Three-dimensional Wentzel–Kramers–Brillouin approach for the simulation of scanning tunneling microscopy and spectroscopy

Abstract

We review the recently developed three-dimensional (3D) atom-superposition approach for simulating scanning tunneling microscopy (STM) and spectroscopy (STS) based on ab initio electronic structure data. In the method, contributions from individual electron tunneling transitions between the tip apex atom and each of the sample surface atoms are summed up assuming the one-dimensional (1D) Wentzel–Kramers–Brillouin (WKB) approximation in all these transitions. This 3D WKB tunneling model is extremely suitable to simulate spin-polarized STM and STS on surfaces exhibiting a complex noncollinear magnetic structure, i.e., without a global spin quantization axis, at very low computational cost. The tip electronic structure from first principles can also be incorporated into the model, that is often assumed to be constant in energy in the vast majority of the related literature, which could lead to a misinterpretation of experimental findings. Using this approach, we highlight some of the electron tunneling features on a prototype frustrated hexagonal antiferromagnetic Cr monolayer on Ag(111) surface. We obtain useful theoretical insights into the simulated quantities that is expected to help the correct evaluation of experimental results. By extending the method to incorporate a simple orbital dependent electron tunneling transmission, we reinvestigate the bias voltage- and tip-dependent contrast inversion effect on theW(110) surface. STM images calculated using this orbital dependent model agree reasonably well with Tersoff-Hamann and Bardeen results. The computational efficiency of the model is remarkable as the k-point samplings of the surface and tip Brillouin zones do not affect the computational time, in contrast to the Bardeen method. In a certain case we obtain a relative computational time gain of 8500 compared to the Bardeen calculation, without the loss of quality. We discuss the advantages and limitations of the 3D WKB method, and show further ways to improve and extend it.

Graphical abstract

Keywords

electron tunneling / STM / STS / WKB / tunneling transmission / spin polarization / metal surfaces / contrast inversion

References

[1] G. Binnig, H. Rohrer, C. Gerber, and E. Weibel, Tunneling through a controllable vacuum gap, Appl. Phys. Lett., 1982, 40(2): 178

[2] G. Binnig, H. Rohrer, C. Gerber, and E. Weibel, Surface studies by scanning tunneling microscopy, Phys. Rev. Lett., 1982, 49(1): 57

[3] W. A. Hofer, A. S. Foster, and A. L. Shluger, Theories of scanning probe microscopes at the atomic scale, Rev. Mod. Phys., 2003, 75(4): 1287

[4] W. A. Hofer, Challenges and errors: Interpreting high resolution images in scanning tunneling microscopy, Prog. Surf. Sci., 2003, 71(5-8): 147

[5] V. A. Ukraintsev, Data evaluation technique for electrontunneling spectroscopy, Phys. Rev. B, 1996, 53(16): 11176

[6] B. Koslowski, C. Dietrich, A. Tschetschetkin, and P. Ziemann, Evaluation of scanning tunneling spectroscopy data: Approaching a quantitative determination of the electronic density of states, Phys. Rev. B, 2007, 75(3): 035421

[7] M. Passoni, F. Donati, A. Li Bassi, C. S. Casari, and C. E. Bottani, Recovery of local density of states using scanning tunneling spectroscopy, Phys. Rev. B, 2009, 79(4): 045404

[8] M. Ziegler, N. Néel, A. Sperl, J. Kröger, and R. Berndt, Local density of states from constant-current tunneling spectra, Phys. Rev. B, 2009, 80(12): 125402

[9] B. Koslowski, H. Pfeifer, and P. Ziemann, Deconvolution of the electronic density of states of tip and sample from scanning tunneling spectroscopy data: Proof of principle, Phys. Rev. B, 2009, 80(16): 165419

[10] K. Palotás, W. A. Hofer, and L. Szunyogh, Simulation of spin-polarized scanning tunneling spectroscopy on complex magnetic surfaces: Case of a Cr monolayer on Ag(111), Phys. Rev. B, 2012, 85(20): 205427

[11] M. Brandbyge, J. L. Mozos, P. Ordejón, J. Taylor, and K. Stokbro, Density-functional method for nonequilibrium electron transport, Phys. Rev. B, 2002, 65(16): 165401

[12] J. Tersoff and D. R. Hamann, Theory and application for the scanning tunneling microscope, Phys. Rev. Lett., 1983, 50(25): 1998

[13] J. Tersoff and D. R. Hamann, Theory of the scanning tunneling microscope, Phys. Rev. B, 1985, 31(2): 805

[14] J. Bardeen, Tunnelling from a many-particle point of view, Phys. Rev. Lett., 1961, 6(2): 57

[15] D. Wortmann, S. Heinze, P. Kurz, G. Bihlmayer, and S. Blügel, Resolving complex atomic-scale spin structures by spin-polarized scanning tunneling microscopy, Phys. Rev. Lett., 2001, 86(18): 4132

[16] W. A. Hofer and A. Garcia-Lekue, Differential tunneling spectroscopy simulations: Imaging surface states, Phys. Rev. B, 2005, 71(8): 085401

[17] T. Kwapiński and M. Jałochowski, Signature of tip electronic states on tunneling spectra, Surf. Sci., 2010, 604(19-20): 1752

[18] H. Ness and F. Gautier, The electronic structure and stability of transition metal nanotips (I), J. Phys.: Condens. Matter, 1995, 7(33): 6625

[19] H. Ness and F. Gautier, The electronic structure of transition metal interacting tip and sample and atomic force microscopy (II), J. Phys.: Condens. Matter, 1995, 7(33): 6641

[20] H. Ness and F. Gautier, Theoretical study of the interaction between a magnetic nanotip and a magnetic surface, Phys. Rev. B, 1995, 52(10): 7352

[21] C. J. Chen, Tunneling matrix elements in three-dimensional space: The derivative rule and the sum rule, Phys. Rev. B, 1990, 42(14): 8841

[22] W. Sacks, Tip orbitals and the atomic corrugation of metal surfaces in scanning tunneling microscopy, Phys. Rev. B, 2000, 61(11): 7656

[23] C. J. Chen, Effects of m ≠ 0 tip states in scanning tunneling microscopy: The explanations of corrugation reversal, Phys. Rev. Lett., 1992, 69(11): 1656

[24] N. Mingo, L. Jurczyszyn, F. J. Garcia-Vidal, R. Saiz-Pardo, P. L. de Andres, F. Flores, S. Y. Wu, and W. More, Theory of the scanning tunneling microscope: Xe on Ni and Al, Phys. Rev. B, 1996, 54(3): 2225

[25] F. Calleja, A. Arnau, J. J. Hinarejos, A. L. Vázquez de Parga, W. A. Hofer, P. M. Echenique, and R. Miranda, Contrast reversal and shape changes of atomic adsorbates measured with scanning tunneling microscopy, Phys. Rev. Lett., 2004, 92(20): 206101

[26] G. Teobaldi, M. Peñalba, A. Arnau, N. Lorente, and W. A. Hofer, Including the probe tip in theoretical models of inelastic scanning tunneling spectroscopy: CO on Cu(100), Phys. Rev. B, 2007, 76(23): 235407

[27] A. Garcia-Lekue, D. Sanchez-Portal, A. Arnau, and T. Frederiksen, Simulation of inelastic electron tunneling spectroscopy of single molecules with functionalized tips, Phys. Rev. B, 2011, 83(15): 155417

[28] L. Vitali, S. D. Borisova, G. G. Rusina, E. V. Chulkov, and K. Kern, Inelastic electron tunneling spectroscopy: A route to the identification of the tip-apex structure, Phys. Rev. B, 2010, 81(15): 153409

[29] C. Sirvent, J. G. Rodrigo, S. Vieira, L. Jurczyszyn, N. Mingo, and F. Flores, Conductance step for a single-atom contact in the scanning tunneling microscope: Noble and transition metals, Phys. Rev. B, 1996, 53(23): 16086

[30] M. Büttiker, Y. Imry, R. Landauer, and S. Pinhas, Generalized many-channel conductance formula with application to small rings, Phys. Rev. B, 1985, 31(10): 6207

[31] J. Cerdá, M. A. Van Hove, P. Sautet, and M. Salmeron, Efficient method for the simulation of STM images (I): Generalized Green-function formalism, Phys. Rev. B, 1997, 56(24): 15885

[32] M. Brandbyge, M. R. Sorensen, and K. W. Jacobsen, Conductance eigenchannels in nanocontacts, Phys. Rev. B, 1997, 56(23): 14956

[33] M. Brandbyge, N. Kobayashi, and M. Tsukada, Conduction channels at finite bias in single-atom gold contacts, Phys. Rev. B, 1999, 60(24): 17064

[34] A. Bagrets, N. Papanikolaou, and I. Mertig, Ab initio approach to the ballistic transport through single atoms, Phys. Rev. B, 2006, 73(4): 045428

[35] A. Bagrets, N. Papanikolaou, and I. Mertig, Conduction eigenchannels of atomic-sized contacts: Ab initio KKR Green’s function formalism, Phys. Rev. B, 2007, 75(23): 235448

[36] M. Polok, A. Bagrets, D. V. Fedorov, P. Zahn, and I. Mertig, Evaluation of conduction eigenchannels of an adatom probed by an STM tip, Phys. Rev. B, 2011, 83(24): 245426

[37] E. M. L. Plumer, J. van Ek, and D. Weller, The Physics of Ultra-High Density Magnetic Recording, Springer Series in Surface Science Vol. 41, Berlin: Springer, 2001

[38] N. Weiss, T. Cren, M. Epple, S. Rusponi, G. Baudot, S. Rohart, A. Tejeda, V. Repain, S. Rousset, P. Ohresser, F. Scheurer, P. Bencok, and H. Brune, Uniform magnetic properties for an ultrahigh-density lattice of noninteracting co nanostructures, Phys. Rev. Lett., 2005, 95(15): 157204

[39] D. Serrate, P. Ferriani, Y. Yoshida, S. W. Hla, M. Menzel, K. von Bergmann, S. Heinze, A. Kubetzka, and R. Wiesendanger, Imaging and manipulating the spin direction of individual atoms, Nat. Nanotechnol., 2010, 5(5): 350

[40] K. Tao, V. S. Stepanyuk, W. Hergert, I. Rungger, S. Sanvito, and P. Bruno, Switching a single spin on metal surfaces by a STM tip: Ab initio studies, Phys. Rev. Lett., 2009, 103(5): 057202

[41] M. Bode, Spin-polarized scanning tunnelling microscopy, Rep. Prog. Phys., 2003, 66(4): 523

[42] R. Wiesendanger, Spin mapping at the nanoscale and atomic scale, Rev. Mod. Phys., 2009, 81(4): 1495

[43] W. Wulfhekel and C. L. Gao, Investigation of non-collinear spin states with scanning tunneling microscopy, J. Phys.: Condens. Matter, 2010, 22(8): 084021

[44] M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S. Heinze, G. Bihlmayer, A. Kubetzka, O. Pietzsch, S. Blügel, and R. Wiesendanger, Chiral magnetic order at surfaces driven by inversion asymmetry, Nature, 2007, 447(7141): 190

[45] C. L. Gao, W. Wulfhekel, and J. Kirschner, Revealing the 120° antiferromagnetic Néel structure in real space: One monolayer Mn on Ag(111), Phys. Rev. Lett., 2008, 101(26): 267205

[46] M. Waśniowska, S. Schröder, P. Ferriani, and S. Heinze, Real space observation of spin frustration in Cr on a triangular lattice, Phys. Rev. B, 2010, 82(1): 012402

[47] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger, G. Bihlmayer, and S. Blügel, Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions, Nat. Phys., 2011, 7(9): 713

[48] M. Takada, P. L. Gastelois, M. Przybylski, and J. Kirschner, A complex magnetic structure of ultrathin Fe films on Rh (001) surfaces, J. Magn. Magn. Mater., 2013, 329: 95

[49] P. Ferriani, K. von Bergmann, E. Y. Vedmedenko, S. Heinze, M. Bode, M. Heide, G. Bihlmayer, S. Blügel, and R. Wiesendanger, Atomic-scale spin spiral with a unique rotational sense: Mn monolayer on W(001), Phys. Rev. Lett., 2008, 101(2): 027201

[50] J. Kudrnovský, F. Máca, I. Turek, and J. Redinger, Substrate-induced antiferromagnetism of a Fe monolayer on the Ir(001) surface, Phys. Rev. B, 2009, 80(6): 064405

[51] A. Deák, L. Szunyogh, and B. Újfalussy, Thicknessdependent magnetic structure of ultrathin Fe/Ir(001) films: From spin-spiral states toward ferromagnetic order, Phys. Rev. B, 2011, 84(22): 224413

[52] L. Balogh, K. Palotás, L. Udvardi, L. Szunyogh, and U. Nowak, Theoretical study of magnetic domain walls through a cobalt nanocontact, Phys. Rev. B, 2012, 86(2): 024406

[53] A. Antal, B. Lazarovits, L. Udvardi, L. Szunyogh, B. Újfalussy, and P. Weinberger, First-principles calculations of spin interactions and the magnetic ground states of Cr trimers on Au(111), Phys. Rev. B, 2008, 77(17): 174429

[54] L. Udvardi, A. Antal, L. Szunyogh, Á. Buruzs, and P. Weinberger, Magnetic pattern formation on the nanoscale due to relativistic exchange interactions, Physica B, 2008, 403(2-3): 402

[55] R. Yang, H. Yang, A. R. Smith, A. Dick, and J. Neugebauer, Energy-dependent contrast in atomic-scale spin-polarized scanning tunneling microscopy of Mn3N2 (010): Experiment and first-principles theory, Phys. Rev. B, 2006, 74(11): 115409

[56] K. Palotás, W. A. Hofer, and L. Szunyogh, Simulation of spin-polarized scanning tunneling microscopy on complex magnetic surfaces: Case of a Cr monolayer on Ag(111), Phys. Rev. B, 2011, 84(17): 174428

[57] K. Palotás, Prediction of the bias voltage dependent magnetic contrast in spin-polarized scanning tunneling microscopy, Phys. Rev. B, 2013, 87(2): 024417

[58] W. A. Hofer, K. Palotás, S. Rusponi, T. Cren, and H. Brune, Role of hydrogen in giant spin polarization observed on magnetic nanostructures, Phys. Rev. Lett., 2008, 100(2): 026806

[59] K. von Bergmann, M. Menzel, D. Serrate, Y. Yoshida, S. Schröder, P. Ferriani, A. Kubetzka, R. Wiesendanger, and S. Heinze, Tunneling anisotropic magnetoresistance on the atomic scale, Phys. Rev. B, 2012, 86(13): 134422

[60] N. Néel, S. Schröder, N. Ruppelt, P. Ferriani, J. Kröger, R. Berndt, and S. Heinze, Tunneling anisotropic magnetoresistance at the single-atom limit, Phys. Rev. Lett., 2013, 110(3): 037202

[61] N. Néel, J. Kröger, L. Limot, K. Palotá, W. A. Hofer, and R. Berndt, Conductance and Kondo effect in a controlled single-atom contact, Phys. Rev. Lett., 2007, 98(1): 016801

[62] K. R. Patton, S. Kettemann, A. Zhuravlev, and A. Lichtenstein, Spin-polarized tunneling microscopy and the Kondo effect, Phys. Rev. B, 2007, 76(10): 100408

[63] T. Uchihashi, J. Zhang, J. Kröger, and R. Berndt, Quantum modulation of the Kondo resonance of Co adatoms on Cu/Co/Cu(100): Low-temperature scanning tunneling spectroscopy study, Phys. Rev. B, 2008, 78(3): 033402

[64] P. Roura-Bas, M. A. Barral, and A. M. Llois, Co impurities on Ag and Cu: Kondo temperature dependence on substrate orientation, Phys. Rev. B, 2009, 79(7): 075410

[65] K. R. Patton, H. Hafermann, S. Brener, A. I. Lichtenstein, and M. I. Katsnelson, Probing the Kondo screening cloud via tunneling-current conductance fluctuations, Phys. Rev. B, 2009, 80(21): 212403

[66] N. Néel, J. Kröger, and R. Berndt, Kondo effect of a Co atom on Cu(111) in contact with an iron tip, Phys. Rev. B, 2010, 82(23): 233401

[67] D. J. Choi, M. V. Rastei, P. Simon, and L. Limot, Conductance-driven change of the Kondo effect in a single cobalt atom, Phys. Rev. Lett., 2012, 108(26): 266803

[68] O. Újsághy, J. Kroha, L. Szunyogh, and A. Zawadowski, Theory of the Fano resonance in the stm tunneling density of states due to a single Kondo impurity, Phys. Rev. Lett., 2000, 85(12): 2557

[69] L. Gao, W. Ji, Y. B. Hu, Z. H. Cheng, Z. T. Deng, Q. Liu, N. Jiang, X. Lin, W. Guo, S. X. Du, W. A. Hofer, X. C. Xie, and H. J. Gao, Site-specific Kondo effect at ambient temperatures in iron-based molecules, Phys. Rev. Lett., 2007, 99(10): 106402

[70] J. A. Aguiar-Hualde, G. Chiappe, E. Louis, and E. V. Anda, Kondo effect in transport through molecules adsorbed on metal surfaces: From Fano dips to Kondo peaks, Phys. Rev. B, 2007, 76(15): 155427

[71] J. A. Aguiar-Hualde, G. Chiappe, E. Louis, E. V. Anda, and J. Simonin, Kondo resonance in the conductance of CoPc/Au(111) and TBrPP-Co/Cu(111), Phys. Rev. B, 2009, 79(15): 155415

[72] A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Fano resonances in nanoscale structures, Rev. Mod. Phys., 2010, 82(3): 2257

[73] Y. Yayon, V. W. Brar, L. Senapati, S. C. Erwin, and M. F. Crommie, Observing spin polarization of individual magnetic adatoms, Phys. Rev. Lett., 2007, 99(6): 067202

[74] B. W. Heinrich, C. Iacovita, M. V. Rastei, L. Limot, J. P. Bucher, P. A. Ignatiev, V. S. Stepanyuk, and P. Bruno, Spin structure of an atomic protrusion: Probing single atoms on cobalt nanoislands, Phys. Rev. B, 2009, 79(11): 113401

[75] L. Zhou, F. Meier, J. Wiebe, and R. Wiesendanger, Inversion of spin polarization above individual magnetic adatoms, Phys. Rev. B, 2010, 82(1): 012409

[76] P. Ferriani, C. Lazo, and S. Heinze, Origin of the spin polarization of magnetic scanning tunneling microscopy tips, Phys. Rev. B, 2010, 82(5): 054411

[77] J. Wiebe, L. Zhou, and R. Wiesendanger, Atomic magnetism revealed by spin-resolved scanning tunnelling spectroscopy, J. Phys. D Appl. Phys., 2011, 44(46): 464009

[78] M. Ternes, A. J. Heinrich, and W. D. Schneider, Spectroscopic manifestations of the Kondo effect on single adatoms, J. Phys.: Condens. Matter, 2009, 21(5): 053001

[79] K. Schouteden, D. A. Muzychenko, and C. Van Haesendonck, Spin-polarized scanning tunneling spectroscopy of self-organized nanoscale Co islands on Au(111) surfaces, J. Nanosci. Nanotechnol., 2008, 8(7): 3616

[80] K. Schouteden, E. Lijnen, E. Janssens, A. Ceulemans, L. F. Chibotaru, P. Lievens, and C. Van Haesendonck, Confinement of surface state electrons in self-organized Co islands on Au(111), New J. Phys., 2008, 10(4): 043016

[81] K. Schouteden, D. A. Muzychenko, P. Lievens, and C. Van Haesendonck, Low-temperature scanning tunneling microscopy and spectroscopy investigation of the electronic surface state of self-organized Cr islands on Au(111), J. Nanosci. Nanotechnol., 2009, 9(11): 6767

[82] K. Schouteden and C. Van Haesendonck, Narrow Au(111) terraces decorated by self-organized Co nanowires: A lowtemperature STM/STS investigation, J. Phys. Condens. Matter, 2010, 22(25): 255504

[83] K. Schouteden, K. Lauwaet, D. A. Muzychenko, P. Lievens, and C. Van Haesendonck, Spin-dependent electronic structure of self-organized Co nanomagnets, New J. Phys., 2011, 13(3): 033030

[84] A. A. Khajetoorians, J. Wiebe, B. Chilian, and R. Wiesendanger, Realizing all-spin-based logic operations atom by atom, Science, 2011, 332(6033): 1062

[85] B. W. Heinrich, C. Iacovita, M. V. Rastei, L. Limot, P. A. Ignatiev, V. S. Stepanyuk, and J. P. Bucher, A spin-selective approach for surface states at Co nanoislands, Eur. Phys. J. B, 2010, 75(1): 49

[86] M. Passoni and C. E. Bottani, Transfer Hamiltonian analytical theory of scanning tunneling spectroscopy, Phys. Rev. B, 2007, 76(11): 115404

[87] F. Donati, S. Piccoli, C. E. Bottani, and M. Passoni, Threedimensional approach to scanning tunneling spectroscopy and application to Shockley states, New J. Phys., 2011, 13(5): 053058

[88] G. Rodary, S. Wedekind, H. Oka, D. Sander, and J. Kirschner, Characterization of tips for spin-polarized scanning tunneling microscopy, Appl. Phys. Lett., 2009, 95(15): 152513

[89] K. Palotás, W. A. Hofer, and L. Szunyogh, Theoretical study of the role of the tip in enhancing the sensitivity of differential conductance tunneling spectroscopy on magnetic surfaces, Phys. Rev. B, 2011, 83(21): 214410

[90] S. Heinze, Simulation of spin-polarized scanning tunneling microscopy images of nanoscale non-collinear magnetic structures, Appl. Phys. A, 2006, 85(4): 407

[91] K. Palotás, G. Mándi, and L. Szunyogh, Orbital-dependent electron tunneling within the atom superposition approach: Theory and application to W(110), Phys. Rev. B, 2012, 86(23): 235415

[92] W. A. Hofer and A. J. Fisher, Simulation of spin-resolved scanning tunneling microscopy: influence of the magnetization of surface and tip, J. Magn. Magn. Mater., 2003, 267(2): 139

[93] A. R. Smith, R. Yang, H. Yang, W. R. L. Lambrecht, A. Dick, and J. Neugebauer, Aspects of spin-polarized scanning tunneling microscopy at the atomic scale: Experiment, theory, and simulation, Surf. Sci., 2004, 561(2-3): 154

[94] H. Yang, A. R. Smith, M. Prikhodko, and W. R. L. Lambrecht, Atomic-Scale Spin-Polarized Scanning Tunneling Microscopy Applied to Mn3N2(010), Phys. Rev. Lett., 2002, 89(22): 226101

[95] N. D. Lang, Spectroscopy of single atoms in the scanning tunneling microscope, Phys. Rev. B, 1986, 34(8): 5947

[96] M. Becker and R. Berndt, Influence of band structure on the apparent barrier height in scanning tunneling microscopy, Phys. Rev. B, 2010, 81(3): 035426

[97] M. Becker, and R. Berndt, Contrast inversion of the apparent barrier height of Pb thin films in scanning tunneling microscopy, Appl. Phys. Lett., 2010, 96(3): 033112

[98] W. Tang, E. Sanville, and G. Henkelman, A grid-based Bader analysis algorithm without lattice bias, J. Phys.: Condens. Matter, 2009, 21(8): 084204

[99] C. J. Chen, Introduction to Scanning Tunneling Microscopy, Chapter 6, Oxford: Oxford University Press, 1993

[100] H. F. Ding, W. Wulfhekel, J. Henk, P. Bruno, and J. Kirschner, Absence of zero-bias anomaly in spin-polarized vacuum tunneling in Co(0001), Phys. Rev. Lett., 2003, 90(11): 116603

[101] A. Tange, C. L. Gao, B. Y. Yavorsky, I. V. Maznichenko, C. Etz, A. Ernst, W. Hergert, I. Mertig, W. Wulfhekel, and J. Kirschner, Electronic structure and spin polarization of the Fe(001)-p(1×1)O surface, Phys. Rev. B, 2010, 81(19): 195410

[102] W. Krenner, D. Kühne, F. Klappenberger, and J. V. Barth, Assessment of scanning tunneling spectroscopy modes inspecting electron confinement in surface-confined supramolecular networks, Scientific Reports, 2013, 3: 1454

[103] G. Kresse and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci., 1996, 6(1): 15

[104] G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B, 1996, 54(16): 11169

[105] J. Hafner, Ab-initio simulations of materials using VASP: Density-functional theory and beyond, J. Comput. Chem., 2008, 29(13): 2044

[106] G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B, 1999, 59(3): 1758

[107] J. P. Perdew and Y. Wang, Accurate and simple analytic representation of the electron-gas correlation energy, Phys. Rev. B, 1992, 45(23): 13244

[108] D. Hobbs, G. Kresse, and J. Hafner, Fully unconstrained noncollinear magnetism within the projector augmentedwave method, Phys. Rev. B, 2000, 62(17): 11556

[109] D. Hobbs and J. Hafner, Fully unconstrained non-collinear magnetism in triangular Cr and Mn monolayers and overlayers on Cu(111) substrates, J. Phys.: Condens. Matter, 2000, 12(31): 7025

[110] H. J. Monkhorst and J. D. Pack, Special points for Brillouinzone integrations, Phys. Rev. B, 1976, 13(12): 5188

[111] A. Kubetzka, P. Ferriani, M. Bode, S. Heinze, G. Bihlmayer, K. von Bergmann, O. Pietzsch, S. Blügel, and R. Wiesendanger, Revealing antiferromagnetic order of the Fe monolayer on W(001): Spin-polarized scanning tunneling microscopy and first-principles calculations, Phys. Rev. Lett., 2005, 94(8): 087204

[112] S. Heinze, S. Blügel, R. Pascal, M. Bode, and R. Wiesendanger, Prediction of bias-voltage-dependent corrugation reversal for STM images of bcc (110) surfaces: W(110), Ta(110), and Fe(110), Phys. Rev. B, 1998, 58(24): 16432

[113] S. Heinze, X. Nie, S. Blügel, and M. Weinert, Electric-fieldinduced changes in scanning tunneling microscopy images of metal surfaces, Chem. Phys. Lett., 1999, 315(3-4): 167

[114] G. Teobaldi, E. Inami, J. Kanasaki, K. Tanimura, and A. L. Shluger, Role of applied bias and tip electronic structure in the scanning tunneling microscopy imaging of highly oriented pyrolytic graphite, Phys. Rev. B, 2012, 85(8): 085433

[115] K. Palotás, and W. A. Hofer, Multiple scattering in a vacuum barrier obtained from real-space wavefunctions, J. Phys.: Condens. Matter, 2005, 17(17): 2705

[116] K. S. Wang, P. M. Levy, S. F. Zhang, and L. Szunyogh, On the calculation of the magnetoresistance of tunnel junctions with parallel paths of conduction, Philos. Mag. B, 2003, 83(10): 1255

[117] M. Ondráček, C. González, and P. Jelínek, Reversal of atomic contrast in scanning probe microscopy on (111) metal surfaces, J. Phys.: Condens. Matter, 2012, 24(8): 084003

[118] L. Szunyogh, B. Újfalussy, P. Weinberger, and J. Kollár, Selfconsistent localized KKR scheme for surfaces and interfaces, Phys. Rev. B, 1994, 49(4): 2721

[119] R. Zeller, P. H. Dederichs, B. Újfalussy, L. Szunyogh, and P. Weinberger, Theory and convergence properties of the screened Korringa-Kohn-Rostoker method, Phys. Rev. B, 1995, 52(12): 8807

[120] G. Rodary, J. C. Girard, L. Largeau, C. David, O. Mauguin, and Z. Z. Wang, Atomic structure of tip apex for spinpolarized scanning tunneling microscopy, Appl. Phys. Lett., 2011, 98(8): 082505

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