Uppsala University Bachelor of Science Degree in physics

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Uppsala University

Bachelor of Science Degree in physics Department of Physics and Astronomy

Division of Materials Theory June 11, 2015

Analysis of a spin-particle tunnelling junction

Author:

Johan Bylin

Supervisor:

Jonas Fransson

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Abstract

This project is to analyse the energy spectrum of a spin-molecular tunnelling junction which is composed of molecules conned between two conducting metallic leads. By letting a continuous stream of electrons ow across the junction the molecules can interact with each other with an indirect force called exchange interaction, and those exchange interactions which are of interest in this project are described by models called the Heisenberg, the Ising and the Dzyaloshinski-Moriya models. The molecules may also interact with themselves anisotropically and if there is an external magnetic eld there will be yet another kind of interaction.

The goal of this project is to see the contribution of all these spin interactions and how they aect the resulting energy spectrum under the variation of the junction's chemical potential and the voltage bias between the leads.

This project is of a theoretical nature where the models are analytically adapted for a restricted scenario and is later on numerically calculated to be graphed and analysed. The models are restricted to only consider molecules of same spin and approximated to only consider interactions between closest neighbouring molecules.

The results are composed of both analytically derived energy values and numerical com- puted values which show that there exists certain critical values of the variation parameters which naturally splits the ground state of the system and that the self-interaction may further split the degenerate ground state. A possible outcome of these result could be the possibility to control the magnetic order of the molecules to either be locked in an anti-ferromagnetic conguration or be easily mixed by manipulating the chemical potential or the voltage bias.

Sammanfattning

Detta projekt handlar om att analysera energispektrumet från en spinn-molekyl-tunnelkor- sning som består av molekyler instängda mellan två ledande metaller. När en kontinuerlig elektronström korsar tunnelkorsningen så kan molekylerna växelverka med varandra via en in- direkt kraft kallad utbytesinteraktion, och de utbytesinteraktioner som är relevanta i denna up- pställning beskrivs av de så kallade Heisenberg-, Ising- och Dzyaloshinski-Moriya-modellerna.

Molekylerna kan också växelverka med sig själva anisotropt och om det nns ett externt mag- netfält så tillkommer ytterligare en interaktionsterm. Målet för detta projekt är att se hur alla dessa spinnbidrag påverkar det slutliga energispektrumet under variation av korsningens kemiska potential och spänningen mellan metalledarna.

Projektet är teoretiskt lagt på så sätt att modellerna är analytiskt anpassade för ett begränsat scenario samt att de är numeriskt beräknade så att energispektrumet kan plottas i grafer och analyseras. Modellerna är begränsade för molekyler av samma spinn och är approximerade så att endast närmsta-granne-interaktioner är beaktade.

Resultaten är uppdelade i både analytiskt framtagna energivärden samt numeriskt beräk- nande energinivåer och båda visar att det nns kritiska värden på variationsparametrarna som automatiskt delar grundtillståndet för systemet samt att självinteraktionerna ytterligare kan dela det degenererade grundtillståndet. Ett möjligt utfall av dessa resultat är att de kan användas till att kontrollera systemets magnetiska ordning på så sätt att det antingen är låst i en antiferromagnetisk konguration eller med enkelhet kan mixas genom att ändra den kemiska potentialen eller spänningen mellan metalledarna.

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1 Introduction:

The research on nano-scale physics is a continuously developing eld where several dis- coveries have led to a better understanding of the applicability of individual atoms and molecules; for example, for molecular magnetic data storage^[1^]. Another eld of study is the theoretical aspects of atomic magnetic interactions where new models are developed to better describe the quantum mechanical eects of these nano-sized interactions.

One of the possible research elds is the study of tunnelling junctions which is usually constructed with two conducting metals separated by some potential gap where electrons may tunnel across. One way to further exploit this basic idea is to introduce molecules within its metallic leads. By letting a continuous stream of electrons ow through the junction one can model an indirect force, an exchange interaction, between the molecules which will result in molecular spin-coupled states aecting the energy level of the junction^[2^]. This project will be to analyse the energy spectrum of such tunnelling junction using a combined Hamiltonian of the form

H = H_S+ Hint

The rst term is an eective spin-coupling Hamiltonian introduced in the paper from Ref. [2]. This Hamiltonian has the form

H_S = X

m6=n

S_m·

J_mnS_n+ I_mn· S_n+ D_mn× S_n

and describes eective spin interactions in terms of so called Heisenberg, Ising and Dzyaloshinski- Moriya models respectively.

The other term is a spin Hamiltonian found in the Ref. [3]. This Hamiltonian describes the anisotropic behaviour of localised magnetic molecules and has the form

H_int=X

m

gµ0B · Sm+ C(S_m^z)²+ E[(S_m^x)²− (S_m^y)²]

where the rst part is a Zeeman term and the other two describes the anisotropy of the molecule.

This project will focus on analysing eigenvalues of the above spin terms and construct the quantized energy levels for the combined Hamiltonian. This will make it possible to see what impact the anisotropic terms, Heisenberg, Ising and Dzyaloshinski-Moriya interactions have on the energy levels for dierent critical values of, for example, the system chemical potential and the junction voltage bias.

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Objective: To construct and set up dierent spin states for various number of molecules to attain the eigenvalues of the Hamiltonians for half spin molecules as well for higher spin molecules.

Goal: To graph the energy levels of the spin states and see how these levels are aected under the variation of certain parameters such as the chemical potential, the spin polarisation of the metallic leads and voltage bias.

2 Background

In order to describe the magnetic interactions of particles one have to use models because there is no closed unied theory of quantum magnetism^[4^]. By developing accurate models for dierent systems a more gathered description of atomic interactions can lead the way to advances in technology, such as particle logic gates^[5^][6^]and molecular data storage devices

[1^].

A crucial concept of quantum magnetism is the theory of exchange interactions which aect intermediate particles as a consequence of the Pauli's principle which states that identical fermions cannot occupy the same quantum state. The result is that when the wave functions of the fermions overlap they experience a repulsive coulomb force which in a particle system can be modelled as the exchange interactions. There exists dierent ways that the exchange can be mediated, either directly by strong coupling of the intermediate particles or indirectly by coupling to delocalized particles which can mediate the exchange.

The symmetry of the exchange interactions can also be of dierent nature where some may exhibit isotropic behaviour and some may express anisotropic behaviour^[4^].

An ordinary linear tunnelling junction is composed of two metallic leads of high conductiv- ity separated by some potential barrier, such as vacuum, an isolator or a semi-conductor.

By means of quantum mechanics there is a probability for particles, such as electrons, to tunnel across from one side to another even though their kinetic energies are lower than the potential. This idea can be further expanded to include particle islands within the junction, and these particles could be for example molecules and such a set-up can look like the one in Figure 1. The interactions of these molecules may aect the potential energy of the junction in such a way that the tunnelling probability of the electrons is increased or decreased^[7^][2^].

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Figure 1: This is a representation of the molecular tunnelling junction where the molecules may interact with the stream of delocalized electrons and give rise to exchange and self interactions. The arrows on the leads represent the leads' spin polarisation and µ is the chemical potential of the junction.

A model that is proposed to describe the exchange interactions of this junction comes from the paper in Ref. [2] that deals with the interactions between the magnetic molecules.

Another eect that is of importance is the self interactions of the individual molecules which might be from an applied magnetic eld or of pure anisotropic behaviour.

2.1 The exchange interactions of the spin-particle tunnelling junction The eective exchange interactions that the molecules within the junction may aect each other with is proposed to be of the form

H_S = X

m6=n

sm·

Jmnsn+ Imn· s_n+ Dmn× s_n

where smis the spin of molecule m, Jmnis an Heisenberg integral, Imn= I_mnz ˆˆz is an Ising integral and Dmn = D_mnzˆis a Dzyloshinski-Moriya (DM) integral^[2^]. These exchanges are mediated between the molecules by an electron ow across the leads of the junction. This enables the molecules to couple to the delocalized electron current and in that manner indirectly mediate the exchange, and this interaction is of RudermannKittelKasuyaYosida (RKKY) nature^[4^]. It is also proposed that the net spin polarisation of the leads play a role in how the delocalized electrons tunnel from lead to molecule and vice versa. This eect creates a coupling to the lead where the polarisation determines its strength, and this coupling aect the Heisenberg and Ising integral such that they scale linearly with its strength while the Dzyloshinski-Moriya scale quadratically^[2^].

The three terms of the model are previously known models themselves and they are:

the Heisenberg model, the anisotropic Ising model and antisymmetric Dzyloshinski-Moriya model.

The Heisenberg model, P

m6=n

Jmnsm· s_n, was developed to describe spontaneous magnetisation in terms of localised magnetic moments. The idea was that there would be a dierence

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in potential energy depending on whether the spin of coupled identical particles were parallel or antiparallel due to the repulsive nature of the Pauli's principle. By assigning this shift in potential energy as an exchange interaction parameter one could describe how the conguration of magnetic moments in solids would spontaneously arrange themselves and give rise to magnetisation^[4^].

The Ising model, P

m6=n

sm· I_mn· s_n, describes the axial anisotropy of the exchange interactions and is one of the few many-particle models which has exact solutions. This model is extensively used in statistical mechanics to describe phase transitions due to the fact that only the z-component of the spin is taken in account which makes the number of state congurations manageable.^[4^]

The Dzyloshinski-Moriya model, P

m6=n

s_m· D_mn× s_n, was proposed for the anti-symmetrical exchange interactions which could be found in weak ferromagnetism. The Dzyloshinski- Moriya interaction would manifest itself in systems of low symmetry and aect neighbouring particles anisotropically^[8^].

It is theoretically proposed that these three models can describe the eective exchange interactions that occurs within a spin-molecular tunnelling junction by covering both the symmetrical and anti-symmetrical contributions of the exchange. But the models do not take in account the eect of external magnetic elds or if the molecules interact with themselves. These contributions are discussed in the following section.

2.2 The anisotropic splitting of single molecules and the Zeeman eect Spin interactions of a quadratic form is a good starting point in modelling quantum magnetic phenomena and a way to write this, without loosing any sense of generality, is as a tensor product of the form

H = S_i· A_ij· S_j

where A is a symmetric tensor. The o-diagonal terms may describe the exchange interactions between particles and can be constructed in the form of the Heisenberg, Ising and Dzyloshinski-Moriya models. But the self-interaction contributions takes its form from the diagonal terms which can be expanded into

H = A_xxS_x²+ A_yyS_y²+ A_zzS_z²

This general expression can be manipulated to represent a physical phenomena by, for example, subtraction or addition of constants which do not change the property of the

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Hamiltonian. By subtracting the constant ¹₂(Axx+ Ayy)(s_x²+ s²_y+ s²_z) and rearranging the terms one may construct a Hamiltonian of the form

H = CS_z²+ E(S_x²− S_y²)

where C = Azz−¹₂(Axx+ Ayy) and E = ¹₂(Axx− A_yy)^[9^]. This Hamiltonian now describe the magnetic anisotropy of atoms and molecules where C governs the axial splitting of the spin-z axis and E mixes the transversal states in terms of spin projection quantum numbers m ^[3^]. Because these terms splits the degeneracy of states without the presence of a magnetic eld the phenomena is also called zero-eld splitting and it is common to restrict the transversal terms as |^E_C| < ¹₃ in order to treat E as a perturbation ^[9^].

If an external magnetic eld is is applied there will be another splitting of degeneracy which is called the Zeeman eect and the Hamiltonian that describes this phenomena has the form:

H = gµ₀B · S

where g is called the g-factor and µ0 is the Bohr magneton ^[3^]. 2.3 The theory of spin for magnetic molecules

The spin of magnetic molecules are dened by partially lled electron shells where the electron's spin can add up to the total spin s = ¹₂, 1,³₂, ... depending on the number of valence electrons. Each of the spin values will have a spin projection quantum number m as m ∈ {−s, −s + 1, ...s − 1, s} meaning that the molecular state can be dened as the ket

|s, mi^[4^].

When there is a system of N molecules their combined spin can be dened by adding the constituents spin values as

min|s₁± s₂± ... ± s_N| ≤ S ≤ (s₁+ s₂+ ... + s_N)

where si is the spin of molecule i. The system's total spin S may assume values in integer steps between the highest and lowest weighted values and just as for the individual molecules the spin projection quantum number for the system will be dened by M ∈ {−S, −S + 1, ..., S − 1, S}. To describe the spin state of such a system in terms of the molecular states one have to dene a linear combination such as

|S, M i = X

m1+m2+...+mn=M

C_m^s¹₁^s_m²^...s₂_...m^N _N|s₁, m₁i|s₂, m₂i...|s_N, m_Ni

where C_m^s¹₁^s_m²^...s₂_...m^N _N are Clebsch-Gordan coecients under the condition that hSi, M_i|S_j, M_ji = δij.

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Because these states represent orthogonal bases of a Hilbert space one may represent ar- bitrary operators, ˆO, as matrices where the matrix elements describes how the molecular states transform. The elements are dened as

O_ij = he_i| ˆO|e_ji

where |eii and |eji are two bases within the Hilbert space ^[7^]. With this notation one can construct the eigenvalues of the molecular system by diagonalising the matrix for certain operators such as the Hamiltonian.

3 Theory

The assumptions and simplications that is made in this project is:

• The individual spins are independent and commute [sn, s_m] = 0

• The molecules are identical and have the same spin

• Only nearest neighbour interactions are taken in account

• Magnetic eld only in ˆz-direction

• The exchange integrals have the same value for every pair (|Jij| = J_1,2 , |Iij| = I_1,2 ,

|D_ij| = D_1,2)

With this in consideration one can subdivide the Hamiltonian into smaller components such that

H = H_S+ Hint= HHeis+ HIsing+ HDM + Hint

and these terms are hereafter individually processed with the statements above.

3.1 Heisenberg contribution H_Heis= X

m6=n

s_m· J_mns_n= X

m6=n

J_mns_m· s_n

Because snand smcommute and only nearest neighbour interactions Jm,m±1are considered the Heisenberg model becomes

H_Heis =X

m

Jm,m+1sm· s_m+1+ Jm,m−1sm· s_m−1

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And as the exchange integral is symmetric Jm,m+1= Jm+1,mthe Hamiltonian will look like H_Heis = 2X

m

Jm,m+1sm· s_m+1

If the Heisenberg exchange has the same value for each pair of molecules, i.e Jm,m+1= J_1,2, then the nal form of the model becomes

H_Heis = 2J_1,2X

m

s_m· s_m+1 = 2J_1,2X

m

s^x_ms^x_m+1+ s^y_ms^y_m+1+ s^z_ms^z_m+1

This can be rewritten with the formulas s^x = ¹₂(s⁺+ s⁻) and s^y = _2i¹(s⁺− s⁻) into^[4^] H_Heis= 2J1,2

X

m

1

2 s⁺_ms⁻_m+1+ s_m⁻s⁺_m+1 + s^z_ms^z_m+1 3.2 Ising contribution

H_Ising = X

m6=n

s_m·

I_mn· s_n

The Ising term has the form Imn= Imnz ˆˆz which leads to H_Ising = X

m6=n

Imn sm· ˆz ˆ

z · sn = X

m6=n

Imns^z_ms^z_n

As in the Heisenberg contribution the Ising exchange integral has symmetric matrix elements and has the same value for each pair of molecules. This leads to the expression

H_Ising = 2I_1,2X

m

s^z_ms^z_m+1

3.3 Dzyaloshinski-Moriya contribution H_DM = X

m6=n

sm·

Dmn× s_n

The Dzyaloshinski-Moriya has the form Dmn= D_mnzˆwhich leads to H_DM = X

m6=n

Dmnsm· ˆz × sn = X

m6=n

Dmn s^y_ms^x_n− s^x_ms^y_n

Because snand smcommute and only nearest neighbour interactions Dm,m±1are considered the model becomes

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H_DM =X

m

D_m,m+1 s^y_ms^x_m+1− s^x_ms^y_m+1 + D_m+1,m s^y_m+1s^x_m− s^x_m+1s^y_m

=⇒

H_DM =X

m

D_m+1,m− D_m,m+1s^x_ms^y_m+1+ D_m,m+1− D_m+1,ms^y_ms^x_m+1,m

Because the Dzyaloshinski-Moriya integral is anti-symmetric the matrix elements should be of the form Dm,m+1 = −D_m+1,m. This gives with the assumption that the elements have equal magnitude |Dm,m+1| = |D_1,2|

H_DM = 2D_1,2X

m

s^y_ms^x_m+1− s^x_ms^y_m+1

With the relations s^x = ¹₂(s⁺+ s⁻) and s^y = _2i¹(s⁺− s⁻) the Dzyaloshinski-Moriya term becomes

H_DM = iD12

X

m

s⁻_ms⁺_m+1− s⁺_ms⁻_m+1

3.4 Anisotropic contribution with Zeeman term H_int=X

m

gµ0B · sm+ C(s^z_m)²+ E[(s^x_m)²− (s^y_m)²]

With the assignment B = B0zˆand with the formulas s^x = ¹₂(s⁺+ s⁻)and s^y = _2i¹(s⁺− s⁻) the Hintmay be rewritten into

H_int=X

m

gµ₀B₀s^z_m+ C(s^z_m)²+ E

2[(s⁺_m)²+ (s⁻_m)²] 3.5 Final form of the Hamiltonian

By combining the eective exchange terms with the self interaction terms one will have the Hamiltonian

H = H_S+ Hint=

=

N −1

X

m=1

J_1,2+ iD_1,2s⁻_ms⁺_m+1+ J_1,2− iD_1,2s⁺_ms⁻_m+1+ 2 J_1,2+ I_1,2s^z_ms^z_m+1+

+

N

X

n=1

gµ₀B₀s^z_n+ C(s^z_n)²+E

2[(s⁺_n)²+ (s⁻_n)²]

What is of particular interest here is that the term (J1,2− iD_1,2)is the complex conjugate of (J1,2+ iD_1,2) and this turns out to be crucial in order to get real eigenvalues out of this Hamiltonian, which can be seen in the section Results.

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3.6 Constructing the states

Because the spins commute one can construct a simultaneous eigenket for N molecules as |S, M; s1, m1; s2, m2; ...; sN, mNi which will be compressed to |m1, m2, ..., mNi due to convenience and because the molecular spins are of the same value (si = s). The number of independent states η spanning the Hilbert space of this system will be given by η = (2s + 1)^N, which in turn is the number of permutations of all the possible values of m₁, m₂, ..., m_N. Each set of the permutation is an orthogonal base in the Hilbert space such that the bases forms the subset {|1i, |2i, ..., |ηi} where |ii = |m1, m₂, ..., m_Niis a base dened by one of the possible permutations. Each base is uniquely dened by the order and magnitude of the possible projection quantum numbers mi and with this one can set up a basis for the molecular system of N = 1, 2, 3, ... molecules each with spin s = ¹₂, 1,³₂, ...

For the half spin molecules, s = ¹₂ =⇒ m_i = ¹₂, −¹₂ and these will be denoted as m = ¹₂ ≡↑

and m = ¹₂ ≡↓for the spin half case. For N = 2 there will be (2 ·¹₂+ 1)² = 4states dening the system. These are shown below to the left.

For spin 1 molecules, s = 1 =⇒ mi = −1, 0, 1and for N = 2 there will be (2 · 1 + 1)² = 9 states and these are shown below to the right.







| ↑, ↑i

| ↑, ↓i

| ↓, ↑i

| ↓, ↓i







≡







|1i

|2i

|3i

|4i













|1, 1i

|1, 0i

|1, −1i

|0, 1i

| − 1, 1i

|0, 0i

|0, −1i

| − 1, 0i

| − 1, −1i







≡







|1i

|2i

|3i

|4i

|5i

|6i

|7i

|8i

|9i







Similar state constructions are done for higher spin molecules and as well for greater number of molecules.

3.7 Constructing the operator matrices of the Hamiltonian

The operators which are relevant to the Hamiltonian are the spin ladder operators s⁺ , s⁻ and the z-component of the spin s^z. The spin ladder operators have the property that

s^±_i |m₁, m₂, ..., m_i, ..., m_Ni =p

s(s + 1) − m_i(m_i± 1)|m₁, m₂, ..., m_i± 1, ..., m_Ni where s^±_i is the ladder operator acting on molecule i. The ladder operators also have the property that there exist certain maximum and minimum values of mi such that the state is terminated. In other words:

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s⁺_i |m₁, m2, ..., m^max_i , ..., mNi = 0, and s⁻_i |m₁, m2, ..., m^min_i , ..., mNi = 0. The z-component of the spin has the eigenvalue property that

s^z_i|m₁, m₂, ..., m_i, ..., m_Ni = m_i|m₁, m₂, ..., m_i, ..., m_Ni and is terminated only if mi= 0.^[7^]

The matrix elements of the ladder operators will be given as (s^±_i )_j,h= hj|s^±_i |hi = hj|p

s(s + 1) − m_i(m_i± 1)|m₁, ..., m_i± 1, ..., m_Ni = ps(s + 1) − m_i(m_i± 1)hj|ki =p

s(s + 1) − m_i(m_i± 1)δ_j,k where |ki = s^±_i |hi = |m₁, ...m_i± 1, ..., m_Ni. For s^z_i the matrix elements will be

(s^z_i)_j,j= hj|s^z_i|ji = m_i

With this one can dene the matrices of the operators s⁺ , s⁻ and s^z acting on the molecular system and by substituting them into the Hamiltonian seen in section 3.5 one will get the matrix dening the spin Hamiltonian. In order to get the energy eigenstates the Hamil- tonian matrix must be diagonalized. This will make the diagonal terms the eigenstates and thus also the energy levels of the system.

3.8 An analytical derivation, two spin-half molecules For the two molecule case with s = ¹₂ the Hamiltonian will have the form

H = J_1,2+ iD1,2s⁻₁s⁺₂ + J1,2− iD_1,2s⁺₁s⁻₂ + 2 J1,2+ I1,2s^z₁s^z₂+ gµ₀B₀(s^z₁+ s^z₂) + C[(s^z₁)²+ (s^z₂)²] +E

2[(s⁺₁)²+ (s⁺₂)²+ (s⁻₁)²+ (s⁻₂)²]

The operators s⁻1, s⁻₂, s⁺₁, s⁺₂, s^z₁ and s^z2 will be 4 × 4 matrices and may be constructed with the formulas described in section 3.7 such that they uniquely denes the operations on the basis shown in section 3.6. An example is the construction of the s⁻₁ matrix.

The matrix elements for s⁻₁ will given by (s⁻₁)j,h= hj|s⁻₁|hi =p

s(s + 1) − mi(mi− 1)hj|m₁− 1, m₂i

Here if m1 = m^min₁ = ↓ the state will be terminated but for m1 = m^max₁ = ↑ the state will have non-zero eigenvalues, and these non-zero states are |1i and |2i. This means that h = 1, 2and these will both have eigenvalue q

1

2(¹₂ + 1) − ¹₂(¹₂ − 1) = 1. This will lead to (s⁻₁)_j,h= hj| ↑ −1, m2i = hj| ↓, m₂i

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In order to full this one needs to nd the states for |ji and because m2 can both be m2 = ↑, ↓one will see that |ji could be both |ji = | ↓, ↑i = |3i and |ji = | ↓, ↓i = |4i. This will lead to the non-zero matrix elements being

(s⁻₁)j,h=

((s⁻₁)_1,3 = 1

(s⁻₁)1,4 = 1 =⇒ s⁻₁ =







0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0







By creating matrices for all the operators in similar fashion and substituting them into the Hamiltonian for this set-up one will arrive after some matrix multiplication and addition to the Hamiltonian matrix

H =







H_1,1 0 0 0

0 H_2,2 H_2,3 0 0 H_3,2 H_3,3 0

0 0 0 H_4,4

















H_1,1 = ^J^1,2^+I₂^1,2^+C + gµ0B0

H_2,2 = H_3,3 = ^C−J^1,2₂^−I^1,2 H_2,3 = J1,2− iD_1,2 H_3,2 = J_1,2+ iD_1,2

H_4,4 = ^J^1,2^+I₂^1,2^+C − gµ₀B0

This matrix can be diagonalized in an analytical fashion and the eigenvalues are presented in the section Results 4.1.

3.9 The execution of the analysis

It is also possible to analytically create the Hamiltonian for the three molecular case (s = ¹₂) and diagonalise it, and its eigenvalues are shown in section Results 4.1 and its matrix is shown in Appendix 6.1. But for the case when s = 1 or above the matrices becomes too tedious and cannot analytically be diagonalised (matrix for the two molecular case s = 1 is shown in Appendix 6.2 ). Here, the best way to tackle the problem is to implement a numerical program which calculates the operator matrices and diagonalises the resulting Hamiltonian. With this one can study more intricate systems and see how the energy levels dier from case to case.

The following results are realised by such a program which calculates the operator matrices and diagonalizes the resulting Hamiltonian. The parameter values of J1,2, I1,2 and D1,2that are used in these results come from the research done in Ref. [2]. The two variation cases that is considered from this research are:

• Variation of J1,2, I_1,2 and D1,2 with respect to the system's chemical potential µ ∈ [−2, 2]meV when temperature T = 1K and voltage bias V = 0.1mV are xed.

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• Variation of J1,2, I1,2and D1,2with respect to the junction's voltage bias V ∈ [−10, 10]mV when temperature T = 1K and µ = −2meV are xed.

Both these cases are also varied with respect to the leads polarisation such that the left lead polarisation PL either have values PL = 0 or PL = 0.8. These can be paired up with dierent values of the right lead polarisation PR, and the cases studied are when (PL= 0, PR = 0)and (PL = 0.8, PR = {−0.8, −0.4, 0, 0.4, 0.8}). For both these variations the Heisenberg integral is nite for all lead polarisations while the Ising and Dzyaloshinski- Moriya integrals are nite only under polarisation and the Ising integral is non-existing for PL = 0.8, PR= −0.8 and the Dzyaloshinski-Moriya is non-existing for equal polarisations P_L= 0.8, P_R= 0.8 ^[2^].

The anisotropic and Zeeman terms are assumed to be constants with respect to variations of both the chemical potential and voltage bias. These terms are varied with no denite values but are restricted such that |gµ0B₀|and |C| are of the order of a few meV . The main restrictions that are made are taken from Ref. [3] where the axial parameter C is assumed to have both positive and negative values and the transversal parameter E are set to be E ≈ ^|C|₅ , which is within the restriction stated in section 2.2.

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4 Results

4.1 Analytical results, spin half 4.1.1 Two molecules







E₁ = ^J^1,2^+I₂^1,2^+C + gµ₀B₀ E₂ = ^J^1,2^+I₂^1,2^+C − gµ₀B₀ E3,4 = ^C−J^1,2₂^−I^1,2 ±q

J_1,2² + D_1,2²

These are the analytical eigenvalues of the two molecular case when s = ¹₂. 4.1.2 Three molecules











E1= J1,2+ I1,2+^6gµ⁰^B₄⁰^+3C E₂= J_1,2+ I_1,2+^3C−6gµ₄ ⁰^B⁰ E₃= ^2gµ⁰^B₄⁰^+3C

E4= ^3C−2gµ₄ ⁰^B⁰

E5,6= ^3C−2gµ⁰^B⁰^−2(J₄ ^1,2^+I^1,2⁾ ±

q J1,2+I1,2

2

+ 2(J_1,2² + D²_1,2) E7,8= ^3C+2gµ⁰^B⁰^−2(J₄ ^1,2^+I^1,2⁾ ±

q J1,2+I1,2

2

+ 2(J_1,2² + D²_1,2) These are the analytical eigenvalues of the three molecular case when s = ¹₂.

The graphs for the analytical eigenvalues are the same as the numerical ones for all variation cases studied here and can be seen in the next section 4.2.

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4.2 Numerical results

4.2.1 Variation with respect to chemical potential

(a)Energy spectrum under the variation of µ as s = ¹₂, N = 2, gµ0B0 = 0 meV and C = E = 0 meV with no lead polarisation PL= 0and PR= 0.

(b) Energy spectrum under the variation of µ as s = ¹₂, N = 2, gµ₀B₀ = 1 meV and C = 1 meV , E = ¹₅ meV with no lead polarisation PL= 0and P_R= 0.

Figure 2: Graph (a) has three fold degenerate ground state within µ ∼ (−1, 1) while graph (b) with applied magnetic eld splits that degeneracy while the parameter C raises the overall energy spectrum.

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(c) Energy spectrum under the variation of µ as s = ¹₂, N = 2, gµ0B0 = 0 meV and C = 0 meV , E = 0 meV with lead polarisation PL= 0.8 and P_R= −0.8.

(d) Energy spectrum under the variation of µ as s = ¹₂, N = 2, gµ0B0 = 1 meV and C = 1 meV , E = ¹₅ meV with lead polarisation PL = 0.8 and PR= −0.8.

Figure 3: Graph (c) has threefold degenerate ground state within µ ∼ (−1, 1) while graph (d) with applied magnetic eld splits the degeneracy while the parameter C raises the overall energy spectrum.

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(e) Energy spectrum under the variation of µ as s = ¹₂, N = 3, gµ0B0 = 0 meV and C = 0 meV , E = 0 meV with lead polarisation PL= 0.8 and P_R= 0.

(f )Energy spectrum under the variation of µ as s = ¹₂, N = 3, gµ0B0 = 1 meV and C = 0 meV , E = 0 meV with lead polarisation PL= 0.8 and PR= 0.

Figure 4: Graph (e) has a twofold degenerate ground states within µ ∼ (−0.8, 0.8) ,with another twofold degenerate level giving it the appearance of being fourfold degenerate.

Graph (f) with applied magnetic eld splits the degeneracy of all degenerate states.

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(g) Energy spectrum under the variation of µ as s = 1, N = 3, gµ0B0 = 1 meV and C = 1 meV , E = ¹₅ meV with no lead polarisation PL= 0and P_R= 0.

(h) Energy spectrum under the variation of µ as s = ³₂, N = 3, gµ0B0 = 0 meV and C = 1 meV , E = ¹₅ meV with lead polarisation PL = 0.8 and PR= 0.

Figure 5: Graph (g) do not have any degenerate states as the magnetic eld and the anisotropic terms splits the levels. Graph (h) shows how the energy scale and the number of levels increases as the values of s and N gets larger.

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Figure 6: This graph shows how the ground state changes with variation of both µ and P_R while PL = 0.8 with s = ¹₂, N = 2, gµ0B0 = C = E = 0 meV. Here AFM means that the region is of anti-ferromagnetic conguration while non-collinear means that the conguration is inconclusive, and in this case it is because a three fold degeneracy. The table depicts the real part of the linear combination of the basis which denes the ground- eigenstate of that region, and each row of the basis should be regarded as its own base where the numbers tell how the projection quantum numbers are ordered.

Figure 7: This graph shows how the ground state of Figure 6 changes when applying a magnetic eld (gµ0B0 = 1 meV). Here FM means that the region is of a ferromagnetic conguration, and AFM still means that it is anti-ferromagnetic.

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4.2.2 Variation with respect to the voltage Bias

(i) Energy spectrum under the variation of V as s = ¹₂, N = 2, gµ₀B₀ = 0 meV and C = 0 meV , E = 0 meV with no lead polarisation PL= 0and PR= 0.

(j)Energy spectrum under the variation of V as s = ¹₂, N = 2, gµ₀B₀ = ¹₂ meV and C = 0 meV , E = 0 meV with no lead polarisation PL= 0and P_R= 0.

Figure 8: Graph (i) has three fold degenerate ground state at around V ≈ ±1 while graph (j) with applied magnetic eld splits that degeneracy.

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(k)Energy spectrum under the variation of V as s = ¹₂, N = 2, gµ0B0 = 2 meV and C = 0 meV , E = 0 meV with lead polarisation PL= 0.8 and P_R= −0.8.

(l) Energy spectrum under the variation of V as s = ¹₂, N = 2, gµ0B0 = 0 meV and C = 1 meV , E = ¹₅ meV with lead polarisation PL = 0.8 and PR= 0.4.

Figure 9: Both the graph (k) and (l) show a much greater span in the energy scale than any previous graphs and this span decreases as PR approaches the value 0.8.

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(m)Energy spectrum under the variation of V as s = 1, N = 2, gµ0B0 = ¹₂ meV and C = 1 meV , E = ¹₅ meV with lead polarisation PL = 0.8 and P_R= 0.8.

(n)Energy spectrum under the variation of V as s = ³₂, N = 3, gµ0B0 = 0 meV and C = 1 meV , E = ¹₅ meV with lead polarisation PL = 0.8 and PR= −0.8.

Figure 10: Graph (m) shows the spectra for equal lead polarisations and graph (n) shows the increasement in the energy span and the number of energy levels as s and N increases.

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Figure 11: This graph shows how the ground state changes with variation of both V and P_R while PL = 0.8 with s = ¹₂, N = 2, gµ0B0 = C = E = 0 meV. Here AFM means that the region is of anti-ferromagnetic conguration. The table depicts the real part of the linear combination of the basis which denes the ground-eigenstate of that region, and each row of the basis should be regarded as its own base where the numbers tell how the projection quantum numbers are ordered.

Figure 12: This graph shows how the ground state of Figure 11 changes when applying a magnetic eld (gµ0B0 = 1 meV). Here FM means that the region is of a ferromagnetic conguration, and AFM still means that it is anti-ferromagnetic.

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5 Discussion

One can see that the energy spectrum under the variation of the chemical potential tend to have a big gap between the upper and lower levels at the regime µ ∼ 0 meV but at some critical value µC ≈ ±0.8 meV the levels get a bit closer to one another, or even intersect, to later fall apart at µ ∼ ±1 meV where the spectrum has its minimum. These behaviours can be understood when consulting the research from Ref.[2] where one can conclude that in the regime µ ∈ (−µC, µC) meV the Heisenberg and Ising integral are the dominant factors while at µ ∼ ±1 meV it's the Heisenberg and the Dzyaloshinski-Moriya integrals that are the main contributions for the spectra's minimum.

For the variation of the voltage bias one can notice that the polarisation play a great role in the span of the energy levels where equal polarisations (graphs (i),(j) and m) give rise to energy scales in the order of ∼ 10 meV while non equal polarisations (graph (k),(l) and n) are of the order ∼ eV . What can be seen here is that for non-equal lead polarisations in the regime V ∈ (−VC, V_C) ≈ (−2, 2) mV it's yet again the Heisenberg and the Ising integrals that are dominant but for |V | ≥ VC it's is almost solely the Dzyaloshinski-Moriya integral that determines the energy spectrum.

Another feature that is apparent in these results for both the variation of µ and V is that the Zeeman and the anisotropic terms, under the assumption that they are constants under the variation, splits only certain levels of the spectrum and the parameter C also raises the overall spectrum.

The degenerate behaviour of these numerical results, especially for the two and three molecular cases (in graphs (a)-(f) and (i)-(l)) when s = ¹₂, can be well understood when compared with the analytical results. For the two molecular case we will have degeneracy when there's no magnetic eld applied but also if I1,2 and D1,2are non-existing which they are for PL= PR= 0(see section 3.9 ). This can be seen in graph (a) and (i) which has a three fold degeneracy in the regime µ ∈ (−µC, µ_C) meV and V ∼ ±4mV, ∼ 0 mV respectively which is in accordance with the analytical formulas which also predicts the splitting in graph (b) and (j) with applied magnetic eld. Another thing that the analytical results can predict is false degeneracies which can be seen in graph (e) where the ground state is two fold degenerate but has an close lying two degenerate level giving the appearance of being four fold degenerate. This can be explained by noticing in Ref.[2] that I1,2 is very small compared with J1,2 and by looking at the analytical formulas one can see that there will be two two degenerate levels close to each other when regarding I1,2 as Taylor expansion parameter.

This means that the analytical formulas can be used in a wider range of analysis to describe the eigenvalues of other set of parameter values for this kind of molecular junction.

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What can be seen in graph (b),(d) and (j) is that the Zeeman term splits the levels such that parallel spin state level (| ↓, ↓i) in the regimes µ ∈ (−µC, µ_C) and V ∈ (−VC, V_C) is lower than the anti-parallel singlet state level (as seen in Figure 6-7 and Figure 11-12 ).

This means that the ground state now will favour a ferromagnetic conguration instead of an anti-ferromagnetic one. Another thing that is of interest is the spontaneous splitting of the ground state in the absence of external magnetic eld, which can be seen in graph (a), (c) and (e). The ground states separates around ±µC ≈ 0.8 meV where it either goes to the spectra's minimum or to a higher level. By again consulting to the research done in Ref.[2] and the analytical formulas in section 4.1 one can notice that the minimums is Dzyaloshinski-Moriya dependent while the intermediate states are purely Heisenberg and Ising dependent (or neither) at the regime |µ| ≥ µC. There is a similar occurrence when looking at the non-equal polarisation cases of the voltage variation, (graph (k),(l) and (m).

Here, the energy level splitting is even more profound and in these energy scales the regime V ∈ (−VC, VC) ≈ (−2, 2) meV seems almost degenerate even with a magnetic eld. The Dzyaloshinski-Moriya interaction also in this case manifest itself in the minimum energy levels and is here the main contributor when |V | ≥ VC meV.

This spontaneous splitting and the fact that the magnetic eld may induce a ferromagnetic ground state can be used to eciently control the magnetic order of the junction. By for example switching on and o the voltage bias one can force states into the ground state by going beyond |V | ≥ VC meV which is an anti-ferromagnetic conguration according to Figure 11. In this regime the levels are suciently separated such that excitations are unfavourable and the state is locked in ground state. By releasing the voltage the level returns to a regime (V ∈ (−2, 2) meV ) where level splitting may be possible with little eort and can produced by a magnetic eld. This generates a ground state favouring a ferromagnetic order which can be seen in Figure 12. This means that by varying the voltage bias and the magnetic eld one can control the magnetic order of the molecules in a simple fashion. This eect can be specied or enhanced by tweaking the lead polarisation. By tuning the polarisation one can decide how much of the Dzyaloshinski-Moriya interaction that aect the system's ground state. This aects the level such that going towards equal polarisations leads to a narrower gap between ground state and intermediate states (Figure 6-7 and Figure 11-12 ).

If it is possible to control the magnetic order of this tunnelling junction as described above it is plausible that this eect may be used to create spin logic transistors which could be used to improve computer hardware. By improving computer hardware the future technologies can be improved in terms of computational powers and energy savings.

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5.1 Conclusion

The conclusions that can be drawn from these results are:

• The ground state degeneracies are naturally split at some critical value µC,VC where it naturally goes towards the spectrum's minimum.

• The self-interaction and the Zeeman terms splits the degeneracy only along certain levels where the axial parameter C and the transversal parameter E only splits states of higher spin s ≥ 1.

• The Zeeman term splits the ground state such that it favours a ferromagnetic conguration in equilibrium.

• The minimums and maximums of the energy spectra are mostly governed by the Dzyaloshinski-Moriya interaction while the intermediate states are purely Heisenberg and Ising dependent.

• By tuning the lead polarisations and either manipulate the chemical potential or the voltage bias one can control the magnetic order of the junction to either be locked in ground state in a anti-ferromagnetic conguration or be easily mixed by applying a magnetic eld.

Uppsala University Bachelor of Science Degree in physics