Quantum transport in strongly interacting one-dimensional nanostructures

(1)

Quantum transport in strongly

interacting one-dimensional

(2)

(3)

Quantum transport in strongly

interacting one-dimensional

nanostructures

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magniﬁcus prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op vrijdag 11 december 2015 om 10:00 uur

door

Ricardo Rodrigo AGUNDEZ MOJARRO

Master of Science

van Technische Universiteit Delft, Nederland, geboren te Tijuana, Mexico.

(4)

Dit proefschrift is goedgekeurd door de promotor: Prof. dr. S. Rogge

promotor: Prof. dr. Y. M. Blanter Copromotor: Dr. M. Blaauboer Samenstelling promotiecommissie:

Rector Magniﬁcus, voorzitter

Prof. dr. S. Rogge Centre for Quantum Computation and Communication Technology, promotor Prof. dr. Y. M. Blanter Technische Universiteit Delft, promotor Dr. M. Blaauboer Technische Universiteit Delft, copromotor Onafhankelijke leden:

Prof.dr.ir. H.S.J. van der Zant Technische Universiteit Delft Prof.dr. J.-S. Caux Universiteit van Amsterdam Prof.dr.ir. H.J.W. Zandvliet Universiteit van Twente

Dr. R. Aguado Instituto de Ciencia de Materiales de Madrid Prof.dr.ir. L.M.K. Vandersypen Technische Universiteit Delft, reservelid

Keywords: Mesoscopic physics, nanoscience, Kondo, atomic chain, slave-boson, spin chain, quantum state transport. Printed by: Proefschriftmaken.nl || Uitgeverij BOXPress

An electronic version of this dissertation is available at

(5)

Para mis padres, Ricardo y Yolanda, quienes durante toda mi vida han apoyado ciegamente mis sueños y aspiraciones.

(6)

(7)

1

Introduction

1.1 Effects studied

In this section we will cover the basic principles of the different effects studied during my research. The Aharonov-Bohm effect [Section1.1.1] and the Fano effect [Section 1.1.2] reveal the wave-particle duality nature of the electron. Both effects shine light onto one of the most intrinsic characteristics of the electron: interference. Interference has allowed physicists to observe a foot-print of the wave like behavior of electrons, and show that at very small scales reality is very different due to quantum mechanics.

At this small scales, especially in one-dimensional systems, interactions play one of the most signiﬁcant roles in the description of electrons. The Kondo eﬀect [Section 1.1.3] demonstrates that interactions can govern the complete physical description of a system and produce unexpected results.

Exciting phenomena can be studied when diﬀerent characteristics of the electron in such small dimensions play a role. Suppose we have a Kondo system, then it turns to be very interesting to check if the wave like like behavior of the electrons is lost, to see if interference is possible between two Kondo systems and if this Kondo system can interfere with another type of electronic system. We can use the Fano-Kondo eﬀect to address this type of questions as it reveals an interference footprint of the electrons.

1.1.1 Aharonov-Bohm effect

In classical mechanics, the motion of electrons is governed by applied electric ( ~E) or magnetic ﬁelds ( ~B). If the electron travels along a path where ~E and ~B

(10)

④ ④

1

2 1. Introduction

are zero, the fields should not affect the electron motion. The Aharonov-Bohm effect demonstrates the opposite. In quantum mechanics, the interference be-tween two paths the electron might take depends on the magnetic flux (Φ) enclosed by the trajectories, even if ~B = 0 in both paths [Fig. 1.1]. The mag-netic flux couples to the electron wavefunction via the phase and can produce constructive or destructive interference depending on Φ.

Figure 1.1: Magnetic Aharonov-Bohm effect. An electron’s wavefunction is split in two,

passing around a shielded magnetic field, then it recombines forming an interference pattern. The effect of the magnetic field is to produce a phase-shift in the electron’s wavefunction. The figure was taken from reference [1].

Historic background

The Aharonov-Bohm effect was first predicted by Ehrenberg and Siday [2] in 1949. A few years later the effect was theoretically re-introduced by Aharonov and Bohm [3] in 1959, and finally experimentally proven in 1960 by Chambers [4]. Solid evidence came in the 1980’s with far more refined experiments by Tonomura et al. [5, 6], where the magnetic field was zero and the magnetic potential was non-zero in the path of the electron.

The discovery of the Aharonov-Bohm effect was one of the cornerstones of the evolution of modern physics. In the 18th and 19th centuries the descrip-tion of modescrip-tion was heavily dominated by Newtonian physics where a force is necessary to create motion. Electromagnetism was first viewed as a spooky phenomenon. Then the description of magnetic and electrical fields came to light and the reconciliation with Newton came with the help of the Lorentz force law ~F = e[ ~E + (~v × ~B)]1_{. Around the same time, the concepts of}

(11)

1.1. Effects studied 3

④ ④

1

netic potential ( ~A) and electrostatic potential (V ) took great importance from

rewriting the famous classical Maxwell equations as a gauge theory. The rela-tions between the potentials and the ﬁelds are expressed as: ~B = ~∇ × ~A and

~

E = −~∇V −δ ~A

δt. The potentials were ﬁrst seen as a mathematical construct

with no physical repercussions until the introduction of quantum mechanics. The Aharonov-Bohm effect showed that the full information of an electromag-netic field is not contained in ~E and ~B but rather in V and ~A. The most conclusive observation was that even if ~E = 0 and ~B = 0 in the whole region where the electron travels, such that there is no Lorentz force applied, the dynamics could still be affected by the electromagnetic field. The electron’s motion couples to the potentials V and ~A, which are not necessarily zero.

Even if the role of potentials in quantum theory remains under debate [7,8], the applications of the effect have exponentially increased over the years. The effect has been observed in a variety of different systems, such as non-superconducting metallic rings [9], carbon nanotubes [10], graphene rings [11,

12], graphene [13] and very recently in photons [14]. It has also created a standard and well understood way of studying interference in diﬀerent systems.

Description

Let us describe the magnetic Aharonov-Bohm eﬀect in its simplest form, so we can grasp the very basics of its dynamics. Suppose we have a magnetic ﬁeld ~B going through the page inside an area S [Fig. 1.2]. ~B 6= 0 inside S and

~

B = 0 elsewhere. A ring in the plane of the page surrounds S. Electrons ﬂow from left to right through the ring [Fig. 1.2]. Classical electrodynamics tells us that since in the electron’s path ~B = 0 there should not be an interaction between the electron and the magnetic ﬁeld. Classically the electron would travel through one arm of the ring and arrive unperturbed on the right side. We start by setting ~B = 0 everywhere (also inside S) and by writing the incoming electron as a traveling plane wave Ψ = eikx_{. Once the electron}

reaches the ring (point P1 in Fig. 1.2), part of the wavefunction will travel

through path 1 and another part through path 2: Ψ = a1eikx+ a2eikx. For

simplicity let us suppose the wavefunction splits equally, a1= a2= a. At the

end of the ring (point P2in Fig. 1.2) the wavefunction will recombine and the

electron’s wavefunction reads Ψ = [eikl1_{+ e}ikl2_]aeikx_{, where l}

1 and l2 are the

lengths of paths 1 and path 2 respectively.

We repeat the same motion but now we turn on the magnetic ﬁeld inside S. In the ring ~B = ~∇ × ~A = 0 but A 6= 0. Then the electron’s wavefunction will obey Schrödinger’s equation:

1

2m(i~~∇ + e ~A)

(12)

④ ④

1

4 1. Introduction

Figure 1.2: Schematic picture of an Aharonov-Bohm ring enclosing a magnetic field ~B.

~

B_{6= 0 inside S and zero elsewhere. An electron travels from left to right inside the ring and}

a finite magnetic potential adds a phase to the electron. The electron’s wavefunction splits in P1 and recombines in P2.

The solution of Eq. (1.1) can be written as Ψ = eig(x)_Ψ′_{, where Ψ}′ _{is the}

solution without the potential ~A. This means that an electron traveling along a path with non-zero potential ~A acquires a phase g(x) given by,

g(x) = e ~ Z x 0 ~ A(x′) · dx′. (1.2)

Then we can write the full solution of Eq. (1.1) at the end of the ring as2

Ψ = [ei(kl1+g(l1))_{+ e}i(kl2+g(l2))_]aeikx

= 2aeik(l1+l2)+g(l1)+g(l2)2 eikxcos2

h

k(l1− l2) +g(l1) − g(l2)

2 i

. (1.3) The magnetic phase diﬀerence g(l1)−g(l2) can be easily calculated using Stokes

theorem3_{and by taking a closer look at the path integrals}4

g(l1) − g(l2) = e ~ Z l1 0 ~ A(x′_{) · dx}′₋e ~ Z l2 0 ~ A(x′_{) · dx}′ = e ~ I Ring ~ A(x′_{) · dx}′₌ e ~ I S ~ B · ds′ ₌eΦ ~ . (1.4)

2_{Here we have used the identities: cos(u) + cos(v) = 2 cos (}u+v

2 ) cos (u−v2 ) and sin(u) + sin(v) = 2 sin (u+v₂ ) cos (u−v₂ ).

3_{Stokes theorem allows us to convert the line integral into a surface integral:}H

S∇ × F · dS =

H rF· r

4_{We can picture the integral −}e

~

Rl2 0 A(x

′_{) · dx}′ _{as the electron going through path 2 in the} opposite direction, then the sum of the paths will give a closed path around the ring B.

(13)

④ ④

1

Interference can thus be tuned by changing the magnetic ﬂux Φ.

Mathemati-cally, the amplitude of the wavefunction is modulated by the length diﬀerence of the arms of the ring but more importantly by the magnetic ﬂux produced by ~B, |Ψ|2_{∝ cos}2h_k(l 1− l2) + e 2~Φ i . (1.5)

The fact that the amplitude of the wavefunction can be modulated by the magnetic ﬁeld means that the transmission of the system can also be modulated. Using the scattering matrix formalism [section1.2.1] we can relate this transmission to conductance, then an extensive amount of setups can be probed in an Aharonov-Bohm structure type by measuring the conductance over the ring. This is the approach we take in chapter1and chapter2, where we study the impact of interference between two systems in an Aharonov-Bohm conﬁguration, on transport.

1.1.2 Fano effect

We can think of a resonance as the response of a system to an external exci-tation at a particular frequency. As the external frequency of the exciexci-tation approaches to the system’s natural frequency the amplitude of the response increases, until it reaches its maximum value when both frequencies are equal. In this situation the system is said to be on resonance. The proﬁle of the re-sponse amplitude versus the external frequency is expected to be symmetric, described by a Lorentzian function. This resonance is known as a Breit-Wigner resonance. The proﬁle of the resonance can also be asymmetric, this reveals an underlying mechanism in the system’s response. Such asymmetric lineshape is called Fano resonance. In particular, Fano resonances in conductance result from the interference between scattering within a continuum of states (back-ground process) and scattering by means of a discrete energy level (resonant process).

The study of scattering of matter has been one of the most important tools to understand the world around us. Since the description of Rydberg spectrum lines of Hydrogen in 1888 which provided the basis for Niels Bohr atomic model in 1913, the understanding of scattering has been crucial. The story of the Fano eﬀect starts with the observation of asymmetric Rydberg spectral lines in the photo-absorption experiments in noble gases performed by Beutler [15] in 1935. It was that same year when Fano [16] suggested a theoretical explanation of this eﬀect. Fano argued that the spectral absorption lines were a result of the interaction of a discrete excited state of an atom with a continuum sharing

(14)

④ ④

1

6 1. Introduction

the same energy level. Since then, what are now called Fano resonances, have been observed in many diﬀerent physical systems and extensively studied in nanoscale systems [17]. Much of the interest in Fano resonances comes from the idea to use them as a tool to study coherence and to provide valuable transport information [18, 19] that may result in future nanoscale device applications [20–22].

Description

A second far more rigorous work from Fano came in 1961 [23]. His derivation can be put into a simple equation expressing the resonant proﬁle of a scattering cross-section as

σ = (ǫ + q)

2

ǫ2_{+ 1} . (1.6)

The parameter ǫ is the detuning between the applied frequency of an external excitation and the natural frequency of the system. The parameter q, also known as the Fano parameter, is the ratio between the transition probability through the system and the transition probability through the continuum. Let us take a look at a speciﬁc simple example related to the work in chapters2

and3 . The response that we are interested in describing is the transmission T of a system with one energy level Eo(natural frequency of the system). The

frequency of the external excitation will then be the energy of the incoming electron (E). In this system ǫ = 2(E − Eo)/Γ where Γ is proportional to

the interaction between the system and the incoming electron. In a transport context, Γ is the width of the resonant energy level. Since we are interested in observing how T responds to an external perturbation E, we rewrite the Fano formula [Eq. (1.6)] as T ∝ (E−Eo+qΓ/2)2

(E−Eo)2+(Γ/2)2. T still needs to be normalized

since the maximum transmission possible is 1. The maximum transmission is Tmax = 1 + q2 when E − Eo = _2qΓ. We then ﬁnd the complete form of

transmission T = 1 1 + q2 (E − Eo+ qΓ/2)2 (E − Eo)2+ (Γ/2)2 . (1.7)

The value of the Fano parameter q reveals the true nature of the scattering process. As we mentioned, q is the ratio between the probability of scattering through the resonant level and the probability of scattering through a back-ground process. If there is no scattering through the backback-ground process we expect q → ∞, and indeed we recover the known Lorentzian transmission through a resonant energy level [magenta line in Fig. 1.3]

T = (Γ/2)

2

(E − Eo)2+ (Γ/2)2

(15)

1.1. Effects studied 7 ④ ④

1

−2 0 2 0 0.2 0.4 0.6 0.8 1 Energy (E − E o) / Γ Transmission T q = 0 q = 1 q = ∞

Figure 1.3: We show the different types of transmission lineshapes predicted by the Fano

formula [Eq. (1.3)]. A Lorentzian transmission when the resonant channel dominates

(ma-genta), a symmetrical dip when the scattering thorugh the continuum dominates (black) and an asymmetric profile when both scattering mechanisms contribute (green).

If both processes contribute to the scattering q takes a ﬁnite value and the shape of the resonance becomes asymmetric with only one maximum and one minimum [green line in Fig. 1.3]. q = 1 corresponds to the special case were both scattering processes contribute equally and the resonant energy level of the system Eo lies in the middle of the maximum and minimum of the

transmission. One interesting characteristic of the Fano interference is the case when q = 0, here the Fano formula describes an antiresonance which corresponds to a completely destructive interference at the resonant energy level [black line in Fig. 1.3]

T = (E − Eo)

2

(E − Eo)2+ (Γ/2)2

. (1.9)

The Fano parameter q permits an easy understanding of the interference between these two scattering processes. It is true that q is not directly an experimentally observable parameter, but if a system’s conductance can be written in a Fano form, we can manipulate our system by understanding the relation between the experimental knobs and q. In chapter 3 we will use the Fano description of conductance to study the interference between a Kondo system and a resonant system in an Aharonov-Bohm ring.

(16)

④ ④

1

8 1. Introduction

1.1.3 Kondo effect

One of the most fascinating manifestations of many-body physics in mesoscopic systems is the Kondo eﬀect. Suppose there are electrons passing through a metal, due to the thermal energy the metal atoms will vibrate, increasing the scattering with the electrons. Such atom vibrations decrease for decreas-ing temperature, makdecreas-ing it easier for electrons to transverse the metal. In a transport measurement the latter translates into a drop of resistance with decreasing temperature. This was observed in most materials until measure-ments in metals with magnetic impurities produced contrasting results. There, it was observed that the resistance increases once the temperature drops under a threshold called the Kondo temperature (TK) [Fig. 1.4(c)]. The increase in

resistance with decreasing temperature in metals containing magnetic impuri-ties is called the Kondo eﬀect.

Decades after the observation of the Kondo effect in metals, it was discov-ered in quantum dots. The physics behind the effect in both type of systems is basically the same, but the manifestation of the effect is completely opposite. Suppose we have an electron trapped in a quantum dot. If temperature is increased the electron will gain thermal energy and the probability of escaping the quantum dot will increase, therefore increasing conductance. Then it fol-lows that conductance should decrease with decreasing temperature [red curve in Fig. 1.5(d)]. This is in general the case, but for temperatures smaller than a threshold (Kondo temperature) the conductance starts increasing with de-creasing temperature, contrary to our intuition [blue curve in Fig.1.5(d)]. The Kondo effect manifests itself in quantum dots as an increase in conductance with decreasing temperature.

Before the observation of the Kondo eﬀect, the picture of resistivity in met-als was incomplete. Electron-phonon scattering was known to be the main dictator of the electrical resistivity in metals. This is the interaction between the conduction electrons and the nuclei as they vibrate around their equilib-rium position. If temperature drops such that there are no vibrations, we are left with a temperature independent residual resistivity given by the scatter-ing of the electrons with any defects or impurities in the metal. However in 1934 de Haas et al. [24] reported the measurement of a resistance minimum in gold. Gold showed a rise in resistivity when the temperature decreased be-low 10 Kelvin. Folbe-lowing de Haas’ observation, other metals showing similar behavior came to light. Surely the understanding of scattering in metals was incomplete. The scattering mechanism behind the strange behavior remained in the shadows for almost 3 decades. It was suspected that magnetic

(17)

impuri-1.1. Effects studied 9

④ ④

1

ties in the metallic hosts were the cause for this eﬀect. Experimental studies

by Van der Berg [25] and Sarachik et al. [26] in 1964 showed the correlation between the resistance minimum and the number of magnetic impurities. Fol-lowing the strong evidence that magnetic impurities were responsible, in that same year Kondo [27] showed in detail how the spin of such magnetic impurities could coupled to the spin of the conduction electrons and lead to a logarith-mic behavior of resistivity. This logarithlogarith-mic dependence log(T ) provided a satisfactory explanation of the resistance minimum. However this extra con-tribution diverges when T → 0 and extensions of the approach where provided by Abrikosov [28], Wilson [29], Costi et al. [30].

More than 60 years since its first observation, the Kondo effect was redis-covered in a new exciting system, the quantum dot[31, 32]. Quantum dots proved to be ideal testbeds for the understanding of mesoscopic physics and the Kondo effect became an intense topic of research [33]. Since its discov-ery in quantum dots in the late nineties, the Kondo effect has been observed and studied in a variety of exciting mesoscopic systems: individual atoms [34], carbon nanotubes [35, 36], molecules [37, 38], bucky balls [39] and graphene [40].

Kondo effect in metals

Here we present a simpliﬁed picture of the dynamics of the Kondo eﬀect. We begin by considering a magnetic impurity in a metal and conduction elec-trons moving around and colliding with the impurity. We describe the system impurity-electron as |k, σe, σIi, where k is the momentum of the electron and

σe(σI) is the spin of the electron(impurity). After a collision we consider two

types of results. The ﬁrst is that the electron is scattered to momentum k′_,

resulting in a ﬁnal system state |k′_{, σ}

e, σIi. We write this Hamiltonian

inter-action as Hk= Jk|k′i hk|. The electron and the impurity can also experience

a spin ﬂip, described by the interaction Hs= Js|k, ¯σe, ¯σIi hk, σe, σI|. We take

Jk = Js= J, the exchange interaction between the conduction electrons and

the impurity. We now calculate the scattering probability of an electron going from k to k′_{. The matrix element of a ﬁrst order process is}

M(1)= hk′, σe, σI| Hk|k, σe, σIi = J (1.10)

The scattering event can also happen in a second order process. The system ﬁrst goes to an intermediate state |k, σe, σIi → |k′′, ¯σe, ¯σIi and then to the

(18)

④ ④

1

10 1. Introduction

(a)

(b)

(c)

Figure 1.4: A magnetic impurity in a sea of conduction electrons. (a) Temperature is

above the Kondo temperature. (b) Temperature is below TK and the conduction electrons

screen the impurity thereby effectively enhancing the scattering cross-section. (c) Resistance increases when temperature is below the Kondo temperature. Figure taken from reference [41].

ﬁnal state |k′_{, ¯}_σ

e, ¯σIi → |k, σe, σIi. We write the matrix element as5

M(2)=X k′′ J21 − Fk′′ ǫk− ǫk′′ = J2ρ Z D ǫF dǫk′′ ǫk− ǫk′′ = J2ρ log ǫk− ǫF ǫk− D . (1.11)

The scattering probability for these two processes is

M(1)_{+ M}(2)

2

. In a transport experiment the state with the incoming energy ǫk needs to be

oc-cupied, and the state with the ﬁnal energy ǫk′ needs to be empty. Therefore

the main contribution comes from energies ǫk near the Fermi level ǫF within

a window of about kBT , then as an approximation we take |ǫk− ǫF| ≈ kBT .

5_{The Fermi function F}

kgives the occupation number of the state with energy ǫk.

We take the intermediate state k′′ _{to have an energy between 0 and D, and the states}

below ǫF to be occupied.

The sum over intermediate states k′′_{has been replaced by an integral using the density of} states ρ(k′′_{) which we take constant in the window of integration.}

(19)

④ ④

1

If we consider an antiferromagnetic exchange interaction J < 0, we ﬁnd the

resistivity of the system as

R ∝ |J|2+ 2 |J|3ρ log D − ǫF− kBT kBT . (1.12)

We notice that only an antiferromagnetic coupling between the conduction electrons and the impurity gives an increment of resistance when the temper-ature decreases. A physical interpretation of this result is that due to this antiferromagnetic coupling, the conduction electrons try to screen the impu-rity spin, forming a singlet system impuimpu-rity-electrons. If temperature is low and does not break this singlet, then electrons couple strongly to the magnetic impurity forming what is called a Kondo cloud and eﬀectively increase the scattering area [Fig. 1.4(b)].

Kondo effect in quantum dots

The Kondo eﬀect in quantum dots follows a similar story but with contrasting results. Imagine a quantum dot with a charging energy U coupled to two electric leads such that the highest energy level (ǫd) in the quantum dot con

only be singly occupied: ǫF − U < ǫd < ǫF (electrons in the left lead do not

have energy enough to pay the charging energy). In this situation the quantum dot is Coulomb blocked [Fig. 1.5(a)]. Since no electron can tunnel into the quantum dot, no current should flow through the structure. One way for the electron to gain additional energy is thermal; that is why current can be finite and increases for increasing temperature. Let us work in the realm of low temperatures and with a zero bias voltage applied across the system. Another way around this expensive energy cost U is by using an intermediate virtual state [Fig. 1.5(b)]. In principle, many configurations are then possible, but we turn our attention to the ones where the spin of the quantum dot is flipped. One of these processes can occur if the electron with the highest energy in the left lead ǫL ≈ ǫF and spin ↓ tunnels onto the quantum dot [Fig. 1.5(a)-(b)].

Immediately afterwards, the electron in the quantum dot with energy ǫd and

spin ↑ tunnels out to the right lead into energy ǫR= ǫL≈ ǫF [Fig. 1.5(b)-(c)].

After this process an electron has been eﬀectively transferred from left to right giving a ﬁnite current. Energy is not conserved in the intermediate state of this process [Fig. 1.5(b)] since the electron from the left lead tunneled to a higher energy level Ev= ǫd+ U . This imposes a restriction over the time the system

can remain in this intermediate state, hence the name virtual state. In order to not violate the energy conservation principle, the system may remain in this virtual state during a time interval dictated by the Heisenberg uncertainty principle, tH= _ǫ ~

(20)

④ ④

1

12 1. Introduction

(a)

Figure 1.5: Kondo effect in quantum dots. (a) The quantum dot is Coulomb blocked

such that an electron cannot be transferred in a first order process. (b) An electron near the Fermi level with spin down enters the quantum dot forming a virtual state. (c) The electron with spin up in the quantum dot tunnels to the right lead within a time given by the Heisenberg uncertainty principle. An electron has effectively been transferred and the spin of the quantum dot has been flipped. (d) Conductance increases when temperature is

below the Kondo temperature (blue curve). Based on figures from reference [33].

There are more processes that eﬀectively ﬂip the spin in the quantum dot. Under a few assumptions for a quantum dot with odd occupancy these interactions can be approximated by an antiferromagnetic coupling

JSe· Sd. (1.13)

Here Sd is the total spin of the quantum dot and Se is the sum of the spin

operators for the electrons in the Fermi leads. The ground state then consists of a singlet with a binding energy equal to the Kondo temperature. Since the Kondo interaction predominantly aﬀects the electrons at the Fermi level, the Kondo singlet appears in the density of states as a narrow resonance pinned at the Fermi energy, if the singlet binding energy is stronger than the thermal energy.

(21)

1.2. Techniques used 13

④ ④

1

explained experimental data with the use of a model consisting of two

inter-fering Kondo transmission channels. In chapter 3 we developed a theoretical approach to describe the interaction of a Kondo scattering process with reso-nant scattering and in chapter4we study the eﬀect of disorder in the context of atomic chains in the Kondo regime.

1.2 Techniques used

1.2.1 Scattering matrix formalism

The scattering matrix formalism allows us to relate the initial state and the final state of a physical system undergoing a scattering process. The purpose of the formalism is to relate the conductance to the reflection and transmission probabilities through the system. Imagine a narrow layer of material with a width W and a length L, such that W << L. Following the classical picture of Ohm’s law we will write the conductance as G = σW_L, where σ is the conductivity of the system. Then we would expect the conductance to increase as we reduce the length. Experiments showed that below a certain length, conductance was quantized as shown below in Fig. 1.6(a) [45]. Clearly the simple classical description at these small scales is not sufficient. The scattering matrix formalism solves this problem with the use of quantum mechanics.

(a)

(b)

Figure 1.6: (a) Conductance versus gate voltage data from the experiment by van Wees

et al. [45]. Conductance is quantized in units of 2e_h2. As gate voltage becomes less negative the width increases allowing more modes to pass through. (b) Schematic representation of electronic transport in the scattering matrix picture. Two leads with chemical potentials

µL(R) are attached via a ballistic conductor containing a scattering region. The electron

travels as a plane wave until it reaches the scattering region where it can be reflected or transmitted.

(22)

④ ④

1

14 1. Introduction

Formalism

Imagine a system that consists of a scattering region between two ballistic conductors. In the ballistic conductors the electron can be described by a traveling plane wave [Fig. 1.6(b)]. The electron travels freely until it reaches the scattering region and is reflected or transmitted. Suppose we work in one-dimension: then reflected wave will have a momentum opposite to the incoming one. The complete wavefunction will then be the sum of the incoming and reflected wave, ΨL= aLeikx+ bLe−ikxand ΨR= bReikx+ aRe−ikx, where

ΨL and ΨR are the wave functions in the left and right side of the scattering

region, respectively [Fig. 1.6(b)]. We can relate the incoming and outgoing amplitudes by the scattering matrix (S) as

bL bR = Sa_aL R =r t ′ t r′  aL aR , (1.14)

where r(r′_{) and t(t}′_{) are the reﬂection and transmission of a wave}

incom-ing from the left(right). These coeﬃcients are determined by solvincom-ing the Schrödinger equation in each of the regions and imposing continuity of the wavefunction and its ﬁrst derivative at the interfaces of the scattering region. S is unitary if we apply particle conservation during the scattering and S is symmetric if our system has time-reversal symmetry.

The S matrix gives us the transmission for a particular incoming wave. We now need to relate this transmission to the current. The Landauer-Büttiker formalism allows us to establish such a relation. As Fig. 1.6(b) shows, elec-trons in the ballistic conductor come from the leads were they are distributed according to the Fermi distribution function

Fα(ǫ) =

1 1 + exp[ǫ−µα

kBT ]

, (1.15)

where α indicates the left (L) or right (R) lead, µαis the corresponding

chem-ical potential and T is the temperature. Imagine there is no scattering region and the leads are attached via a ballistic conductor. Then the current going into the right lead will just be the diﬀerence between the number of elec-trons going into the lead minus the number of elecelec-trons going out of the lead: IR = _heR dǫ[FL(ǫ) − FR(ǫ)]. We now add a scattering region, then

each electron has an associated transmission and we rewrite the current as IR = ehR |t(ǫ)|

2

dǫ[FL(ǫ) − FR(ǫ)]. If the width of such a scattering region is

small then the wavefunction inside the region will be quantized such that only a few modes can go through. Hence, only these available modes add to the

(23)

④ ④

1

current. Then the the current is [44]:

IR= e h X n Z |tn(ǫ)|2[FL(ǫ) − FR(ǫ)]dǫ. (1.16)

If we consider small temperatures T → 0 the current becomes IR= _hePn

RǫF+eVL

ǫF−eVR |tn(ǫ)|

2

dǫ. For a small bias voltage, all energies ǫ will be near the Fermi level, and the

transmission in the energy integration window can be considered to be constant tn(ǫF). Performing the integration we ﬁnd IR = _he(eVL+ eVR)Pn|tn(ǫF)|2.

Then the conductance G = I

V becomes G =e 2 h N X n |tn(ǫF)|2. (1.17)

Eq. (1.17) is known as the Landauer-Büttiker formula. In the experiment of van Wees et al. [45] shown in Fig. 1.6(a), the scattering region consisted of a ballistic conductor with variable width (W ). The width was controlled by an applied gate voltage. Since the scattering region is a ballistic conductor we have unitary transmission. From Eq. (1.17) the conductance is G = e2

hN ,

where N is the number of available modes given by N = Integer[2W_λ_F ]. There-fore by changing the width we can include or exclude one mode, and counting for spin degeneracy this results in steps of 2e2

h = e2

π~ [Fig. 1.6(a)].

By writing the electron as an incoming and reﬂected traveling wave, we solve the Schrödinger equation of our system and use the scattering matrix formalism to calculate the conductance. Together with the slave-boson tech-nique [Sec. 1.2.2] we applied this formalism in chapter2, chapter3and chapter

4to calculate conductance.

1.2.2 Slave-boson mean field theory

Strongly interacting systems cannot be solved using perturbation theory, there-fore special techniques need to be applied in order to find an accurate solution to the wavefunction of the system. The Kondo effect [Sec. 1.1.3] is a result of strong interactions. The main features in quantum dots are the enhancement of conductance at zero bias, its specific temperature dependence, a conductance peak splitting in a magnetic field and the zero-bias maximum in the differential conductance. Different numerical and analytical methods have been developed to explore its physical properties. Some examples are: Bethe-ansatz [46], the equation of motion method [47], second order perturbation theory [48, 49], the non-crossing approximation [50], renormalization group methods [30] and

(24)

④ ④

1

16 1. Introduction

slave-boson mean-field theory (SBMFT) [51]. In this thesis the study of the Kondo effect has been done by applying the slave-boson formalism, which we describe below. There are different types of SBMFT. In this section I present the Coleman approach [52] and the Kotliar-Ruckenstein approach [53]. The Kotliar-Ruckenstein formalism takes the Coulomb interaction to be finite, the Coleman approach can be seen as a simplified version by taking the limit U → ∞. The SBMFT describes the features of the Kondo effect and also allows us to study systems out of equilibrium in the Kondo regime [51].

The SBMFT introduces bosonic operators to describe the states of the quantum dot and establishes strict rules between these boson operators and the fermion operators. The boson operators act when the fermion operators act, hence the name slave-boson. The original Hamiltonian is then written as a mix of these two types of operators plus the respective constraints over the bosonic operators. Then the boson operators are replaced by their expectation values, arriving at a non-interacting Hamiltonian. This non-interacting Hamiltonian depends on the boson expectation values which can be obtained by solving their equations of motion. With all the information in place it is possible to diagonalize such a non-interacting Hamiltonian and ﬁnd the solution for the strongly interacting system.

In the following we present a simple model of a QD attached to two leads described by the Anderson single impurity model:

H = HL+ X σ ǫdσc†dσcdσ+ V X σ,kα [c†_dσckασ+ c † kασcdσ] + U nd↑nd↓. (1.18)

Here HLis the Hamiltonian of the leads, ǫdσis the energy level in the quantum

dot, V is the coupling between the quantum dot and the reservoirs, U is the Coulomb interaction in the quantum dot, cdσ(c†dσ) and ckασ(c

†

kασ) are the

creation(annihilation) operators in the quantum dot and the leads respectively. nσ is the occupation operator for for spin σ. In the following we treat the

simplest structure, one quantum dot, but the formalism is easily expandable to more complicated systems.

Coleman formalism: U → ∞

In this formalism we consider the Coulomb interaction to be inﬁnite: double occupation in the quantum dot is then forbidden. We start by introducing the boson operator e(e†_{) where the state e}†_{|0i represents the empty state in the}

quantum dot. We then have to create a boson when removing an electron from the quantum dot to leave it in the empty state, this means that in Eq. (1.18) we need to make the substitution cdσ→ cdσe†. To impose a forbidden double

(25)

1.2. Techniques used 17 ④ ④

1 (a)

(b)

Coleman Kotliar-Ruckenstein

(c)

Figure 1.7: (a) Calculations using the Kotliar-Ruckenstein formalism for U = 7. Linear

conductance at zero temperature (orange line) and the electron occupation number in the quantum dot (blue) versus the position of the energy level in the quantum dot. Green dots

correspond to experimental data from Goldhaber-Gordon et al. [54]. (b) Comparison

be-tween the infinite U Coleman formalism and the finite U = 4 Kotliar-Ruckenstein formalism. (c) Zero temperature differential conductance as a function of the external bias voltage ap-plied for different strengths of the magnetic field h ∝ |B|. Figures adapted from references [55,56].

occupation we set the constraint e†_{e +}P

σc †

dσcdσ = 1. Then our Hamiltonian

[Eq. (1.18)] is written as H = HL+ X σ ǫdσc†dσcdσ+ V X σ,kα [ec†_dσckασ+ e †_c† kασcdσ] + γ[e†e +X σ c†dσcdσ− 1]. (1.19)

Here the constraint has been added by means of a Lagrange multiplier γ. We then use the slave-boson mean-ﬁeld approximation and treat the boson oper-ators as real numbers. By doing this we arrive at an eﬀective non-interacting

(26)

④ ④

1

18 1. Introduction Hamiltonian, Hef f = HL+ X σ ˜ ǫdσc†dσcdσ+ ˜V X σ,kα [c†_dσckασ+ c † kασcdσ] + γ(e2_{− 1),} _(1.20)

where ˜ǫdσ = ǫdσ + γ and ˜V = V e. In Eq. (1.20) the energy level of the

quantum dot and its coupling to the leads has been renormalized. In order to diagonalize Eq. (1.20) we need the values γ and e, and we then obtain their equations of motion by minimizing the energy. This leads to the following coupled equations, ∂hHef fi ∂γ = e 2₊X σ hc†dσcdσi − 1 = 0 ∂hHef fi ∂e = 2γe + V X σ,kα [hc†dσckασi + hc † kασcdσi] = 0. (1.21)

As we see in Eqs. (1.21) the fermion expectation values are needed to solve for γ and e. Then we need to self-consistently solve Eqs. (1.21) and diagonalize Eq. (1.20) via the scattering matrix formalism [Sec. 1.2.1]. From the scattering matrix formalism transmission can be calculates which then can be related to conductance. The Coleman approach captures some of the essence of the Kondo effect, Fig. 1.7(b) shows how the conductance increases to unity for the quantum dot in the Coulomb blockade regime. In Fig. 1.7(b) we see some difference with the more sophisticated Kotliar-Ruckenstein approach. In particular, features related to the nature of a finite U [as in Fig.1.7(a)] cannot be captured by this approach.

In chapters2and3we use the Coleman formalism to diagonalize the system Hamiltonian and calculate the conductance of the system in the Kondo regime.

Kotliar-Ruckenstein formalism: Finite U

Here we present a simple derivation of the Kotliar-Ruckenstein formalism. A complete description can be found in reference [56]. This formalism follows a similar procedure as the Coleman formalism but in this case since U is ﬁnite we need more bosonic operators to represent the other states in the quantum dot. In fact this formalism uses a total of 4 boson operators acting on the quantum dot: e(e†_{), p}

σ(p†σ) and d(d†). Similar to the Coleman description,

the state e†_{|0i represents the empty state in the quantum dot. p}†

σ|0i

(27)

④ ④

1

now the doubly occupied state. We immediately notice that we can

substi-tute nd↑nd↓ → d†d in the original Hamiltonian Eq. (1.18). Now when we

create or annihilate a fermion in the quantum dot we also need to create or annihilate bosons. This is accomplished by the substitution cdσ → cdσzσ,

where zσ = e †_p σ+p†¯σd √ 1−d†_d−p† σpσ p 1−e†_e−p† ¯ σp¯σ

. In this formalism the constraints for the bosonic operators are the completeness relation and the correspondence condition between fermions and bosons:

e†_{e +}P

σp†σpσ+ d†d = 1

c†_dσcdσ = p†σpσ+ d†d σ =↑, ↓ . (1.22)

Then our original Hamiltonian [Eq. (1.18)] can be written as

H = HL+ X σ ǫdσc†dσcdσ+ V X σ,kα (zσc†dσckασ+ z † σc†kασcdσ) + U d †_d +γ(e†e +X σ p†σpσ+ d†d − 1) + X σ λσ(c†dσcdσ− p†σpσ− d†d). (1.23)

Here the constraints have been added by means of three Lagrange multipliers γ and λσ. We then use the slave-boson mean-ﬁeld approximation and treat

the boson operators as real numbers. By doing this we arrive at an eﬀective non-interacting Hamiltonian, Hef f = HL+ X σ ˜ ǫdσc†dσcdσ+ X σ,kα ˜ Vσ(c†dσckασ+ c † kασcdσ) + γ(e2₊X σ p2 σ+ d2− 1) − X σ λσ(p2σ+ d2) + U d2, (1.24)

where ˜ǫdσ = ǫdσ + λσ and ˜Vσ = V zσ. The energy level of the quantum dot

and its coupling to the leads has been renormalized. In order to diagonalize Eq. (1.24) we need the Lagrange multipliers and the boson expectation values. These are obtained from the equations of motion [Eq. (1.25)] by minimizing the energy. This leads to the so-called KR equations:

∂ ∂e, ∂ ∂pσ , ∂ ∂d, ∂ ∂γ, ∂ ∂λσ hHef fi = 0. (1.25)

We solve Eq. (1.25) and diagonalize Eq. (1.24) in self-consistent way. In con-trast to the Coleman approach this formulation can give qualitative solutions for the whole range of the quantum dot energy level as seen in Fig. 1.7(a).

(28)

④ ④

1

20 1. Introduction

Since the Lagrange multiplier associated with the renormalization of the en-ergy level (λσ) is spin dependent, it can also be used to described the spin level

splitting by magnetic ﬁelds [Fig. 1.7(c)] [55]. This formalism can also treat a system out of equilibrium as observed in Fig. 1.7(c) where a bias voltage is applied.

In chapter4 the Kotliar-Ruckenstein formalism is used to study the inﬂu-ence of disorder on conductance in the Kondo regime. In that study we set the energy level of the quantum dot in the middle of the Coulomb blockade and proceed to introduce disorder. We thus need a description of the eﬀect for the whole spectrum of energy levels.

1.3 Systems studied

1.3.1 Hubbard model

One of the most important models in condensed-matter physics is the Hub-bard model. It started the theoretical treatment of strongly correlated systems by explaining several unresolved phenomena. The most outstanding result, that gave great importance to the model, was the prediction of the Mott insulator-metal transition, which could not be understood in terms of conven-tional band-theory [57]. Since its appearance and study in 1963 by Gutzwiller [58], Kanamori [59] and Hubbard [60], whom arrived to the model indepen-dently, it has had numerous applications thanks to its simplicity which allows to translate a physical system into the model and also because it captures the essence of the phenomenon. The Hubbard model is an extension of the non-interacting tight-binding model, where electrons can hop between lattice sites with a hopping energy −t without feeling each other. The Hubbard model introduces an additional term that takes into account the Coulomb repulsion (U ), for electrons occupying the same lattice site. One of the simplest Hub-bard Hamiltonians we can write is that of a chain of hydrogen atoms. The hydrogen atom has only one electron in the s orbital and this orbital can be occupied by at most two electrons. Then if the atoms are close together elec-trons can easily hop from one atom to another and the system is expected to be a conductor as conventional band theory predicts. Now imagine that we increase the distance between the atoms, then the hopping energy |t| de-creases. From some separation onwards the repulsion energy of neighboring electrons should be larger than the hopping energy, blocking the hopping of electrons and changing the system into an insulator. The former picture is easily captured by the Hubbard Hamiltonian

H = −tX

i,σ

(c†_i,σci+1,σ+ c†i+1,σci,σ) + U

X

i

(29)

1.3. Systems studied 21

④ ④

1

where c†_i,σ(ci,σ) is the creation(annihilation) operator of an electron with spin

σ on lattice site i. Hamiltonian Eq. (1.26) exempliﬁes the simplicity of the Hubbard model and shows how a complicated strongly interacting system can be easily written down as a mathematical model. Such simplicity is deceptive: the Hubbard model turn out to be a mathematically complicated problem, an exact solution exists so far only for a simple one-dimensional case. For more complicated systems the model is normally treated with numerical simulations, as is the case in this thesis. In chapters2,3and4we use the Hubbard model to describe a system consisting of quantum dots with strongly interacting electrons and use the Hubbard model to calculate the wavefunction of the system.

1.3.2 Spin chains as qubit buses

Any computer consists of three fundamental parts. 1) the central processing unit (CPU) that performs operations on the data, 2) a memory to hold and store the data and 3) a variety of peripherals (mouse, keyboard, touchscreen, etc.) that serve as an interface to the user. The communication between these components is one of the most important mechanism of a computer. They communicate via a so-called "bus" which routes data from one device to another. For example, suppose you double click in a folder in your computer, then the data of the click has to be sent to the CPU and the CPU needs to access the memory to know the contents of that folder and then send them to your screen. This simple task requires the movement of data through several devices. In the architecture vision of a quantum computer using qubits, things are not much different. Indeed, the way to perform calculations is drastically different from normal computers but the need to transport data is unavoidable. These quantum data buses should be able to perform a quantum state transfer (QST) with high accuracy. The level of accuracy of a QST is measured by means of fidelity which ranges from zero to one, with one being a perfect QST. Several types of quantum data buses have been proposed: phonon modes for trapped ions [67], cavity photon modes for superconducting qubits and spin qubits [68,69], and spin chains for spin qubits [70,71]. In chapter5we study the use of spin chains as quantum data buses.

Heisenberg Hamiltonian

Heisenberg spin chains are a viable candidate for a quantum data bus since the Heisenberg couplings appear in diﬀerent solid state systems and the Heisenberg chains map a system of interacting fermions, The success of a spin chain as a quantum data bus depends on the availability and controllability of the chain couplings. In Fig. 1.8(a) a schematic representation of a spin chain is shown,

(30)

④ ④

1

22 1. Introduction

where the chain ends are the sender (L) and the receiver (R) qubits. Between the sender and receiver qubits the proposed spin bus (in this case made of 5 qubit sites). Since the control of individual Heisenberg couplings in the chain is experimentally challenging, the goal is to transfer the state encoded in qubit L through the chain to qubit R with the least intervention possible, i.e. trying to keep the chain couplings JCand the qubit positions in the chain ﬁx during

the transfer. The Hamiltonian of the system can be written as follows,

(a)

Figure 1.8: (a) Diagram for a Heisenberg spin chain of N = 5 sites [Eq. (1.27)]. We want

to transfer the state in the L-qubit to the R-qubit. (b) Diagram of the effective Hamiltonian Eq. (1.28) for an odd size bus. (c) Diagram of the effective Hamiltonian Heven= J_LR(2)σL·σR

for an even size bus.

H = HCQ+ HC= JLσL· σ1+ JRσN· σR+ JC N −1

X

n=1

σn· σn+1. (1.27)

In Eq. (1.27) HC= JCPN −1n=1 σn· σn+1is the Hamiltonian of the chain, where

JCis the Heisenberg coupling between elements of the chain, N is the number

of sites in the chain and σn = (σnx, σny, σnz) are the Pauli matrices operating

on site n. HCQ = JLσL · σ1+ JRσN · σR in Eq. (1.27) is the interaction

(31)

1.3. Systems studied 23

④ ④

1

JL(JR) is the the Heisenberg coupling between the sender(receiver) qubit and

the chain.

It has been theoretically shown that by engineering the individual couplings QST can be achieved [72, 73]. One setup consists of qubit-chain couplings much smaller than the gap between the ground state and the excited. In this setup, QST can be achieved with arbitrary accuracy [74],

• The sender qubit is initially decoupled from the chain.

• Then a spin-spin interaction between the sender and the chain is switched on and kept constant,

• This will give rise to a non-eigenstate that is evolving in time with phases given by the energies of the eigenstates.

• As a result of the phases, an oscillating ﬁdelity is found such that the lineshape of ﬁdelity has many maximums and minimums.

In chapter 5 we study adiabatic protocols where unitary ﬁdelity can be achieved which does not oscillate in time but stays at is maximum value.

Even and odd spin chains

In this section we present an approximation to the Hamiltonian (1.27) that will allow us to study the low energy states of a long chain with an analytical treatment not only numerical. In the above mentioned studies by Friesen et al. [71] an eﬀective Hamiltonian is derived under the condition that JR, JL≪ ∆C,

where ∆C≈JCπ 2

N is the gap between the ground state and the excited state of

the chain. This can be experimentally achieved if the separation between the chain elements is smaller than the separation between the chain and the sender, and receiver qubits. The physical argument behind the approximation is that if the system is in the ground state, and the couplings between the chain-qubits are small, the state of the system can be viewed as a perturbation of the ground state of the isolated chain. Therefore we can project the complete Hamiltonian [Eq. (1.27)] to states containing only the ground state of the chain. Under this approximation the eﬀective Hamiltonian reads [71,74]

Hef f = JL(1)σL· σC+ JR(1)σC· σR+ JLR(2)σL· σR, (1.28)

where J_L(1) = JLm1, JR(1) = JRmN and JLR(2) ≈ JLJR

2∆C (m1mN − ss1N), with

mi the local magnetic moment of site i in the chain and ss1N the spin-spin

correlation function between the ﬁrst and last site. We know that for an even number of sites N in the chain, the local magnetic moments are zero (mi= 0)

(32)

④ ④

1

24 1. Introduction

yielding the eﬀective Hamiltonian for an even spin bus as Heven= JLR(2)σL· σR

[illustrated in Fig. 1.8(c)]. For an odd number of sites all the terms in Eq. (1.28) are non-zero, such that Hodd = Hef f [Eq. 1.28] as illustrated in Fig.

1.8(b). Since the size of our system scales very rapidly with size, as 2N_{, the}

study of the eﬀective Hamiltonian is crucial in order to treat some part of our system analytically, even though the complete dynamics of the spin chain in general needs to be treated numerically.

Figure 1.9:Schematic picture of the evolution of a state. The green and red lines represent the instantaneous eigenstates of the system, and the blue arrow represents the evolved state. An adiabatic evolution happens if the state (blue) follows the ground state (green) through the complete evolution. Figure adapted from reference [75].

Adiabatic quantum state transfer

A QST of a qubit state across a chain of spins presents several problems to overcome. The most relevant problem in quantum communication is deco-herence, even though there have been many advances in this ﬁeld, such as quantum error correction which makes the control fault tolerant, it is of great importance to have a precise and stable quantum state transfer. An adia-batic QST presents two important advantages in comparison with other types of protocols: adiabatic QST is robust against weak variations of the system which makes it experimentally viable and the transfer time of the evolution does not need to be precisely controlled since once the state is transferred the system stays in a steady state (ﬁdelity does not oscillate). Mathematically, during a complete adiabatic evolution the state of the system stays as close as possible to an eigenstate. This imposes two basic requirements to achieve adiabaticity; First, the initial state of the system (|Ψii) needs to be an

(33)

eigen-References 25

④ ④

1

state and second the evolution of the system needs to be slow enough such

that there is a gap between the corresponding eigenvalue and the rest of the energy spectrum. The latter condition is generally written as

T ≫ _w~, (1.29)

where T is the total time of the evolution and w is the energy gap between the states. In Fig. 1.9we show a speciﬁc case were the state to follow is the ground state, such that the evolution time T is limited by the energy gap w shown.

In summary, in chapter5 we look for a quantum protocol that: • Transfers a state from the sender to the receiver with high ﬁdelity.

• Does not require much control leaving the spin chain untouched such that only JRor JL vary.

• Is as adiabatic as possible so that it is precise and stable versus weak variations.

• Is much faster than the decoherence time to minimize errors.

References

[1] A. Tonomura and F. Nori, Quantum physics: Disturbance without the force, Nature 452, 298 (2008).

[2] W. Ehrenberg and R. E. Siday, The Refractive Index in Electron Op-tics and the Principles of Dynamics, Proceedings of the Physical Society. Section B 62, 8 (1949).

[3] Y. Aharonov and D. Bohm,Significance of Electromagnetic Potentials in the Quantum Theory, Phys. Rev. 115, 485 (1959).

[4] R. G. Chambers, Shift of an Electron Interference Pattern by Enclosed Magnetic Flux, Phys. Rev. Lett. 5, 3 (1960).

[5] A. Tonomura, T. Matsuda, R. Suzuki, A. Fukuhara, N. Osakabe, H. Umezaki, J. Endo, K. Shinagawa, Y. Sugita, and H. Fujiwara, Ob-servation of Aharonov-Bohm Effect by Electron Holography, Phys. Rev. Lett. 48, 1443 (1982).

(34)

④ ④

1

26 References

[6] A. Tonomura, N. Osakabe, T. Matsuda, T. Kawasaki, J. Endo, S. Yano, and H. Yamada,Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave, Phys. Rev. Lett. 56, 792 (1986).

[7] S. Popescu, Dynamical quantum non-locality, Nature Physics 6, 151 (2010).

[8] L. Vaidman, Role of potentials in the Aharonov-Bohm effect, Phys. Rev. A 86, 040101 (2012).

[9] R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, Obser-vation of h

e Aharonov-Bohm Oscillations in Normal-Metal Rings, Phys. Rev. Lett. 54, 2696 (1985).

[10] A. Bachtold, C. Strunk, J.-P. Salvetat, J.-M. Bonard, L. Forro, T. Nuss-baumer, and C. Schonenberger, Aharonov-Bohm oscillations in carbon nanotubes, Nature 397, 673 (1999).

[11] S. Russo, J. B. Oostinga, D. Wehenkel, H. B. Heersche, S. S. Sobhani, L. M. K. Vandersypen, and A. F. Morpurgo, Observation of Aharonov-Bohm conductance oscillations in a graphene ring, Phys. Rev. B 77, 085413 (2008).

[12] J. Schelter, P. Recher, and B. Trauzettel,The Aharonov-Bohm effect in graphene rings, Solid State Communications 152, 1411 (2012).

[13] F. de Juan, A. Cortijo, M. A. H. Vozmediano, and A. Cano, Aharonov-Bohm interferences from local deformations in graphene, Nature Physics 7, 810 (2011).

[14] E. Li, B. J. Eggleton, K. Fang, and S. Fan, Photonic Aharonov-Bohm effect in photon-phonon interactions, Nature Communications 5 (2014).

[15] H. Beutler, Über Absorptionsserien von Argon, Krypton und Xenon zu Termen zwischen den beiden Ionisierungsgrenzen2 P 3 2/0 und2 P 1 2/0, Zeitschrift für Physik 93, 177 (1935).

[16] U. Fano,Sullo spettro di assorbimento dei gas nobili presso il limite dello spettro d’arco, Il Nuovo Cimento 12, 154 (1935).

[17] A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar,Fano resonances in nanoscale structures, Rev. Mod. Phys. 82, 2257 (2010).

(35)

References 27

④ ④

1

[18] J. Verduijn, G. C. Tettamanzi, G. P. Lansbergen, N. Collaert, S.

Biese-mans, and S. Rogge, Coherent transport through a double donor system in silicon, Applied Physics Letters 96, 072110 (2010).

[19] L. E. Calvet, J. P. Snyder, and W. Wernsdorfer,Fano resonance in elec-tron transport through single dopant atoms, Phys. Rev. B 83, 205415 (2011).

[20] D. M. Cardamone, C. A. Staﬀord, and S. Mazumdar,Controlling Quan-tum Transport through a Single Molecule, Nano Letters 6, 2422 (2006).

[21] D. Darau, G. Begemann, A. Donarini, and M. Grifoni,Interference effects on the transport characteristics of a benzene single-electron transistor, Phys. Rev. B 79, 235404 (2009).

[22] C. A. Büsser and A. E. Feiguin,Designing a symmetry-protected molecular device, Phys. Rev. B 86, 165410 (2012).

[23] U. Fano, Effects of Configuration Interaction on Intensities and Phase Shifts, Phys. Rev. 124, 1866 (1961).

[24] W. de Haas, J. de Boer, and G. van dën Berg,The electrical resistance of gold, copper and lead at low temperatures, Physica 1, 1115 (1934). [25] G. J. Van der Berg, Progess in Low Temperature Physics, vol. Vol. IV

(American Physical Society, North Holland, Amsterdam, 1964).

[26] M. P. Sarachik, E. Corenzwit, and L. D. Longinotti,Resistivity of Mo-Nb and Mo-Re Alloys Containing 1Fe, Phys. Rev. 135, A1041 (1964).

[27] J. Kondo,Resistance Minimum in Dilute Magnetic Alloys, Prog. of Theor. Phys. 32, 37 (1964).

[28] A. A. Abrikosov, Influence of impurity ferromagnetism and of the exter-nal magnetic field on the resistance of metals with magnetic impurities, Physics. Physique. Fizika 2, 61 (1965).

[29] K. G. Wilson, The renormalization group: Critical phenomena and the Kondo problem, Rev. Mod. Phys. 47, 773 (1975).

[30] T. A. Costi, A. C. Hewson, and V. Zlatic,Transport coefficients of the An-derson model via the numerical renormalization group, Journal of Physics: Condensed Matter 6, 2519 (1994).

(36)

④ ④

1

28 References

[31] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, and M. Kastner, Kondo effect in a single-electron transistor, Nature 391, 156 (1998).

[32] S. Cronenwett, T. Oosterkamp, and L. Kouwenhoven, A tunable Kondo effect in quantum dots, Science 281, 540 (1998).

[33] L. Kouwenhoven and L. Glazman, Revival of the Kondo effect, Phys. World 14, 33 (2001).

[34] A. F. Otte, M. Ternes, K. von Bergmann, S. Loth, H. Brune, C. P. Lutz, C. F. Hirjibehedin, and A. J. Heinrich, The role of magnetic anisotropy in the Kondo effect, Nature Phys. 4, 847 (2008).

[35] J. Nygard, D. Cobden, and P. Lindelof, Kondo physics in carbon nan-otubes, Nature 408, 342 (2000).

[36] P. Jarillo-Herrero, J. Kong, H. van der Zant, C. Dekker, L. Kouwenhoven, and S. De Franceschi, Orbital Kondo effect in carbon nanotubes, Nature 434, 484 (2005).

[37] W. Liang, M. Shores, M. Bockrath, J. Long, and H. Park, Kondo reso-nance in a single-molecule transistor, Nature 417, 725 (2002).

[38] J. Park, A. Pasupathy, J. Goldsmith, C. Chang, Y. Yaish, J. Petta, M. Rinkoski, J. Sethna, H. Abruna, P. McEuen, et al.,Coulomb blockade and the Kondo effect in single-atom transistors, Nature 417, 722 (2002). [39] L. Yu and D. Natelson,The Kondo effect in C-60 single-molecule

transis-tors, Nano Lett. 4, 79 (2004).

[40] J.-H. Chen, L. Li, W. G. Cullen, E. D. Williams, and M. S. Fuhrer,

Tunable Kondo effect in graphene with defects, Nature Phys. 7, 535 (2011). [41] M. Grobis, G. Rau, R. M. Potok, and D. Goldhaber-Gordon, Kondo Effect in Mesoscopic Quantum Dots, arXiv:cond-mat/0611480 [cond-mat.mes-hall] (2006).

[42] S. De Franceschi, S. Sasaki, J. M. Elzerman, W. G. van der Wiel, S. Tarucha, and L. P. Kouwenhoven, Electron Cotunneling in a Semi-conductor Quantum Dot, Phys. Rev. Lett. 86, 878 (2001).

[43] D. V. Averin and Y. V. Nazarov,Virtual electron diffusion during quantum tunneling of the electric charge, Phys. Rev. Lett. 65, 2446 (1990).

(37)

References 29

④ ④

1

[44] Y. V. Nazarov and Y. M. Blanter, Quantum Transport: Introduction to

Nanoscience (Cambridge University Press, 2009).

[45] B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven, D. van der Marel, and C. T. Foxon,Quantized con-ductance of point contacts in a two-dimensional electron gas, Phys. Rev. Lett. 60, 848 (1988).

[46] A. Tsvelick and P. Wiegmann, Exact results in the theory of magnetic alloys, Advances in Physics 32, 453 (1983).

[47] C. Lacroix,Density of states for the Anderson model, Journal of Physics F: Metal Physics 11, 2389 (1981).

[48] K. Yosida and K. Yamada, Perturbation Expansion for the Anderson Hamiltonian, Progress of Theoretical Physics Supplement 46, 244 (1970).

[49] B. Hobvtić and V. Zlatić, Perturbation Expansion for the Asymmetric Anderson Hamiltonian, physica status solidi (b) 99, 251 (1980).

[50] N. E. Bickers, Review of techniques in the large-N expansion for dilute magnetic alloys, Rev. Mod. Phys. 59, 845 (1987).

[51] R. Aguado and D. C. Langreth,Out-of-Equilibrium Kondo Effect in Dou-ble Quantum Dots, Phys. Rev. Lett. 85, 1946 (2000).

[52] P. Coleman, New approach to the mixed-valence problem, Phys. Rev. B 29, 3035 (1984).

[53] G. Kotliar and A. E. Ruckenstein, New Functional Integral Approach to Strongly Correlated Fermi Systems: The Gutzwiller Approximation as a Saddle Point, Phys. Rev. Lett. 57, 1362 (1986).

[54] D. Goldhaber-Gordon, J. Göres, M. A. Kastner, H. Shtrikman, D. Ma-halu, and U. Meirav, From the Kondo Regime to the Mixed-Valence Regime in a Single-Electron Transistor, Phys. Rev. Lett. 81, 5225 (1998).

[55] B. Dong and X. L. Lei,Kondo-type transport through a quantum dot under magnetic fields, Phys. Rev. B 63, 235306 (2001).

[56] B. Dong and X. L. Lei,Kondo-type transport through a quantum dot: a new finite- U slave-boson mean-field approach, Journal of Physics: Con-densed Matter 13, 9245 (2001).

Quantum transport in strongly interacting one-dimensional nanostructures

Quantum transport in strongly

interacting one-dimensional

Quantum transport in strongly

interacting one-dimensional

nanostructures

Proefschrift

Ricardo Rodrigo AGUNDEZ MOJARRO

Contents

1

Introduction

1.1

Effects studied

1.1.1

Aharonov-Bohm effect

1

1

1

1

1.1.2

Fano effect

1

1

1

1.1.3

Kondo effect

1

1

(a)

(b)

(c)

1

1

(a)

1

1.2

Techniques used

1.2.1

Scattering matrix formalism

(a)

(b)

1

1

1.2.2

Slave-boson mean field theory

1

1

(a)

(b)

(c)

1

1

1

1.3

Systems studied

1.3.1

Hubbard model

1

1.3.2

Spin chains as qubit buses

1

(a)

1

1

1

References

1

1

1

1