Non-equilibrium dynamics ofa single spin in a tunnel junction

Februari 2014

Non-equilibrium dynamics of a single spin in a tunnel junction

Henning Hammar

Teknisk- naturvetenskaplig fakultet UTH-enheten

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Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

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Box 536 751 21 Uppsala

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018 – 471 30 03

Telefax:

018 – 471 30 00

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http://www.teknat.uu.se/student

Non-equilibrium dynamics of a single spin in a tunnel junction

Henning Hammar

Making spintronic devices is a hot topic for future technical development. In this work the non-equilibrium dynamics of a single spin in a tunnel junction is analyzed and numerically simulated. This is done in order to understand the dynamics of e.g. a magnetic molecule between two metal contacts for future spintronic devices. The work starts with looking at the system in a many-body theory picture in order to derive the interesting properties of the system. An initial solution for the system is

analytically calculated as well as for the dynamic case. The dynamic has then been numerically simulated in order to get the time evolution of the system.

The results showed that the dynamics of the molecular spin induced a spin dependent charge and spin currents in the system and that the currents could be used to control the molecular spin.

It showed qualitatively how different parameters, for example coupling strength, effect the system and what to consider when designing a system similar to this.

Ämnesgranskare: Anders Bergman Handledare: Jonas Fransson

Contents

1 Introduction 2

1.1 Molecular spintronics . . . 2

1.2 Description of the system . . . 2

1.3 Aim . . . 3

2 Theory 3 2.1 Many-body theory and Greens functions . . . 3

2.2 Dening the system . . . 4

2.3 Equation for the local spin moment . . . 4

2.4 Quantum dot Greens function . . . 6

2.5 Tunneling current . . . 8

3 Analysis 8 3.1 Initial solution . . . 9

3.2 The dynamic Greens function . . . 10

3.3 Equations for the tunneling current . . . 13

4 Method 15 4.1 Ordinary dierential equation . . . 15

4.2 Integrals . . . 16

4.3 Resources . . . 16

5 Results 16 6 Discussion 24 7 Conclusion 25 A Appendix 26 A.1 Code . . . 26

1 Introduction

1.1 Molecular spintronics

Electronic devices and circuits are getting smaller and smaller and more eective and has for many years followed the prediction of Moore's law [1]. Conventional electronic devices as silicon integrated circuits are now entering a level where quantum eects starts to come into the picture. In order to continue this progress dierent kind of electronic devices is of great interest. Molecular spintronics is one of the emerging new elds where one uses the properties of magnetic molecules in order to create electronic devices. Here, one uses the spin instead of the charge of the electron when performing the operations. By using the magnetic properties of electrons instead of their charge it enables to create dissipationless electronics, as it is not longer needed to move charges. Therefore less energy is needed which is both environmental friendly and has a great impact on for example mobile devices where the need of battery reduces.

In spintronic devices the magnetic molecules can be used as nanomagnets and single-molecule magnets are among the smallest devices that could be used [2]. Molecular spintronics using single-molecule magnets has many interesting features that can be used in order to create devices for information storage and processing [3]. One of the key properties of single-molecule magnets is that they have a strong magnetic anisotropy leading to magnetic bistability which is good for making nanoscale memory cells [4]. Another key property is that they have non-trivial quantum dynamics which is good for making quantum computers [5]. Quantum computers are computers that uses the properties of a quantum state when performing computations and therefore can outperform classical computers in factoring numbers and search in databases. Because of the applications it is of great interest to understand the dynamics and properties of single-molecule magnets.

In this work the non-equilibrium dynamics of a local spin moment in a tunnel junction is analyzed. This could be a magnetic molecule between two metal contacts where one is interested in controlling either the spin polarization of the current or the magnetic moment of the molecule. This could be used for a memory cell or for performing operations on the local spin moment for quantum information processing, although here we treat the spin moment of the molecule classically.

Previous works on studying similar systems has been done. In one work the authors study what happens when pumping the spin of electrons into an excited state with a spin-polarized scanning tunneling microscope and then analyze the spin relaxation time for each excited state [6]. In another paper the authors study the magnetic anisotropy of magnetic molecules and show that higher order spin system coupled to ferromagnets can be used and controlled by electrical current in the same way as simple spin-half systems [7]. Also work has been done to analyze the non-equilibrium transport through magnetic vibrating molecules [8]. The most similar work is by Filipovic et al [9] where they analyze the spin transport and tunable Gilbert damping in a single-molecule junction using the same idealized picture and the same approach as in this work. Their conclusions was that they get that the coupling between the spin and the molecular spin induces inelastic spin currents and a Gilbert damping on the molecular spin and a modication of its frequency.

1.2 Description of the system

The system to be analyzed is a system consisting of a local spin moment in a tunnel junction interacting with the current going through the junction. In the idealized picture it is described by a quantum dot connected to two metal contacts and interacting with the local spin moment. The quantum dot is a quantum island consisting of a single level, see gure 1. When applying a voltage over the two contacts the electrons will start tunneling to the quantum dot and then to the other contact. As the electrons carries both charge and a magnetic spin this will create a charge and a spin current going through the quantum dot. The currents will then interact with the local spin moment. The local spin moment can be for example the magnetic moment

of the molecule. The local spin moment will then interact with the current going through the quantum dot.

This is creating the dynamics of the system and what is going to be studied in the thesis.

1.3 Aim

The aim of the thesis is to analyze the non-equilibrium dynamics of a single spin in a tunnel junction. This includes an analytical derivation of the equations of the system from existing theory in order to numerically calculate the dynamics of the system. Here it is interesting to see how to describe such a system and where the dynamics comes from. The dynamics of the system is then calculated for dierent input parameters in order to understand how they aect the system and what conclusions that can be made.

2 Theory

In order to analyze the system we rst need to build up the theory we need. For understanding the theory it is recommended to have an insight in quantum mechanics and solid state physics. First we will consider how to work with a many-body quantum system and what mathematical tools that can be used. Then we regard the specic system that is going to be considered. When doing so we rst develop the equation of motion for the local spin moment. Then we analyze the quantum dot Greens function and lastly the tunneling current going through the dot. The theory regarded in this section is based on previous works done by J. Fransson [10].

2.1 Many-body theory and Greens functions

Due to the vast amount of particles treated in a solid state system we need to dene some mathematical concepts in order to treat the physics of the many-body system. We start by dening an operator ˆψ that acts in our system. This represents the quantization of the quantum wave function for the electron. The electrons are created at time t' by ˆψ^†(t⁰) and they are annihilated at time t by ˆψ(t). We then dene the propagator for the electrons. The one-electron Greens function is dened as the propagation of an electron created at time t' and annihilated at time t and is calculated for fermionic particles as

G(t, t⁰) = (−i)D

T ˆψ(t) ˆψ^†(t⁰)E

=







−iD ˆψ(t) ˆψ^†(t⁰)E t > t⁰ iD ˆψ(t) ˆψ^†(t⁰)E

t < t⁰

, (1)

where T is the time-ordering operator. This is propagator is dened on the complex time contour. In this work we will treat a non-equilibrium problem and need to convert our propagators into the real time domain. This is done by dening new propagators, G^<(t, t⁰) and G^>(t, t⁰), called the lesser and greater Greens function respectively. The lesser/greater superscript is used to indicate if the time t is less/larger than t' on the time contour, called the Keldysh contour. We can also dene the retarded/advanced Green function

G^r(t, t⁰) = θ(t − t⁰)G^>(t, t⁰) − G^<(t, t⁰) , (2)

G^a(t, t⁰) = θ(t⁰− t)G^<(t, t⁰) − G^>(t, t⁰) , (3) which describes the connection between the lesser and greater Greens function.

ȝ

ȝ

S(t) İ

J

ī

ī

R

eV

Figure 1: A quantum dot with a single electron level, ε0, coupled to a two metallic contacts with chemical potential µL and µR. The quantum dot also couples to an outer local spin moment, S(t).

2.2 Dening the system

The system that we want to describe is a single spin moment S in a tunnel junction interacting with a single level quantum dot, see gure 1. The quantum dot is coupled through tunneling interactions to metal contacts, electron reservoirs, with the respective chemical potential µχ, χ = L, R, where L/R denotes the left/right reservoir. The Hamiltonian for the system becomes

H = HL+HR+HT+HQD+HS+Hint. (4)

We write the electron operator in the reservoirs as ckσ and in the quantum dot as dσ, where c^†kσ creates an electron in the reservoir with momentum k and spin σ and ckσ annihilates it. These operators are analogue to ˆψ. The same goes for the quantum dot, but as we only have one energy level we do not need to regard the momentum and it is simplied as d^†σ and dσ. The Hamiltonian for the reservoirs thus becomes Hχ =P

kσ(εkσ− µχ)c^†_kσckσand represents the energy for the reservoir χ. Here is ε the energy levels of the reservoir, k denotes the momentum vector and σ the spin. The tunneling Hamiltonian is HT =HT L+HT R, where HT χ = TχP

kσc^†_kσdσ+ H.c., i.e. it creates an electron in the reservoir and annihilates it in the quantum dot and vice versa. The quantum dot is represented by HQD =P

σεσd^†_σdσ and the spin system by HS = −gµBS · B, where g is the g-factor, µB the Bohr constant, S the spin vector and B an external magnetic eld. The interactions between the spin and the quantum dot is given by Hint= −J s · S, where J is the interacting strength and s = Pσσ⁰d^†_σσσσ⁰dσ⁰/2.

2.3 Equation for the local spin moment

First we want to derive the equation for the local spin moment. It is derived from the eective action. For the local spin moment we can write the eective action as

S = 1 S²

ˆ

S^q(t) · [S^c(t) × ˙S^c(t)]dt − gµBB(t) · S^q(t) +1 e

ˆ

S^q(t) · j(t, t⁰) · S^c(t⁰)dtdt⁰ (5) where the spin superscripts q and c denotes if the spin is fast or slow varying, respectively, depending on the frame of reference. This equation follows from previous works [10]. The current j(t, t⁰)will be calculated by

the approximate expression j(t, t⁰) ≈ie

2J²θ(t − t⁰)spσ G^<(t⁰, t)σG^>(t, t⁰) − G^>(t⁰, t)σG^<(t, t⁰)

(6) where G^</>(t, t⁰)is the lesser/greater Greens function for propagation of a quantum dot electron.

From the eective action we can derive the equation of motion. We take the functional derivative of the eective action and set it to zero as

0 = δS δS^q(t) = 1

S²S^c(t) × ˙S^c(t) − gµBB(t) + 1 e

ˆ

j(t, t⁰) · S^c(t⁰)dt⁰. (7) We can drop the superscript and take the cross product from the left with S(t)× to get

0 = 1

S²S(t) ×h

S(t) × ˙S(t)i

− gµBS(t) × B(t) +1 e

ˆ

S(t) × j(t, t⁰) · S(t⁰)dt⁰. (8) Assuming that the length of the spin is constant and that we can treat it classically we can set ∂t|S(t)|²= 0 such that the rst term can be simplied to − ˙S(t) and moved to the left hand side. The resulting equation of motion becomes

S(t)˙ = −gµ_BS(t) × B(t) +1 e

ˆ 

j^(H)(t, t⁰)S(t) × S(t⁰) +S(t) × j^(I)(t, t⁰)·S(t⁰) − S(t) × j^{(DM )}(t, t⁰)·S(t⁰)

dt⁰. (9)

Here, we decomposed the product S(t) × j(t, t⁰) · S(t⁰)into an isotropic Heisenberg, anisotropic Ising and anisotropic Dzyaloshinsky-Moriya (DM) like interaction according to

S(t) × j(t, t⁰) · S(t⁰) = j^(H)(t, t⁰)S(t) × S(t⁰) + S(t) × j^(I)(t, t⁰) · S(t⁰) + S(t) × j^{(DM )}(t, t⁰) × S(t⁰). (10) The interaction parameters are dened through

j^(H)(t, t⁰) = ieJ²θ(t − t⁰) G^<₀(t⁰, t)G^>₀(t, t⁰) − G^>₀(t⁰, t)G^<₀(t, t⁰)

−G^<₁(t⁰, t)G^>₁(t, t⁰) + G^>₁(t⁰, t)G^<₁(t, t⁰)
, (11)

j^(I)(t, t⁰) = ieJ²θ(t − t⁰) G^<₁(t⁰, t)G^>₁(t, t⁰) − G^>₁(t⁰, t)G^<₁(t, t⁰) +G^<₁(t⁰, t)G^>₁(t, t⁰) + G^>₁(t⁰, t)G^<₁(t, t⁰)t

, (12)

j^{(DM )}(t, t⁰) = −eJ²θ(t − t⁰) G^<₀(t⁰, t)G^>₁(t, t⁰) − G^>₀(t⁰, t)G^<₁(t, t⁰)

−G^<₁(t⁰, t)G^>₀(t, t⁰) + G^>₁(t⁰, t)G^<₀(t, t⁰)
, (13) where G(t, t⁰) = G0(t, t⁰)σ⁰+ σ · G1(t, t⁰) is the quantum dot Greens function which is separated into a spin-dependent and a spin-independent part. The quantum dot Greens function is dependent of the spin and its evolution. Thus the interaction parameters gives feedback from changes of the spin from the changes in the quantum dot Greens function. This gives a loop such that the evolution of the spin depends on the quantum dot Greens function and vice verca.

2.4 Quantum dot Greens function

We now derive the quantum dot Greens function. We start of by taking the time derivative of equation 1 which gives

∂_tG_σσ⁰(t, t⁰) = −iD

{d_σ(t)d^†_σ0(t⁰)}E

− iD

T [d_σ(t),H ]d^†_σ⁰(t⁰)E

, (14)

where T is the time-ordering operator. Inserting the Hamiltonian for the quantum dot we get

(i∂_t− εσ)G_σσ⁰(t, t⁰) = δ_σσ⁰δ(t − t⁰) + ˆ

V_σ(t, τ )G_σσ⁰(τ, t⁰)dτ − J hS · σ_σσ⁰i Gσσ⁰(t, t⁰), (15) where the second term on the left hand side comes from the quantum dot Hamiltonian, the second term on the right hand side comes from the tunneling Hamiltonian and the third term from the interaction with the spin. From this operation follows that

V_σ^</>(t, t⁰) = X

k

T_χ²gkσ(t, t⁰), (16)

where gkσ(t, t⁰) = (−i)T e^{−iεkσ(t−t0)−ie´t0tVχ(τ )dτ} is the lead Greens function. Here is e the elementary charge and Vχ(τ )the applied bias voltage over the two metal contacts. We Fourier transform equation 16 and get

V_σ^</>(t, t⁰) = (±i)X

k

T_χ²f (±εkσ)e^{−εkσ(t−t0)−ie}

´_t

t0V_χ(τ )dτ

= (±i)X

χ

Γ^χ_σI_χ^</>(t, t⁰), (17)

where we dened Γ^χσ=P

kT_χ²δ(ω − εkσ)and I_χ^</>(t, t⁰) =

ˆ

f (±ω)e^{−iω(t−t0)−ie´t0tVχ(τ )dτ}dω

2π (18)

where f(ω) is the Fermi function and f(−ω) = 1 − f(ω). We write the coupling matrix V=v0σ⁰+ σ · v₁ in terms of spin-independent and spin-dependent components. Then we have

v^</>₀ (t, t⁰) = (±i)X

χσ

Γ^χ_σI_χ^</>(t, t⁰) = (±i)X

χ

Γ^χ₀I_χ^</>(t, t⁰), (19)

v₁^</>(t, t⁰) = (±i)X

χσ

σ^z_σσΓ^χ_σI_χ^</>(t, t⁰)ˆz = (±i)X

χ

Γ^χ_SI_χ^</>(t, t⁰), (20) where we have introduced the parameters Γ0=P

σΓσ and ΓS=P

σσ^z_σσΓσˆz.

We introduce a zero Greens function gσ(t, t⁰) as the solution to the equation (i∂t− εσ)gσ(t, t⁰) = δ(t − t⁰) +´

Vσ(t, τ )gσ(τ, t⁰)dτ. We can write the retarded/advanced zero Greens function as

g_σ^r/a(t, t⁰) = (±i)θ(±t ∓ t⁰)e^{−i(εσ∓iΓσ)(t−t0)}. (21) Writing it in its spin-independent and spin-dependent components, g(t, t⁰)=g₀(t, t⁰)σ⁰+ σ · g₁(t, t⁰), it be- comes

g₀^r/a(t, t⁰) = (±i

2)θ(±t ∓ t⁰)X

σ

e^{−i(εσ∓iΓσ)(t−t0)}, (22)

g^r/a₁ (t, t⁰) = (±i

2)θ(±t ∓ t⁰)X

σ

σ_σσ^z e^{−i(εσ∓iΓσ)(t−t0)}ˆz. (23)

The lesser/greater forms of g becomes

g^</>(t, t⁰) = ˆ

g^r(t, τ )V^</>(τ, τ⁰)g^a(τ⁰, t⁰)dτ dτ⁰ = g^</>₀ (t, t⁰)σ0+ σg^</>₁ (t, t⁰) (24) where we divided into the spin-independent and spin-dependent components

g^</>₀ (t, t⁰) = ˆ 

g₀^rv^</>₀ g^a₀+ g₁^rv₀^</>g^a₁+ g^r₀v₁^</>g₁^a+ g^r₁v₁^</>g₀^a

dτ dτ⁰ (25)

g^</>₁ (t, t⁰) = ˆ 

g₀^rv₁^</>g^a₀+ g^r₁v₁^</>g₁^a+ g^r₀v^</>₀ g₁^a+ g₁^rv^</>₀ g^a₀

dτ dτ⁰ˆz. (26) In terms of the zero Greens function we write the approximate solution to 15 as

G^</>(t, t⁰) = g^</>(t, t⁰) − J ˆ

g(t, τ ) hS(τ )i ·σg(τ, t⁰)dτ

= g^</>₀ (t, t⁰)σ⁰+ σ · g₁^</>(t, t⁰) − J ˆ

g₀(t, τ )σ⁰+ σ · g₁(t, τ )
hS(τ )i · σ g0(τ, t⁰)σ⁰+ σ · g1(τ, t⁰)
dτ

= G^</>₀ (t, t⁰)σ⁰+ σ · G^</>₁ (t, t⁰), (27)

where we dened G^</>0 (t, t⁰) = g^</>₀ (t, t⁰) + δG^</>₀ (t, t⁰)and G^</>1 (t, t⁰) = g^</>₁(t, t⁰) + δG^</>₁ (t, t⁰). Here is

δG^</>₀ (t, t⁰) = −J ˆ 

g₀^r(t, τ ) hS(τ )i ·g^</>₁ (τ, t⁰) + g^</>₀ (t, τ ) hS(τ )i ·g₁^a(τ, t⁰) +g^r₁(t, τ ) · hS(τ )i g^</>₀ (τ, t⁰) + g^</>₁ (t, τ ) · hS(τ )i g^a₀(τ, t⁰) + i [g₁^r(t, τ ) × hS(τ )i] ·g^</>₁ (τ, t⁰) + ih

g₁^</>(t, τ ) × hS(τ )ii

·g^a₁(τ, t⁰)

dτ, (28)

δG^</>₁ (t, t⁰) = −J ˆ 

g^r₀(t, τ ) hS(τ )i g^</>₀ (τ, t⁰) + g₀^</>(t, τ ) hS(τ )i g^a₀(τ, t⁰) +i [g^r₁(t, τ ) × hS(τ )i] g₀^</>(τ, t⁰) + ih

g^</>₁ (t, τ ) × hS(τ )ii

g^a₀(τ, t⁰) +ig₀^r(t, τ )h

hS(τ )i × g^</>₁ (τ, t⁰)i

+ ig₀^</>(t, τ ) [hS(τ )i × g^a₁(τ, t⁰)]

+ i [g₁^r(t, τ ) × hS(τ )i] ×g₁^</>(τ, t⁰) + ih

g^</>₁ (t, τ ) × hS(τ )ii

×g^a₁(τ, t⁰)

dτ (29) where we have considered the Keldysh contour.

2.5 Tunneling current

The tunneling current owing between the electrodes represents the change of particles in the two contacts.

We begin from the charge current in the left electrode, I_L^C(t) = −e∂_tD P

pσn_pσE

= iesp∂_tP

pg_p^<(t, t⁰), which leads to the expression

I_L^C(t) = −2e

~

spImΓ^L ˆ t

−∞

I_L^>(t)G^<(t⁰, t) + I_L^<(t)G^>(t⁰, t)
dt⁰

= I₀^C(t) + I₁^C(t), (30)

where Γ^χ= Γ^χ₀σ⁰+ σ · Γ^χ_S and

I₀^C(t) = −4e

~Im ˆ t

−∞

I_L^>(t)Γ^L₀G^<₀(t⁰, t) + I_L^<(t)Γ^L₀G^>₀(t⁰, t)
dt⁰, (31)

I₁^C(t) = −4e

~ Im

ˆ t

−∞



I_L^>(t)Γ^L_S· G^<₁(t⁰, t) + I_L^<(t)Γ^L_S· G^>₁(t⁰, t)

dt⁰. (32)

The spin current follow from IL^S(t) = −e∂t

DP

pσσ⁰c^†_pσσσσ⁰cpσ⁰

E

= iespσ∂tP

pg_p^<(t, t⁰) which leads to the expression

I_L^S(t) = −2e

~spImσΓ^L ˆ t

−∞

I_L^>(t)G^<(t⁰, t) + I_L^<(t)G^>(t⁰, t)
dt⁰

= I₀^S(t) + I₁^S(t) (33)

where

I₀^S(t) = −4e

~ Im

ˆ t

−∞

I_L^>(t)Γ^L₀G^<₁(t⁰, t) + I_L^<(t)Γ^L₀G^>₁(t⁰, t)
dt⁰, (34)

I₁^S(t) = −4e

~Im ˆ t

−∞

 I_L^>(t)h

Γ^L_SG^<₀(t⁰, t) + iΓ^L_S × G^<₁(t⁰, t)i

+ I_L^<(t)h

Γ^L_SG^>₀(t⁰, t) + iΓ^L_S× G^>₁(t⁰, t)i

dt⁰. (35)

3 Analysis

The aim of the project is to solve the time evolution of the system regarded in the theory section numerically.

This is done by solving the ordinary dierential equation in equation 9. To do this we need to rst nd an initial solution for a time t0and then nd the dependence ˙S(t, S) for each time step. This is done by solving the Greens function for the initial case and the dynamic case, regarded in the following subsections. The

nal product of the calculation is the tunneling currents which is derived from the Greens function for the initial and dynamic case.

3.1 Initial solution

First we solve the Greens functions for the system without any applied voltage and solve it in equilibrium.

We regard the symmetric case where there is no χ dependence and we write the tunneling parameters as ΓS =pΓ0where p is a polarization factor between 0 and 1. We also dene Γ↑= ^Γ₂⁰(1+p)and Γ↓=^Γ₂⁰(1−p). We do the same for the electron level of the quantum dot and write it as εS =pε0, ε↑=^ε₂⁰(1 + p⁰)and ε↓=

ε0

2(1 − p⁰). As we have no applied voltage we can simplify Iχ^</>(t, t⁰) =´

f (±ω)e^{−iω(t−t0)−ie´t0tVχ(τ )dτ dω}_2π =

´f (±ω)e^{−iω(t−t0) dω}_2π. This gives that we can solve the components, g(t, t⁰)=g0(t, t⁰)σ⁰+ σ · g1(t, t⁰), for the zero Greens function

g^</>₀ (t, t⁰) = ∓ ˆ

Λ₀(ω)f (±ω)e^{−iω(t−t0)}dω

2π, (36)

g^</>₁ (t, t⁰) = ∓ ˆ

Λ₁(ω)f (±ω)e^{−iω(t−t0)}dω

2πˆz, (37)

where

Λ₀(ω) = i

2 Γ₀ 1

Γ²_↑+ (ω − ε↑)² + 1 Γ²_↓+ (ω − ↓)²

!

+ Γ_S 1

Γ²_↑+ (ω − ε↑)² − 1 Γ²_↓+ (ω − ↓)²

!!

, (38)

Λ1(ω) = i 2 ΓS

1

Γ²_↑+ (ω − ε_↑)² + 1 Γ²_↓+ (ω − _↓)²

! + Γ0

1

Γ²_↑+ (ω − ε_↑)² − 1 Γ²_↓+ (ω − _↓)²

!!

. (39)

From this we can derive the Greens function and its components, G^</>(t, t⁰) = G^</>₀ (t, t⁰)σ⁰+σ·G^</>₁ (t, t⁰). In order to do this we need to solve the equation for the spin. We solve equation 9 assuming a constant magnetic eld in the z-direction, B = Bˆz, and no interaction. This gives the spin components Sx = Sxysin(ωLt), Sy= Sxycos(ωLt)and Sz= Szwhere |S|²= S_xy² + S_z²and ωL= gµB|B|. The components for the Greens function thus becomes

G^</>₀ (t, t⁰) = ∓ ˆ

f (±ω)e^{−iω(t−t0)}



Λ0(ω) + J

2SzC0(ω) dω

2π, (40)

G^</>_1x (t, t⁰) = ±i 2J S_xy

ˆ

f (±ω)e^{−iω(t−t0)}

A(ω)e^iωLt− A^∗(ω)e^−iωLt0+ B(ω)e^−iωLt− B^∗(ω)e^iωLt0dω 2π,

(41)

G^</>_1y (t, t⁰) = ∓1 2J Sxy

ˆ

f (±ω)e^{−iω(t−t0)}

A(ω)e^iωLt+ A^∗(ω)e^−iωLt0− B(ω)e^−iωLt− B^∗(ω)e^iωLt0dω 2π,

(42)

G^</>_1z (t, t⁰) = ∓ ˆ

f (±ω)e^{−iω(t−t0)}



Λ1(ω) + J

2SzC1(ω) dω

2π, (43)

where

A(ω) = Λ1(ω) − Λ0(ω)

ε_↑− ω + ω_L− iΓ_↑, (44)

B(ω) = Λ1(ω) + Λ0(ω)

ε_↓− ω − ωL− iΓ_↓, (45)

C0(ω) = 2 Λ0(ω) ω − ε↑

Γ²_↑+ (ω − ε_↑)² − ω − ε↓

Γ²_↓+ (ω − _↓)²

!

+ Λ1(ω) ω − ε↑

Γ²_↑+ (ω − ε_↑)² + ω − ε↓

Γ²_↓+ (ω − _↓)²

!!

, (46)

C1(ω) = 2 (Λ0(ω) + Λ1(ω)) ω − ε↑

Γ²_↑+ (ω − ε_↑)²+ ω − ε↓

Γ²_↓+ (ω − _↓)²

!

. (47)

3.2 The dynamic Greens function

At time zero we apply a constant voltage V between the two contacts and turn on the interaction. When extending our analysis from a constant spin to a changing discrete spin we need to make use of an ap- proximation. It is that for suciently small time steps the expectation value of the spin in the integral in equation 27 can be assumed to be a constant value, which is the spin calculated from the previous time step, hS(t)i = S^k−1. Then equation 27 simplies to integrals over the zero Greens functions, as for the initial solution.

When performing the integral over time, as when calculating the current, we need to regard the correct time ordering. In order to do so we need to divide equation 18 into three parts, no applied voltage and where t and t' is less than t0, with applied voltage and where t' is less than t0 and then one part with applied voltage and where both t and t' is greater than t0. Then equation 18 becomes

I_χ^</>(t, t⁰) =







´f (±ω)e^{−iω(t−t0) dω}_2π t⁰< t < t0

´f (±ω)e^{−iω(t−t0)−ieVχt dω}_2π t⁰< t₀< t

´f (±ω)e^{−i(ω+eVχ)(t−t0) dω}_2π t0< t⁰< t

. (48)

The rst part is already derived as the initial solution while we need to regard the second and third part when we extend the analysis for the dynamic case. We simplify the derivation assuming the symmetric case V = VL= VR. For the case where t' is less than t0 we get the zero Green functions

g₀^</>(t, t⁰) = ∓ ˆ

α0(ω)f (±ω)e^{−iω(t−t0)−ieV t}dω

2π, (49)

g^</>₁ (t, t⁰) = ∓ ˆ

α1(ω)f (±ω)e^{−iω(t−t0)−ieV t}dω

2πˆz, (50)

where

α₀(ω) = −i 4

 Γ₀

 1

ω + eV − ε_↑+ iΓ_↑ + 1

ω + eV − _↓+ iΓ_↓

  1

ω − ε_↑− iΓ↑

+ 1

ω − _↓− iΓ↓



+

 1

ω + eV − ε↑+ iΓ↑

− 1

ω + eV − ↓+ iΓ↓

  1

ω − ε↑− iΓ↑

− 1

ω − ↓− iΓ↓



+ΓS

 1

ω + eV − ε_↑+ iΓ_↑ + 1

ω + eV − _↓+ iΓ_↓

  1

ω − ε_↑− iΓ_↑ − 1 ω − _↓− iΓ_↓



+

 1

ω + eV − ε_↑+ iΓ_↑ − 1

ω + eV − _↓+ iΓ_↓

  1

ω − ε_↑− iΓ_↑ + 1 ω − _↓− iΓ_↓



, (51)

α1(ω) = −i 4

 ΓS

 1

ω + eV − ε_↑+ iΓ_↑ + 1

ω + eV − _↓+ iΓ_↓

  1

ω − ε_↑− iΓ_↑ + 1 ω − _↓− iΓ_↓



+

 1

ω + eV − ε_↑+ iΓ_↑ − 1

ω + eV − _↓+ iΓ_↓

  1

ω − ε_↑− iΓ_↑ − 1 ω − _↓− iΓ_↓



+Γ0

 1

ω + eV − ε_↑+ iΓ_↑ + 1

ω + eV − _↓+ iΓ_↓

  1

ω − ε_↑− iΓ↑

− 1

ω − _↓− iΓ↓



+

 1

ω + eV − ε↑+ iΓ↑

− 1

ω + eV − ↓+ iΓ↓

  1

ω − ε↑− iΓ↑

+ 1

ω − ↓− iΓ↓



. (52) When we go to the case where t' is greater than t0 we get the zero Green functions

g^</>₀ (t, t⁰) = ∓ ˆ

β₀(ω)f (±ω)e−i(ω+eV )(t−t⁰)dω

2π, (53)

g^</>₁ (t, t⁰) = ∓ ˆ

β₁(ω)f (±ω)e−i(ω+eV )(t−t⁰)dω

2πˆz, (54)

where

β₀(ω) = −i

2 Γ₀ 1

Γ²_↑+ (ω + eV − ε↑)²+ 1

Γ²_↓+ (ω + eV − ↓)²

!

+Γ_S 1

Γ²_↑+ (ω + eV − ε↑)² − 1

Γ²_↓+ (ω + eV − ↓)²

!!

, (55)

β1(ω) = −i 2 ΓS

1

Γ²_↑+ (ω + eV − ε_↑)² + 1

Γ²_↓+ (ω + eV − _↓)²

!

+Γ0

1

Γ²_↑+ (ω + eV − ε_↑)² − 1

Γ²_↓+ (ω + eV − _↓)²

!!

. (56)

This in turn will give the total Green function. For the case where t' is less than t0we get

G^</>₀ (t, t⁰) = ∓ ˆ

f (±ω)e^{−iω(t−t0)−ieV t}



α₀(ω) + S_z(t)J

2D₀(ω) dω

2π, (57)

G^</>_1x (t, t⁰) = ∓J ˆ

f (±ω)e^{−iω(t−t0)−ieV t}Dx(ω, t)dω

2π, (58)

G^</>_1y (t, t⁰) = ∓J ˆ

f (±ω)e^{−iω(t−t0)−ieV t}Dy(ω, t)dω

2π, (59)

G^</>_1z (t, t⁰) = ∓ ˆ

f (±ω)e^{−iω(t−t0)−ieV t}D_z(ω, t)dω

2π, (60)

where

D0(ω) = α0

 1

ω + eV − ε_↑+ iΓ_↑ − 1

ω + eV − _↓+ iΓ_↓ + 1

ω − ε_↑− iΓ_↑ − 1 ω − _↓− iΓ_↓



+α1

 1

ω + eV − ε_↑+ iΓ_↑ + 1

ω + eV − _↓+ iΓ_↓ + 1

ω − ε_↑− iΓ_↑ + 1 ω − _↓− iΓ_↓



, (61)

Dx(ω, t) = 1 2

 (α0− α1)(Sx(t) − iSy(t))

ω + eV − ε_↑+ iΓ_↑ +(α0+ α1)(Sx(t) + iSy(t)) ω + eV − _↓+ iΓ_↓ +(α₀− α₁)(S_x(t) + iS_y(t))

ω − ε↑− iΓ↑

+(α₀+ α₁)(S_x(t) − iS_y(t)) ω − ↓− iΓ↓



, (62)

Dy(ω, t) = 1 2

 (α₀− α1)(S_y(t) + iS_x(t))

ω + eV − ε_↑+ iΓ_↑ +(α₀+ α₁)(S_y(t) − iS_x(t)) ω + eV − _↓+ iΓ_↓ +(α0− α1)(Sx(t) + iSy(t))

ω − ε_↑− iΓ_↑ +(α0+ α1)(Sy(t) + iSx(t)) ω − _↓− iΓ_↓



, (63)

Dz(ω, t) = α1+^J₂ Sz(t)

 1

ω + eV − ε_↑+ iΓ_↑ − 1

ω + eV − _↓+ iΓ_↓ + 1

ω − ε_↑− iΓ_↑ − 1 ω − _↓− iΓ_↓



+α₁

 1

ω + eV − ε_↑+ iΓ_↑ + 1

ω + eV − _↓+ iΓ_↓ + 1 ω − ε_↑− iΓ↑

+ 1

ω − _↓− iΓ↓

 (64). For the case where t' is greater than t0 we get

G^</>₀ (t, t⁰) = ∓ ˆ

f (±ω)e−i(ω+eV )(t−t⁰)



β0(ω) + Sz(t)J

2E0(ω) dω

2π, (65)

G^</>_1x (t, t⁰) = ∓J ˆ

f (±ω)e−i(ω+eV )(t−t⁰)Ex(ω, t)dω

2π, (66)

G^</>_1y (t, t⁰) = ∓J ˆ

f (±ω)e−i(ω+eV )(t−t⁰)E_y(ω, t)dω

2π, (67)

G^</>_1z (t, t⁰) = ∓ ˆ

f (±ω)e−i(ω+eV )(t−t⁰)Ez(ω, t)dω

2π, (68)

where

E₀(ω) = 2 β₀(ω) ω + eV − ε_↑

Γ²_↑+ (ω + eV − ε↑)²− ω + eV − ε_↓ Γ²_↓+ (ω + eV − ↓)²

!

+β₁(ω) ω + eV − ε_↑

Γ²_↑+ (ω + eV − ε_↑)² + ω + eV − ε_↓ Γ²_↓+ (ω + eV − _↓)²

!!

, (69)

E_x(ω, t) = (β₀(ω) − β₁(ω))Sx(t)(ω + eV − ε_↑) + Sy(t)Γ_↑ Γ²_↑+ (ω + eV − ε_↑)² +(β₀(ω) + β₁(ω))S_x(t)(ω + eV − ε_↓) − S_y(t)Γ_↓

Γ²_↓+ (ω + eV − ↓)² , (70)

Ey(ω, t) = (β0(ω) − β1(ω))Sy(t)(ω + eV − ε_↑) + Sx(t)Γ_↑

Γ²_↑+ (ω + eV − ε_↑)² (71)

+(β0(ω) + β1(ω))Sy(t)(ω + eV − ε_↓) − Sx(t)Γ_↓

Γ²_↓+ (ω + eV − _↓)² , (72)

Ez(ω, t) = β1(ω) + J Sz(t) (β0(ω) − β1(ω)) ω + eV − ε↑

Γ²_↑+ (ω + eV − ε_↑)² +(β0(ω) + β1(ω)) ω + eV − ε_↓

Γ²_↓+ (ω + eV − _↓)²

!

. (73)

3.3 Equations for the tunneling current

When deriving the equations for the tunneling current we need to divide our derivation into rst taking the case when t < t0, with no applied voltage, and then derive the current for t > t0, with applied voltage. For the case when t < t0, when we have no applied current, we get

I₀^C(t) = −4e

~Γ0Im ˆ

i(f (ω) − f (ω⁰)) Λ₀(ω⁰) +^J₂S_z(t)C₀(ω⁰)
ω⁰− ω + iδ

dω 2π

dω⁰

2π, (74)

I₁^C(t) = −4e

~ ΓSIm

ˆ

i(f (ω) − f (ω⁰)) Cz(ω⁰, t) ω⁰− ω + iδ

dω 2π

dω⁰

2π, (75)

I_0x^S(t) = −4e

~

J Γ0Im ˆ

i(f (ω) − f (ω⁰)) Cx(ω⁰, t) ω⁰− ω + iδ

dω 2π

dω⁰

2π, (76)

I_0y^S(t) = −4e

~

J Γ₀Im ˆ

i(f (ω) − f (ω⁰)) C_y(ω⁰, t) ω⁰− ω + iδ

dω 2π

dω⁰

2π, (77)

I_0z^S(t) = −4e

~ Γ0Im

ˆ

i(f (ω) − f (ω⁰)) Cz(ω⁰, t) ω⁰− ω + iδ

dω 2π

dω⁰

2π, (78)

I_1x^S(t) = −4e

~

J ΓSIm

ˆ (f (ω) − f (ω⁰)) Cy(ω⁰, t) ω⁰− ω + iδ

dω 2π

dω⁰

2π, (79)

I_1y^S(t) = +4e

~

J ΓSIm

ˆ (f (ω) − f (ω⁰)) Cx(ω⁰, t) ω⁰− ω + iδ

dω 2π

dω⁰

2π, (80)

I_1z^S(t) = −4e

~ΓSIm ˆ

i(f (ω) − f (ω⁰)) Cz(ω⁰, t) ω⁰− ω + iδ

dω 2π

dω⁰

2π, (81)

where δ is an innitesimal constant and

Cx(ω, t) = (Λ0(ω) − Λ1(ω))Sx(t)(ω − ε_↑) + Sy(t)Γ_↑

Γ²_↑+ (ω − ε_↑)² + (Λ0(ω) + Λ1(ω))Sx(t)(ω − ε_↓) − Sy(t)Γ_↓

Γ²_↓+ (ω − ε_↑)² , (82)

Cy(ω, t) = (Λ0(ω) − Λ1(ω))Sy(t)(ω − ε_↑) + Sx(t)Γ_↑

Γ²_↑+ (ω − ε_↑)² + (Λ0(ω) + Λ1(ω))Sy(t)(ω − ε_↓) − Sx(t)Γ_↓

Γ²_↓+ (ω − ε_↑)² , (83)

Cz(ω, t) = Λ1(ω) + J Sz(t) (Λ0(ω) − Λ1(ω)) ω − ε↑

Γ²_↑+ (ω − ε_↑)² + (Λ0(ω) + Λ1(ω)) ω − ε↓

Γ²_↓+ (ω − ε_↑)²

!

. (84) When calculating the current for the case t > t0, with an applied constant voltage, we need to have in mind the cases where t⁰ < t0 and t⁰ > t0 which yields dierent Green functions. By doing so we get the currents for t > t0

I₀^C(t) = −4e

~Γ0Im ˆ

i (f (ω) − f (ω⁰))

"

β0(ω⁰) +^J₂Sz(t)E0(ω⁰) ω⁰− ω + iδ + α0(ω⁰) +^J₂Sz(t)D0(ω⁰)

ω⁰− ω + eV + iδ e^{−ieV (t+t0)}−β0(ω⁰) +^J₂Sz(t)E0(ω⁰) ω⁰− ω + iδ

!

e^{−i(ω−ω0)(t−t0)}

#dω 2π

dω⁰ 2π(85),

I₁^C(t) = −4e

~ ΓSIm

ˆ

i (f (ω) − f (ω⁰))

 Ez(ω⁰, t) ω⁰− ω + iδ +

 Dz(ω⁰, t)

ω⁰− ω + eV + iδe^{−ieV (t+t0)}− Ez(ω⁰, t) ω⁰− ω + iδ



e^{−i(ω−ω0)(t−t0)} dω 2π

dω⁰

2π, (86)

I_0x^S(t) = −4e

~

J Γ₀Im ˆ

i (f (ω) − f (ω⁰))

 Ex(ω⁰, t) ω⁰− ω + iδ +

 Dx(ω⁰, t)

ω⁰− ω + eV + iδe^{−ieV (t+t0)}− Ex(ω⁰, t) ω⁰− ω + iδ



e^{−i(ω−ω0)(t−t0)} dω 2π

dω⁰

2π, (87)