosphorus atom. This seems to suggest thatthis ultra-low energy 5:5:8s defect is indeed bent and stable under lateral strain which canbecome important in actual working nanodevices. V. ELECTRONIC STRUCTURE OF ONE-DIMENSIONAL LINEDEFECTS
Having identified these very low energy 1D line defect structures in single-layered phos-phorene, we now turn to their electronic structures. It was recently reported that intrin-sic point defects and grain boundaries in single-layered phosphorene are electronically be-nign/inactive, i.e. preserving the semiconducting nature of phosphorene. The authorsbelieve that unlike other heteronuclear 2D nanomaterials (e.g. the metal dichalcogenides),the so-called homoelemental bonding in phosphorene accounts for this “absence of chemicaldisorder”, and thus mitigates the formation of deep gap states. Starting with single-layered defect-free phosphorene, we calculate the total electronicdensity-of-states (DOS) using the HSE06 hybrid functional, and the E g is found to be 1.60 eV10 Density-of-states (arb. unit) E n e r gy ( e V ) FIG. 4. (Color online) Electronic density-of-states (DOS) for (a) 4:8p, (b) 4:8c, (c) 4:4s, (d) 4:4t,(e) 5:5:8t, and (f) 6:5:8:4:8:5 line defect structures. For comparison, the DOS of pristine single-layered phosphorene is indicated by the gray shaded area. Horizontal dotted line at 0 eV denotesthe Fermi energy for metallic structures, and valence band maxium for semiconducting structures. which is in good agreement with other theoretical values.
Surprisingly, in contrary tothe grain boundary defects studied in Ref. 13, we find that the 4:4s, and 4:4t line defectstructures exhibit a metallic behavior (Figs. 4c and 4d) while 4:8p, 4:8c, 5:5:8s, 5:5:8t, and6:5:8:4:8:5 yield a semiconducting band gap, E g (Figs. 4a, 4b, 4e, and 4f). All consideredline defect structures doe not possess a net magnetic spin moment.Taking a closer look at the proposed low energy line defects in this work, we find thatthose structures containing the square-shaped motifs (hence, 4-fold which deviates from thepristine 3-fold coordination) are metallic (i.e. with no gap). This lack of topological bondpreservation (with respect to the 3-fold coordination in pristine phosphorene) explains theobserved metallic behavior that was not seen for the grain boundary defects in Ref. 13.On the other hand, those defect structures preserving the 3-fold P centers (including thosewith pentagon-shaped motifs) are indeed “electronically benign/inactive”, maintaining afairly similar band gap energy with pristine phosphorene. Notwithstanding, the remarkablylow energy defect 5:5:8s structure is found to have a calculated E g of 2.03 eV – somewhatunexpectedly wider than that of defect-free phosphorene. We rationalize the increase in the11 IG. 5. (Color online) Partial electron density and density-of-states of the ultra-low energy 5:5:8sline defect structure. Partial electron density is plotted at the isosurface value of 2 × − e/˚A ,for both the (a) conduction band and the (b) valence band edges. E g to the newly formed localized defect states near the valence band edge, as seen in Fig. 5c,as a direct consequence of the structural modifications and atomic rearrangements along thedefect fault lines.As can be seen from Tab. I, the metallic line defect structures do tend to have a slightlyhigher line defect formation energy than that of the semiconducting ones. At a first glance,these low energy line defects in single-layered phosphorene may provide a clue into thepossibility of modulating the electronic properties by carefully engineering these line defectsfor new optoelectronic nanodevices.To further understand the origin of the newly formed defect states near the conductionband minimum and the valence band maximum, we plot the partial electron densities ofthe ultra-low energy line defect 5:5:8s structure in Figs. 5a and 5b, respectively. Comparingto pristine phosphorene where each P atom forms covalent bonds with three neighboring Patoms via the 3 p electrons, the overall chemical bonding character of this defect structureis follows the similar inter-mixing of the P 3 p orbitals.Specifically, near the conduction band edge (cf. Fig. 5a), the partial electron density12 IG. 6. (Color online) HSE06-predicted band alignment between phosphorene (with fault linedefects) and various metal contacts. The valence band maximum (VBM) and conduction bandminimum (CBM) of the semiconducting defects are shown in red and blue horizontal bars, respec-tively while the Fermi level of the metallic defects are shown in black. The work functions of thevarious metals are taken from Refs. 32–35. is found to be largely located between the P atoms located in the same plane (i.e. alongthe “ xy ” direction) and a π ∗ -like orbital character at the P-P dimer along the fault line.However, near the valence band edge (cf. Fig. 5b), the partial electron density is largelyconfined between the P atoms in different planes (i.e. along the “ z ” direction) with a σ -likebond forming between the P atoms in the P-P dimer parallel to the fault line.13 I. PHOSPHORENE/METAL CONTACTS: BAND GAP ALIGNMENT
It is known that controlling the relative positions of the band edges (with respect to theFermi level of the metal contact) is crucial to effectively design and tailor the electronicproperties in nanoscale FET devices. Recently, it was found that varying the number ofphosphorene layers in face-contact with various metal electrodes drastically modulates theposition of the valence band maximum and thus modifies the Schottky barrier height. Tocompare, we calculate the corresponding work functions of the fault line defect structuresand align their band edges with respect to the vacuum level, as shown in Fig. 6. Next, webenchmark the relative band edge positions with the work functions of various commonly-used metal contacts (as obtained from Refs. 32–35).In particular, the low-energy line defect 5:5:8s is found to shift valence band edge to lowerenergies by almost 0.5 eV, as compared to that of pristine phosphorene. On the other hand,the next low-energy line defect 5:5:8t moves the conduction band edge to higher energies, andthus slightly broadens energy band gap. Along these lines, by carefully applying line defectengineering in phosphorene, the generation of selected low-energy line defect structures mightact as an effective Ohmic contact with certain metals of high work function values such astungsten, platinum, irridium, and palladium. This atomic-scale control directly allows oneto extend the tunability of p -type phosphorene-based FET device performance. VII. SUMMARY
To conclude, by using the fault method, various structures of phosphorene with 1D faultline defects were designed and studied using first-principles DFT calculations. We investi-gated the energetics and electronic structure of these fault line defects, and identified newlow-energy fault line defects (i.e. even much lower fault defect formation energy when com-pared to those found on graphene and h -BN). We also showed how these low-energy faultline defects could exhibit a range of electronic structure (from metallic to semiconducting).The lowest fault line defect is termed 5:5:8s which comprises P-P dimer pairs along the faultline, and this new ultra-low energy defect structure (with a formation energy of 0.01 eV/˚A)is bent at a small angle of 136 . ◦ with a energy band gap of 2.03 eV. We propose thatmany, if not all, of these 1D line defects may co-exist during large-scale synthesis of poly-14rystalline 2D phosphorene, even under growth conditions. Finally, our calculations extendsour understanding of the impact of 1D fault line defects in p -type phosphorene-based FETnanodevices, going beyond that of the commonly studied point defects. ACKNOWLEDGMENTS
We gratefully acknowledge support from the Basic Science Research Program by theNRF (Grant No. 2014R1A1A1003415). Computational resources have been provided bythe KISTI supercomputing center (KSC-2015-C3-009).
REFERENCES X. Zou and B. I. Yakobson, Acc. Chem. Res. , 73 (2015). X. Zou and B. I. Yakobson, Small , 4503 (2015). J. Wang, S. N. Li, and J. B. Liu, J. Phys. Chem. A , 3621 (2015). J. M. Carlsson, L. M. Ghiringhelli, and A. Fasolino, Phys. Rev. B , 165423 (2011). W. Zhou, X. Zou, S. Najmaei, Z. Liu, J. Kong, J. Lou, P. M. Ajayan, B. I. Yakobson, andJ.-C. Idrobo, Nano Lett. , 2615 (2013). Q. H. W. K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and M. S. Strano, Nat. Nanotechnol. , 699 (2012). K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V.Grigorieva, and A. A. Firsov, Science , 666 (2004). K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, andA. K. Geim, Proc. Natl. Acad. Sci. U.S.A. , 10451 (2005). D. Jariwala, K. K. Sangwan, L. J. Lauhon, T. J. Marks, and M. C. Hersam, ACS Nano , 1102 (2014). F. Schwierz, J. Pezoldt, and R. Granzner, Nanoscale , 8261 (2015). L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, Nat.Nanotechnol. , 372 (2014). H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tom´anek, and P. D. Ye, ACS Nano , 4033(2014). 15 Y. Liu, F. Xu, Z. Zhang, E. S. Penev, and B. I. Yakobson, Nano Lett. , 6782 (2014). J. R. Brent, N. Savijani, E. A. Lewis, S. J. Haigh, D. J. Lewis, and P. O’Brien, Chem.Commun. , 13338 (2014). M. Buscema, D. J. Groenendijk, S. I. Blanter, G. A. Steele, H. S. J. van der Zant, andA. Castellanos-Gomez, Nano Lett. , 3347 (2014). S. P. Koenig, R. A. Doganov, H. Schmidt, A. H. C. Neto, and B. ¨Ozyilmaz, Appl. Phys.Lett. , 103106 (2014). Y. Cai, G. Zhang, and Y.-W. Zhang, Sci. Rep. , 6677 (2014). L. Kou, C. Chen, and S. C. Smith, J. Phys. Chem. Lett. , 2794 (2015). Q. Wei and X. Peng, Appl. Phys. Lett. , 251915 (2014). A. Singh and U. V. Waghmare, Phys. Chem. Chem. Phys. , 21664 (2014). Y. Liu, X. Zou, and B. I. Yakobson, ACS Nano , 7053 (2012). M. U. Kahaly, S. P. Singh, and U. V. Waghmare, Small , 2209 (2008). G. Kresse and J. Hafner, Phys. Rev. B , 558 (1993). G. Kresse and J. Hafner, Phys. Rev. B , 14251 (1994). G. Kresse and J. Furthm¨uller, Phys. Rev. B , 11169 (1996). G. Kresse and D. Joubert, Phys. Rev. B , 1758 (1999). J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. , 3865 (1996). O. V. Yazyev and S. G. Louie, Phys. Rev. B , 195420 (2010). W. Hu and J. Yang, J. Phys. Chem. C , 20474 (2015). V. Wang, Y. Kawazoe, and W. T. Geng, Phys. Rev. B , 045433 (2015). J. Qiao, X. Kong, Z.-X. Hu, F. Yang, and W. Ji, Nat. Commun. , 4475 (2014). J. H¨olzl, F. K. Schulte, and H. Wagner,
Solid Surface Physics (Springer-Verlag, Berlin,1979) Chap. Work Functions of Metals. J. C. Riviere,
Solid State Surface Science , Vol. 1 (Dekker, New York, 1969) Chap. WorkFunction: Measurements and Results. H. B. Michaelson, J. Appl. Phys. , 4729 (1977). H. L. Skriver and N. M. Rosengaard, Phys. Rev. B46