MAT-VET-F 20012 Examensarbete 15 hp Maj 2020 Formulation of Model Problem for Chirality Induced Spin Selectivity Effect Henrik Bruce Abstract Formulation of Model Problem for Chirality Induced Spin Selectivity Effect Henrik Bruce Teknisk- naturvetenskaplig fakultet UTH-enheten Spin dependent electron transport in chiral molecules, the so-called chirality induced spin selectivity (CISS) effect, have attracted much Besöksadress: attention over the past few years. Experimentally the spin polarization Ångströmlaboratoriet Lägerhyddsvägen 1 has been detected, and there is a theoretical consensus on the Hus 4, Plan 0 necessity of both spin-orbit coupling and geometrical helicity in order to get a non-vanishing spin polarization. Several model Hamiltonians Postadress: has been proposed to describe the CISSS effect, and while they can Box 536 751 21 Uppsala yield spin polarization agreeing with the experimentally observed magnitudes, they are relying on unrealistic values of the spin orbit Telefon: interaction parameters. In recent years the importance of electron 018 – 471 30 03 correlation has been emphasized. Thus, this thesis presents the Telefax: general theory on how to treat the CISS effect as a many body problem, 018 – 471 30 00 taking electron correlation into account. The Hamiltonian modelling is described and one approach on how to treat the helical structure of the Hemsida: molecule and the spin-orbit coupling is presented. Building on this http://www.teknat.uu.se/student thesis, further studies will hopefully lead to a first principle understanding of the CISS effect. Handledare: Jonas Fransson Ämnesgranskare: Chao Xu Examinator: Martin Sjödin ISSN: 1401-5757, MAT-VET-F 20012 Popul¨arvetenskaplig sammanfattning CISS, kort f¨orchirality induced spin selectivity, ¨arett fenomen som uppst˚ar n¨arelektroner transporteras genom s˚akallade kirala molekyler, t.ex. en helix. Elektroner har en inneboende egenskap, spin, som kan vara riktad antingen upp eller ner i n˚agonriktning. I vanliga fall prioriteras ingen av riktningarna ¨over den andra n¨arelektroner transporteras genom n˚agotmedium, men i fal- let med kirala molekyler har man uppm¨attatt den ena av spinnriktningarna prioriteras ¨over den andra. D˚adetta fenomen relativt nyligen uppt¨acktes har man ¨annu ingen full¨andad f¨orklaringtill varf¨ordet sker. Forskare v¨arlden¨over ¨ar¨overens om vissa kriterier, s˚asom helicitet och spinn-bankoppling, som m˚astevara n¨arvarande f¨oratt f˚aen spinn-polariserad str¨om. Ett s¨attman tror ska leda till en b¨attref¨orklaring¨argenom att modellera molekylen med en flerkroppsmodell d¨arman tar h¨ansyntill v¨axelf¨orh˚allandenmellan flera elektroner samtidigt. Detta ¨arn¨odv¨andigtd˚aden ¨omsesidigp˚averkan mellan elektroner f¨or¨andrar elektronernas m¨ojliga energiniv˚aer. Den h¨arrapporten behandlar CISS fenomenet som ett flerkroppsproblem. Den grundl¨aggandekvantmekaniska teorin f¨oratt f¨orst˚afenomenet tas upp och ett s¨attatt modellera molekylen beskrivs. Den h¨arrapporten ¨ar inte n˚agotf¨ardigtarbete i den mening att fenomenet f¨orklaras till fullo, utan ska snarare betraktas som ett inledande steg i riktningen mot en fullst¨andigteori. Contents 1 Introduction1 1.1 Background............................1 1.2 Project Description........................1 2 Theory2 2.1 Postulates of quantum mechanics................2 2.2 Spin and classification of particles................4 2.3 Identical Particles.........................4 2.4 Schr¨odingervs Heisenberg picture................5 3 Simplified model7 3.1 Hubbard operators........................7 3.2 Eigenenergies and transition energies..............7 4 Model Hamiltonian 10 4.1 Tight Binding model....................... 10 4.2 Necessary conditions for CISS.................. 11 4.3 Second quantization....................... 12 4.4 Field operators.......................... 14 5 CISS model 15 5.1 CISS Hamiltonian......................... 15 5.2 Reformulation of Hamiltonian.................. 17 6 Summary 19 Bibliography 19 1 Introduction 1.1 Background The capability of chiral molecules to spin-polarize electrons has been named chirality induced spin selectivity (CISS). When electrons are transported through chiral molecules, one spin direction is preferred above the other. Which spin direction that is preferred depends on the chirality of the molecule and on the direction of motion. There is a general understanding of what causes the spin polarization, e.g. the necessity of chirality and so-called spin orbit coupling. However, the CISS effect still lacks a first principle understanding. A chiral molecule is a molecule that can be distinguished from its mirror image. Thus, chiral molecules come in pairs. The only thing separating the two molecules from each other is the three dimensional spatial arrangement of the atoms. One example of a chiral object outside the molecular world is the human hand. One of the hands can never be made to coincide with its own mirror image with translations and rotations alone. But many organic molecules are also chiral, making the CISS effect relevant for many biophys- ical processes such as electron conduction through helical proteins. Helices can be either left handed or right handed, with the handedness being deter- mined by the right hand rule. Helices are also chiral, with the convention in this thesis being positive (negative) chirality corresponding to right (left) handed helicity. The ability of chiral molecules to spin-polarize electrons was first reported in 1999 [1]. Since then, numerous experimental studies have been conducted and many theories with model Hamiltonians have been proposed. However, there is not a theoretical consensus on how to explain the phenomenon yet. Although theories based on model Hamiltonians are able to explain the spin polarization qualitatively, they rely on unrealistic values of the spin-orbit interaction parameters. Considering the importance of electron correlations has been suggested as a way to address this problem. 1.2 Project Description This project aims to study the CISS effect as a many-body problem tak- ing electron correlation into account. One approach on how to model the molecule and the Hamiltonian describing the different interactions is pre- 1 sented. By doing this a model Hamiltonian for the CISS effect is formulated, from which the equations of motion are calculated. This thesis thus presents a first step towards what eventually is a theoretical first principle under- standing of the CISS effect. The thesis is structured as follows: (i) first, the fundamental theory of quantum physics, as relevant to this paper, is derived and described. (ii) the theory is demonstrated on a simplified Hamiltonian consisting of only two sites. It is important to note that this Hamiltonian is not relevant for the CISS phenomenon since it neither accounts for the chirality nor the spin- orbit interactions. However, it serves a purpose as an illustrative example of how to interpret the model and what it means to solve the system. This is followed up by (iii), a more rigorous description and derivation of the CISS Hamiltonian in the second quantization formalism. Lastly, (iv), the CISS Hamiltonian is expanded in terms of Hubbard operators and and the Heisenberg equation of motion is calculated. 2 Theory 2.1 Postulates of quantum mechanics In classical mechanics, the total energy of system can be determined by the p2 sum of all kinetic energies 2m plus all potential energies V (r). If these values are known for every part of the system at all times, one knows how the system will evolve in time. In quantum mechanics on the other hand, the Heisenberg uncertainty principle tells us that both the position and momentum of a particle cannot be determined with certainty at the same time. This forces us to deal with probabilities, and to not consider a particle as a geometric point, but rather as an entity spread out in space. All quantum systems are represented by a wave function, denoted , from which all measurable information about the system can be obtained. The mathematical framework of quantum mechanics is then built around the concept of operators. To get information about a system, so-called operators are used, working on the wave function. The quantum analogue of the classical way of determining the energy of a system is the Hamiltonian operator, H^ 2 H^ = − ~ r2 + V (r; t) (2.1) 2m 2 2 −~ 2 where 2m r is the kinetic energy and V (r; t) can be any potential. In the Dirac formalism of quantum mechanics, the Hamiltonian is an operator in Hilbert space, a complex vector space spanned by the eigenkets, jφi, of the Hamiltonian. The spectrum of allowed energy levels of the system is given by the set of eigenvalues, Ei, solving the equation, ^ H jφii = Ei jφii (2.2) One of the postulates of quantum mechanics is that any observable, e.g. any quantity that can be observed in a physical experiment, can be represented by a self-adjoint linear (Hermitian) operator. Hermitian operators always have real eigenvalues, in accordance with them being operators corresponding to observables. Before a measurement has been made, the system is described by its wave function which can be expanded in terms of its eigenstates, X j i = ci jφii (2.3) i An operator acting on the wave function, initially a superposition of eigen- states, causes the wave function to collapse into one of its eigenstates. Since the eigenstates are orthonormal, i.e., hφijφji = δi;j, the coefficients ci = hφij i can be found. Hence equation 2.3 can be rewritten, X X j i = hφij i jφii = jφiihφij j i (2.4) i i from which one can conclude that, X jφiihφij = 1 (2.5) i This is known as the completeness relation, implying that a wave function representing any physical system can be expressed as a linear combination of the eigenkets of any observable of the system. The completeness relation can be used to derive matrix representations of any operator. By inserting the completeness twice, for an arbitrary operator O^, the operator can be written as, ^ X ^ O = jφiihφij O jφjihφjj (2.6) i;j 3 2 ^ Altogether, there are N numbers of form hφijOjφji where N is the dimen- sionality of the ket space.