Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Cheung, Alfred Ka Chun
2015-08-13T19:11:53Z
2015
Master of Science - MSc
University of British Columbia
Using density functional theory (DFT) we demonstrate that the adsorption of large cations such as potassium or cesium facilitates the epitaxial growth of polar LaAlO₃ (LAO) on SrTiO₃ (STO). The low ionization potential of K favors efficient electron transfer to the STO conduction band and results in a 2D electron gas which exactly compensates for the diverging potential with increasing layer thickness. For large cations like K or Cs, DFT total energy considerations show that they remain adsorbed on the LAO surface and do not enter substitutionally into LAO. These results suggest a novel scheme for growing clean LAO/STO interface systems, and polar systems in general, by performing the growth process in the presence of large, low ionization potential alkali metal ions.
https://circle.library.ubc.ca/rest/handle/2429/54405?expand=metadata
Large Electropositive Cations asSurfactants for the Growth of PolarEpitaxial FilmsbyAlfred Ka Chun CheungB.Sc., The University of British Columbia, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2015c© Alfred Ka Chun Cheung 2015AbstractUsing density functional theory (DFT) we demonstrate that the adsorptionof large cations such as potassium or cesium facilitates the epitaxial growthof polar LaAlO3 (LAO) on SrTiO3 (STO). The low ionization potential ofK favors efficient electron transfer to the STO conduction band and resultsin a 2D electron gas which exactly compensates for the diverging potentialwith increasing layer thickness. For large cations like K or Cs, DFT totalenergy considerations show that they remain adsorbed on the LAO surfaceand do not enter substitutionally into LAO. These results suggest a novelscheme for growing clean LAO/STO interface systems, and polar systemsin general, by performing the growth process in the presence of large, lowionization potential alkali metal ions.iiPrefaceThe results of this thesis have been published in Physical Review B as A.K. C. Cheung, I. Elfimov, M. Berciu, and G. A. Sawatzky, Phys. Rev. B91, 125405 (2015). The calculations were performed by the author, AlfredCheung, and the interpretation and analysis of the results were done by Al-fred Cheung, Ilya Elfimov, Mona Berciu, and George Sawatzky. All densityfunctional theory calculations were performed using the Vienna ab initiosimulation package (VASP).iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 The polar catastrophe . . . . . . . . . . . . . . . . . . . . . . 32.2 Compensation mechanisms . . . . . . . . . . . . . . . . . . . 42.2.1 Electronic reconstruction . . . . . . . . . . . . . . . . 42.2.2 Compensation by surface oxygen vacancies . . . . . . 72.2.3 Other compensating mechanisms . . . . . . . . . . . . 82.3 Our proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Density Functional Theory . . . . . . . . . . . . . . . . . . . . 103.1 The Hohenberg-Kohn Theorems . . . . . . . . . . . . . . . . 113.1.1 The First Theorem . . . . . . . . . . . . . . . . . . . 113.1.2 The Second Theorem . . . . . . . . . . . . . . . . . . 123.2 The Kohn-Sham Method . . . . . . . . . . . . . . . . . . . . 133.2.1 Approximating the exchange-correlation functional . 163.3 Application to periodic systems . . . . . . . . . . . . . . . . 163.4 Plane wave basis and the motivation for pseudopotentials . . 173.4.1 Pseudopotentials versus all-electron methods . . . . . 184 Calculations and Results . . . . . . . . . . . . . . . . . . . . . 194.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Compensation by potassium electron donation . . . . . . . . 21ivTable of Contents4.3 Cohesive energies . . . . . . . . . . . . . . . . . . . . . . . . 224.4 Potassium coverage beyond 1/2 per unit cell . . . . . . . . . 254.5 Undesired substitutions of K into LAO . . . . . . . . . . . . 275 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . 28Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30vList of Figures2.1 The simple cubic unit cell of ABO3 perovskite. The structureconsists of alternating AO and BO2 layers. . . . . . . . . . . 42.2 Simplified electrostatics view of the polar catastrophe in LAO.(a) The electric field is sketched as a function of the positionalong the polar direction, normal to the layers. The fieldalternates between 0 and a non-zero value. When integratedto get the electric potential, a net potential difference existsacross the film which increases with the film thickness, asshown in (b). The divergence in potential as film thickness isincreased is known as the polar catastrophe. Its high energeticcost forces the system to reconstruct in one way or another inorder to negate the potential build up. Figure adapted fromRef. [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Simplified electrostatics view of electronic reconstruction asa compensation mechanism for the polar catastrophe at ann-type LAO/STO heterojunction. In this simple picture, halfan electron per unit cell is transferred from the top AlO2 layerto the interface. This results in the electric field as a functionof the normal position as shown in (a). The electric fieldalternates between opposite direction. The electric potentialobtained when the field is integrated is shown in (b). Incontrast to the unreconstructed system of Fig. 2.2, there isno longer a net potential difference across the LAO film. Inthis theory, the transferred electrons enter into the empty3d orbitals of the interface TiO2 layer, forming the q-2DEGobserved in experiment. Figure adapted from Ref. [3]. . . . . 6viList of Figures4.1 (a) Kads(LAO)m(STO)4 for m = 2 (initial structure). Katoms are placed on the surface AlO2 layer in the middleof the squares formed by oxygen atoms. This is illustrated inpanel (b) which shows a top view of the K adsorption sites onthe surface AlO2 layer. The simulated square unit cell has aside length of√2a, where a = 3.913 A˚ is the lattice constantof bulk STO obtained from DFT. . . . . . . . . . . . . . . . . 204.2 (a) Layer and element projected DOS for Kads(LAO)3(STO)4.The Fermi energy is at 0. The scale is the same for all panels.Upon adsorption of K, the TiO2 layers become conducting.The conduction electron density is greatest for the TiO2 layerclosest to the interface. The xy-planar-average and macro-scopic average electric potentials are plotted versus the z-position along the (001) direction of the supercell, within theLAO region, for the system (b) without and (c) with K. Apotential buildup of the order of 2 eV exists for the bare sys-tem without K. This is eliminated in the system with the Ksurfactant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Total DOS for (LAO)m(STO)4, m = 1− 6, with (black line)and without (shaded gray) K adsorption. The Fermi energyis at 0. The scale is the same for all panels. The critical thick-ness at which the system without K undergoes electronic re-construction and becomes conducting is 4 unit layers of LAO.In contrast, the system with K is conducting at all LAO thick-nesses simulated. . . . . . . . . . . . . . . . . . . . . . . . . . 244.4 Cohesive energy per adsorbed K, defined in Eq. (4.3), as afunction of the thickness of LAO. The system with K becomesmore stable as the thickness increases. The cohesive energyis approximately 1 eV for 1 unit layer of LAO, increasing tomore than 2 eV as LAO thickness is increased. . . . . . . . . 254.5 K PDOS near the Fermi energy, at three cases of K coverage,for LAO thickness of two unit layers. The density of conduc-tion electrons retained by K increases significantly going from1/2 to 3/4 per unit cell coverage, and then from 3/4 to 1 perunit cell coverage. Hence, K adsorption beyond the criticalamount needed for the ideal exact compensation does not re-sult in extra electrons being transferred to the interface; theextra electrons remain in the K overlayer. . . . . . . . . . . . 26viiAcknowledgementsI would like to express my deepest gratitude to my two research supervisorsProfessors Mona Berciu and George Sawatzky. It is hard to overstate howmuch I have learned and grown as a young physicist under their guidancethese past two years. They are truly amazing scientists and people. I amespecially indebted to Mona who first introduced me to the fascinating worldof condensed matter physics and gave me my first taste of research way backin my undergraduate days.I would also like to thank Dr. Ilya Elfimov who often acted as a thirdresearch supervisor to me during my degree. I owe all my knowledge andinsight of density functional theory to Ilya.Finally, I would like to thank all my fellow group members. I am par-ticularly grateful for the interesting and useful discussions I’ve had with Dr.Robert Green and Mirko Mo¨ller.viiiChapter 1IntroductionOne of the fascinating aspects of the study of transition metal and rare earthoxides is the essentially boundless possibilities for synthesizing new mate-rials with novel properties. Of particular interest are the perovskite classof materials which are the focus of this thesis. The perovskites are knownto exhibit rich phase diagrams with a myriad of electronic, structural, andmagnetic phase transitions. The properties of these materials are highly sen-sitive to very slight changes in geometrical parameters such as bond lengthsand angles. Consequently, changing the identity of the cationic species,which directly affect the lattice structure, can lead to drastic changes in theproperties of the perovskite.As if the potential for discovering novel phases from bulk perovskitematerials is not enough, the advent of epitaxial layer-by-layer growth ofmaterials through molecular beam epitaxy (MBE) and pulsed laser deposi-tion (PLD) has further extended the horizon in this field. Epitaxial growthmethods have allowed for the growth of materials that otherwise do not existin bulk form. Furthermore, stress and strain, which affect bond angles andbond lengths, can be applied to epitaxially grown materials by controllingthe identity of the substrate. Finally and most importantly for the topic ofthis thesis, it is found that at the interface of two oxide materials engineeredby epitaxial growth, novel properties emerge that are often completely dif-ferent from the bulk properties of the constituent materials. This leads tothe new frontier of oxide heterostructures.One of the most famous examples in recent years of how the interfacecan differ completely from the bulk is the case of polar LaAlO3 (LAO) epi-taxially grown on top of a non-polar SrTiO3 (STO) substrate. LAO andSTO are both large band gap insulators in their bulk forms. However, itwas discovered by Ohtomo and Hwang in 2004 that when polar LAO isgrown epitaxially on top of STO, a high mobility quasi-2 dimensional elec-tron gas (q-2DEG) is found at the n-type (TiO2/LaO) interface [1]. Sincethe discovery of this q-2DEG, enormous efforts have been made to under-stand its origins and its relation to the polar, crystallographic orientationinduced electric potential which diverges with the LAO film thickness – the1Chapter 1. Introductionso-called “polar catastrophe” [2–4]. Indeed, while the mechanism for theformation of this q-2DEG is still highly controversial, what is agreed uponis how intriguing this system is for both basic and applied science. Froma more fundamental point of view, superconductivity and magnetism andtheir coexistence have been found in this q-2DEG [5–7]. From a more appli-cations oriented perspective, it has been found that the conductivity of thissystem can be controlled through external electric potentials and by varyingthe LAO film thickness [8].Before the LAO/STO interface system can be understood and utilizedin any application, clean growth must first be achieved. The importance ofhaving a controlled scheme for growing clean interfaces is reflected by howsamples grown by different groups can exhibit very different properties [9].The daunting challenge of achieving clean growth is a natural outcome ofthe polar nature of the system: there are simply too many ways in whichthe system can compensate for the polar diverging potential, and each ofthese changes the properties of the resulting epitaxial film. Thus, the polarcatastrophe is somewhat of a mixed blessing – without it, there would beno q-2DEG at all and this system would not be interesting; with it, we losecontrol over the quality of the interface.In this thesis, we propose a novel solution for this challenge. Our pro-posal is to purposely introduce a minute concentration of potassium atomsinto the epitaxial growth chamber, which are then adsorbed onto the AlO2surface as a surfactant. Our reasoning is that potassium, being an alkalimetal, very readily gives up an electron. Thus, when it is adsorbed, elec-trons from the potassium surfactant layer can be donated to the interfaceat an energy cost significantly lower than other competing compensationmechanisms. Using density functional theory (DFT) calculations, we showthat this is this is indeed the case. Furthermore, we also show that due tothe large size of potassium ions, substitution into the LAO film is highlyunlikely, a necessary property for an effective surfactant.The thesis is organized as follows. In Chapter 2, we give an overviewof the background of the LAO/STO interface system and the concept ofpolar catastrophe. We also motivate our proposal for clean epitaxial growthby the use of large electropositive cations as surfactants. In Chapter 3, weprovide a review of density functional theory. We then present our findingsin Chapter 4. Finally, we conclude in Chapter 5 with a summary and furtherdiscussions.2Chapter 2Background2.1 The polar catastropheBecause the polar catastrophe is essentially an electrostatic effect that arisesdue to the charge distribution and spatial termination of a crystal, to un-derstand its role in the LAO/STO interface system we must first considerseparately the crystal structure of LAO and STO. As stated in the previ-ous chapter, both LAO and STO crystallize in a simple ABO3 (A and Bare cations) perovskite structure consisting of alternating layers of AO andBO2. The idealized simple cubic unit cell is shown in Fig. 2.1.For SrTiO3, the A cation is Sr2+ and the B cation is Ti4+. The structurethus consists of alternating charge neutral SrO and TiO2 planes (oxygenhas a formal oxidation number of 2+). For LaAlO3 however, the situationis different with the A cation being La3+ and the B cation Al3+. In thiscase, the LaO plane has a net charge of +1 and the AlO2 plane has a netcharge of −1 per unit cell. Hence in LAO, we obtain a system consisting ofalternating, oppositely charged layers.For LAO, what electric potential profile is obtained within such a systemof alternating charged layers? To understand this at a basic conceptual level,let us make a further simplification and assume that each charged layeris a uniformly charged sheet with charge density ±σ. From elementaryelectrostatics, we know that each sheet produces a constant electric field(σ/2)nˆ where is the dielectric constant of the medium, and nˆ is the unitvector normal to and pointing away from the sheet.With this setup, we can easily deduce the electric field as a function ofthe position z normal to the charged sheets. For a system of six layers, thisis shown in Fig. 2.2a. The electric field alternates between 0 and +(σ/)zˆin the regions between the sheets. To get the electric potential as a functionof z, we simply integrate the electric field. The result is shown in Fig. 2.2b.A non-zero potential drop exists between the top and bottom layers. Fur-thermore, this potential drop increases as more layers are added. Takingthis example back to the case of LAO, it can be shown that over nanometerlength scales, this potential drop is of the order of tens of electron volts.32.2. Compensation mechanismsFigure 2.1: The simple cubic unit cell of ABO3 perovskite. The structureconsists of alternating AO and BO2 layers.Such a system is clearly unstable, and it must change in some way to re-move this diverging potential. This highly unstable build up of a potentialdifference over microscopic length scales, essentially due to crystallographicproperties, is known as the polar catastrophe and it is expected to play acritical role in systems such as (001)-terminated LAO.2.2 Compensation mechanisms2.2.1 Electronic reconstructionIf a system is strongly polar, what are the available mechanisms for thesystem to remove the electric potential build up? One option, proposedby Ohtomo and Hwang to explain the existence of the q-2DEG at theLAO/STO interface, is electronic reconstruction [3, 4, 10]. Consider ann-type LAO/STO interface, again in the simplified picture of viewing eachcharged layer as a uniformly charged sheet. In the non-compensated systemwith alternating charge +1/-1 layers, we saw that the potential differencegrows as the LAO thickness increases. However, what happens if half anelectron per unit cell is transferred from the top AlO2 layer to the interface?In this case, the top AlO2 layer is left with a net charge of +1/2 per unit42.2. Compensation mechanismsFigure 2.2: Simplified electrostatics view of the polar catastrophe in LAO.(a) The electric field is sketched as a function of the position along the polardirection, normal to the layers. The field alternates between 0 and a non-zero value. When integrated to get the electric potential, a net potentialdifference exists across the film which increases with the film thickness, asshown in (b). The divergence in potential as film thickness is increased isknown as the polar catastrophe. Its high energetic cost forces the system toreconstruct in one way or another in order to negate the potential build up.Figure adapted from Ref. [3].cell, while the interface between LAO and STO acquires a net charge of -1/2per unit cell. By the same elementary electrostatics analysis done earlier,it can be shown that the electric field now alternates between +(σ/2)zˆand −(σ/2)zˆ in the regions between the sheets, as shown in Fig. 2.3a.Integrating to get the electric potential, we find that it no longer divergeswith film thickness as shown in Fig. 2.3b. In this way, the polar catastrophehas been negated. Moreover, in this theory, the -1/2 electrons per unit celldonated to the interface forms the q-2DEG seen in experiments.The electronic reconstruction picture is appealing in its simplicity. Un-fortunately, it fails to explain a few key experimental observations. Firstly,the energy cost associated with the transfer of 1/2 an electron per unit cell,essentially the band gap between the valence band of LAO and the conduc-tion band of STO, predicts the onset of electronic reconstruction to occur ata LAO thickness of 1-2 layers (the thickness dependent potential build upmust exceed this band gap for a Zener breakdown to occur and the electronsto be transferred). However, the critical thickness is experimentally mea-sured to be at 4 unit layers [4]. Secondly, the electron transfer mechanism52.2. Compensation mechanisms1/2 e-per u.c.ElectronicReconstructionFigure 2.3: Simplified electrostatics view of electronic reconstruction as acompensation mechanism for the polar catastrophe at an n-type LAO/STOheterojunction. In this simple picture, half an electron per unit cell is trans-ferred from the top AlO2 layer to the interface. This results in the electricfield as a function of the normal position as shown in (a). The electric fieldalternates between opposite direction. The electric potential obtained whenthe field is integrated is shown in (b). In contrast to the unreconstructedsystem of Fig. 2.2, there is no longer a net potential difference across theLAO film. In this theory, the transferred electrons enter into the empty3d orbitals of the interface TiO2 layer, forming the q-2DEG observed inexperiment. Figure adapted from Ref. [3].62.2. Compensation mechanismspredicts that the number of electrons transferred should increase graduallyfrom zero at the critical thickness. In contrast, experimentally, the carrierdensity at the interface is observed to jump to a non-zero value at the crit-ical thickness and then remain constant as the LAO thickness is furtherincreased [4]. Finally, electronic reconstruction predicts the existence of aconducting hole gas at the top AlO2 surface in addition to the q-2DEG atthe interface. No such surface conductivity has been observed.2.2.2 Compensation by surface oxygen vacanciesClearly, electronic reconstruction cannot be the whole story. Other compen-sating mechanisms must exist to account for what is observed experimen-tally. One particularly promising proposal is compensation via the formationof surface oxygen vacancies [6, 8, 12–24]. In this proposal, 1 out of 8 oxygenatom sites are vacant on the top AlO2 surface. In other words, the sur-face formula unit is AlO1.75 leading to a net charge of +1/2. Additionally,the missing O2− anions must have left behind 2 electrons each. These aretransferred to the interface and form the q-2DEG. Overall, the top AlO1.75surface acquires an effective net charge of +1/2 per unit cell while the inter-face acquires a net charge of -1/2 per unit cell. Hence, the polar problem issolved and the q-2DEG at the interface is accounted for, as in the electronicreconstruction scenario. A key difference exists in this case however – theoxygen vacancies are immobile on the surface and therefore do not lead toany surface conductivity. This is in agreement with experiment.Besides accounting for the insulating behaviour of the AlO2 surface, thesurface oxygen vacancy picture also explains the abrupt increase in interfacecarrier density at the critical thickness and its constant value as more LAOlayers are grown. Consider that at thicknesses below the critical thickness,the energy cost for forming a surface oxygen vacancy must be positive (itcosts energy to form a lattice vacancy). As the thickness of LAO is increasedand the polar electric potential builds up, the energy cost must decrease.Finally, at the critical thickness, the vacancy formation energy becomes neg-ative [24]. Forming an oxygen vacancy now stabilizes the system and surfaceoxygen vacancies spontaneously form at (in an ideal case) a concentrationof 1/8. Once the vacancies have formed and electrons are transferred tothe interface, the polar problem no longer exists and increasing the LAOthickness further makes no change. Hence, compensation by surface oxy-gen vacancies can be said to be a “one-off” process, thereby explaining theabrupt increase in the interface carrier density at the critical thickness andits constant value for higher thicknesses.72.3. Our proposal2.2.3 Other compensating mechanismsAlthough we have discussed in detail only two compensating mechanisms,many credible alternatives have been proposed, each often explaining certainaspects of the experimental observations while failing to rationalize others.For example, Pickett and Pentcheva used DFT calculations to demonstratethat within the LAO film, buckling of the lattice can partially reduce thepolar potential [25, 26]. In their proposal, La3+ and Al3+ cations are dis-placed relative to the O2− anions. The relative displacement occurs suchthat internal dipoles are formed which create electric fields opposite in di-rection to the electric field formed by the overall polar crystal orientation.The net effect is to partially reduce the polar potential build up and increasethe critical thickness required for electronic reconstruction to become ener-getically favourable. Other proposed mechanisms include cation intermixingbetween STO and LAO at the interface as well as interface defects [11, 12].Which one of these mechanisms is truly responsible for the compensationof the polar catastrophe and the existence of the q-2DEG? There really is noanswer to this question because there is no reason why these compensationmechanisms should be mutually exclusive. Although the issue is still underdebate, there is now consensus that compensation of this diverging electricpotential (the polar catastrophe) is achieved through a combination of pureelectronic reconstruction, interface defects, lattice distortions, and oxygenvacancies. The relative contributions of these mechanisms to removing thepolar catastrophe is expected to depend critically on the specific details ofthe growth conditions. This likely explains why samples grown by differentgroups can show widely dissimilar properties. The important role played bydefects and vacancies [3] also explains why the growth of clean LAO/STOinterfaces, necessary for device applications, has proven to be such a chal-lenge.2.3 Our proposalIn this thesis the main problem we aim to solve can thus be framed asfollows: how can clean epitaxial growth be achieved for the LAO/STO sys-tem and more generally with any strongly polar epitaxial film? From thepreceding discussion, we have established that this is basically a questionof competing energy costs for various compensation mechanisms. Becauseno particular compensation mechanism that is available to the system issubstantially more energetically efficient compared to the others, they allcome into play to varying degrees. The outcome is the lack of control over82.3. Our proposalsample quality seen in experiments. Therefore, if the growth conditions canbe modified such that a particular compensation mechanism becomes muchmore thermodynamically efficient, then this problem would be solved.Our proposal to purposely introduce a minute amount of potassium intothe growth chamber is based precisely on this principle. We use DFT calcu-lations to show that a coverage of 1/2 K per unit cell of the AlO2 surface ofLAO acts like a surfactant that stabilizes the epitaxial growth of the polarLAO on top of STO. Each K donates an electron to the LAO/STO interface,generating a q-2DEG while keeping the top LAO surface insulating. Thiscompensates for the polar catastrophe for any LAO film thickness, removingthe reason for the appearance of defects, vacancies or distortions. Moreover,substitution of the large K ions into the LAO film is unlikely for epitaxialtype growth. As a result, after the deposition of half a monolayer of K, theLAO film will grow cleanly under this surfactant.We emphasize that we use the term surfactant as is customary in thecrystal growth community [27, 28]: a surfactant facilitates the growth ofthe film while “floating” on its surface [27–31]. An effective surfactant isenergetically most favored on the film’s surface [29]; when an adatom (La,Al, and O) arrives onto the surface, the adatom and a surfactant atom (K)exchange positions such that the surfactant re-emerges on the surface andthe adatom is buried underneath. The process is then repeated [27–32].Our proposal is supported by past successes in using surfactants to stabi-lize crystal surfaces [27–35]. For example, (111)-terminated MnS, otherwiseimpossible to grow due to the polar problem, was stabilized by adsorptionof I− on its surface [33]. The use of alkali metals as electron donors is alsocommon: K deposition onto YBa2Cu3O6+x was used to tune its doping level[36, 37]. Indeed, although we focus in this thesis on LAO/STO, there is noobvious reason for this method to not work for other polar materials. Webelieve that, in general, the use of low ionization energy surfactants canfacilitate the growth of ionic materials with strongly polar orientations.9Chapter 3Density Functional TheoryWe claim that our proposed scheme for clean epitaxial growth should beapplicable to the entire class of materials with strongly polar orientations.Nevertheless, because we largely illustrate the physics using the LAO/STOinterface system as a specific material example, the theoretical technique wechoose must naturally be one of the ab initio methods. The best moderntechnique for such purposes is density functional theory. Hence, we devotethis chapter to an overview of the formalism behind DFT. We will aim toonly stress the main ideas without delving into rigorous proofs and details.In the spirit of ab initio methods, the goal is to solve for the many-electron wavefunction Ψ which satisfies the Schro¨dinger equation HˆΨ =EΨ, given the positions and charges of all the positive nuclei. The general(non-relativistic) electronic Hamiltonian Hˆ (in atomic units) assumes thefollowing form:Hˆ = −12N∑i=1∇2i +N∑i=1N∑j>i1|ri − rj |−N∑i=1M∑A=1ZA|ri −RA|, (3.1)where N is the total number of electrons, M is the total number of nuclei, riis the position vector of electron i, RA is the position vector of of nucleus A,and ZA is the charge of nucleus A. Note that we have already used the Born-Oppenheimer Approximation in which the positions of the positive nuclei arefrozen and they provide only a fixed external potential felt by the electrons.The first term gives the electron kinetic energy, the second term gives theelectron-electron repulsion, and the last term gives the external potentialfrom the positive nuclei. Note that the first two terms are universal to allelectron system, while the last term is unique to any given system.Solving for Ψ(r1, ..., rN ) is a truly challenging if not impossible task. Stu-dents of undergraduate level quantum mechanics know that there is alreadyno analytical solution for N > 1. Either approximations must be made, ornumerical methods must be employed. For larger N , even numerical effortsbecome futile as computation times scale exponentially with increasing N .Considering that in condensed matter systems, the number of electrons is of103.1. The Hohenberg-Kohn Theoremsthe order 1023, it is easy to dismiss the entire problem as impossible! At thispoint, two options are available: (1) instead of working with the full Hamil-tonian of Eq. 3.1, simpler effective model Hamiltonians can be derived whichare much easier to handle but still capture the essential physics; (2) retainthe full Hamiltonian of Eq. 3.1 but reformulate the problem such that it istractable numerically. It is not inaccurate to claim that much of theoreticalcondensed matter physics is based on using one of these two approaches tothe many-body problem.3.1 The Hohenberg-Kohn TheoremsIn DFT, the second approach is taken. Consider the ground state of amany-body system described by the wavefunction Ψ0(r1, ..., rN ). The basicidea in DFT is that, instead of working with the wavefunction Ψ0 with3N degrees of freedom, we can instead work with the ground state electrondensity n0(r). Clearly, this is a drastic simplification as we have reducedthe degrees of freedom to just 3: x, y, and z! But, can the ground stateproperties of a system, as properly described by the wavefunction Ψ, reallybe extracted from only the electron density n0(r)? The answer is yes1, andit is based on the two Hohenberg-Kohn Theorems derived in 1964 [38].3.1.1 The First TheoremThe first Hohenberg-Kohn theorem states that n0(r) uniquely determinesthe external potential U(r), and hence the Hamiltonian, of a fully-interactingsystem of electrons in the ground state. The proof is by contradiction. Con-sider two systems described by Hamiltonians Hˆ and Hˆ ′ with two (different)external potentials V (r) and V ′(r). The ground state wavefunctions of thesesystems are Ψ and Ψ′, respectively. Assume for now that these wavefunc-tions have the same ground state electron density n(r) = n′(r) = n0(r).Let E0 = 〈Ψ|Hˆ|Ψ〉 be the eigenenergy of state Ψ and E′0 = 〈Ψ′|Hˆ ′|Ψ′〉be the eigenenergy of state Ψ′. By the variational principle,E0 < 〈Ψ′|Hˆ|Ψ′〉, (3.2)1One might object that other properties, such as whether a material is insulating ormetallic, are also considered “ground state properties”, which the formalism here failsto describe. However, strictly speaking, there really are no “ground state properties”other than the ground state electron density and the total energy. Whether a material isinsulating/semiconducting or metallic, for example, is determined by the conductivity gap,which is related to electron addition and removal energies, i.e. excited state properties.113.1. The Hohenberg-Kohn Theoremssince the state Ψ′ is the ground state of Hamiltonian Hˆ ′ and not Hˆ. We canexpress the right hand side of the inequality in Eq. 3.2 as:〈Ψ′|Hˆ|Ψ′〉 = 〈Ψ′|[Hˆ − Hˆ ′]|Ψ′〉+ 〈Ψ′|Hˆ ′|Ψ′〉= 〈Ψ′|[V (r)− V ′(r)]|Ψ′〉+ E′0=∫d3r[V (r)− V ′(r)]n(r) + E′0=∫d3r[V (r)− V ′(r)]n0(r) + E′0 (3.3)Hence, the inequality becomes:E0 <∫d3r[V (r)− V ′(r)]n0(r) + E′0. (3.4)We can repeat the same steps with Ψ′ as the reference system, i.e. con-sider E′0 < 〈Ψ|Hˆ ′|Ψ〉. In this case, we have:〈Ψ|Hˆ ′|Ψ〉 = −∫d3r[V (r)− V ′(r)]n′(r) + E0= −∫d3r[V (r)− V ′(r)]n0(r) + E0. (3.5)This leads to the inequality:E′0 < −∫d3r[V (r)− V ′(r)]n0(r) + E0. (3.6)Finally, adding the inequalities of Eq. 3.4 and Eq. 3.6, we obtain thatE0+E′0 < E0+E′0 – a contradiction. Therefore, the assumption n(r) = n′(r)cannot hold; the ground state electron densities from two different externalpotentials must be different. In other words, there is a unique Hamiltonianfor a given ground state density. The converse statement, that there is aunique ground state density for a given Hamiltonian, is trivial assuming anon-degenerate ground state. Thus, we conclude that there is a one-to-onecorrespondence between n(r) and Hˆ.3.1.2 The Second TheoremThe second Hohenberg-Kohn Theorem establishes a variational principlein terms of the electron density n(r) and it follows closely from the First123.2. The Kohn-Sham MethodTheorem. From the First Theorem, we know that all the information aboutthe ground state, such as the ground state energy, can be obtained from n(r)in place of Ψ(r1, ..., rN ). This means that there exists a functional Ψ[n(r)]which maps n(r) to the corresponding Ψ. There is also a functional of theground state density for the ground state energy. This justifies the name ofDensity Functional Theory.Denote the functional which maps the ground state electron densityn0(r) to the corresponding ground state energy E0 as E[n0(r)] such thatE[n0(r)] = E0. Consider a trial state Ψt with a corresponding trial elec-tron density nt(r). The conventional variational principle in terms of statesgives 〈Ψt|Hˆ|Ψt〉 ≥ E0. But, again by the First Theorem, instead of workingwith complicated states and wavefunctions, we can simplify the problem andwork instead with the electron density n(r). In terms of electron density, thevariational condition becomes: E[nt(r)] ≥ E0. Thus, the Second Theoremis basically a reformulation of the variational principle and tells us that tofind the true ground state electron density, we must minimize the energyfunctional with respect to n(r).3.2 The Kohn-Sham MethodWhile the Hohenberg-Kohn Theorems firmly establish the theoretical justi-fication for working with electron densities rather than wavefunctions, theydo not offer a practical prescription for the calculation of the ground stateelectron density. This came in 1965 with the introduction of the Kohn-Sham(KS) Method which provided a clear recipe for the self-consistent determi-nation of the ground state density [39]. The KS method remains the onlypractical implementation of DFT; as such, while DFT, strictly speaking,refers to the much broader formalism in terms of electron density as de-scribed by the Hohenberg-Kohn Theorems, in all modern applications ofDFT, it is implicit that the KS method has been applied.In the Kohn-Sham Method, the key simplification is that instead of work-ing with the full interacting system of electrons, we consider a reference non-interacting system. This reference non-interacting system is chosen suchthat the ground state electron density is equal to that of the actual systemof interacting electrons. Provided that this choice is made, the ground stateelectron density of the reference system can be easily calculated becausethe system is non-interacting. The wavefunction takes the form of a Slaterdeterminant obtained from filling up the N lowest energy single-particle or-bitals ψi(r). The electron density is then given by n(r) =∑Ni=1 |ψi(r)|2. Of133.2. The Kohn-Sham Methodcourse, the question now is how can a suitable non-interacting reference sys-tem be chosen? The answer is that we do not know how to do this exactly,but we can group all the unknown parts into a term of the non-interactingreference Hamiltonian called the exchange-correlation energy.Consider the total energy functional E[n(r)] consisting of contributionsfrom kinetic energy T [n(r)], the electron-electron interaction Uee(r), and theexternal potential Vext[n(r)] =∫V (r)n(r)d3r:E[n(r)] = T [n(r)] + Uee[n(r)] +∫V (r)n(r)d3r. (3.7)Let Ts[n(r)] be the kinetic energy of the reference system. It is in generalnot equal to the kinetic energy of the original interacting system. We canexpress the true kinetic energy as the sum of Ts[n(r)] and the difference inthe reference and true kinetic energies, denoted here as ∆T [n(r)]:T [n(r)] = Ts[n(r)] + ∆T [n(r)]. (3.8)The electron-electron interaction term can also be divided into a classicalHartree term and a non-classical exchange term Uex[n(r)]:Uee[n(r)] =12∫n(r)n(r′)|r− r′|d3rd3r′ + Uex[n(r)]. (3.9)Applying Eq. 3.8 and Eq. 3.9 to Eq. 3.7, we obtain:E[n(r)] =Ts[n(r)] + ∆T [n(r)]+ 12∫n(r)n(r′)|r− r′|d3rd3r′ + Uex[n(r)]+∫V (r)n(r)d3r. (3.10)Observe that the only two functionals which we do not know the form of are∆T [n(r)], the “correlation energy”, and Uex[n(r)], the exchange energy. Wegroup these two unknown functionals together into a single unknown func-tional Exc[n(r)] – the famous exchange-correlation energy. Hence, Eq. 3.10becomes:E[n(r)] = Ts[n(r)] +12∫n(r)n(r′)|r− r′| d3rd3r′ +∫V (r)n(r)d3r+ Exc[n(r)].(3.11)We have now succeeded in expressing the total energy functional of theoriginal interacting system in terms of the kinetic energy of the reference143.2. The Kohn-Sham Methodnon-interacting system, Ts[n(r)] and the three other terms which we canidentify as coming from the effective single-particle potential of the referencesystem. The form of this effective potential Veff [n(r)] is given by taking thefunctional derivative of these three terms with respect to n(r). This yieldsVeff [n(r)] = V (r) +∫n(r′)|r− r′|d3r′ + Vxc[n(r)], (3.12)where Vxc[n(r)] is defined to be δExc[n(r)]δn(r) .Eq. 3.12 represents the single-particle potential that the electrons in thereference non-interacting system experience. By construction, the groundstate electron density of this reference system is the same as that of theoriginal interacting system. Therefore, assuming that the form of Vxc[n(r)]is known, we now have a self-consistent recipe for the calculation of theground state density n(r) of the original interacting system:1. Choose an initial guess for n(r).2. Construct the effective single-particle potential Veff [n(r)] = V (r) +∫ n(r′)|r−r′|d3r′ + Vxc[n(r)] for the reference non-interacting system.3. Solve for the ground state of the reference system. The solution is aSlater determinant formed by occupying the N lowest energy single-particle eigenstates ψi(r).4. Recalculate a new n(r) by taking n(r) =∑Ni=1 |ψi(r)|2 and repeatsteps until convergence of E[n(r)].The formalism outlined above was based on static positions of the nuclei(the Born-Oppenheimer Approximation). The formalism can be extendedto allow for structural relaxation. This is done by first assuming a fixednuclei configuration and calculating the electronic ground state via the recipeabove. Next, the net forces on all the nuclei are determined by using theHellmann-Feynman Theorem. Based on these forces, the nuclei positionsare updated and this process if repeated until the forces on all nuclei arebelow a certain threshold value.Of course, one major issue remains: we do not know the form of theexchange-correlation energy functional! Various approximations have beenmade for its form. Much of ongoing theoretical work on the DFT formalismis focused on finding better forms for Vxc. While this is a difficult task, thegood news is that the exchange-correlation energy is a universal functionalthat applies to all interacting electron systems: the positions of the positivenuclei do not affect it.153.3. Application to periodic systems3.2.1 Approximating the exchange-correlation functionalThe most important approximation to the exchange-correlation functionalis the Local Density Approximation (LDA). In addition to being widelyused in its own right, LDA is also the starting point on which many otherimprovements are based. In LDA, Vxc[n(r)] is assumed to take the form:Vxc[n(r)] =∫d3rn(r)xc(n(r)). (3.13)That is, it is a function only of the local electron densities at each point.An approximation to the function xc(n) can be computed numerically fora uniform electron gas [40]. This form is then applied to general systemswhen n(r) is not uniform. Consequently, LDA works very well for systemswith electron densities that vary slowly in space.The generalized gradient approximation (GGA) is another popular ap-proximation. In GGA, Vxc[n(r)] is allowed to depend not only on the localelectron densities, but also on the local gradient of the electron density,∇n(r) [41]:Vxc[n(r)] =∫d3rf(n(r),∇n(r)). (3.14)The GGA tends to allow for stronger spatial variation in the electron density.The difficulty with using GGA, however, is that unlike in LDA there is noreference system such as the uniform electron gas which can be used todetermine f(n(r),∇n(r)). Thus, various forms of GGA exist. The mostwidely used one used in solid state physics is the Perdew-Burke-Ernzerhof(PBE) functional [41]. The advantage of the PBE functional is that itdepends only on fundamental constants – there are no parameters that mustbe fitted according to the system being modeled.3.3 Application to periodic systemsSo far, we have presented DFT and the KS Method in a general form suitablefor application to any system of interacting electrons and nuclei. Indeed,DFT is used extensively for the study of molecular systems in chemistry, inaddition to its traditional applications to crystalline systems in solid statephysics. However, since we are concerned here with periodic systems, in thissection we will mention some of the important aspects of applying DFT toperiodic structures.As usual for periodic systems, due to the discrete translational symme-try of the lattice, we work within a Brillouin zone defined by the crystal163.4. Plane wave basis and the motivation for pseudopotentialsstructure, and crystal momentum k is a good quantum number. Hence,the single-particle eigenstates of the KS Hamiltonian take the form of Blochfunctions labeled by k and a band index n:ψnk(r) = unk(r)eik·r, (3.15)where unk(r) is a function with the same periodicity as the lattice. Theenergies Enk of these eigenstates of the Kohn-Sham reference system definea band structure, which can be summed over to obtain a density of statesρ(E) =∑nk δ(E − Enk). A very important point is that the band struc-ture and density of states obtained do not actually represent the systemof interacting electrons! They only describe the electronic structure of thenon-interacting Kohn-Sham reference system. The only property of the ref-erence system which does have physical meaning is the ground state electrondensity. Nevertheless, in DFT, the KS band structure and density of statesare often interpreted as describing the electronic structure of the originalsystem in some way. In this work, we will also use this interpretation of theKS density of states.3.4 Plane wave basis and the motivation forpseudopotentialsThe choice of the basis set is also a significant part of applying DFT ef-ficiently. Whereas for isolated molecular systems, it is natural to adoptatomic orbitals centred about each atom as the basis set, for periodic sys-tems, the natural choice is plane waves eip·r. Plane waves can be chosento have arbitrarily high energies (frequencies). Therefore, it is necessary topick a cutoff energy and only keep in the basis plane waves with energiesless than this cutoff. The higher the cutoff, the more complete the basis,and the more accurate the results, at the cost of longer computation times.One challenge of working with a plane wave basis comes from the shapeof the attractive Coulombic potential in regions close to a nucleus. In theseregions, the potential goes to very negative values, which in turn meansthat the eigenfunctions oscillate very rapidly in space. Hence, to describethese parts of the eigenfunctions, very high energy plane waves are needed,thereby greatly increasing computation time. A way around this problem isthrough the use of pseudopotentials.173.4. Plane wave basis and the motivation for pseudopotentials3.4.1 Pseudopotentials versus all-electron methodsIn the pseudopotential method, only the valence electrons are included ex-plicitly as electrons in the calculation. The core electrons are taken intoaccount only in an effective manner through their screening of the poten-tial from the nuclei felt by the valence electrons. The screened effectivepotential – called the pseudopotential – no longer approaches negative in-finity in regions close to the nuclei, removing the need for a large planewave energy cutoff. Thus, computation time is greatly reduced in two ways:(1) a lower energy cutoff can be used, and (2) the number of electrons issmaller since the core electrons are not explicitly included. In general, theform of a pseudopotential of an atom is determined by requiring that thepseudo-eigenfunctions (obtained from solving Schro¨dinger’s equation withthe pseudopotential in place) match the true eigenfunctions for radial dis-tances beyond some cutoff. For a given DFT implementation, there aretypically several pseudopotentials available for each element differing in thenumber of electrons placed in the core.Pseudopotential methods are particularly useful when studying proper-ties which are dictated by the valence electrons. This is actually not toosevere of a restriction since it is the valence orbitals, both occupied or unoc-cupied, which determine important aspects such as the low energy responseto applied electromagnetic fields (i.e. all forms of spectroscopy) and chem-ical reactivity. In this work, since we deal with very large supercells ofthe LAO/STO heterojunction, it is computationally prohibitive to use all-electron methods. Therefore, we employ the pseudopotential method for allour calculations.18Chapter 4Calculations and Results4.1 MethodAll DFT calculations reported in this thesis are performed with the Viennaab initio simulation package (VASP) [42] using the projector augmentedplane wave method [43, 44]. The PBE functional is used for the exchange-correlation energy. The energy cutoff for the plane-wave basis functions is400 eV. For structural optimization calculations, a Γ-centered (7,7,1) k-pointmesh is used. For density of states (DOS) calculations on the optimizedstructures, a Γ-centered (17,17,1) k-point mesh is used.The structures simulated are comprised of m unit layers of LAO on topof 4 unit layers of STO substrate. The interface is n-type (TiO2/LaO). Katoms are adsorbed on the surface AlO2 layer of LAO at a concentrationof 1 adsorbed atom per 2 lateral AlO2 unit cells, i.e. the concentrationneeded for exact compensation of the polar problem. We refer to this asKads(LAO)m(STO)4. Coverages of 3 K per 4 lateral unit cells and 1 Kper lateral unit cell are also considered. The m = 2 structure is shown inFig. 4.1(a). We placed the K atoms above the center of squares formed bythe oxygen atoms in the surface AlO2 layer, as shown in Fig. 4.1(b). Thisposition should be the most stable configuration because a K+ cation isattracted to each O2− anion. Indeed, alternate positions were found to beunstable towards relaxing back into this position, thus validating our choice.The lateral lattice constant is fixed at√2a, where a = 3.913 A˚ is thecalculated DFT-PBE lattice constant for bulk STO. Atoms of the bottom-most STO unit layer are kept fixed so as to simulate the effect of the infinitelythick substrate. All other atoms are allowed to relax along the z-directionuntil the force on each atom is less than 0.02 eV/A˚. 15 A˚ of vacuum isplaced above each slab to minimize interactions between periodic copiesof the slab. Dipole corrections to the total energy and electric potentialare used to remove any remaining spurious contributions due to periodicboundary conditions [45].194.1. MethodFigure 4.1: (a) Kads(LAO)m(STO)4 for m = 2 (initial structure). K atomsare placed on the surface AlO2 layer in the middle of the squares formed byoxygen atoms. This is illustrated in panel (b) which shows a top view of theK adsorption sites on the surface AlO2 layer. The simulated square unit cellhas a side length of√2a, where a = 3.913 A˚ is the lattice constant of bulkSTO obtained from DFT.204.2. Compensation by potassium electron donation4.2 Compensation by potassium electrondonationLayer and element projected partial densities of states (PDOS) are shownin Fig. 4.2(a) for m = 3. Upon K adsorption, electrons are donated into theTi 3d conduction bands. The electron density is greatest for the titanatelayer near the interface and decays for layers further away, forming a q-2DEG as in pure electronic reconstruction. In both cases, the extent of theelectron transfer is limited by an associated energy cost. For pure electronicreconstruction, this is the band gap between the valence band of LAO andthe conduction band of STO. In the present case, this is the binding energyof the 4s electron of the adsorbed K. This parameter controls how muchof the diverging potential across LAO is compensated. To evaluate theirefficiency, we look for residual potential buildup across the LAO film.If V (x, y, z) is the electric potential function, then the planar-averagepotential along z, V¯ (z), is defined by:V¯ (z) = 1S∫SdxdyV (x, y, z), (4.1)where S is area of the lateral unit cell. It is also useful to define the macro-scopic average potential, V¯ (z) [46]:V¯ (z) = 1az∫ z+az/2z−az/2dz′V¯ (z′). (4.2)Here, az refers to the lattice constant of LAO in the z-direction. In otherwords, V¯ (z) averages out oscillations within one unit cell. Because of re-laxation, especially in the uncompensated film, a constant az is actuallyill-defined. For convenience, we take as az the relaxed thickness of LAOdivided by the number of unit cells. One can also generalize the notionof the macroscopic average to account for the interface with STO [46], butsince we are only interested in the potential buildup across LAO, we use thedefinition in Eq. 4.2.In Fig. 4.2b, we plot V¯ (z) and V¯ (z) within the LAO region for the systemwithout adsorbed K. The same is shown for the system with adsorbed K inFig. 4.2c. No potential buildup is observed across LAO in the system withK, whereas a potential buildup of ∼ 2 eV is present in the system without K.This is evident looking at the overall slopes in the macroscopic potentials,as well as the positions of the minima in the planar-average potentials.In an effort to explain why no residual potential is actually observedexperimentally, Ref. [47] simulated the placement of metallic overlayers on214.3. Cohesive energiestop of LAO/STO after growth; this has subsequently been done with Co[48]. The main conclusion was that the presence of metallic contacts used formeasurements explains why residual potential is absent in experiment. Onemight extend these results of Ref. [47] as another way to remove the potentialbuildup. However, in this case the top layer is metallic and separating itsconductivity from that of the q-2DEG is difficult. This is not an issue forour proposal, where the K surfactant layer is insulating after donating itselectrons to the interface. Moreover, since the metal capping is done aftergrowth, the tendency for defects and vacancies to form during growth is notmitigated. In contrast, as we argue below, the crystal grows underneath theK surfactant layer free of such defects.It has been shown that in the absence of oxygen vacancies or otherdefects, the onset of electronic reconstruction is delayed through formation ofpolar distortions within the LaO layers, which create internal compensatingdipoles that screen the electric potential. Below a critical thickness of 4-5unit LAO layers, this suffices to partially compensate the polar potential.In thicker films with larger potential buildup, the compensation requireselectronic reconstruction, which in turn removes the need for these polardistortions. It is thus worthwhile to consider whether a critical thickness,for similar reasons, exists for the transfer of the K 4s electron to the Ti 3dbands.Fig. 4.3 shows DOS for 1 ≤ m ≤ 6, with and without adsorbed K.The critical thickness at which the system without K becomes conducting isfound to be of 4 LAO unit layers (Fig. 4.3(d)). In contrast, the system withadsorbed K is conducting for all thicknesses: even at one unit layer of LAO,where the potential across the film is smallest, K gives up its 4s electron tothe Ti 3d bands and the polar distortions within LAO are eliminated.4.3 Cohesive energiesFigures 4.2 and 4.3 show that K adsorption negates the polar catastropheand stabilizes the LAO/STO heterojunction. The energetic stability can bequantified by calculating the cohesive energy Ecoh(m):Ecoh(m) = EKadsLAOmSTO4 − ELAOmSTO4 − EK , (4.3)where m is the number of unit layers of LAO, EKadsLAOmSTO4 is the totalenergy of the system with K adsorption, ELAOmSTO4 is the total energy ofthe system without K adsorption (using the same size of lateral unit cell),and EK is the energy per atom of metallic K (body-centered cubic). Thus224.3. Cohesive energies0246PDOS[states/eVcell]K (x 8)(a)AlOLaOAlOLaOAlOLaOTiOSrO-6 -4 -2 0 2 4Energy [eV]TiO15 20 25-16-808Potential[V]Planar averageMacroscopic average(b)15 20 25Planar averageMacroscopic average(c)K KAlO2AlO2AlO2LaOLaOLaOSrOSrOSrOSrOTiO2TiO2TiO2TiO2A3A2A1L3L2L1T1S1KKA3L3A2L2A1L1T2T2T1S1L1 A1 L1 A1 L2 A2L2 A2 L3 A3 L3 A315 20 25 15 20 25Position along (001), z[Å] Position along (001), z[Å]PDOS [states/eV cell]Energy [e ]Potential [V]Position along (00 [Å] P sition along (001), z[Figure 4.2: (a) Layer and element projected DOS for Kads(LAO)3(STO)4.The Fermi energy is at 0. The scale is the same for all panels. Upon ad-sorption of K, the TiO2 layers become conducting. The conduction electrondensity is greatest for the TiO2 layer closest to the interface. The xy-planar-average and macroscopic average electric potentials are plotted versus thez-position along the (001) direction of the supercell, within the LAO region,for the system (b) without and (c) with K. A potential buildup of the or-der of 2 eV exists for the bare system without K. This is eliminated in thesystem with the K surfactant.234.3. Cohesive energies0102030DOS [states/eV cell](a) m = 1(b) m = 2(c) m = 3(d) m = 4(e) m = 5-2 -1 0 1 2Energy [eV](f) m = 6Figure 4.3: Total DOS for (LAO)m(STO)4, m = 1 − 6, with (black line)and without (shaded gray) K adsorption. The Fermi energy is at 0. Thescale is the same for all panels. The critical thickness at which the systemwithout K undergoes electronic reconstruction and becomes conducting is 4unit layers of LAO. In contrast, the system with K is conducting at all LAOthicknesses simulated.Ecoh is the cohesive energy per adsorbed K with respect to metallic K. It isplotted as a function of m in Fig. 4.4.Two observations can be made: (1) The K-adsorbed system becomesmore stable relative to the original system as the number of LAO unit layersis increased. This is a trivial consequence of the potential buildup acrossLAO increasing with m—the greater the potential buildup, the greater the244.4. Potassium coverage beyond 1/2 per unit cellenergy reduction when it is eliminated by electron transfer from adsorbedK; (2) For the thicknesses considered, |Ecoh| ranges from 1 eV to over 2 eV,a very substantial energetic stabilization.1 2 3 4 5 6Number of LAO unit layers-2.2-2-1.8-1.6-1.4-1.2-1Ecoh[eV]Number of LAO unit layersEcoh [eV]Figure 4.4: Cohesive energy per adsorbed K, defined in Eq. (4.3), as afunction of the thickness of LAO. The system with K becomes more stableas the thickness increases. The cohesive energy is approximately 1 eV for1 unit layer of LAO, increasing to more than 2 eV as LAO thickness isincreased.4.4 Potassium coverage beyond 1/2 per unit cellWe can also consider what happens if more potassium is adsorbed than theminimum of 1/2 per unit cell needed for compensation of the polar problem.In Figs. 4.5, we plot the K projected densities of states (PDOS) near EF atthree values of K coverage: 1/2 per unit cell, as well as 3/4 and 1 per unitcell. The thickness of the LAO layer is m = 2.For the latter two coverages, Fig. 4.5 shows that although the K over-layer now has sufficient electrons to transfer more than 1/2 electron per unit254.4. Potassium coverage beyond 1/2 per unit cell-1 -0.5 0 0.5Energy [eV]00.10.20.30.4PDOS [states/eV cell]1/2 K per unit cell3/4 K per unit cell1 K per unit cellFigure 4.5: K PDOS near the Fermi energy, at three cases of K coverage,for LAO thickness of two unit layers. The density of conduction electronsretained by K increases significantly going from 1/2 to 3/4 per unit cellcoverage, and then from 3/4 to 1 per unit cell coverage. Hence, K adsorptionbeyond the critical amount needed for the ideal exact compensation does notresult in extra electrons being transferred to the interface; the extra electronsremain in the K overlayer.cell to the interface, the extra electrons are retained in the K overlayer forthe higher coverages beyond 1/2 K per unit cell, as expected. This indi-cates that K donates just enough electrons to compensate for the divergingpotential. Anything beyond that would result in energetically expensiveovercompensation.The energetic favorability for further K adsorption can be quantified bycalculating the additional cohesive energy upon adsorption of extra K atoms.In particular, we take as a reference 4 lateral unit cells of LAO/STO with2 adsorbed K atoms (1/2 K per unit cell coverage), whose total energy wedenote by EK2[LAO2STO4]4 . We can now further adsorb 1 or 2 more K atomsonto this system to attain 3/4 and 1 per unit cell K coverage. We denotethe total energies of these systems by EK3[LAO2STO4]4 and EK4[LAO2STO4]4 ,respectively. The extra cohesive energies upon adsorption of each additionalK can then be defined as:Ecoh,3/4 = EK3[LAO2STO4]4 − EK2[LAO2STO4]4 − EK , (4.4)264.5. Undesired substitutions of K into LAOEcoh,4/4 = EK4[LAO2STO4]4 − EK3[LAO2STO4]4 − EK . (4.5)We find that |Ecoh,3/4| = 0.39 eV and |Ecoh,4/4| = 0.56 eV. These areto be contrasted with the cohesive energy per adsorbed K for the first twoadsorbed K atoms, which was calculated to be 1.2 eV (Fig. 4.4). The co-hesive energy as defined is approximately 0.7-0.8 eV smaller for K atomsadsorbed beyond the critical 1/2 per unit cell coverage, i.e. it is much lessenergetically favorable for extra K atoms to be adsorbed onto the system.This observation is critical for preventing any undesired accumulation of Konto the surface. In particular, the large cohesive energy difference betweenK atoms adsorbed up to 1/2 coverage and K atoms adsorbed beyond 1/2coverage suggests that if the substrate temperature is kept higher than somecritical threshold, then any extra K will evaporate off. This would preventa thick layer of K metal from forming on the surface and hence fundamen-tally changing the system—the surfactant density will stabilize at exactlythe density needed to compensate for the diverging potential.4.5 Undesired substitutions of K into LAOOur proposal to use a K surfactant to stabilize the epitaxial growth ofLAO/STO(001) hinges on K remaining on the AlO2 surface instead of enter-ing into LAO by substitution of La or Al ions. Substitution is highly unlikelygiven the large differences in ionic radii between K+ (151 pm), La3+ (116pm) and Al3+ (54 pm) [49]. We confirm this by calculating the total energychange if K exchanges position with an La or Al ion close to the surface, ina m = 5 film (results are expected to be representative for all m).Consider first the exchange of K with a La ion in the top LaO layer.The energy cost per substituted K is 1.93 eV if K exchanges positions withthe La beneath it, and 2.27 eV for the other La (cf. Fig. 4.1). In a strikingdisplay of how energetically unfavorable is the substitution of K for Al, ourrelaxation of the substituted structure resulted in the K pushing its way backabove the rest of the structure. Substitution into deeper layers of LaO/AlO2is expected to be just as, if not more, energetically costly. This proves thatthe energy cost for K substitution of La/Al within the bulk of LAO is indeedvery large.27Chapter 5Summary and ConclusionsTo summarize, we have shown that K adsorbed on the AlO2 surface of ann-type LAO/STO(001) heterojunction compensates the diverging electricpotential across LAO by donating its 4s electron to the Ti 3d conductionband. The compensation is highly efficient, with very little residual potentialbuildup across LAO. Also, substitution of K into layers of LAO by exchangewith La or Al is demonstrated to be extremely unfavorable.Taken together, these results suggest an elegant scheme for growing cleanLAO/STO heterojunctions and polar films in general. By executing thegrowth in the presence of alkali metals with low ionization energies, thediverging potential is eliminated without appealing to the myriad of other– often uncontrollable – compensation mechanisms. On a practical level,this requires only a small surface concentration of K, of ∼ 1014/cm2. Thecohesive energy for K adsorbed beyond the 1/2 per unit cell (ideal) concen-tration is significantly smaller than for the ideal concentration. Hence, ifthe substrate temperature is sufficiently high, any extra K will evaporate,preventing the formation of an undesired thick metallic overlayer of K onthe surface. The large size of K also prevents the alkali metal cations frombeing incorporated into the film during growth, guaranteeing their role assurfactants. In constrast, smaller 3d transition metal ions (TM) would notbe suitable because most of them form LaTMO3 perovskite structures withsimilar lattice constants as LAO so it is reasonable to expect that they wouldsubstitute for Al during growth.In principle, it is possible that K may undergo oxidation with the result-ing oxide accumulating on the surface, impeding further growth. Whetherthis really happens can only be ascertained through experiment. If it does, apotential solution is to grow the first two or three layers without K, allowingthe internal buckling to compensate for the potential buildup [25]. The O,La, and Al sources are then shut off, after which K is absorbed onto theexposed AlO2 surface. This will remove the buckling since the K will takeover in compensating for the polar problem. The K source is then turnedoff and the growth of the LAO is restarted, with the film growing cleanlyunderneath the layer of K surfactant.28Chapter 5. Summary and ConclusionsMost importantly, our proposal can become a new paradigm for growingLAO/STO(001) samples that no longer have properties highly sensitive togrowth conditions. The problem of electronic and structural properties be-ing attributed to different defects in samples prepared differently has beenidentified as a major potential pitfall for the development of applicationsbased on the LAO/STO interface [9]. 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Thesis/Dissertation
2015-09
10.14288/1.0166534
eng
Physics
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University of British Columbia
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Large electropositive cations as surfactants for the growth of polar epitaxial films
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http://hdl.handle.net/2429/54405