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Ab initio study of the electronic structure of Co adatom on Cu(111) surface 2005
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Title | Ab initio study of the electronic structure of Co adatom on Cu(111) surface |
Creator |
Hossain, Arefa |
Date Created | 2009-12-16T21:05:01Z |
Date Issued | 2009-12-16T21:05:01Z |
Date | 2005 |
Description | We present the ab initio Density Functional study of the electronic structure of Co adatoms on Cu (111) surface. A number of Scanning Tunneling Spectroscopy (STS) experiments performed on the systems which contain magnetic adatoms on the noble metal surfaces (e.g., Co, Ce on Cu, Ag, Au (111) planes) [1, 6, 7] report the observation of intriguing resonances at the Fermi energy. However, the interpretation of these experiments in terms of the Kondo physics is not totally conclusive and gave rise to many questions that are still to be answered. The first step towards understanding the meaning of these experiments is to understand the electronic structure of such systems. The so called supercell approximation is used to simulate a Cu slab consisting of Cu (111) planes. The electronic properties of Co adatom on Cu (111) surface is studied in a symmetric 2 x 2 supercell in the xy directions with fifteen Cu (111) and two Co planes in the z direction. It is observed that Co is in the d⁸ high spin state with the magnetic moment of 1.7 μB per Co atom. The width of Co d DOS is a result of the coupling with the Cu sp surface and bulk states. At least the first three Cu layers of the bulk Cu slab have significant influence on the electronic properties of Co. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/] |
Date Available | 2009-12-16T21:05:01Z |
DOI | 10.14288/1.0092221 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 2005-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/16829 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0092221/source |
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Ab Initio Study of the Electronic Structure of Co adatom on C u ( l l l ) Surface by Arefa Hossain B.Sc , The University of Dhaka, 2001 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S FOR T H E D E G R E E OF M A S T E R OF SCIENCE in The Faculty of Graduate Studies (Physics) T H E U N I V E R S I T Y OF BRITISH C O L U M B I A October 17, 2005 © Arefa Hossain, 2005 11 Abstract We present the ab initio Density Functional study of the electronic structure of Co adatoms on C u ( l l l ) surface. A number of Scanning Tunneling Spectroscopy (STS) experiments performed on the systems which contain magnetic adatoms on the noble metal surfaces (e.g., Co , Ce on C u , A g , A u (111) planes) [1, 6, 7] report the obser- vation of intriguing resonances at the Fermi energy. However, the interpretation of these experiments in terms of the Kondo physics is not totally conclusive and gave rise to many questions that are s t i l l to be answered. The first step towards under- standing the meaning of these experiments is to understand the electronic structure of such systems. The so called supercell approximation is used to simulate a C u slab consisting of C u ( l l l ) planes. The electronic properties of Co adatom on C u ( l l l ) surface is studied in a symmetric 2 x 2 supercell in the xy directions wi th fifteen C u ( l l l ) and two C o planes in the z direction. It is observed that Co is in the d8 high spin state wi th the magnetic moment of 1.7 \IB per Co atom. The width of Co d D O S is a result of the coupling wi th the C u sp surface and bulk states. A t least the first three C u layers of the bulk C u slab have significant influence on the electronic properties of Co. I l l Contents Abstract ii Contents iii List of Figures v Acknowledgements viii 1 Introduction 1 1.1 The Anderson impurity model and the Kondo effect 2 1.2 Introduction to Scanning tunneling microscopy 4 1.3 Spectroscopic observation of the Kondo resonance 6 1.4 The quantum mirage experiment 7 1.5 Current theoretical interpretation of S T M results 8 1.6 Problems with the interpretation of S T M results 9 1.7 Motivation for the ab-initio study 10 2 Density Functional methods 12 2.1 Thomas-Fermi theory for electron density . . . . 12 2.2 The first Hohenberg-Kohn theorem 13 2.3 The second Hohenberg-Kohn theorem 14 2.4 The self-consistent Kohn-Sham equations 16 2.5 Local density approximation 19 2.6 Tight binding-Linear muffin tin orbital and atomic sphere approxima- tion method 20 2.6.1 Projecting orbital character using Fatband plots 21 2.7 Linear augmented plane wave method or Full potential method (WIEN2k) 22 3 C u Surface Electronic Structure Study 23 3.1 Cu bulk electronic structure study 23 3.2 Supercell Calculations and relative error minimizations 24 3.3 A single C u ( l l l ) plane 27 3.4 Electronic structure as a function of the number of layers 28 3.5 Slab consisting 15 C u ( l l l ) planes 29 Contents iv 4 Electronic Structure of Co Adatoms on a Cu(l l l ) Surface . . . . 41 4.1 Optimal Co-Cu distance 47 5 Outlook 57 5.1 Tight Binding Hamiltonian for 1 layer of C u ( l l l ) plane 57 5.2 Summary and outlook 58 Bibliography 61 List of Figures 3.1 Crys ta l structure of C u bulk and the corresponding Br i l l ou in zone. . . 3.2 B a n d structure picture for C u bulk crystal structure from Fu l l potential and L M T O calculation. Lines represents L M T O bands and points represents Fu l l potential bands in the first figure. Density of states (total and partial , calculated wi th in full potential) are shown for C u bulk in the lower panel. The zero of energy is at Fermi energy 3.3 Fatband plots of 4s, 4p and 3d bands for C u bulk calculated wi th in L M T O method. The first panel shows that 4s bands are at the lowest energy level. A l l the 3d bands are below Ep and bands crossing Ep have mainly 4p character 3.4 One layer of C u ( l l l ) plane has triangular symmetry and these trans- lat ion vectors (a, = 4.18322x - 2.41518t/, a 2 = 4.83037y and a 3 = 20.947625) map the crystal structure as shown in the above figure. Here a l l the lengths are in atomic units (a.u.). Hexagonal Br i l lou in zone for C u ( l l l ) plane and the corresponding reciprocal lattice vectors (61 = 2TT(0.23905X), b2 = 27r(0.11953x + 0.20702y) and 6 3 .= 27r(0.04774,S) ). 3.5 B a n d structure of a single layer of C u ( l l l ) plane by L M T O and Fu l l potential. Lines represents L M T O bands and points represents Fu l l potential bands in the above figure 3.6 Fatband plot for Copper 4s, 4p z and empty sphere Is bands. The wid th of al l the bands have been decreased by sinificant amount and band crossing the Fermi energy has mostly px + py orbital characters. Also one can see in the middle panel that 4p 2 band is above the Fermi energy at T point 3.7 Self consistent full potential band structures of three C u ( l l l ) planes wi th unit cell length 14.6 a.u., 19.2 a.u., 24 a.u., 28.8 a.u. and 33.6 a.u. along c axis. Band structure in the first panel shows dispersion in kz along the high symmetry line of T to A. B u t wi th the increase of the length of the empty layer this kz dispersion decreases. In the fifth panel we notice that for the empty layer of distance 20.95 a.u., the bands along the high symmetry line of Y to A are almost dispersionless. Therefore wi th this length of the vacuum layer, slabs are successfully decoupled List of Figures v i 3.8 B a n d structures of 15 C u ( l l l ) plane by F u l l potential and L M T O method where lines represents L M T O bands and points represents full potential bands. Bo th the band structures are almost similar accept the surface states at T point which are found to be at 0.6 eV and 0.13 eV below Fermi energy respectively in full potential and L M T O method. 39 3.9 L M T O band structure of 15 C u ( l l l ) planes with 28.9e charge on the outer most C u plane 40 3.10 B a n d structure plot of 15 C u ( l l l ) plane wi th -0.3 R y potential applied on the Is orbital of the nearest empty sphere plane 40 4.1 Density of states (DOS) plots for Co 3d electrons in the case of mono- layer of C o on a C u slab that contains 15 (111) planes. The upper panel shows D O S for the spin up electrons and the lower panel shows the spin up case. The zero energy is at Fermi energy. Most of the 'spin up' states are found below the Fermi energy while the 'spin down' states are centered around Ep 42 4.2 Density of state (DOS) plots of Co 3d 'spin up' (the upper panel) and 'spin down' (the lower panel) states for asymmetric (one Co atom on one side of C u slab) and symmetric (two C o atoms, one on each side of the outer most C u planes) slabs calculated wi th in full potential method. A 2 x 2 supercell is considered in these calculations. The zero of energy is at Fermi energy 44 4.3 Density of state (DOS) plots of Co 3d spin up and down states for 2 x 2 and 3 x 3 supercell structures (with two Co atoms per unit supercell of 11 layer C u slab, calculated wi th in full potential method) are shown in the upper and lower panel, respectively. The zero of energy is at Fermi energy. There is no significant difference in the D O S for both spin up and down states 46 4.4 Tota l energy as a function of the distance between Co adatom and C u ( l l l ) surface. The zero of energy corresponds to the total energy of a configuration when the distance between the Co adatom and C u surface is 4.83 a.u. We notice that, total energy has a min imum for the system when the distance between Co adatom and C u ( l l l ) surface is 4.79 a.u 48 4.5 Par t ia l D O S of A\ state in the case of three distances between Co adatom and C u ( l l l ) surface. D O S for optimal distance is shown in the middle panel. The zero of energy is at Fermi energy 49 List of Figures vii 4.6 The upper panel shows total and partial density of states for Co As, Ap and 3d spin up and spin down electrons in the case of optimal Co adatom and Cu surface distance calculated within full potential method. The lower panel shows partial density of states (PDOS) of different 3d states. One can see from the lower panel PDOS that all the spin up states are filled (centered below Ep) and spin down states are partially filled (PDOS centered at the Ep). The zero of energy is at Fermi energy 50 4.7 L M T O band structure of Co /Cu (111) calculated for optimized Co-Cu distance. Co d 3z2 — r2 coefficient squared is plotted in the z axis. . . 51 4.8 This is a plot of electron spin density of half of the slab with 2 x 2 supercell structure. The number of unit cells is doubled in x and y directions. 0.005 and -0.005 spin density isosurfaces are represented by red and blue colors, respectively (calculated within L M T O method). Large red spheres at the top most layer represents Co atoms and the remaining seven layers are C u ( l l l ) planes. As one can see magnetism propagates through the whole slab since we observe some (small or large) amount of both positive and negative spin density present in all layers of Cu 53 4.9 Three panels with cuts.through three different Cu layers are shown here. Isolines show only positive spin density and this is only one unit cell. In the first panel the horizontal cut shows the Cu spin density in the first Cu plane. Similarly in the. second and third panels cuts show the spin densities in the second and third Cu planes, respectively. One can see that in the second plane the spin density is quite spread all over the plane while they are relatively concentrated around Cu atom in other planes. The second Cu layer is therefore different from any other Cu layers 54 4.10 The above picture shows the Co spin down density for all Co states in the energy window of -0.2 to +0.2 Ry around Fermi energy (calculated within L M T O method). The isolines corresponding to different spin densities are shown in this picture. There is only one Co per unit cell. The part of density on the left of Co is coming from the neighboring cell. The horizontal cut shows Co density in the top Cu layer 56 V l l l Acknowledgements First , I would like to thank my supervisor Prof. George A . Sawatzky for giving me a very interesting problem to work on and for sharing his deep insight whenever I had difficulties. For many personal reasons this period of t ime has been one of the hardest in my life and it would not be possible for me to finish the project without his k ind considerations. It is hard to overemphasize the help I got from Dr . Ilya Elfimov. I thank h i m for al l the help he provided in every aspect of the project. I am grateful to Prof. Andrea Damascelli for k indly agreeing to be the second reader of my thesis and for his valuable suggestions. I would also like to thank all the members of Sawatzky-Damascelli group, in particular Bayo L a u . A major part of the calculations presented in the thesis has been done using Westgrid: Western Canada Research G r i d at Cortex cluster. I thank the facility for playing an indispensable role in the calculations. Final ly, I would like to thank Suman and Ryan for their continuing support and inspiration. Chapter 1 i Introduction Crystal surfaces are very special because they break the translational symmetry and thereby introduce different potentials at the crystal boundary as compared to the bulk. This changes the electronic structure near the surface, leading to new, poten- tially interesting material properties. The special surface environment of a metallic crystal can be pushed to a more exotic limit by introducing magnetic impurities on them. Interaction of the quasi two-dimensional electron gas at the surface with the net spin of the magnetic impurity can bring the system into novel ground states that are currently a topic of intense research, both theoretical and experimental. Apart from their fundamental scientific importance, these systems are also of great technological importance. The miniaturization and increase in speed of electronic devices is fast approaching the limit where quantum effects interfere with the normal operation of the device. At this limit the future of the electronic industry is in great need of exploiting the fundamental interactions that govern the behavior of electrons in the solid. A recent series of scanning tunneling spectroscopy experiments with clever ma- nipulations of magnetic atoms on various metal surfaces [1, 6, 7, 8] has produced some very intriguing results. Among these experiments, most notable is the 'quan- tum mirage' experiment [1] where Manoharan et al studied an elliptical quantum confinement or corral built with magnetic atoms (Co) placed on a C u ( l l l ) surface. The issues addressed by the experiments are directly related to the general nature of electron correlations between the conduction electrons of a metal and their interac- tions with surface impurities. Therefore, it is important, for both fundamental science and technologies, to carefully examine the results and the current interpretations of these experiments. In reality the results of the experiments which we shall discuss at length in the later part of the chapter, generated more questions than they answered. This motivated us to look at the electronic structure of Co impurities deposited on C u ( l l l ) surfaces starting from a very detailed study of the electronic structure of pure C u ( l l l ) plane. Chapter 1 of the thesis starts with a very general introduction to the nature of the interactions of conduction electrons with the localized impurity states and Kondo physics. This is followed by a discussion of how to interpret S T M spectra, in partic- ular, the relation between dl/dV spectra and densities of states of the system. This discussion helps us to understand the justification behind the earlier S T M observa- tion of the Kondo resonance. After setting up the ground work we discuss the very Chapter 1. Introduction 2 important S T M experiments done by Manoharan et al [1] and some problems with the current interpretation of the experimental results. 1.1 T h e A n d e r s o n i m p u r i t y m o d e l a n d t h e K o n d o e f f ec t The simplest model of a local magnetic moment associated with a spin S = 1/2 impurity in a metallic host is known as the Anderson impurity model [2]. The An- derson model starts with the Hamiltonian for the system of local magnetic moment in a metallic host: HAnderson = ^crC^C^^ + ^ e^a^da + Ud\^da^d\^dal k,cr a + Y,^ac{tada + h.C.) (1.1) k,<T where the first term represents the energy of the noninteracting electrons of the Fermi sea, the second term represents the energy of an electron in the localized d or / level, the third term represents an on-site repulsion if two electrons try to occupy the localized level and-the last term represents coupling between the d-level and the conduction electrons. Also c k C T (c k ] ( r ) is the creation(annihilation) operator of an electron in the Fermi sea with wave vector k and spin a and d)a(da) is the creation(annihilation) operator of an electron in the localized d or f level with spin o~. U is the energy required for adding a second electron of opposite spin to the impurity level and V^a is the k dependent hopping parameter for connecting the electrons of the fermi sea to the localized impurity level. In the literature a single magnetic impurity in a nonmagnetic host is often referred to as a Kondo impurity. At sufficiently low temperature, the spin of the Kondo impurity interacts with the spin of the surrounding conduction electrons (causing spin-flip scattering) which results in a change of the temperature dependence of the resistivity notably a minimum in the resistivity at a characteristic temperature. This phenomenon is known as the Kondo effect. In other words, the Kondo effect is the many body response of the free electrons in the Fermi sea to the magnetic impurity. From the Anderson Hamiltonian one finds out that even for a small value of Vkd<rT the singly and doubly occupied states of the local level broadens by an amount F « 27T|Vk̂ CT|2/o0 as given by Fermi's Golden rule where po is the density of conduc- tion electrons at the Fermi energy. In the regime e</ < Ep and e<i + U > Ep, the ground state of the impurity is always singly occupied therefore one has a magnetic impurity here. This regime of magnetic impurity is of central interest for the Kondo Chapter 1. Introduction 3 problem. Anderson showed that Eq. 1.1 will lead to local moment formation when r « 2ir\Vkdt<T\2p0 < \ed\,ed + U. Using a canonical transformation in the local moment regime of the Anderson model, Schrieffer and Wolff [5] showed that the Anderson Hamiltonian (Eq. 1.1) can be transformed to the Kondo Hamiltonian with an energy-dependent exchange inter- action Jkk' • The Kondo Hamiltonian describes the scattering of conduction electrons by a localized impurity spin in terms of the exchange interaction J k k / which is —exchange X ) 4 , k ' ( ^ 5 * k ) - ( ^ 5 * d ) . (1.2) k,k' where field operators are \I>k = ^ ^ , = ^ and 25 = a are the Pauli matrices. The form of J k k / is also calculated by Schrieffer and Wolff [5]: Jkk' = V£V k , d { 1 — — + - — — ) . (1.3) I et - ed - U tk, - ed - U ek - ed ek, - ed J At sufficiently low temperature (i.e., temperatures below a characteristic Kondo temperature TK , defined below) the spin of the conduction electrons together with the local moment tend to form a many body spin singlet ground state that collectively screens the local spin of the Kondo impurity. In this situation, if the spin of the magnetic ion is 1/2, the Kondo effect completely screens the ion spin at sufficiently large distances. This spin compensation in the conduction electron gas is usually called the 'spin compensation cloud' or 'Kondo cloud'. This means that the spin-flip scattering is frozen-out and we can treat the scattering as purely potential scattering. The impurity density of states (i.e. the density of states of the atomic d or / levels that give rise to the magnetic moment) develops a narrow DOS peak near the Fermi energy and this is called the Abrikosov-Shul resonance or Kondo resonance [3, 4]. The impurity density of states of the Anderson model in the local-moment regime is shown in Fig. 9 of [9]. Now we should emphasize the following points which have central importance to our work: 1. In the Kondo model a characteristic temperature TR-, known as Kondo tem- perature, is given by TK cx e x p ( - - ^ ) (1.4) where J(J > 0) is the antiferromagnetic coupling between the impurity spin and the spin of the host electrons that has been approximated as k independent, po is the density of states at the Fermi level. We already mentioned that at T w TK a spin compensation cloud is built up around the magnetic impurity. By noticing that Chapter 1. Introduction 4 only those electrons are strongly scattered by the impuri ty whose energy is not much further from the Fermi level than the characteristic Kondo energy ksTx, one can realize that the spatial extension of this compensation cloud may be characterized by the 'coherence length' of the Kondo problem: KBJ-K where vp is the Fermi velocity and kg is the Bol tzmann constant. In the Kondo effect, the narrow resonance appearing in the scattering amplitude has the width /csT/f. Since ksTK can be small, the corresponding coherence length t\vrK is rather large. We should note that because of the large spatial extend of the Kondo cloud of electrons the effective impur i ty has become very large. It is this extra scattering at low temperatures that causes the resistivity to increase wi th decreasing temperature below the K o n d o temperature. 2. We have seen that, in case of Anderson Hamil tonian, even for a small value of Vkd the singly and doubly occupied states of the local level broadens by an amount T « 2ir\Vkd\2Po. The broadening of the localized states is originated from the one-electron scat- tering processes between the localized states and conduction electrons. This can be described by the localized state Green function Gdt(7(uj) renormalized by the Coulomb repulsion. Thus the scattering (non-spin-flip) amplitude of the conduction electrons wi th incoming and outgoing momenta k and k', respectively can be expressed as: tkw,a(u) = VkdGdtlJ(u)Vdk, (1.6) Taking the imaginary part of Eqn . 1.6 wi th k — k' one can arrive at: lmtkki<T(uj) = ^\Vwd\2Pd,o{^)- (1.7) This equation plays a central role in our approach to the problem because according to the 'optical theorem' the total scattering cross section is proportional to the imaginary part of the forward scattering amplitude tkk>a(uj), and therefore it is proportional to pd,o{w)- Th i s connection is the reason for choosing to calculate the density of states of the localized states. We shall say more about this in Chapter 4. 1.2 I n t r o d u c t i o n t o S c a n n i n g t u n n e l i n g m i c r o s c o p y Several S T M experiments done on metal surfaces wi th embedded magnetic atoms have observed a sharp suppression in the differential conductance dl/dV when the S T M t ip is in the immediate vicini ty of the magnetic impuri ty and the voltage is close Chapter 1. Introduction 5 to zero [6, 7, 1, 13, 15]. This sharp suppression has been attributed as a manifestation of the Kondo resonance around the magnetic atom. To understand 'how' and 'what' is being observed by the dl/dV spectra one first needs to know about the tunneling mechanism in the S T M measurements. The treatment presented here closely follows Fiete et al [9]. The basic tunneling geometry and energy diagram is shown in Fig. 1 of [15]. One can take the S T M data in two different ways. One is called the 'constant current' method when the total current is kept constant by changing the height of the tip above the surface with the applied voltage. The data taken in this method is called the 'topographic image' of the sample that is being scanned. The other type of mea- surements are called 'constant height' method where one keeps the distance between the tip and the surface constant thereby observes changes in the tunneling current with the applied voltage. For a system with very low temperature and small voltages applied in the tunnel- ing current loop one can apply perturbation theory to compute the tunneling current in terms of the unperturbed tip and surface states. According to Fermi's golden rule, the current at position r and S T M bias voltage V is (the formula given here can be found in [9]) where e is the charge of the electron, t{y) is the tip(surface) states, / is the Fermi function and Mt^(r) is the matrix element from the tip to the surface state at r. The above equation illustrates that the tunneling current is proportional to the square of the matrix element between an occupied tip state and an empty surface state times a factor which gives the probability that the tip state is occupied and the surface state is empty. If one treats the tip as a point source then he/she gets |Mt)W(r)|2 oc |t/>„(r)|2 where ^(r) are the eigenfunctions of the surface. At very low temperatures (~ 10 K) one may substitute the Fermi functions by step functions and therefore use the relation J duj5(et + eV — OJ)5(LU — ev) — 5(et + eV — e„) to get where gt(e) is the density of states of the tip states. In Eq. 1.9 _ ^ \^Pu{r)\25{e-Ev) is the local density of states of the surface states at position r and energy e. Therefore replacing the sum over tip states by an integral one can write 7(r) = ~T E |M t ,,(r) | 2 / ( £ i) [ l - f(ev)]8{et + eV - e„) (1.8) (1.9) eV (1.10) o Chapter 1. Introduction 6 Assuming gt(e) is constant one gets, eV 7(r)oc j LDOS{r,e)de, (1.11) o and ^ « L O O S ( r , £ ) . (1.12) The above calculation is one of the ways to interpret S T M dl/dV data that relates it to the L D O S of the surface states. 1.3 Spectroscopic observation of the Kondo resonance Madhavan et al [7] first reported the measurement of local electronic structure of an isolated Kondo impuri ty on a metallic surface. S T M was used to obtain spectroscopic data on individual C o atoms deposited onto A u ( l l l ) surface at 4 K . F i g . 2 and F i g . 3 of [7] shows the differential conductance dl/dV spectra measured (Fig. 2) without and wi th a C o atom on A u ( l l l ) surface and (Fig. 3) wi th the S T M tip held at various distances from the center of the C o atom. It clearly shows a sharp suppression in the differential conductance dl/dV near the Fermi energy when the S T M tip is in the immediate vic in i ty of the magnetic impurity. The Kondo resonance has also been observed by L i et al [6] for the system of a single Ce impuri ty on A g ( l l l ) surface at 5 K . F i g . 1 and 2 [6] shows the presence of Kondo resonance on Ce adatom atom which is absent for nonmagnetic A g adatoms. Manoharan et al [1] reported observing the same sharp dip near the Fermi energy in the dl/dV spectra at the location of Co adatom on C u ( l l l ) surface as shown in F i g . 1 of [1]. K n o r r et al [8] also observe a Kondo resonance for Co adatoms on C u ( l l l ) surface (Fig . 2 of [8]) and the effect of surface state on the dl/dV spectra. The dip observed in the various experimetns just mentioned has been interpreted as a Kondo resonance. Madhavan et al [7] have done a detailed comparison of the experimental dip wi th the expected theoretical dip that one can expect due to Kondo effect. The explanation behind measuring the Kondo resonance as a dip in the dl/dV spectra and how good the experimental results fit wi th theory are the topics of Sec. 1.5. It should be noted that among the large number of experiments performed on similar systems and conditions we have mentioned only the relevant ones for this particular work. Chapter 1. Introduction 7 1.4 T h e q u a n t u m m i r a g e e x p e r i m e n t A very interesting property of a surface is that it can introduce energy levels in the crystal band gaps (often known as surface states). Th is is because the wave functions do not obey Bloch 's theorem in the direction perpendicular to the surface and can decay exponentially away from the surface towards both the bulk material and the vacuum. Therefore electrons are free to move in the plane of the surface which forms a type of two-dimensional electron gas there. Often, the surface-state band is only part ial ly filled, giving a low density on the surface, and a nearly quadratic dispersion relation wi th a constant effective mass. In case of the C u ( l l l ) surface the electrons in the quasi two-dimensional electron gas belong to the surface state band with an energy min imum at 0.45 e V below Fermi energy [18]. Now, i t is well known that the presence of defects in a metal like C u wi l l introduce charge screening that has an oscillatory component known as Friedel oscillations [23]: 4>{v) = ^cos(2UFr). ' Crommie et al [10] reported confining the surface electrons by using a closed struc- ture (corrals) built from iron adatom on C u ( l l l ) surface. Thei r S T M topographic image of a circular electronic resonator constructed wi th Fe adatoms on C u ( l l l ) sur- face shows beautiful standing wave pattern (Fig . 2 of [10]) which are interpreted as the Friedel type of charge oscillations. Manoharan used the same idea of confined electrons but in an elliptical corral instead of a circular one and then use it to project the Kondo resonance. They first used a scanning tunneling microscope ( S T M ) to position cobalt atoms in an elliptical r ing, and then placed another cobalt atom at one of the two focal points of the ring. Now, while a circle has only one center an ellipse has two 'focal points' or foci. Just like i n an el l ipt ical room where a faint sound generated at either of the two foci can be heard clearly far across the chamber at the other foci Manoharan et al used a quantum state that concentrated large electron densities at each focus point of the ell iptical corral. W h e n they placed a cobalt atom at one focus, a mirage appeared at the other focus: the same electronic states in the surface electrons surrounding the cobalt atom were detected on the other foci by the spectroscopic measurement of S T M even though no magnetic atom was actually there. In other words, in the dl/dV spectra the same resonance line shape, width and zero-bias shift were detected on the empty foci as produced by the real atom only attenuated by a factor of about eight. Chapter 1. Introduction 8 1.5 C u r r e n t t h e o r e t i c a l i n t e r p r e t a t i o n o f S T M r e s u l t s In the S T M experiments on metallic surfaces wi th magnetic impurities [1, 6, 7] one observes a dip in the dl/dV spectra at Fermi energy. This dip has been interpreted as a Kondo resonance exhibiting a Fano profile. To explain this observation it is impor- tant to note that, i n systems like these, below TK, there exists a roughly Lorentz ian ' shaped L D O S of the 3d orbital near Ep. There is also a conduction electron L D O S of the substrate due to the Abrikosov- Shul resonance discussed in Sec. 1.1. The width of the resonance, AE, is proportional to the Kondo temperature, AE oc kgT^. M a d - havan et al [7] pointed out that for a tunneling experiment into the resonance, one should expect di/dV to reflect the d orbital spectral density and reveal a Lorentzian like peak about Ep- B u t this can only happen when one assumes that electrons can only tunnel into the d orbital of a magnetic impuri ty and ignores electron tunnel into the surrounding continuum of conduction band states. Th is situation is far away from reality where an electron tunneling form a S T M t ip to the Kondo resonance has two possible channels, i.e., d orbital and the continuum, and this leads to an additional quantum interference term. Effect of such interference for transitions from an arbi- t rary in i t ia l state to a noninteracting discrete state in resonance wi th a continuum has been calculated by Fano [12]. The dl/dV spectra that has a Fano line shape varies wi th the lateral t ip-adatom distance r as shown in F i g . 2 of [8]. A t this point we want to illustrate the theory of this line shape which has been developed by Fano [12]. Let us consider an atomic system containing one discrete state (j) and a continuum of states tp£- Each of these states is non-degenerate. For such a composite system let the final state to be ^E- The ratio of the transition probabili ty from any ini t ia l state i through any transition operator T to the final state \&E and the transition probability from i to the in i t ia l unperturbed continuum state IJJE c a r i be represented by a single family of curves. These curves are represented by \^E\T\i)\2 (q + e)2 q2-l+2qe \WE\T\I)\2 l+e2 1+e2 { • ' where q is the so called Fano line shape parameter and e is an energy variable. According to P l i h a l and Gadzuk [15] the tunneling conductance can be written as D I , T r\ , Mrf ~l + 2q(r)e where e = (eV + AE)/(kBTx), c is the background dl/dV signal and AE is the small shift of the resonance from the Fermi energy [8], The Fano line shape parameter q(r) is given by Chapter 1. Introduction 9 where G(r) is the modified conduction electron Green's function as seen by the tip and t(r) is a function proportional to the matr ix element for direct tunneling into the localized state. t(r) depends on the overlap of the t ip wave function wi th the localized state and it falls off rapidly wi th r following the relation t(r) = to exp[—d(r)/a] where d(r) is the t ip-adatom distance and a is the decay length. F i g . 1 of [12] shows line shapes for different values of q. From E q . 1.15 one can see that even for no direct tunneling to the localized state i.e., when t = 0, a Fano line shape can result due to the indirect tunneling from the t ip to the adsorbate by conduction electron propagation represented by ReG(r)/lmG(r). Knor r et al [8] reported for an on-atom (r = 0) position of the t ip from the fitting of the measured dI/dV(V) curves that the Kondo temperature TK for C o / C u ( l l l ) is (54 ± 2 )K and the value of the Fano parameter q = 0.18 ± 0.03. Manoharan et al [1] reported a similar value of TK = (53 ± 5)K of for the C o / C u ( l l l ) adatom-substrate system. This line shape is also detectable only for r < 10 A. Hence the dip seen in the S T M experiments are interpreted as the Kondo resonance [14] showing up as Fano resonance [12] that has a width that determines the Kondo temperature. Gunnarsson et al [25] tried to interpret the experimental line shape by neglecting the direct coupling between the tip and the adsorbate 3d orbital and with the sub- strate d orbital . Jones-Jennings-Jepsen potential is used for the surface wave function and the adsorbate is modeled by a single d orbital and the momentum dependence of the hybridizat ion matr ix elements is considered explicitly. Results from the study matched quite well wi th the experimental data even when the surface states are not included in the calculation. 1.6 P r o b l e m s w i t h t h e i n t e r p r e t a t i o n o f S T M r e s u l t s After specifying the features of the 'quantum mirage' experiment we address the unanswered questions that arise from it. One crucial issue is that once one knows how to get Kondo Hamil tonian then he/she can solve it for spin S = 1/2 systems. In Co we have 5 d orbitals (some of them are degenerate) wi th the number of electrons yet to be fixed for C o adatoms embedded in C u ( l l l ) surface. Therefore this problem of C o adatom is much more complicated and it is not clear what k ind of model Hamil tonian one should use in this case. In the above discussion we have always assumed that there is only one kind of substrate state that can be tunneled to. B u t the eigenstates of the substrate can be linear combination of atomic states wi th coefficients that are strongly energy depen- Chapter 1. Introduction 10 dent. This is why, in general, the matrix elements for tunneling from the t ip to the various atomic basis states can be very different because the tunneling matrix ele- ments are strongly energy dependent. For example for Co on top of C u the tunneling to atomic d states w i l l have smaller matr ix elements than that to the more extended p and s states . Also because of the broken symmetry at the surface the p and s states mix w i th the d states and we have in general a very complicated problem. However the interpretation of the tunneling experiments is generally done wi th the assumption that Eq.s 1.11 and 1.12 are correct. Because the resonance at Fermi energy depends on temperature these experimen- tal results are interpreted wi th the help of Kondo physics. Kondo cloud is related to Kondo temperature which is determined by the width of the resonance at Ep. Fano line shape is detected only wi th in the range r < 10 Awhere r is the lateral distance between the tip and adatom. B u t Kondo temperature estimated from the experiments are ~ 50K corresponds to a much bigger Kondo cloud compare to 10 Aa t least i f we use a simple interpretation that the energy scale determined by the width would tel l us what the range of states at the Fermi energy is that are involved. This would result in only a small number of k states and therefore a large cloud radius. In the interpretation of dl/dV spectra as a quantity proportional to L D O S one assumes the S T M t ip as a point source (represented by s orbital) wi th constant density of states. B u t this might not be the case and the orbitals present at the t ip might be playing an important role as the tunneling current is proportional to the tunneling matr ix element from the t ip to the surface. From E q . 1.15 one can see that Fano parameter depends on the lateral distance between the t ip and adatom. Therefore a change in this distance wi l l change the value of q which therefore expect to result in the dl/dV spectra. B u t no such change in the dl/dV spectra is noticed as one can see from F i g . 2 of [7], Since TK has a very sensitive dependence on electron density at a certain site we expect a variation of Tx if the C o adatom is placed near a step edge. B u t experiments [16, 22] d id not found any dependence of TK on the position of Co adatom on C u ( l l l ) , A g ( l l l ) or A u ( l l l ) . 1.7 M o t i v a t i o n f o r t h e a b - i n i t i o s t u d y A s mentioned in the previous section that dl/dV measures local density of states ( L D O S ) times the tunneling matr ix element squared, it is important to know the possible tunneling states present near the vicini ty of S T M t ip in real space. We also mentioned that the total scattering cross section of the conduction electrons scat- tered by the impur i ty localized states is proportional to its (localized state) L D O S . Therefore one needs to know the electronic distribution of this Co adatom on C u ( l l l ) surface system. Also one wants to know the form of an effective hamiltonian for the system wi th numerical values for most important hopping parameters. To explain ex- Chapter 1. Introduction 11 perimental observations such as the width of Kondo resonance near Fermi energy, one must know the coupling between different states of the system. Prerequisite condition for that is to know the local density of states near Fermi energy. Therefore we want to study the electronic structure of Co adatom on C u ( l l l ) surface. Density functional theory ( D F T ) is one of the most efficient and accurate method for calculating the ground state electronic properties of a system of interacting electrons. W i t h i n D F T methods one can calculate different physical properties of the system wi th high level of accuracy. Therefore we treat this problem in hand wi th D F T method. The rest of the thesis is organized as follows: Chapter 2 gives a very brief in- troduction to our main tool of investigation, namely the density functional theory. Results of our study are given in chapters 3 and 4 and they form the heart of the thesis. In chapters 3 and 4 we wanted to simulate the system of our interest, a single C o adatom on C u ( l l l ) surface. Chapter 3 only deals wi th how to make a slab made of C u ( l l l ) plane that effectively simulates the substrate used in the S T M experi- ments. In chapter 4 we placed Co atoms on top of the slab and studied the electronic structure of Co . The results are discussed in chapter 5. 12 Chapter 2 Density Functional methods Density functional theory ( D F T ) is one of the most successful approaches to quantum mechanical many-body electronic structure calculations of molecular and condensed matter systems. It is essentially a variational method whose success not only en- compasses standard bulk materials but also complex materials such as proteins and carbon nanotubes. Tradi t ional methods for calculating electronic structure of a system of interacting electrons, like Hartree-Fock theory are based on the complicated many-electron wave function. The main objective of the density functional theory is to replace the many- body electronic wave function with the ground state electronic density as the basic quantity. D F T is, therefore, derived from the ./V-particle Schrodinger equation and is entirely expressed in terms of the density distr ibution of the ground state, pcs(r), and the single particle wave function <j>j. Whereas the many-body wave function is dependent on 3 N variables, three spatial variables for each of the N electrons, the density is only a function of three variables and is a simpler quantity to deal wi th both conceptually and practically, i.e. D F T reduces the calculations of the ground state properties of systems of interacting particles exactly to the solution of single-particle Hartree-type equations. This is why it has been most useful for systems of very many electrons and therefore we decided to use D F T methods for calculating the electronic structure of a cobalt adatom on copper (111) surface. In particular, we used two powerful approximation methods such as Linear Muffin T i n Orb i ta l approximation ( L M T O ) and Linear Augmented Plane Wave method ( L A P W ) to do our calculations. The L A P W program has the name W I E N 2 k and the method is also known as the Fu l l Potential Method . In this chapter we briefly discuss the formulation of D F T and important features of L M T O and full potential methods. 2.1 T h o m a s - F e r m i t h e o r y f o r e l e c t r o n d e n s i t y One can start from Thomas-Fermi theory that says for interacting electrons moving in an external potential v(r) due to the ion cores, the relation between v(r) and the Chapter 2. Density Functional methods 13 density distr ibution p(r) is: p(r) = 7 ( M - ^ / / ( r ) ) 3 / 2 (2.1) veff(v) EE v(r) + J J ^ d r ' (2.2) where 7 = 5 ^ iwY^ a n d M is the r independent chemical potential. The second term in E q . 2.2 is the classical electrostatic potential generated by the density p(r) of al l the valence electrons. E q . 2.1 works best for systems of slowly varying density [26, 27].- 2 .2 T h e f i r s t H o h e n b e r g - K o h n t h e o r e m Hohenberg and K o h n started from Thomas-Fermi theory and establish the connection between the electron density and the many-electron Schrodinger equation (which is expressed in terms of ip(r%, r 2 , . . , TN))- We start wi th Hohenberg-Kohn theorems which are at the heart of the density functional theory. The first Hohenberg-Kohn theorem states that The ground state density pcs(r) of a bound system of interacting electrons in some external potential v(r) determines this potential uniquely [28],[29]. Proof. Th i s proof is valid for a non-degenerate ground state. Let PGS{*) be a non- degenerate ground state density of N electrons in the potential f i(r) corresponding to the ground state ipi and the energy E\. Then £1 = v & l f f i h M = j vi(r)pGS(r)dv+ ( ^ i l T + V y ^ ) (2.3) where Hi is the total Hamil tonian corresponding to v\, T and Vee are the kinetic and interaction energy operators for the electrons. Now one can assume that there exists a second potential i>2(r), not equal to t>i(r) + constant, wi th ground state 1/% necessarily ip2 e^ipi which gives rise to the same PGs(r)- Thus E2 = (V 2 | W 2 > = j v2(r)pGS(v)dv + (il>2\T+ Vee\ip2) (2.4) Since ip is assumed to be non-degenerate, the Rayleigh-Ri tz minimal principle gives Chapter 2. Density Functional methods 14 E1 < <̂ 2|ffl|V>2> E2 < ( V i l W i ) Addi t ion of E q . 2.5 and 2.6 leads to the contradiction. Ei + E2 < Ei + E2. Therefore it is proved that the existence of a second potential which is not equal to t>i(r) + constant and gives the same pGsiy) must be wrong. Also , pGs(r) determines the number of electrons, N N = j pGS(r)dr. (2.7) Since pGs(r) determines both N and v(r), it gives the full hamiltonian and all properties derivable from the hamiltonian through the solution of time independent or time dependent Schrodinger equation (even in the presence of the additional perturbation like electromagnetic fields). For example, the many body eigenstates 4>°(Vii r2, T N ) , V , 1( ri> r2, •••! FN),---) the 2 particle Green's function G ( r i , t\] r 2 , t2) and so on. This theory is extended later in the case of degenerate ground state[30] and is also val id for the special case of non interacting electrons. 2 .3 T h e s e c o n d H o h e n b e r g - K o h n t h e o r e m The most important property of an electronic ground state is its energy Ecs- One can calculate i t by variational principle: EGS = mini,(il>\H\il>) (2.8) Hohenberg and K o h n expressed the minimum energy using density. The derivation given below follow the steps of R . O . Jones[33] (but Levy[31] and Lieb[32] first showed the derivation in this way which is simpler than the original derivation by Hohenberg and Kohn) . One considers N electrons moving in an external potential vext(r), i.e., the hamiltonian is J vx(r)PGs(r)dr + (^2\T + Vee\^2) E2 + y"(fi(r) - t>2(r))pG S(r)dr j v2(r)PGS(r)dr + (^\T + Vee\^) Ei + J(v2(r) - ui(r))/9Gs(r)dr (2.5) (2.6) Chapter 2. Density Functional methods 15 N H = T + VEE + YJve*t(ri), (2.9) i=i where H and VEE are the kinetic and electron-electron interaction operators respec- tively. Now Levy defined a universal functional (since the functional does not refer to any specific system and any specific potential) F\p] = min^P{^\T + V e e | V ) , (2-10) or F{p\ = (rmm\T+vee\rmin) ( 2 - i i ) where the min imum is taken over al l IJJ that give p. The density p at any position r i is defined to be p{ri) = N J dr2... j drNV>*(ri, r 2 , T N ) ^ ^ ! , r 2 , r N ) . (2.12) The second Hohenberg-Kohn theorem states E[p] = J dvvext(r)p(v) + F[p] > EGS, (2.13) J dvvext(r)PGs(r) + F[PGS] = EGS. (2.14) and Proof. Wr i t i ng v = 2~2iLi vext{r\) one gets / drvext(r)p(r) + F[p\ = ( O + T + K e I C J > EGs, (2.15) according to the min imum property of the ground state. Use of the min imum property once more gives EGs = Woslv + T + Vee|V>GS> < + T + V e e | « > . (2.16) Now subtracting the interaction wi th the external potential gives WGS\T + VEE\^GS) < (^SN\T + V y « > • • ( 2 - 1 7 ) The above equation is true only when ( V G S I T + VJtfcs) = W O T + V^IC*). (2.18) Then one has Chapter 2. Density Functional methods 16 Eos = j drvext(r)pGS(r) + {ipGs\T + Vee\ipGs) = j drvext(r)PGs(r) + ( « T + = J drvext{r)PGS{r) + F[pGs}- (2.19) Hence the second Hohenberg-Kohn theorem is proved. It follows from E q . 2.18 that if the ground state is non-degenerate, i/Cin = V'GS- If the ground state is degenerate V£Sn i s e c l u a l to o n e °f the ground state wave functions, and the others can also be obtained. The ground state density then determines the ground state wave function(s), from which all ground state properties can be calculated. These properties are therefore func t iona l of the density which the Hohenberg-Kohn theorem has stated before. B u t these theorems do not specify the form of the functional dependence of energy on the density. Hohenberg and K o h n only states that to get back to the Thomas-Fermi theory, (Vee) wi th respect to the ground state can be written as felKefe) = \ J d r d r ' ^ ^ . (2.20) However they d id not give the density representation of the kinetic energy part of the electrons. A t this point Kohn-Sham gives a set of single particle equations which largely remedied the problem involving the form of kinetic energy and is the next major step in the development of D F T . 2 .4 T h e s e l f - c o n s i s t e n t K o h n - S h a m e q u a t i o n s Kohn-sham self-consistent equations are very similar to the Hartree self-consistent single particle equations for the approximate description of the electronic structure of atoms (Hartree equations are based on Thomas-Fermi theory where every electron is regarded as moving in an effective potential generated by the average charge density of al l the other electrons). Hartree equations are the following Chapter 2. Density Functional methods 17 | _ I V 2 + ^ ( r ) } ^ ( r ) = £ ; ^ ( r ) (2.21) N p(r) = J > , - ( r ) | 2 (2.22) 3 = 1 vH(r) = - - + f dr'r^- (2.23) r J |r — r ' | where in E q . 2.21 j denotes both spatial and spin quantum numbers, p(r) in E q . 2.22 is the mean density (for which, in the ground state, the sum runs over N lowest eigenvalues) and VH(T) in E q . 2.23 is the effective single particle potential. In the expression for vj/(r) the first term represents the potential due to a nucleus of atomic number Z and the second term represents the potential due to the average density distr ibution p(r). To solve these equations one may start from a first approximation (e.g., Thomas- Fermi theory), construct VH(T), solve E q . 2.21 and recalculate p(r) from E q . 2.22, which should be the same as the ini t ia l p(r). If it is not one iterates appropriately unti l it is. The Hartree differential E q . 2.21 takes the form of the Schrddinger equation for non-interacting electrons moving in the external potential u e / / - So for such a system the H K variational principle becomes Ev{T)[p] = J drv(v)p(r)+T{p(r)} > EGS, (2.24) where T[p(r)} is the kinetic energy of the ground state of non-interacting electrons wi th density distr ibution p(r). One wants E q . 2.22 to be stationary wi th respect to the variations of p(r) which leave the total number of electrons unchanged, and the Euler-lagrange equation for this purpose is 6Ev[p(r)} = j M r ) | « ( r ) + ^ ^ U o s - ^ dr = 0, (2.25) where PGS is the exact ground state density for v(r) and e is a Lagrange multiplier to assure particle conservation. In this case the ground state energy and density can be obtained by solving the single particle equations Chapter 2. Density Functional methods 18 ^ - I v 2 + U ( r ) - £ ? ^ i ( r ) = 0 (2.26) JV p G S ( r ) = £ | ^ . ( r ) | 2 (2.27) £ G S = X ; ^ - ( 2- 2 8) To match this description wi th the case of interacting electrons Kohn-Sham write the functional F[p(r)] in the following form F[p(r)} = T[p(v)} + I J d r d r ' ^ ^ - + ^ ( r ) ] , _ (2.29) where Exc[p(r)] is the so called exchange-correlation energy functional. The H K variational principle for interacting electrons is therefore Ev(r)[p] = J drv(v)p(v) + T[p(v)} d r d v ' ^ ^ + Exc[p(r)} > EGS. (2.30) The corresponding Euler-lagrange equation is 8Ev[p(*)} = j6p(r){v(v) + 5-^^\p=pcs + J < k ^ - p \ + . S p { r ) U P G S - 6 | d r = 0. (2.31) Now let one write and . . . l - ) ^ U , (2.32) U « „ ( r ) = „(r) + J + " . c M - (2.33) Substi tut ing these expressions in E q . 2.31 one finds that it has the same form as E q . 2.25 for non-interacting particles moving in an effective external potential veff(r). Therefore the minimizing density pGs(r) can be found by solving the single particle equation Chapter 2. Density Functional methods 19 ( ~ ^ V 2 + veff(r) - E^j <fc(r) = 0 (2.34) wi th N P G 5 ( r ) - ^ | ^ ( r ) | 2 (2.35) Eos = E ^ + E*dPGs(r)} - j drvxc(r)PGS(r) - \ J d v d v ' 9 0 8 ^ 0 ^ . (2.36) These are the so called Kohn-Sham self consistent equations. If one neglects Exc and vxc altogether, the K S equations (2.34)-(2.36) reduce to the self consistent Hartree equations. Aga in to solve these K S equations self consistently one can start wi th a guess of the charge density PGS- B y using some approximate form for the functional dependence of Exc on density, one must compute Vxc as a function of r. The set of K S equations are then solved to obtain an ini t ia l set of K S orbital . Th is set of orbital is used to compute an improved density from E q . 2.35 and the process is repeated unt i l the density and exchange correlation energy converge to wi th in some tolerance. After getting the self consistent density, the electronic energy can be computed from E q . 2.36. The K S orbital on each iteration can be computed numerically or they can be expressed in terms of a set of basis functions. Therefore by solving the K S equation one can find the coefficients in the basis set expansion. The choice of these basis sets comes wi th experience. 2 .5 L o c a l d e n s i t y a p p r o x i m a t i o n Several different schemes have been developed for obtaining approximate forms for the functional for the exchange correlation energy. The main source of error in D F T usually arises from the approximate nature of Exc. The most widely used and most simple approximation for Exc is the local density approximation ( L D A ) in which ELXD = j PGs(v)exc[PGs(r)]dv (2.37) where ex c[p&s(r)] is the exchange-correlation energy per electron in a homogeneous electron gas of constant density. Also for spin polarized systems the local spin density approximation ( L S D A ) ap- Chapter 2. Density Functional methods 20 pears to be the most accurate scheme for D F T calculations. In such a case Exc is given by ELSD = J PGs(r)exc[PGs^r),PGs,i(r)]dv (2.38) where exc[pGs,i(r), pGS,[{_r)] is the exchange-correlation energy per particle in a homo- geneous, spin -polarized electron gas wi th spin-up and spin-down densities pGS,i(r) and pcs,i(r) respectively. These expressions for exchange-correlation energies are clearly approximations because neither positive charge nor electronic charge are uniformly distributed in actual systems. To account for the inhomogeneity of the electron density, a nonlocal correlation involving the gradient of P G S ( F ) is often added to the exchange energy (this is the so called generalized gradient approximat ion(GGA)) . There are many other amendments to Exc relative to the system that one wants to solve in practice. Two standard programs known as Tight binding-Linear muffin t in orbital-Atomic sphere approximation ( T B - L M T O - A S A ) and Linear augmented plane wave (also known as Fu l l potential method or W I E N 2 k ) are used to calculate electronic proper- ties of the system of Cobalt impurity deposited on metallic C u ( l l l ) surface. These two programs basically solve the Kohn-Sham self consistent equations and deter- mines the band structure, density of states, total energy and so on. Since Co has a net spin in this system, we used L S D A for the D F T calculations wi th in L M T O and full potential methods. 2.6 T i g h t b i n d i n g - L i n e a r m u f f i n t i n o r b i t a l a n d a t o m i c s p h e r e a p p r o x i m a t i o n m e t h o d The tight binding-Linear muffin t in orbital method is a specific implementation of density functional theory wi th in the local density approximation ( L D A ) . In this method each atom in the unit cell is replaced by a sphere of volume of a single atomic cell. Th is is the so called atomic sphere approximation. In a crystal, atomic states feel the corresponding potential together wi th some additional potentials from the neighboring atoms. Therefore this problem is avoided by considering a constant potential outside the sphere and this is the muffin t in approximation of the atomic potential. In this program, the Wigner Seitz spheres are considered as atomic spheres. The crystal is divided up into regions inside muffin-tin spheres and an interstitial region. Inside the muffin-tin spheres the energy independent basis functions are constructed from linear combination of the functions <f>i(Eu, r) (solution of the radial Schrodinger equation at the fixed arbitrary energy Ev) and their energy derivatives 4>t(Ev,r). Therefore L M T O method is no longer exact for muffin-tin potential but the error in Chapter 2. Density Functional methods 21 energies and wave functions are of fourth and second order in the difference between eigenvalues of the secular equations (secular equations that one gets for using varia- tional principle for Schrodinger's equation wi th energy independent basis functions) and E„. Another important feature of L M T O is that, in the interstitial region, the value of the potential is equal to the energy of the state being considered. Therefore the state has zero kinetic energy in that region which give rise to an additional error of order (Eigenvalue of the secular equation - Vmtz), where Vmtz is the potential in the interstit ial region. The program considers a set of s, p, d L M T O s per atom. For the system of Co impuri ty embedded on C u ( l l l ) surface, only s,p and d orbitals are considered for C u and Co and s and p orbitals for empty spheres wi th in L M T O calculation. Complete description of the theory of T B - L M T O - A S A method can be found in [35] and [36]. 2.6.1 Projecting orbital character using Fatband plots The amplitude of a particular basis function in a given eigenvector present in a particular band of the band structure can be projected and displayed wi th the help of so called fatband plots. In these plots the amplitude of the projection is represented by a energy and k dependent width added to the band structure. The L M T O method expands the wave function in the atomic orbital basis set {Xi(k)} in the following form: where i is the atomic orbital index and n is the band index. Therefore at certain k point one can plot | c j ] n | 2 for a specific orbital basis function that gives the amount of that orbital character present in each bands of the band structure picture. This is the so called fatband plotting. Following is a simple example of fatband plotting. Let us consider 3 atoms having s orbital only sit t ing in a row. If we consider the nearest neighbor hopping parameter t and the crystal field spl i t t ing for the middle atom to be e then the hamiltonian can be writ ten as where these 3 s orbital wave functions make the basis set. For e = - 2 eV and t = - 1 eV the eigenvalues are p,N (2.39) H = 0 t 0 t e t 0 t 0 E= [-2.73,0.00,0.73] and the corresponding eigenvectors are the following Chapter 2. Density Functional methods 22 " -0.33 0.71 0.63 -0.89 0.00 -0.46 -0.33 -0.71 0.63 where i — th column represents i — th eigenvector. Now c*d of the 1st basis orbital in these three eigenvectors are •0.11,0.50,0.39 Also c*Ci for the 2nd and 3rd orbital in the eigenvectors are 0.79,0.00,0.21 0.11,0.50,0.39. In a fat band plot the width of a level in this example represents the amount of selected orbital character in the eigenstate corresponding to that energy. Hence by plotting fatband one can project different orbital characters in each bands. 2 .7 L i n e a r a u g m e n t e d p l a n e w a v e m e t h o d o r F u l l p o t e n t i a l m e t h o d ( W I E N 2 k ) This program uses the same basis functions defined with respect to a muffin-tin potential inside the atomic sphere as the LMTO program. The difference of this DFT scheme with LMTO is that in the interstitial region the trial wave functions are chosen to be a linear combination of energy independent augmented plane waves. Therefore it is free from the additional error that is present in LMTO for choosing vanishing kinetic energy for the MTOs in the interstitial region. This program is also known as the most accurate method for calculating the total energy of a system. But we can no longer simply project the orbital characters present in a specified band of the band structure because the basis set contains large number of plane waves in the interstitial region. Description of the full potential program can be found in [37]. We shall use both LMTO and WIEN2k programs simultaneously to calculate electronic properties for the system under consideration and then compare results calculated within both methods to interpret the experimental findings. 23 Chapter 3 Cu Surface Electronic Structure Study 3.1 C u b u l k e l e c t r o n i c s t r u c t u r e s t u d y To study properties of the adatoms on C u ( l l l ) surface one needs to learn first about the electronic structure of a C u bulk and (111) surface. C u bulk has face centered cubic ( F C C ) lattice and the space group Fm — 3m. The Br i l lou in zone (BZ) cor- responding to this lattice is shown in F i g . 3.1. The electronic configuration of C u atom is [Ar ]4s 2 4p°3d 9 . Therefore only C u 4s, 4p and 3d orbitals are relevant for the electronic structure of C u bulk near Fermi level. A solid material contains 6.022 x 10 2 3 electrons and ion cores per c m 3 , but the three dimensional translational symmetry allows to reduce this problem to just one unit cell. According to the Bloch theorem, the solution of Schrodinger equation of an electron i n a periodic potential V r (r+R)= V(r) (where R is a lattice vector) has the form where the function u^r) has the same translational symmetry as the lattice,i.e., u k (r + R) = iik(r). Here the vector k is called the momentum vector of the electron since hk is the momentum. Replacing \I/k(r) in the Schrodinger equation by the above function we get a wave equation for u k(r). The eigenfunctions and eigenvalues of this wave equation are explicit functions of k as well as the energy is now a multi-valued function of k. Tha t is for each value of k, there is a large number of solutions, giving a set of discrete energies E\^,Eiy., ••• Since these energies depend on k, they vary continuously as k is varied over its range of values, forming energy bands for each level. Therefore we write energy eigenvalues as - E n > k , and refer to the subscript n as band index. Hence in the simple case such as C u bulk the electronic properties of the infinite crystal are determined solely by the electronic structure wi th in the first B Z . The electronic structure of C u bulk calculated along high symmetry lines in the first Br i l l ou in zone is shown in F i g . 3.2. Note that Fermi level is at zero energy. A s one can see the agreement between both methods, L M T O and F u l l potential, in this case is remarkable. The character of eigen functions at given energies is depicted by so-called fatbands calculated wi th in L M T O method. From F i g . 3.3 one can see that * u ( r ) = e x p i k r u k (r), (3.1) Chapter 3. Cu Surface Electronic Structure Study 24 Figure 3.1: Crys ta l structure of C u bulk and the corresponding Br i l lou in zone. atomic sp orbitals are highly mixed and form almost free electron like bands. However, at the lowest energies (from -10 to -5 eV), these bands bear mostly s character. As the band disperses toward Fermi energy, it gains more and more p character and near the Fermi energy it has mostly 4p orbital character. Strong overlap between the s and p orbital wave functions results in such large dispersion for the free electron like band in the band structure. O n the other hand the d orbi tal wave functions are more compact. Therefore direct d-d orbital overlap between neighboring atoms is smaller resulting in only a 4 eV bandwidth for 3d bands compare to basically infinite eV for sp bands. In contrast to simple atomic configuration of C u , the d band of the bulk material is fully occupied resulting in a dw configuration. 3.2 S u p e r c e l l C a l c u l a t i o n s a n d r e l a t i v e e r r o r m i n i m i z a t i o n s To study the electronic structure of the surface one has to brake the periodicity of the lattice at least in one direction. The simplest way is to use a so called supercell approximation. In the supercell approximation, several neighboring unit cells are combined into one new cell using translation vectors of the original lattice. Th is 'bigger' unit cell is called a supercell. For example, an F C C crystal can be represented by three F C C Chapter 3. Cu Surface Electronic Structure Study 25 Full potential ami LMTO band structure 2 h 0 i i i_i i J-̂ I i i i r — i — i i i <—i—i—i—i—i—i—i—i—i -10 -5 0 5 10 Energy (eV) Figure 3.2: Band structure picture for Cu bulk crystal structure from Full potential and LMTO calculation. Lines represents LMTO bands and points rep- resents Full potential bands in the first figure. Density of states (total and partial, calculated within full potential) are shown for Cu bulk in the lower panel. The zero of energy is at Fermi energy. Chapter 3. Cu Surface Electronic Structure Study 26 translation vectors and one atom per unit cell. A t the same time it can also be represented by three translation vectors of simple cubic crystal and four atoms per unit cell which is a supercell representation of F C C crystal for the same unit cell. To consider a supercell that combines several unit cells the translation vectors are chosen accordingly. For example, the following choice of the translation vectors for an F C C crystal a i = O.Ox - 0.5y + 0.55 a 2 = 0.5£ - 0.5y + O.Oz a 3 = nx + ny + nz (3.2) (where n is an integer) together wi th one atom per unit cell represents a supercell that consists of n F C C unit cells in the (111) direction. This new choice of a larger unit cell is also an approximation because everything is s t i l l periodic in the crystal. For example, if one removes one of the four atoms in the F C C supercell (one that is represented by three S C translation vectors together wi th four atoms per unit cell), one would form a vacancy that w i l l then form a simple cubic lattice i n the crystal structure. The same idea is used to simulate vacuum in the crystal structure by removing several adjacent atomic planes. This is often called the slab approximation in which the system is modeled by a sequence like ...\vacuum\Cuslab\vacuum\Cuslab\... If we then make the vacuum layer extremely large so that the C u slabs have vir tual ly no interaction and at the same time make the C u slabs also infinitely thick the influence of the surfaces at the top and bottom of the cu slabs would give an exact representation of the influence of breaking the translational symmetry at the surface. In this approximation there are two possible sources of error as far as the properties of the surface are concerned. First type of error arise when the spatial separation between two adjacent slabs is not large enough. In that case the layers of the slabs couple wi th each other. In the supercell geometry, required for calculations of the surface electronic structure the momentum components parallel to the surface are well defined quantum numbers, whereas the perpendicular component is formally also conserved because our slabs are periodic but for each k perpendicular vector now wi th in the very large super cell there would be a large number of bands. The number of bands would be equal to the number of C u layers in the C u slab. W i t h i n each such band the dispersion in the perpendicular direction w i l l go to zero as the distance between the slabs is increased so they no longer interact. In this l imit the state representing these band are discrete and have dispersion only in a direction parallel to the slab. Hence we do not expect to find any continuous dispersion of the energy bands along the direction perpendicular to the surface. B u t when the slabs are not separated enough, because of the coupling between slabs some dispersion i n Chapter 3. Cu Surface Electronic Structure Study 27 band structure along k±_ is present. The first step in this calculation is to determine the spatial separation between two slabs needed to decouple the surface layers of the adjacent slabs. The second type of error occurs when the slab is too th in . In contrast to the real material where one has only one surface, in the slab approximation there are two 'physical ' surfaces. Therefore in the resulting band structure there wi l l be at least two sets of bands associated wi th different surfaces. If there is no coupling between them they should be degenerate. O n the other hand when the slab is too thin, the two surfaces of the same slab can couple to each other and the degeneracy on the surface states is lifted. Another effect is the change in the band width as a function of the number of layers in a slab. In a th in slab, atoms do not have all the hopping possibilities between the planes compare to bulk. This significantly reduces the band width for bands with different orbi tal characters compare to bulk band widths. Therefore it is important to make sure that errors due to slab approximation are minimized before doing any calculation. To do this we simply carry out calculations as a function of the slab thickness and the vacuum layer thickness to determine at which min imum thickness the errors introduced are small enough that they wi l l not influence the physics we are after. 3.3 A single C u ( l l l ) plane The simplest starting point to understand the electronic structure of C u ( l l l ) surface is to look at the electronic structure of one single layer. Note that a C u ( l l l ) plane has tr igonal symmetry as shown in F i g . 3.4. The space group is P — 3 m l . The corresponding Br i l l ou in zone is shown in F ig . 3.4. We shall show later that in the full potential method a spatial separation of 20.95 a.u. between slabs is sufficient to decouple the layers of different slabs. Hence in the full potential calculation of a single layer, the lattice constant along direction perpendicular to the (111) plane has been taken to be 20.95 a.u. The slab approximation in L M T O method is introduced by replacing C u atoms wi th the empty spheres for the number of layers corresponding to the thickness of the vacuum layer we are t rying to simulate. We find that six layers of empty spheres between two C u ( l l l ) planes effectively serves as the vacuum between adjacent slabs in this case. Th i s corresponds to the distance of 26.63 a.u. between C u layers and is close to what we found for the full potential method. A comparative study of the band structure from L M T O and Wien2k-Ful l potential methods is shown in F i g . 3.5. Fig.(3.5) shows that bandwidths of bands bearing characteristics of different or- bitals become significantly narrower than compared to bulk C u . For example, the bandwidth of 3d bands is 1.5 eV compare to 4 e V in bulk C u . This is expected since a single C u ( l l l ) plane is essentially two dimensional and it does not have the Chapter 3. Cu Surface Electronic Structure Study 28 hopping possibilities between the planes compare to C u bulk. The nearest neighbor coordination number is only 6 as compared to 12 in the bulk F C C structure. Very naively one would guess that the band width would be reduced by about a factor of two. Because of the symmetry of the C u ( l l l ) plane, the degeneracy of 4p orbitals is lifted resulting in different bands bearing different orbital characters of px,py and pz. One finds from fatband plots that the band crossing Ep has mostly px +py (about 60 percent) orbi tal character and the band at 2 eV above Ep at the T point has mostly pz character. 3.4 Electronic structure as a function of the number of layers In this section the electronic structure of a C u ( l l l ) slab is studied as a function of the number of layers in the slab. To make the slab, more C u ( l l l ) planes are gradually added to the single C u ( l l l ) plane. The study of important features of the corresponding band structures fixes the number of layers in the slab. As was discussed in the previous section, for a single layer the band width of 3d band gets significantly narrower than that of C u bulk. The most immediate change to notice from the band structure of three C u ( l l l ) planes is that most of the band width of the 3ri bands has been restored. Min imiza t ion of the error due to coupling between the layers of two adjacent slabs requires the study of the band structure wi th different slab separations. Therefore, wi th in full potential method, self consistent band structure calculations have been carried out for several spatial separations between adjacent slabs. The resulting band structures are shown in F i g . 3.7. One can clearly see that for 6.75 a.u. separation between two slabs, bands along the high symmetry line of T to A disperses strongly because of the presence of kz (k± to the surface for this case) dispersion. Upon increasing the spatial separation the bands become less dispersive along this T to A line. For a separation of more than 20.95 a.u. there is a negligible dispersion in the band structure along the specified line and the bands are almost completely flat. Hence for this spatial separation slabs are successfully decoupled. One therefore fixes this spatial separation between slabs in al l the later calculations by Wien2k- full potential method. In L M T O method the same purpose is served by a spatial separation of 26.63 a.u. between two adjacent slabs. In the band structure calculated within full potential method for 7 C u ( l l l ) layer slab, Shockley surface states appear in the sp-band gap at T point. Because of the presence of two 'physical ' surfaces in the simulated crystal structure and the coupling between them, the degeneracy of the two surface states is lifted. Therefore two surface states show up below Ep at T point in the band structure. This is a result of choosing Chapter 3. Cu Surface Electronic Structure Study 29 a slab that is not thick enough to decouple the two surfaces of the same slab. Several band structures with different number of layers in the slab are studied in this respect. With the increase of the number of layers in the slab, the decrease in splitting between these two surface states at T point is observed. At the same time, together with the increasing number of energy bands, the bandwidths of different bands are restored to that of bulk Cu. Fig. 3.8 shows that in the band structure of a 15 layer slab, there are 15 almost two-dimensional free electron like bands in the lowest energy (-10 to -5 eV) region bearing mostly Cu 4s character and the bandwidth for 3d bands is ~ 4 eV as in bulk Cu. Also the surface states are now almost degenerate at 0.6 eV below Ep at the T point are obtained. Therefore one can take the 15 layer slab as the system that is thick enough so that the two outer surfaces do not feel each other and this system forms the basis for our further investigation. 3 .5 S l a b c o n s i s t i n g 15 C u ( l l l ) p l a n e s In order to study the electronic structure of Cu( l l l ) slab, band structure and density of states of the slab containing 15 (111) planes are investigated within LMTO and Full potential methods. The crystal structure is represented by a unit cell of length 76.16 a.u. including the spatial separation of 20.95 a.u. between two adjacent slabs within the full potential method. Position of the different atoms in Cu( l l l ) planes in the structure file are fixed according to the symmetry of the structure and the length of the unit cell. In LMTO 15 Cu( l l l ) planes and 6 layers of empty spheres are considered and the length of the unit cell is 82.824 a.u with 26.63 a.u. thick empty slab (a layered structure consisting of empty spheres only). The band structures calculated within Wien2k and LMTO are shown in Fig. 3.8. Fig. 3.8 shows the surface state at T point is found 0.6 eV below Fermi energy in full potential. This is quite consistent with the experimental result since Shock- ley type surface states are detected by STM experiments [18] 0.4 ± 0.02 eV below the Fermi energy at T point. Recent Angle Resolved Photo Emission Spectroscopy (ARPES) experiment [38] observed surface state at 0.435 ±0.001 eV below the Fermi energy at T point. However band structure calculated within the LMTO method the Shockley state is found at 0.13 eV below the Fermi energy at the T point. The rest of the band structure looks almost the same in both cases. Since the full potential method is very time consuming and we will have to increase our unit cell even further in the study of Co on Cu we try to correct the LMTO method so that we can use this in the future. Therefore we try to correct LMTO and bring the surface state at the same energy position as it is in the full potential band structure. One can approach the problem in different ways. Our starting approach is to change the spatial separation between the surface Cu layer and the rest of the slab. The magnitudes of the coupling parameters between layers depend directly on their Chapter 3. Cu Surface Electronic Structure Study 30 spatial separation. Therefore by moving a layer towards or away from the slab one can effectively change the band structure. In Table 3.1 we show different separations (compare to the 4.83 a.u. separation of the layers corresponding to the bulk crystal structrure) between the layers and the energies of several features in the resulting band structures. In the L M T O method a finite amount of overlaps between atomic spheres are considered. The program is designed to allow for less than 15 percent overlap between two atomic spheres while an atomic and an interstitial sphere are allowed to overlap by 20 percent. Therefore if the outermost plane is moved towards or away from the slab by a considerable amount, the radii of the atomic spheres in a plane should change. A s the L M T O basis set depends on the size of these atomic spheres, their changes are expected to make considerable difference in the band structure. W i t h i n L M T O we find that an increase of 0.4 a.u. length between the two outer most C u ( l l l ) planes and the rest of the slab changes both the radii of atoms in the outermost C u ( l l l ) plane and the next empty sphere layer. Our next approach is to change the nuclear charge of the outer most C u ( l l l ) plane. Changing the nuclear charge means changing the potential which can simulate a shift in the band s. B u t we found that by changing the charge by only 0.1 the whole band structure of the outer most C u plane is quite different as compared to the full potential band structure F i g . 3.9. Next we t ry to adjust the band structure by applying an external potential to the top most C u ( l l l ) layer. App ly ing a positive or negative potential means decreasing or increasing the electron density in that C u layer. Table 3.2 shows the potentials applied on different orbitals of C u plane and their effects in the band structure. Our results in Table 3.1 and 3.2 justifies the reason behind our next step of applying a potential to the empty spheres to try to get a similar band structure compare to full potential. From fatband plots one can see that C u Apz and the empty sphere Is band couple strongly wi th each other. Therefore, by applying a potential to the Is orbi tal of the empty sphere layer next to the C u ( l l l ) plane one should be able to change the position of the surface state in the band structure without very much affecting the rest of the band structure. In table 3.3 the amount of potential applied to the empty sphere layer and the resulting changes noticed in the band structure are presented. In this case, by applying an attractive potential to the empty sphere plane, we transfer some of the electron density of C u ( l l l ) plane to this empty layer. Results show that, wi th in L M T O method, -0.3 R y potential applied to the s orbital of the empty layer gives a band structure almost similar compare to that of full potential together wi th the surface state at -0.55 eV. Therefore we considered applying an external potential of -0.3 R y to the s orbital of the nearest empty sphere plane in our later L M T O study of the electronic structure of the system. In conclusion of this section we have found that the surface electronic structure can be very well simulated with a slab thickness of 82.824 a.u. and a vacuum Chapter 3. Cu Surface Electronic Structure Study 31 Table 3.1: Lis t of different distances between the outer most two C u ( l l l ) planes (symmetry of the crystal structure gives that the separation between any two C u ( l l l ) planes is d = 4.83 atomic unit (a.u.) along c axis) and changes in the corresponding band structures. Length added Posit ion of the Change in radii Changes noticed to the separation surface state of atoms and in the band between planes at T point empty spheres structure (in a.u.) (in eV) -0.5 -0.05 no change Same as in the case of -0.4 a.u. distance addition -0.4 -0.09 no change 3d bands are different compare to Fu l l potential -0.3 -0.1 no change Rest of the band structure is almost similar to that of F u l l potential -0.2 -0.15 no change 3d band near M point looks similar to that of Fu l l potential -0.1 -0.15 no change 3d band near M point changes 0 -0.13 no change Almost similar compare to F u l l potential band structure accept a 3d band at M point 0.1 -0.1 no change N o change in the rest of the band structure 0.2 -0.08 no change 3d band near M point looks similar compare to F u l l potential 0.3 -0.02 no change The band at M point crossing Ep starts to move away from EF 0.4 0 radii change The band at M point crossing Ep moves further away from EF 0.5 -0.2 radii change B a n d at M point crossing EF is different compare to Fu l l potential Chapter 3. Cu Surface Electronic Structure Study 32 Table 3.2: A table wi th different potentials applied on the last layer of C u ( l l l ) plane and their outcome in the band structures. Potential applied O n orbital Posit ion of surface Changes noticed in in Rydberg (Ry) state at T point (in eV) the band structure 0.4 4s and 4p -0.39 4s bands change around K and M point 0.2 4s and 4p -0.39 4s bands change again around K and M point 0.1 4s and 4p -0.25 4s bands start to change around K and M point 0 none -0.13 Almost similar compare to F u l l potential band structure accept a 3d band at M point -0.1 4s and 4p -0.12 3d band changes near M point -0.1 Ap only -0.12 3d band near M point changes -0.1 4s only 0 B a n d structure looks almost similar to that of Fu l l potential layer of 26.63 a.u. In order to get the surface state in L M T O and full potential at the same energy we can apply an external potential of -0.3 R y to the empty spheres adjacent to the outermost C u plane. Chapter 3. Cu Surface Electronic Structure Study 33 Table 3.3: A table wi th different potentials applied on the empty sphere plane next to the outer most C u ( l l l ) plane in the slab and their outcome in the band Potential O n orbital Posit ion of surface M a i n features of applied in state at T point the band structure Rydberg(Ry) (in eV) 0.4 Is and 2p 0.4 No change in the rest of the band structure 0 none -0.13 A 3d band around M point i n F u l l potential band structure is not present in L M T O -0.4 Is only -0.7 A 3d band develops near M point which looks the same as in F u l l potential -0.3 Is only -0.55 Almos t similar band structure compare to Fu l l potential Chapter 3. Cu Surface Electronic Structure Study 34 Figure 3.3: Fatband plots of 4s, Ap and 3d bands for Cu bulk calculated within LMTO method. The first panel shows that 4s bands are at the lowest energy level. All the 3d bands are below EF and bands crossing EF have mainly 4p character. Chapter 3. Cu Surface Electronic Structure Study 35 Figure 3.4: One layer of C u ( l l l ) plane has triangular symmetry and these trans- lat ion vectors (Sx = 4.18322f - 2.41518y,a 2 = 4.83037y and a 3 = 20.947625) map the crystal structure as shown in the above figure. Here a l l the lengths are in atomic units (a.u.). Hexagonal Br i l lou in zone for C u ( l l l ) plane and the corresponding reciprocal lattice vectors (6i = 27 r ( 0 . 2 3 9 0 5 i ) , 6 2 = 2TT(0.11953X + 0.20702y) and 6 3 = 2TT(0.04774Z) Chapter 3. Cu Surface Electronic Structure Study 36 Pull potential mill LMTO band structure Figure 3.5: B a n d structure of a single layer of C u ( l l l ) plane by L M T O and Ful l potential. Lines represents L M T O bands and points represents Fu l l po- tential bands in the above figure. Chapter 3. Cu Surface Electronic Structure Study 37 M K Figure 3.6: Fatband plot for Copper 4s, 4p2 and empty sphere Is bands. The width of all the bands have been decreased by sinificant amount and band crossing the Fermi energy has mostly px +py orbital characters. Also one can see in the middle panel that 4p2 band is above the Fermi energy at T point. Chapter 3. Cu Surface Electronic Structure Study 38 Cu_supcrcell atom 0 size 0.20 Cu_superceH atom 0 size 0.20 Cu_supercell atom 0 size 0.20 VI ; A izv Figure 3.7: Self consistent full potential band structures of three C u ( l l l ) planes wi th unit cell length 14.6 a.u., 19.2 a.u., 24 a.u., 28.8 a.u. and 33.6 a.u. along c axis. B a n d structure in the first panel shows dispersion in kz along the high symmetry line of T to A. B u t wi th the increase of the length of the empty layer this kz dispersion decreases. In the fifth panel we notice that for the empty layer of distance 20.95 a.u., the bands along the high symmetry line of T to A are almost dispersionless. Therefore with this length of the vacuum layer, slabs are successfully decoupled. Chapter 3. Cu Surface Electronic Structure Study 39 Full potential and L M T O band structure G M K G Figure 3.8: Band structures of 15 Cu( l l l ) plane by Full potential and LMTO method where lines represents LMTO bands and points represents full potential bands. Both the band structures are almost similar accept the surface states at T point which are found to be at 0.6 eV and 0.13 eV below Fermi energy respectively in full potential and LMTO method. Chapter 3. Cu Surface Electronic Structure Study 40 TDLMTO. test 7„ Cu7 7.=28.9 G K M CA I. II A Figure 3.9: L M T O band structure of 15 C u ( l l l ) planes wi th 28.9e charge on the outer most C u plane. -0.3 Ry pot mi IL* K M CA I, 11 A Figure 3.10: B a n d structure plot of 15 C u ( l l l ) plane wi th -0.3 R y potential applied on the Is orbital of the nearest empty sphere plane. 41 Chapter 4 Electronic Structure of Co Adatoms on a C u ( l l l ) Surface W h e n Co (a transit ion metal) is deposited on the non-magnetic metallic C u ( l l l ) surface, the interaction between the electrons in the part ia l ly filled d shell of C o and the electrons of the metal results in interesting magnetic phenomena. To try to understand the magnetic properties of a Co impuri ty in this system we study the electronic structure of Co adatoms on a C u ( l l l ) surface. Let us start wi th a monolayer of Co deposited on a C u ( l l l ) slab. A s discussed in the previous chapter, a slab consisting of 15 C u ( l l l ) planes has been chosen so that it is thick enough to decouple its two surfaces. A t the same time it is possible to do calculations wi th our available C P U power and memory. F i g . 4.1 shows the various spin polarized L D O S of one monolayer of Co on C u ( l l l ) surface. We find for this system that Co is ferromagnetic. The total number of 3d electrons which is obtained by integrating the part ia l density of C o 3d states up to the Fermi energy is 7.23 in this case wi th the magnetic moment of 1.72 \IB per Co . The majority of the states in the vicini ty of Fermi energy are C o 'spin down' states wi th most of the 'spin up ' states below it . Even though the central layers of the C u slab basically feels a cubic crystal field as in the bulk, the crystal field of the outermost C o layer is not cubic but trigonal. The point group symmetry of this trigonal crystal is D^d- The eg and t2g orbitals in the cubic coordinate system are not the basis function of the irreducible representation of Du- One therefore chooses a new coordinate system to represent the basis functions of the irreducible representation of this group. In this new coordinate system the z axis is the threefold rotation axis chosen along the (111) direction, (z,x) is the mirror plane and y is the twofold rotation axis as shown in F i g . 11 of [40]. The five d orbitals in the new coordinate system are given by the following combination of the d orbitals in the cubic coordinate system (written in terms of x',y', z' axes) Chapter 4. Electronic Structure of Co Adatoms on a Cu(lll) Surface 42 8 0.5 ° 0 0 ' ' ' ' ' 1 1 1 1 1 1 111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 j 1 1 1 1 1 1 1 1 1 1 1 Spin up DOS Total 3d DOS i • • i I 1 3d A_l state 1 1 Zk M / v \ / I 3d E(2) states - T T - T - T - J - ^ 1 1 3d E(l) states 1 1 1 Energy (eV) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 " , 1 i i i i i 1 i i i i i 1 i i i i i 1 i i i i i Spin down DOS Total 3d DOS . . A_l state \^^r-^-\^y^T7\.,, . . i 3d E(2) stales 1 , .Tr-r-t-, • • i i i i i i 1 i -Pi r i 4-rrn t I n n 3d E(l) stales . . , , / h - ^ 1 -4 - 3 - 2 - 1 0 1 2 3 4 Energy (eV) Figure 4.1: Density of states (DOS) plots for Co 3d electrons in the case of monolayer of Co on a Cu slab that contains 15 (111) planes. The upper panel shows DOS for the spin up electrons and the lower panel shows the spin up case. The zero energy is at Fermi energy. Most of the 'spin up' states are found below the Fermi energy while the 'spin down' states are centered around Ep. Chapter 4. Electronic Structure of Co Adatoms on a Cu(lll) Surface 43 |3z 2 - r2) \V*) \zx) \xy) \x2 - y2) These five d orbitals are categorized into three groups according to the irreducible representations of Dza- \3z2 — r 2 ) is called A\, \yz) and \zx) are called E(l) and \x2 — y2) and \xy) are called £'(2). Fatband plots show that in this crystal field, the degeneracy of the Co 3d orbital is partly lifted and we get a non-degenerate Ai band and doubly degenerate bands for E{1) and E{2). Our goal is to study the electronic properties of a Co adatom on C u ( l l l ) . To do this and still make use of the powerful band structure software we need to still have a periodic structure of Co adatoms in the x, y plane. These atoms though should be far enough apart that interactions between them is negligible. We start with a Co adtom layer in a 2 x 2 supercell in x and y directions. Together with the 15 C u ( l l l ) layers in each slab and two Co atom, there are altogether 95 atoms in a unit cell. This is already a very large calculation which slows down the computational speed by a considerable amount. Apparently the problem of spending memory and C P U time can be avoided by trying to preserve symmetry of the system as much as possible. Ideally one would like to have one Co atom in a supercell to study its electronic structure on the Cu surface. But Co on both sides of the slab is the most symmetric case for the system under consideration. At the same time, the number of C u ( l l l ) planes in the slab is reduced from 15 to 11 for similar reasoning. The crystal structure is simulated by making a slab of 13 C u ( l l l ) planes using P — 3ml symmetry of the C u ( l l l ) slab and then replacing the outermost layers of C u ( l l l ) planes by a 2 x 2 superstructure of Co atoms with less number of Co atoms per plane compare to Cu. The calculation is done for the two cases a Co layer only on one side giving an antisymmetric situation and the more symmetrical case of a Co layer on both sides. The results are presented in Fig. 4.2. Fig. 4.2 shows no significant difference in the projected DOS plots of the sym- metric and asymmetric combinations. Since the self consistent density functional calculation for asymmetric slab takes much more computer time, we choose to work = ~{\x'y') + \y'z') + \z'x'))- = -±=(2\(x'f-(y>)2)-\y'z>) + \z'x')), = /I |3(z') 2 - r2) - ^ ( 2 | * V > - \y'z') - | z V » , = - ^ H ( z ' ) 2 - (y')2) - + \z'x')), = - ^ l 3 ^ ) 2 - r 2 ) - ^ ( 2 | x y ) - | y V ) - | z V ) ) . (4.1) Chapter 4. Electronic Structure of Co Adatoms on a Cu(lll) Surface 44 ; 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Spin up DOS 3d A _ l state t . . . . 1 . . . . t . - — asym sym J | :. A 3d E(2) states 1 , , , . 1 . . . . 3d E(l) states . . . . 1 . . . . 1 . . . >f W . . I • • • T\l , I - 3 - 2 - 1 0 1 2 3 4 Energy (eV) Figure 4.2: Density of state (DOS) plots of Co 3d 'spin up' (the upper panel) and 'spin down' (the lower panel) states for asymmetric (one Co atom on one side of Cu slab) and symmetric (two Co atoms, one on each side of the outer most Cu planes) slabs calculated within full potential method. A 2 x 2 supercell is considered in these calculations. The zero of energy is at Fermi energy. Chapter 4. Electronic Structure of Co Adatoms on a Cu(lll) Surface 45 wi th a symmetric slab wi th Co impuri ty on C u ( l l l ) surface. Prom F ig . 4.2 we see that the overall width of the C o Ai states (in the local coordinate system) is about 0.5 eV together wi th some dips and peaks in the D O S . The wid th of the Co d states can result from two main mechanisms. The coupling of the C o - r i states wi th a C u sp bands, resulting in a k ind of v i r tua l bound state with a width given by 27r] V ^ d | 2 times D O S as described in chapter 1 for the Anderson model. Secondly the d states can attain a dispersional width because we have placed them in a lattice. The coupling between Co-d orbitals on neighboring Co sites can lead to a dispersional band wid th which could be important because the supercell under consideration is relatively small . B u t this is quite unlikely because d orbitals are quite compact and the coupling parameter between two d orbital is inversely proportional to the fifth power of the spatial separation between the atoms [39]. Of course one should also take into account the indirect hoping v i a C u states but also these are probably quite small . Following Harrison [39] the hopping parameters between sp-d orbital and d-d are given by h2r3J2 _ h2r3d *ldm Vldm 17/0 > *ddm ^?fidm Tc V ^ ' ^ J md''z may where I is any of the 5 and p orbital, rn stands for a or n bonding, 77's are dimensionless universal constants also mentioned in [39] for different cases of I and m and d is the distance between atoms. The most important point to notice here is that these cou- pling parameters are inversely proportional to the spatial separation between atoms. It decreases rapidly wi th the increase of d in the above expression. To make sure the origin of the widths for Co d states, a larger (3 x 3) supercell has been investigated and the resulting D O S calculated wi th in self consistent full potential method are compared to the 2x2 super cell in Fig.(4.3). . The calculation wi th a 2 x 2 supercell shows that the total number of 3d electrons is 7.24 wi th the magnetic moment of 1.97/^5. In a 3 x 3 supercell the total number of 3d electrons is 7.24 wi th the magnetic moment of 2.00/ig. These numbers together wi th the D O S plots indicate that there is no significant difference between 2 x 2 and 3 x 3 supercell cases. One can therefore say that coupling between Co-d orbitals of two different Co atoms is not very strong even in the case of 2 x 2 supercell. The wid th in the D O S of Co d bands is coming from the coupling between C o d and the C u sp surface and bulk states. Hence our final choice for studying the electronic structure of Co impuri ty on C u ( l l l ) surface is the slab having 2 x 2 supercell geometry wi th 15 C u ( l l l ) planes along z axis and two C o atom per supercell (one on each side of the slab). Chapter 4. Electronic Structure of Co Adatoms on a Cu(lll) Surface 46 2 4 - — 2x2 — 3x3 , i j T—r— i—|—i—i—i—r— t i i i i i—i i i i | — i i i i — [ — i — i — i — i — Spin up DOS 3d A _ l state • . . . . 1 . . . . 1 . . . . - — 2x2 — 3x3 t .VOL 1 , , . , 3d E(2) states . . . . 1 . . . . 1 - — 2x2 3x3 3dE(l) states . . . . 1 , . , , 1 , , , , 1 , , , , Energy (eV) - 1 1 . 2x2 3x3 . . 1 . 1 1 1 j 1 1 1 1 1 1 1 1 1— ., J . —1—1—1—1—| 1 1 1—1—pT—T—1—1 1 1 1 1 1 Spin down DOS { , U - L J . . . . I . . . < I . I . . - 2x2 — 3x3 , , 1 , i , . i , . . . i , V , 1 1 3d E(2) states l - — 2x2 3x3 , , , 1 ^ > 3dE(l) states ^ J . . . . i ^ • ' • . . i T x — . I I . . . . I -4 - 3 - 2 - 1 0 1 2 3 4 Energy (eV) Figure 4.3: Density of state (DOS) plots of Co 3d spin up and down states for 2 x 2 and 3 x 3 supercell structures (with two C o atoms per unit supercell of 11 layer C u slab, calculated wi th in full potential method) are shown in the upper and lower panel, respectively. The zero of energy is at Fermi energy. There is no significant difference in the D O S for both spin up and down states. Chapter 4. Electronic Structure of Co Adatoms on a Cu(lll) Surface 4.1 Optimal C o - C u distance So far we have been using the distance between Co adatom wi th the C u surface which is same as the distance between two C u ( l l l ) planes in the slab. The width of Co d states is coming mainly from the coupling of these states wi th C u sp surface and bulk states. Coupl ing parameters change wi th the distance between atoms under consideration. Therefore a change i n the vertical distance between C o adsorbate and C u surface can result in significant change in the Co d density of states. It can also change the widths of different D O S together wi th their shapes. The ab initio density functional calculations provide the total ground state energy which can be used to find the equil ibrium vertical position of the Co atom on the C u ( l l l ) surface. Density functional method is based on the Born-Oppenheimer approximation that says, due to their heavy masses nuclei moves much slower than the electrons, therefore one can consider the electrons as moving in a field of fixed nuclei wi th zero nuclear kinetic energy and a constant nuclear potential energy. In this situation one can treat each configuration in the adiabatic l imit to find the configuration wi th lowest total energy. Therefore one looks for the lowest total energy configuration for a slab that contains 15 C u ( l l l ) planes together with two Co atoms in a 2 x 2 supercell. Fu l l potential calculation is the most accurate method for calculating the total energy of the system. Al though L M T O is very similar to full potential, once Co is moved from the equil ibrium position due to atomic sphere approximation, the total energy becomes completely different compare to that of full potential. F i g . 4.4 is a plot of different total energies as a function of the change in spatial separation between C o adatom and C u ( l l l ) surface. We find wi th in full potential method that a spatial separation of 4.79 a.u. between the C o adatom and C u surface yields the lowest total energy configuration for our system. The difference between the total energy of the configuration wi th equil ibrium distance between Co adatom and. C u surface and the configuration wi th lowest total energy is found to be ~ 1 eV per unit cell and so is very significant. Since there are 2 C o atoms per unit cell, the change i n energy associated wi th each C o atom is 0.5 eV . F i g . 4.5 shows the D O S of spin down C o d A\ states for different distances between the C o adatom and C u surface and we chose the second case of lowest total energy configuration for further investigation. From L M T O calculations of this system wi th the opt imal distance between Co adatom and C u surface we find that there are 7.56 electrons in the 3d orbitals with the magnetic moment of 1.70 PB per Co . Fu l l potential calculation gives that there are 6.47 electrons in the 3d orbitals wi th the magnetic moment of 1.71 ps per Co . One notice that in the full potential calculation of the C o / C u ( l l l ) opt imal distance, due to the proper treatment of the atomic potential inside and outside of the muffin-tin sphere, the radius of the muffin-tin sphere decreases. Hence the amount of d electron present inside the atomic sphere (we integrate the electron density inside the sphere) Chapter 4. Electronic Structure of Co Adatoms on a Cu(lll) Surface 48 • * — , 1 , 1 1 1 1 r 0.2 \ - 0 - -<> i i i i * m i« m i i i -0.1 -0.08 -0.06 -0.04 -0.02 0 Change in spatial separation (a.u.) Figure 4.4: Tota l energy as a function of the distance between C o adatom and C u ( l l l ) surface. The zero of energy corresponds to the total energy of a configuration when the distance between the C o adatom and C u surface is 4.83 a.u. We notice that, total energy has a min imum for the system when the distance between Co adatom and C u ( l l l ) surface is 4.79 a.u. is smaller compare to that of the previous full potential calculation. F i g . 4.6 shows D O S plots for bands carrying different orbi tal characters calculated wi thin Wien2k full potential method. We find that the D O S for spin up electrons are centered below the Fermi energy and that for spin down electrons are mostly centered at the Fermi energy. Therefore we can comment that for spin up electrons al l five 3d orbitals are almost filled together with some unfilled orbitals for spin down electrons. Al though the shapes of the density of states calculated wi th in these two methods differ, the band widths look almost comparable in case of 4s, 4p and 3d bands. Comparison of D O S plots from L M T O and full potential method shows that the width of the C o d A\ is 0.5 eV in both cases. One cannot definitely say that i n the experiment one directly observes the D O S found from these single particle pictures. The dl/dV spectra is interpreted to measure the local density of states times the tunneling matr ix element squared. Therefore presence of an eigenstate that is a linear combination of atomic states wi th coefficients (that are strongly energy dependent) makes the tunneling matr ix element (which also becomes energy dependent) much more complicated. In such a situation it is Chapter 4. Electronic Structure of Co Adatoms on a Cu(lll) Surface 49 1 1 1 1 111 1 1 1 1 ' 1 ' ' 1 1 1 - a) :: J 1 1 1 . -̂i-T . ' I I I i I | I I I i i j l i i""i--r"|' i i i i l Spill down DOS of A _ l slate 1 4.83 a.u. Co adatom-Cu surface distance 1 ~ i 1 i i i 1 • 1 1 • 1 • * 1 * 1 * 1 1 : J 1 1 l_̂ lTV i 4.79 a.u. Co adatom-Cu surface distance v. . c) - 1 1 I ^ - K T V 4.734 a.u. Co adatom-Cu surface distance I W T 4 J - I - I ] 1 ] 1 1 1 1 1 1 J 1 1 l i -4 - 3 - 2 - 1 0 1 2 3 4 Energy (eV) Figure 4.5: Partial DOS of Ax state in the case of three distances between Co adatom and C u ( l l l ) surface. DOS for optimal distance is shown in the middle panel. The zero of energy is at Fermi energy. important to know which atomic states are mostly responsible for tunneling and also the corresponding tunneling strengths. Here the radial parts of the orbital wave functions are equally important together with the angular symmetry. A n orbital character study of the band structure is carried on in this situation. Fig. 4.7 is a three dimensional plot of the L M T O band structure and the projection of the eigenfunction onto Co d A\ state. That is we plot the wave vector k and energy eigenvalues in the x,y plane and Mg = X^Li^kK^2 — r2)i)((3z2 — r2)^^) along z axis. It is like fatbands but without the scaling factor multiplied that is used in plotting them. This 3D plot shows that near the Fermi energy at the T point Co d A\ state strongly mixes with Cu sp surface and bulk states present in the vicinity of Co atom. At K point the height of this band is almost 0.8 meaning that at this point, the eigenfunction has mostly Co d A\ character that does not mix with Cu states. At this point Co d electron spin density in real space is studied. The density distri- bution for spin up and down states can be written as Pyi(r) = niV ll&y l(r)\2 • Here is the occupation number and fa is the eigenfunction of Kohn-Sham Eq. Electron spin density is defined as ps = p^ — p\ where p\ and pi represents density of spin up and spin down electrons, respectively. The electron spin density of half of the slab is shown in Fig. 4.8 where we consider Chapter 4. Electronic Structure of Co Adatoms on a Cu(lll) Surface 50 2 2 0 i i I I r j " t i i 11 j i i i T i I I 1 M | 1 I ! 1 1 | 1 I I M Spin up DOS 3d A _ l state 1 1 1 t 1 1 1 1 1 t 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 I ' j T r T T T d) ; j 1 1 1 1 1 | 1 1 1 1 1 | M 1 1 1 Spin down DOS 3d A _ l state_ V. • .1 1 b) 3d E(2) states e) : ) i i - f . • 3d E(2) states •Vv—l 1 c) 3d E(l) states , 0 : } 3d E(l) states Energy (eV) Energy (eV) Figure 4.6: The upper panel shows total and partial density of states for Co 4s, 4p and 3d spin up and spin down electrons in the case of optimal Co adatom and Cu surface distance calculated within full potential method. The lower panel shows partial density of states (PDOS) of different 3d states. One can see from the lower panel PDOS that all the spin up states are filled (centered below Ep) and spin down states are partially filled (PDOS centered at the Ep). The zero of energy is at Fermi energy. Chapter 4. Electronic Structure of Co Adatoms on a Cu(lll) Surface 51 Energy (eV) Figure 4.7: L M T O band structure of C o / C u (111) calculated for optimized C o - C u distance. Co d 3z2 - r2 coefficient squared is plotted in the z axis. two 2 x 2 supercells in both x and y directions. The isosurfaces represented by red and blue show the surfaces of constant spin density of 0.005 and -0.005 electrons per unit volume, respectively. Since C o has rather large magnetic moment, the C u states are spin polarized too which leads to the spin density extended throughout the whole C u slab. A s described i n chapter 3, one needs four C u atoms per plane to map a 2 x ' 2 supercell structure. These four C u atoms can be categorized into two groups wi th respect to the position of the Co atom. Firs t type of C u atoms have mult ipl ici ty 3 that belongs to those triangles which has Co atom sit t ing on the threefold rotation axis on the plane above the triangle. Second type of C u has mul t ip l ic i ty 1 and sits at the center of the 2 x 2 supercell in the x, y plane. The magnetic moments of these different C u atoms induced by C o are listed in Table 4.1. The values of the magnetic moments together wi th F ig . 4.8 shows clearly that magnetism propagates throughout the whole slab. However one needs to look at the spin density distr ibution at this point since these numbers don't tel l us how and where exactly the spin states are present in the real space. From F i g . 4.8 we notice that the spin density is asymmetric in shape and also there is a large negative spin density blanket (the large blue shapes present between the Co adatoms and the first C u layer) between Co and first C u layer. Also the coupling between C o d states and C u conduction electron states is anti-ferromagnetic Chapter 4. Electronic Structure of Co Adatoms on a Cu(lll) Surface 52 Table 4.1: Lis t of the magnetic moments on different C u atoms in the C u layers of half of the slab. Magnetic moment on Magnetic moment on C u plane number C u wi th mult ipl ici ty 3 C u wi th mult ipl ic i ty 1 (u-B per Cu) (U-B per Cu) Firs t -0.0071 0.0114 Second -0.0136 -0.0126 T h i r d -0.0041 -0.0075 Fourth 0.0117 0.0082 Fi f th 0.0007 -0.0019 Six th -0.0052 -0.0069 Seventh -0.0002 -0.0028 Eigh th (middle most) 0.0065 0.0044 which results in a small magnetic moment on C u in the first layer. From F i g . 4.8 one can immediately see that the large spin down density compensates almost all of the spin up density in the muffin t in sphere of C u in the first layer. The density distr ibution is so anisotropic that we get a small negative magnetic moment for C u atoms wi th mult ipl ic i ty 3 (which is the integral of the spin density over the muffin t in sphere) and a small positive number for that of the C u atoms wi th mult ipl ici ty 1 in the first layer. Chapter 4. Electronic Structure of Co Adatoms on a Cu(lll) Surface 53 Co adatom 1st Cu plane 2nd Cu plane 3rd Cu plane Figure 4.8: This is a plot of electron spin density of half of the slab with 2x2 supercell structure. The number of unit cells is doubled in x and y directions. 0.005 and -0.005 spin density isosurfaces are represented by red and blue colors, respectively (calculated within L M T O method). Large red spheres at the top most layer represents Co atoms and the remaining seven layers are C u ( l l l ) planes. As one can see magnetism propagates through the whole slab since we observe some (small or large) amount of both positive and negative spin density present in all layers of Cu. 4.9: Three panels with cuts through three different Cu layers are shown here. Isolines show only positive spin density and this is only one unit cell. In the first panel the horizontal cut shows the Cu spin density in the first C u plane. Similarly in the second and third panels cuts show the spin densities in the second and third C u planes, respectively. One can see that in the second plane the spin density is quite spread all over the plane while they are relatively concentrated around C u atom in other planes. The second Cu layer is therefore different from any other Cu layers. Chapter 4. Electronic Structure of Co Adatoms on a Cu(111) Surface 55 In Fig. 4.9 the isolines show only positive spin density. We notice that the second layer of Cu has the highest magnetic moment per Cu among all other layers although the its magnitude is quite comparable in other layers. As one can see from Fig. 4.9, the spin density is spread over the whole second Cu plane but it is highly concentrated around Cu sites in all other layers. Therefore we conclude that it is important to consider the spin density distribution since it is very anisotropic and gives rise to almost comparable magnetic moments through the whole Cu slab. Fig. 4.10 shows the Co electron spin down density for all Co spd states integrated in the energy window -0.2 to +0.2 Ry around the Fermi energy. As expected, the spin density has highest value at the center of the Co atom and the value decreases as one moves away from Co. This picture tells us that Co spd states mix with the first Cu layer states pretty strongly since we see some Co spin density in that layer. Cu states are not present in this picture because the occupation number for all those states are set to zero. The isolines in this picture correspond to different spin densities as indicated in the thermometer. The above investigation shows that a 2 x 2 symmetric supercell consisting of 15 Cu( l l l ) planes with 2 Co atoms one on each side of the slab is sufficient for studying the electronic properties of Co adatoms on a Cu( l l l ) surface. We find that Co 3d states are split into non-degenerate A\ and two doubly degenerate E(l) and E(2) states by the surface potential of trigonal symmetry. Note that the Co adatom is not at a inversion center. Therefore Co p and d states can mix. In addition, we find that the width of the Co d states mainly arise from the coupling to Cu sp surface and bulk states. The calculated number of Co d electrons is 7.56 with the magnetic moment of 1.7 PB per Co which is consistent with a ds configuration of Co in the high spin state. The spin density calculated within the LMTO scheme shows a much longer range as compared to the part of the charge density associated with Co states. It is interesting to note that the induced magnetic moment in the second Cu layer is even bigger than the magnetic moment in the first layer. Chapter 4. Electronic Structure of Co Adatoms on a Cu(lll) Surface 56 Figure 4.10: The above picture shows the Co spin down density for al l Co states in the energy window of -0.2 to +0.2 R y around Fermi energy (calculated wi th in L M T O method). The isolines corresponding to different spin densities are shown in this picture. There is only one Co per unit cell. The part of density on the left of C o is coming from the neighboring cell. The horizontal cut shows Co density in the top C u layer. Chapter 5 Outlook 57 5.1 T i g h t B i n d i n g H a m i l t o n i a n f o r 1 l a y e r o f C u ( l l l ) p l a n e With the knowledge of a basis set and relative hopping parameters one can write down the Hamiltonian matrix for a specific system in the tight binding method. Therefore solving the Hamiltonian one gets the eigenstates and eigenvalues hence the band structure. Our initial interest was to write a model Hamiltonian with approximate numerical values for important hopping parameters between orbitals of the magnetic system of Co adatom on Cu( l l l ) surface. Therefore we started with a single Cu( l l l ) plane. Density functional calculation for Cu( l l l ) slab shows that sp type states are present near the Fermi energy. Hence only 4s and 4p orbitals are chosen to represent each atoms in plane and the nearest neighbor hopping is considered between atomic or- bitals. If the nearest neighbor hopping is mainly important for the system under consideration, almost similar band structure is expected from tight binding model compare to that of LMTO. For a single Cu( l l l ) plane, band structure calculated in tight binding method seemed to match pretty well with the band structure that we get from LMTO. Following is a description of the model Hamiltonian that we have for a single Cu( l l l ) plane. As mentioned, 4s and 4p orbitals are chosen to represent each atoms in the plane together with the nearest neighbor hopping only. Even though this is not a complete description of the system, it can help us getting approximate numerical values for hopping parameters. A mathematical package (MAPLE) is used to get the eigenvalues and the eigenvectors. The eigenvalues are plotted as a function of the wave vector to get the associated band structure. The basis set is written as {|4s) = |1),\Apx) = |2), |4py) = |3), \Apz) = |4)}. The 4 x 4 Hamiltonian matrix is written in terms of this basis set is given by: ( Hn # 1 2 # 1 3 Hu ^ £j _ # 2 1 # 2 2 # 2 3 # 2 4 # 3 1 H32 # 3 3 # 3 4 \ # 4 1 # 4 2 # 4 3 # 4 4 / where #^ stands for (i\H\j). The nearest neighbor hopping parameter between Chapter 5. Outlook 58 different orbitals are represented by tsaa = a hopping between s — s orbitals, tspa = a hopping between s — p orbitals, tn>a = a hopping between p - p orbitals and tpjm — hopping between p — p orbitals. In the (111) plane one Cu atom has six nearest neighbors. The coordinates of an atom (O) and its six nearest neighbors {AUA2,A3,A4,A5 and A6) are (0,0,0) and (a,0,0), (a/2,\/3a/2,0), (-a/2,v /3a/2,0),(- a,0,0),(-a/2,-\/3~a/2,0) and (a/2,-\/3a/2,0), respectively. Elements of this symmetric Hamiltonian matrix are the following Hn = 2ia ? ( 7{cos(k • ri) + cos(k • r 2 ) + cos(k • r 3 ) } Hn = 1.732ii s p f f{sin(k-r2) + sin(k-r3)} ' H13 = itspa{2 sin(k • r x ) + sin(k • r2) - sin(k • r 3 ) } u Ep + 2tppv cos(k • ri) + (1.5tppa + 0.5ipp7r){cos(k • r 2 ) + cos(k • r 3 ) } 0.866(£p p C T - £pp7r){cos(k • r 2 ) - cos(k • r 3 ) } Ul2 i..iG4u,3p<T\&iu.\is. • i2) T s i n ^ R . • i 3 ; / ' i3 spa x 2 ) 3 Hu = 0 H22 — p H23 = O.J H24 = 0 7/ 3 3 = Ep + 2£p p ( 7cos(k • r i ) + (0.5i p p C T + 1.5tppw){cos(k • r 2 ) + cos(k • r 3 ) } Hu = 0 Hu = EPz + 2£pp7r{cos(k • r i ) + cos(k • r 2 ) + cos(k • r 3 ) } where k is the reciprocal lattice vector in k space. Here r i = A\ and — vx = A 4 and so on. Also in this symmetric matrix H2i — H{2, H3i — H^... Ep and EPz are the crystal field splitting for (px, py) and pz states. The resulting band structure is then compared with that of L M T O . Fitting of these two band structures give us relatively acceptable numerical values for the hopping parameters from tight binding method. Next we tried adding more C u ( l l l ) planes according to the symmetry of the crystal. But as more C u ( l l l ) layers were added to the single C u ( l l l ) plane, we found it quite difficult to fit the band structure from our model hamiltonian and that from the L M T O . Therefore we conclude that the next-nearest, next-next-nearest neighbor and so on hopping should be considered for the proper treatment of the system. 5.2 S u m m a r y a n d o u t l o o k We have presented a rather detailed ab initio study of the electronic structure of Co adatoms on C u ( l l l ) surface. A careful look at the band structure and orbital charac- ters of Cu bulk showed that 4p bands are dominantly present near Fermi energy. The band structure of a single C u ( l l l ) plane shows that the band widths corresponding to bands with different orbital charaters are reduced by considerable amount. With the increase of number of layers in the slab this band width is regained. Finally we Chapter 5. Outlook 59 find that a slab thickness of 82.824 a.u. and a vacuum layer of 26.63 a.u. is our choice for studying the surface electronic properties of C u ( l l l ) slab. A n external potential of -0.3 R y applied to the nearest empty sphere layer wi th in L M T O method gives surface state at the same energy (-0.55 eV) compare to F u l l potential. In chapter 4 the electronic structure of C o adatoms on C u ( l l l ) surface is inves- tigated. Self consistent density functional calculation of C o monolayer on C u ( l l l ) surface shows that C o is magnetic in this system. We find that a 2 x 2 supercell in x,y direction is sufficient for studying electronic structure of C o impurity. Trigonal crystal field at the surface of this simulated C u ( l l l ) slab lifted the degeneracy on the Co d states and non-degenerate A\ state and doubly degenerate E(\) and E(2) states are observed. The lowest total energy of the system is calculated when the distance between Co adatom and C u surface is 4.79 a.u. The change in energy (difference between total energies of the configurations wi th 4.83 a.u. and 4.79 a.u. distances between C o adatom and C u surface) associated wi th each C o atom is found to be 0.5 eV. Par t i a l density of states ( P D O S ) plots for different C o d states show that al l the spin up D O S ic concentrated below the Fermi energy. Therefore spin up states are almost filled. For spin down states the D O S is centered at the Fermi energy indicating that they are part ial ly filled. A self consistent density functional calculation shows that there are 7.56 electrons in 3d states wi th a total magnetic moment of 1.70 JJLB per Co which is consistent wi th a d8 configuration of Co in the high spin state.. Near the Fermi energy at T point the C o d Ai (3z2 — r2) state strongly mixes wi th C u sp states. The trigonal crystal field at the surface allows the mixing between Co d and p states since C o is not at the inversion center. Th is is the difference between the C o / C u ( l l l ) case and the A g / A g ( l l l ) case where a l l the 3d states are filled for A g atoms. Al though d (especially 3z2 — r2) states cross the Fermi level, they are more compact compare to p states which are also present at EF . Therefore tunneling could be into p states rather than d states. The wid th of the Co d Ai state is a result of the coupling wi th the C u sp surface and bulk states. Also from the spin density plots we notice that C o adatoms polarize nearest C u layers of the slab by considerable amount. According to L M T O results the second C u layer has the highest magnetic moment per C u among all other layers. The spin density distr ibution is quite anisotropic and it is extended throughout the whole C u slab. Also at least three C u layers have significant influence on the electronic properties of C o . To outline the future direction of this project, we begin by noting that the amount of different orbi ta l characters present in the basis wave function is not known. There- fore this subject needs to be investigated further. Once we can find out important orbi tal states present i n the wave function, a tight binding treatment of the basis set together wi th at least nearest and next nearest neighbor hopping may help us wri t ing an effective Hamil tonian for the system. Chapter 5. Outlook 60 We note that the Co 3d spin down states centered at the Fermi energy and espe- cially the Co d 3z2 — r2 states are narrow in energy and the density of states exhibits very sharp structures including a strong dip just below the Fermi energy. This sharply structured density of states would of course also be seen in a STM measurement and so from the present study it is not that clear that a more sophisticated interpreta- tion such as in terms of the Kondo effect are needed. On the other hand we should also realize that the very small width of the d states will cause strong changes in the density of states if we add real correlations to the problem as in the Anderson impurity model including a Hubbard U. If large enough this would again split the d states at the Fermi energy into occupied and unoccupied regions with a gap and then we would be back to a Kondo like problem but very important would then be to take into account the sharp structure in the density of band states needed to produce the sharp structures in the Co d density of states. In other words the problem will be much more complicated than the simple Kondo models usually used. There is another important issue though seen in these calculations and that is the very sharp structure seen in the Co s and p density of states right at the Fermi energy. As has been mentioned in the introduction we expect much larger tunneling matrix elements from a tip to a solid for tunneling involving the Co 4s and 4p states than for the much more compact 3d states. The density of states of these seen in Fig 4.6 exhibit a sharp dip very close to the Fermi energy with a width of only about 50 meV. This could indicate a different origin of the dip seen in STM than that of the Kondo problem. Al l these issues should now be studied with appropriate models. LDA+U could be used to see what happens if U is switched on. One should look at the local projected density of states of other adatoms such as Cu or Mn to see if the sharp structure seen very close to Ep is characteristic of Co or is a general effect. 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