Tunable Fano Resonance in Double Quantum
Dot systems
Seif Alwan
February 20, 2019
Contents
1 Introduction 3 2 The Model 63 Evaluation of the Green’s function 11
4 Probability Current 24
Abstract
We investigate the electronic transport through a double quan-tum dot device embedded in the tunnel junction between two metallic leads. We find that transitioning from the serial to the parallel con-figurations is associated with the progressive reduction of tunneling through one quantum dot. As a result the asymmetric Fano lineshape appears, which we interpret as the formation of an anti-bonding and a bonding state in the junction. The bonding state facilitates the majority of the tunneling as the geometry approaches symmetric cou-pling to the leads. At the limit of the transition towards the parallel configuration, the anti-bonding state is completely localized from the continuum of the leads and only the bonding state is left to participate in the transmission.
1
Introduction
In recent years, there has been major progress in the fabrication techniques for electronic devices in the nanometer scale. This progress has occurred in correlation with the reduction in the measurements of electronic components to the degree where even dimensions are reduced to two, one and even zero dimensions, i.e., films, wires and point clusters (or dots), respectively. Re-ducing the dimensions of a system is associated with the motion of charge carriers becoming more confined, resulting in quantization of their states. In our work, we will be looking at how the confinement affects the electronic transport through these Quantum Dot (QD) systems. The transmission of the electron current is observed as a discreet spectrum as it passes the one dimensional QD junction, where junction refers to the central device of the QDs positioned between two metallic leads. These metal leads are consid-ered to be a continuum in their states, and are therefore able to provide a steady flow of electrons through the junction. In fact, these QD systems have become a reoccurring theme in the theoretical condensed matter community as ”toy” systems for studying quantization effects in transport processes.
Interesting characteristics, such as quantum interference effects that arise in the electron transport, which is related to the discretization of the energy structures are owed primarily to the scale of the these systems. Multiple QD systems have in much of the current literature been treated much like an artificial molecule, and indeed this is a reasonable description as the states representing the QDs can easily be seen as a combination of atomic orbitals and treated much like bonding and anti-bonding states.
A main feature of the transport through QD systems is that the electron coherence (its phase) is preserved under transmission. The coherence of the transport allows for a better description of the transport characteristics. This allows the transmission to be governed entirely by the coupling to the leads, but also by the coupling parameter connecting the QDs.
In our work, we study the Double Quantum Dot (DQD) system, which includes the serial coupled DQD and the parallel coupled DQD, as well as the transition in between. In the serial coupled DQD the leads couple to the QDs asymmetrically, where each lead may only interact with a single QD. Whilst in the parallel coupled system, we find that both leads couple equally to both QDs. This in turn allows us to gain a clear picture of the quantum transport phenomena under different coupling conditions by simply varying the coupling parameters.
As the junction consists of two QDs, the electron current is allowed multi-ple pathways through either QD. This gives rise to variations in the transmis-sion spectra and the formation of a well known asymmetric lineshape which emerges due to quantum interference effects made possible by the existence of multiple QDs. The asymmetric lineshape in the transmission spectra is well known as the Fano lineshape [6], which indicates in simple terms as a complete loss of transmission through a single pathway and the increase through another.
In addition to the quantum transport properties, the interaction effects of electrons play an important role when considering the correlation effects, such as the Coulomb blockade and Kondo effect at low temperatures. However interesting, these effects are beyond the scope of this work, yet in literature are often associated with similar studies. In the present work, we limit our-selves to the study the non-equilibrium transport through the DQD system, where we focus on the characteristics of the transmission to identify inter-ference effects in the electronic spectrum. The equilibrium of our system is allowed to be broken across the junction originating at the leads. This can be done in a number of ways, by variations in the temperature or the chemical potential of the leads but also by simply creating a biased current over the junction. We will avoid temperature variations in our system and instead create a non-equilibrium over the junction through a misalignment of the chemical potential of the leads as well as biasing the voltage over the junction.
We also consider the associated Density Of State (DOS) as well as the local DOS. The local DOS gives us an emphasizes on the mixing of densi-ties between the QDs, which further strengthens the claim that the DQD system may function as an artificial molecule. As the electronic structure of a molecule consists of a combination of atomic like states, similarly we can describe the junction as a combination of QD states.
We generate different transport conditions through the junction by vary-ing the different couplvary-ing conditions. These range from serial (or asymmetric) coupling to the leads to a parallel (or symmetric) coupling to the leads. These configurations are explained in more detail in the following section, section 2, where we introduce the model of the DQD system. In section 3, we derive the
necessary Green’s function from which we construct the Landauer-B¨uttiker
formula and the DOS. When transitioning from serial to parallel, we are able to locate the Fano lineshape as it is first formed until it fully vanishes, leaving one state in the junction to carry the entirety of the charge transport. These
2
The Model
The system of interest is illustrated in figure 1, where the two states of
en-ergies εAσ and εBσ represent the two QDs in the junction with spin σ. The
states couple together through the interdot coupling constant tc, which
de-termines the coupling strength between the QDs, for finite coupling strength,
writing the energy difference as εAσ− εBσ = 0 is insufficient to attain
degen-eracy of the system.
We have chosen to use the indices A and B to represent the anti-bonding and bonding states of the levels in conjunction to the interpretation of the DQD system as an artificial molecule. This is also repeated in literature and provides a more convenient and perhaps more insightful overview of the system.
The coupling between the junction and the leads are given by the coupling
constant Γα
σ where α = L, R denote the coupling from the Left (L) and right
(R) lead. They represent the flow of electrons from the leads, that function as source and drain. The leads are assumed to be a continuum of states
and therefore Γα denotes the line-width of the distribution interacting with
the QDs, and the strength of the coupling determines the strength of the interaction from the continuum.
Figure 1: Simplified schematics of the two state QD molecule coupled to a left and right leads.
For the purpose of this study the leads are considered to be non-polarized, where due to the nature of the interference phenomenon, the polarization of the leads are considered optional as the total transmission of the current through the DQD junction is unaffected by it. We will proceed to develop the framework for lead polarization for future studies, but retain a simple picture within the scope of this report where the leads are characterize by
the chemical potential µα around the Fermi level denoted by µ0.
We have also included in figure 1 localized spin moments denoted by
SA and SB which again we choose not to fully implement but will develop
the mathematical framework for future work, we therefore only give a short description here.
The SA and SB each couple to the localized states εAσ and εAσ
respec-tively by the exchange coupling parameters JA and JB. These localized spin
moments may then interact with each other by the effective spin-spin
interac-tion which we write as JAB. To conceptualize this in simpler terms the DQD
forms an artificial molecule allowing passage of current from the surrounding leads, the localized spin moments are then calculated externally facilitating the exchange interaction between the QDs. In orbital terminology, we can draw the picture of two aromatic groups limited to s- and p-orbitals and their
hybrids, these providing a flow of electrons with a metal trapped in the center of each molecule that donates a d-orbital that in turn provides a localized spin moment. The exchange interaction originates from the coupling of the localized spin moments, the evaluation of the exchange interaction is received for a later time and we limit ourselves to studying the transport properties. To model the system we write the following Hamiltonian composed of a
lead Hamiltonian ˆHleads, a central component ˆHCen and the tunneling
com-ponents ˆHT unneling i.e,
ˆ
H = ˆHleads+ ˆHT unneling+ ˆHCen (1)
We begin by dissecting each component starting with ˆHleads,
ˆ Hleads = X kσ εkσc † kσckσ+ X pσ εpσc†pσcpσ (2)
where c and c† are the creation and annihilation operators of the left
(right) lead at some momentum vector k(p) with energy ε and spin σ.
We represent the central region Hamiltonian ˆHCen as,
ˆ
HCen= ˆHCen,0+ ˆHCen,int
= X m={A,B},σ εmσd†mσdmσ+ X m16=m2,σ γm1,m2d†mσσσ1σ2dmσ + X m,σ1,σ2 Jmd†mσ1dmσ2 · Sm(t) + X m Umnm↑nm↓ (3)
We partition the Hamiltonian by first defining a non interacting compo-nent given by the first term in equation 3. And compacting the residual 2-4
terms in the interacting component ˆHCen,int. We begin by dissecting the first
term corresponding to ˆHCen,0 where εmσ = εm,0+12gµBBσzσσ gives the energy
of the electron level of the state m. The term 12gµBBσzσσ arises due to the
Zeeman splitting from spin polarization in the leads, where g is the g-factor
µB the Bohr constant and B is the magnetic field in the z-direction, B = Bˆz.
In the scope of this project we will not need to consider polarized leads as we are only interested in the interference effects that arise from the transmission. Polarization will come to play when we investigate the exchange interactions which is omitted from this report.
The creation and annihilation operators for the quantum dots are given
by d†mσdmσ for the state m ∈ A, B with spin σ. The intra-dot tunneling
is given by the second term in 3, where γm1,m2 gives the intra-dot coupling
strength between the states and the summation is constructed in such a way to avoid self-tunneling. The interaction between the spin moments and
the quantum dots are given by Jmd†mσ1dmσ2 · Sm(t) where Jm is the effective
exchange interaction strength between the spin moment and the quantum dot. As this term does not effect the interferenace effects which is within
the order of the transmission we set Jm = 0. The final term gives the well
known Hubbard on-site coulomb repulsion term, where nm↑ = d
†
m↑dm↑ and
nm↓ = d
†
m↓dm↓ are the pair body particle operators. As a large body of the
scientific publications involve the use of the U parameter, we make a small digression to justify why we will omit it (letting U = 0).
Figure 2: The Coulomb blockade for a two state tunnel junction, where εF
is the Fermi energy of the leads.
From figure 2, the QD level ε1 is occupied by a spin-up electron, the
spin-down level is then elevated by a Coulomb energy of U to ε2 = ε1+ U .
Assuming this state is empty, then for an electron from the leads to hop
over to the level of energy ε2 it first needs enough energy to overcome the
Coulomb energy U . This is known as the Coulomb blockade. Similarly the
spin-up electron occupying the ε1 energy level can not jump to the leads due
to energy conservation. As direct single electron tunneling is forbidden, we therefore have no flow of current through the junction. However, a higher order process still occurs where a spin-up electron jumps out of the QD level into the lead and is instantly replaced by a spin-down electron from the surrounding lead. This is a virtual process but occurs rapidly and results in
a measurable current. In turn the rapid spin flips results in the formation of a singlet in the QD spin state as the average spin vanishes. This is in fact analogous to the Kondo phenomenon in a magnetic impurity in a metallic system, where the impurity forms a singlet state with the itinerant electrons of the conduction band at lower temperatures resulting in scattering centers that in turn raise the resistivity in the system with decreased temperature.
The main argument here is that QD systems are immensely small in comparison to the environment they are embedded in (in this case a left and right lead). As such it would warrant a large energy to move from one state to the next.
However, for our choice of Hamiltonian stems from our study in interfer-ence effects in DQD systems, quantum interferinterfer-ence is a wave phenomenon and hence our choice of U should not matter. In fact the preservation of coherence as described by the higher order tunneling simplifies the detection of the Fano lineshape due to the added control over the tunneling process.
The components of the Hamiltonian that contributes to the cotunneling process described in figure 2 is given by,
ˆ
HAleads =X
Lσ
(TLσ(A)c†LσdAσ+ ¯TLσ(A)d†AσcLσ)
+X Rσ (TRσ(A)c†RσdAσ+ ¯T (A) Rσd † AσcRσ) ˆ HB leads = X Lσ (TLσ(B)c†LσdBσ + ¯T (B) Lσ d † BσcLσ) +X Rσ (TRσ(B)c†RσdBσ + ¯T (B) Rσ d † BσcRσ) (4)
where Tα∈L,Rm∈A,B is the tunneling matrix element relating the left and right
lead to the QD levels A and B. The tunneling process is represented by
the creation and annihilation, i.e. Tασ(m)c†ασdm gives the annihilation of an
electron in the level m and creation of an electron in the lead α, the conjugate ¯
Tασ(m)d†mcασ then gives the opposite process.
The possibilities to tunnel an electron between the leads and through the DQD could occur through a number of pathways. The leads may tunnel electrons to the two QDs in the central region, but as the QD levels are
provides further routes.
The main interest of this work is to investigate the effects of blocking the tunneling process through one of the QDs forcing the electron current to only tunnel through the single QD. As the transmission is analogous to a probability current the different pathways are best described as probability events. This leads us to consider the possibility of interference effects in the system.
We want to characterize the current transport through the junction, hence we need a method that gives us the expression of the Green’s function in the junction coupled to the leads. More specifically we need the Green’s function that represents the DQD with a term that represents the flow of electrons from and to the leads.
To achieve this we derive through the means of the Heisenberg equation of motion (eom) method, which relates the different Green’s functions of the system to a set of equations. This hierarchy of equations must be truncated to some degree to generate a closed set of equations.
3
Evaluation of the Green’s function
The purpose of this section is to evaluate the Green’s function for the system using the Heisenberg eom to generate a closed set of equations. From there we can proceed to write their matrix representations which will allow us to
separate the terms in belonging to ˆH and GRfrom the fundamental condition
ˆ
HGR =1. We begin by looking without proving the Heisenberg eom for the
single particle Green’s function which is given by, i¯h∂Gmnσσ0(t, t 0) ∂t = δmnδσσ0δ(t, t 0 ) − i ¯ hD ˆT h d(t)mσ, ˆH i d(t0)†nσ0 E (5) As the following procedure is dominated by the evaluation of commutation relations and straight forward algebra the main complexity is contained in the length of the Hamiltonian itself. Therefore it is most convenient to evaluate
the eom for ˆH in its components,
ˆ
H = ˆHleads+ ˆHAT unneling+ ˆH
B
This equation is a component of the device Hamiltonian evaluated in its commutation relation with the annihilation operator of site m with spin
sigma dmσ
As the ˆHleadis evaluated in its commutation relation with the annihilation
operator of site m with spin sigma dmσ of some QD, we can quickly assess
that the commutation relation hd(t)mσ, ˆHleads
i
vanishes because ˆHleads is a
function of the lead field operators cσ of the leads. We therefore only have
to consider expanding three terms, h dmσ, ˆH i =hdmσ, ˆHAT unneling+ ˆH B T unneling+ ˆHCen i (7) We begin by evaluating h dmσ, ˆHT unnelingA i and h dmσ, ˆHBT unneling i as only
one component may commute with dmσ i.e. d
†
Aσ0 the rest will vanish.
[dmσ, ˆHAT unneling] = [dmσ, X kσ0 (Tkσ(A)0c † kσ0dAσ0 + ¯T(A) kσ0d † Aσ0ckσ0)] + [dmσ, X pσ0 (Tpσ(A)0c † pσ0dpσ0 + ¯T(A) pσ0d † Aσ0cpσ0)] =X kσ0 ¯ Tkσ(A)0{dmσ, d † Aσ0}ckσ0 +X pσ0 ¯ Tpσ(A)0{dmσ, d † Aσ0}cpσ0 =X k ¯ Tkσ(A)ckσ + X p ¯ Tpσ(A)cpσ (8) In a similar fashion, [dmσ, ˆHT unnelingB ] = X k ¯ Tkσ(B)ckσ+ X p ¯ Tpσ(B)cpσ (9) ˆ
HCen is not as easily evaluated as ˆHmT unneling, so for bookkeeping purposes it
is convenient to separate the commutation relation into components of the central Hamiltonian and evaluate them separately.
h dmσ, ˆHCen i = [dmσ, X m0{A,B},σ0 εm0σ0d† m0σ0dm0σ0] + [dmσ, X m16=m2,σ0 γm1,m2d † m0σ0dm0σ0] + [dmσ, X m0,σ 1,σ2 Jm0d† m0σ 1σσ1σ2dm0σ2 · Sm0(t)] (10)
We begin by evaluating relation between dmσ and the isolated representations
of the two QDs, we should find that only the hermitian conjugate represen-tations of the field operators produce a finite result similar to the evaluation of ˆHA
T unneling and ˆHBT unneling.
[dmσ, X m0{A,B},σ0 εm0σ0d† m0σ0dm0σ0] = X m0{A,B},σ0 εm0σ0[dmσ, d† m0σ0dm0σ0] = X m0{A,B},σ0 εm0σ0({dmσ, d† m0σ0}dm0σ0 − d† m0σ0 (((( ((( {dmσ, dm0σ0}) = X m0{A,B},σ0 εm0σ0δmm0δσσ0dm0σ0 = εmσdmσ (11)
The following relation represents the inter dot tunneling between
quan-tum dots with a tunneling coefficient gamma γ. Setting the site basis γm16=
γm2 will ensure that self tunneling is avoided.
[dmσ, X m16=m2,σ0 γm1,m2d † m0σ0dm0σ0] = X m1,m2,σ0 γm1,m2({dmσ, d † m1σ0}dm2σ0 − d † m1σ0(((( ((( {dmσ, dm2σ0}) = X m1,m2,σ0 γm1,m2δmm1δσσ0dm2σ0 = X m1 γm,m1dm1σ (12)
For convenience we define the hybrid Green’s function that represents the propagation of fermions between the leads and QDs, the purpose of doing
this will become clear. gknσσ0(t, t0) = −i ¯ h D ˆT ckσ(t)d † nσ0(t0) E (13) To find the closed set of equations we follow a similar procedure and
differ-entiate gknσσ0(t, t0) with respect to t,
i∂gknσσ0(t, t 0) ∂t = i ∂ ∂t h (−i)Θ(t − t0)Dckσ(t)d † nσ0(t 0 )E+ (+i)Θ(t0− t)Dd†nσ0(t 0 )ckσ(t) Ei + (−i) ˆ T (i)∂ckσ(t) ∂t d † nσ0(t0) = δ(t − t0) D {ckσ(t), d †(t0) nσ0 } E + (−i) ˆ T (i)∂ckσ(t) ∂t d † nσ0(t0) = (−i) ˆ T (i)∂ckσ(t) ∂t d † nσ0(t0) = (−i)D ˆT [ckσ, ˆH]d†nσ0(t0) E (14)
Which leads to a new commutation relation [ckσ, ˆH] through the Heisenberg
relation. To evaluate this we follow the similar procedure as before the dif-ference being that we are now evaluating the commutation relation of the total Hamiltonian with the annihilation operator of a particle in the lead
with momentum k ckσ. Meaning [ckσ, ˆHCen] vanishes.
[ckσ(t), ˆH] = [ckσ, ˆHleads+ ˆH (A) T + ˆH (B) T ] = [ckσ, X k0σ0 εk0σ0c† k0σ0(t)ck0σ0(t)] + [ckσ, X k0σ0 Tk(A)0σ0c † k0σ0(t)dAσ0(t)] + [ckσ, X k0σ0 Tk(B)0σ0c † k0σ0(t)dBσ0(t)] =X k0σ0 εk0σ0δkk0δσσ0ck0σ0(t) + X k0σ0 Tk(A)0σ0δkk0δσσ0dAσ0(t) + X k0σ0 Tk(B)0σ0δkk0δσσ0dBσ0(t) = εkσckσ(t) + T (A) kσ dAσ(t) + T (B) kσ dBσ(t) (15) i∂gknσσ0(t, t 0) ∂t = εkσgknσσ0(t, t 0
) + (−i)Tkσ(A)D ˆT dAσ0(t)d†
nσ0(t 0 )E + (−i)Tkσ(B)D ˆT dBσ0(t)d† nσ0(t 0 )E (16)
Rearranging by placing gknσσ0(t, t0) to one side,
(i∂
∂t− εkσ)gknσσ0(t, t
0
) = (−i)Tkσ(A)D ˆT dAσ0(t)d†
nσ0(t 0 )E+ (−i)Tkσ(B)D ˆT dBσ0(t)d† nσ0(t 0 )E = Tkσ(A)GAnσσ0(t, t0) + T(B) kσ GBnσσ0(t, t 0 ) (17)
Where GAnσσ0(t, t0) and GBnσσ0(t, t0) are hybrid Green’s functions, describing
the interaction between the central region and the leads.
(i ∂
∂τ − εkσ)gknσσ0(τ, t
0
) = Tkσ(A)GAnσσ0(τ, t0) + T(B)
kσ GBnσσ0(τ, t0) (18)
Define gkσ−1(τ, t0) = (i∂τ∂ − εkσ) such that (i∂τ∂ − εkσ)gkσ(τ, t0) = δ(τ − t0).
Where gkσ(τ, t0) now has the same property as the Green’s function with the
non-interacting free Hamiltonian given by (i∂τ∂ − εkσ). gkσ(τ, t0) is known as
a bare or non-interacting Green’s function, where we make good use of the
property gkσ(t, τ)(i∂τ∂ − εkσ) = δ(t − τ) by multiplying with gkσ(t, τ) to get,
gkσ(t, τ)(i ∂ ∂τ − εkσ)gknσσ0(τ, t 0 ) = δ(t − τ)gknσσ0(τ, t0) (19) Integrating over τ, Z d(τ)δ(t − τ)gknσσ0(τ, t0) = gknσσ0(t, t0) = Z d(τ)Tkσ(A)gkσ(t, τ)GAnσσ0(τ, t0) + Z d(τ)Tkσ(B)gkσ(t, τ)GBnσσ0(τ, t0) (20)
Plugging these new expressions for the hybrid Green’s functions back into our original expression for the Green’s function we can rewrite equation 5 as,
i∂Gmnσσ0 ∂t = δmnδσσ0δ(t − t 0 ) + εmσGmnσσ0(t, t0) + γmAGAnσσ0(t, t0)δmB+ γmBGBnσσ0(t, t0)δmA +X k ( ¯Tkσ(A)gknσσ0δmA+ ¯T(B) kσ gknσσ0δmB)Gmnσσ0 +X p ( ¯Tpσ(A)gpnσσ0δmA+ ¯T(B) pσ gpnσσ0δmB)Gmnσσ0 (21)
(i∂ ∂t− εmσ)Gmnσσ0(t, t 0 )δmnδσσ0δ(t − t0) + γmAGAnσσ0(t, t0)δmB+ γmBGBnσσ0(t, t0)δmA +X k gknσσ0( ¯T(A) kσ δmA+ ¯T (B) kσ δmB) + X p gpnσσ0( ¯T(A) pσ δmA+ ¯Tpσ(B)δmB) = δmnδσσ0δ(t − t0) + γmAGAnσσ0(t, t0)δmB+ γmBGBnσσ0(t, t0)δmA +X kα ( Z
dτ Tkσα(A)gkσαGAnσσ0(τ, t0)( ¯T(A)
kσαδmA+ ¯T (B) kσαδmB) + Z dτ Tkσα(B)gkσαGBnσσ0(τ, t0)( ¯T(A) kσαδmA+ ¯T (B) kσαδmB)) = δmnδσσ0δ(t − t0) + γmAGAnσσ0(t, t0)δmB+ γmBGBnσσ0(t, t0)δmA + Z dτΣ(A,A)σσ (t, τ )GAnσσ0(τ, t0)δmA+ Z dτΣ(B,B)σσ (t, τ )GBnσσ0(τ, t0)δmB + Z dτΣ(A,B)σσ (t, τ )GAnσσ0(τ, t0)δmB+ Z dτΣ(B,A)σσ (t, τ )GBnσσ0(τ, t0)δmA (22) (i∂ ∂t− εmσ)Gmnσσ0(t, t 0 ) = δmnδσσ0δ(t − t0) + γmAGAnσσ0(t, t0)δmB+ γmBGBnσσ0(t, t0)δmA + Z dτΣ(A,A)σσ (t, τ )GAnσσ0(τ, t0)δmA+ Z dτΣ(B,B)σσ (t, τ )GBnσσ0(τ, t0)δmB + Z dτΣ(A,B)σσ (t, τ )GAnσσ0(τ, t0)δmB+ Z dτΣ(B,A)σσ (t, τ )GBnσσ0(τ, t0)δmA (23)
Let t → τ such that (i∂τ∂−εmσ) = g−1mσ(t, τ) and gmσ(t, τ)(i∂τ∂ −εmσ) = δ(t−τ).
This is very much similar to what we did earlier, we have yet to account for
any interaction terms, multiplying the LHS with gmσ and integrating over τ
we get, Z dτgmσ(i ∂ ∂t− εmσ)Gmnσσ0(τ, t 0 ) = Gmnσσ0(t, t0) (24)
Doing the same with the RHS of the equation is a bit more work but doable or rather unavoidable, Gmnσσ0(t, t0) = δmnδσσ0gmσ(t, t0) Z dτgmσ(t, τ)Gmn¯σσ0(τ, t0) + γmAδmB Z dτgmσ(t, τ)GAnσσ0(τ, t0) + γmBδmA Z dτgmσ(t, τ)GBnσσ0(τ, t0) + Z dτ0 Z dτgmσ(t, τ)Σ(A,A)σσ (τ, τ 0 )GAnσσ0(τ0, t0)δmA + Z dτ0 Z dτgmσ(t, τ)Σ(B,B)σσ (τ, τ 0 )GBnσσ0(τ0, t0)δmB + Z dτ0 Z dτgmσ(t, τ)Σ(A,B)σσ (τ, τ 0 )GAnσσ0(τ0, t0)δmB + Z dτ0 Z dτgmσ(t, τ)Σ(B,A)σσ (τ, τ 0 )GBnσσ0(τ0, t0)δmA
Fourier transforming the above,
Gmnσσ0(ω) = δmnδσσ0gmσ(ω)Gmn¯σσ0(ω)
+ γmAδmBgmσ(ω)GAnσσ0(ω) + γmBδmAgmσ(ω)GBnσσ0(ω)
+ gmσ(ω)Σ(A,A)σσ (ω)GAnσσ0(ω)δmA+ gmσ(ω)Σ(B,B)σσ (ω)GBnσσ0(ω)δmB
+ gmσ(ω)Σ(A,B)σσ (ω)GAnσσ0(ω)δmB+ gmσ(ω)Σ(B,A)σσ (ω)GBnσσ0(ω)δmA
We further simplify by multiplying with gmσ−1(ω) to produce on the LHS,
gmσ−1(ω)Gmnσσ0(ω) = (ω − εmσ)
and on the RHS a considerable reduction in terms,
(ω − εmσ) = δmnδσσ0
+ γmAδmBGAnσσ0(ω) + γmBδmAGBnσσ0(ω)
+ Σ(A,A)σσ (ω)GAnσσ0(ω)δmA+ Σ(B,B)
σσ (ω)GBnσσ0(ω)δmB
+ Σ(A,B)σσ (ω)GAnσσ0(ω)δmB+ Σ(B,A)
σσ (ω)GBnσσ0(ω)δmA
We now have the final form that we can use to create the set of closed equations, we now need to evaluate the different cases that arises from the
site indices m, n and the spin indices σ, σ0. To do this we just need to plug and play so to speak until we have a set of four equations.
For m = A, n = A and σ = σ0 =↑
(ω − εA↑)GAA↑↑(ω) = 1 + γABGBA↑↑(ω) + Σ
(A,A)
↑↑ (ω)GAA↑↑(ω) + Σ
(B,A)
↑↑ (ω)GBA↑↑(ω)
(ω − εA↑) − Σ(A,A)↑↑ (ω))GAA↑↑(ω) − γABGBA↑↑(ω) − Σ(B,A)↑↑ (ω)GBA↑↑(ω) = 1
(25) .
For m = A, n = A and σ = σ0 =↓
(ω − εA↓)GAA↓↓(ω) = 1 + γABGBA↓↓(ω) + Σ
(A,A) ↓↓ (ω)GAA↓↓(ω) + Σ (B,A) ↓↓ (ω)GBA↓↓(ω) (ω − εA↓− Σ (A,A) ↓↓ (ω))GAA↓↓(ω) − γABGBA↓↓(ω) − Σ (B,A) ↓↓ (ω)GBA↓↓(ω) = 1 (26) . For m = B, n = B and σ = σ0 =↑ (ω − εB↑)GBB↑↑(ω) = 1 + γBAGAB↑↑(ω) + Σ (B,B) ↑↑ (ω)GBB↑↑(ω) + Σ (A,B) ↑↑ (ω)GAB↑↑(ω) (ω − εB↑− Σ (B,B) ↑↑ (ω))GBB↑↑(ω) − γBAGAB↑↑(ω) − Σ (A,B) ↑↑ (ω)GAB↑↑(ω) = 1 (27) . For m = B, n = B and σ = σ0 =↓
(ω − εB↓)GBB↓↓(ω) = 1 + γBAGAB↓↓(ω) + Σ(B,B)↓↓ (ω)GBB↓↓(ω) + Σ(A,B)↓↓ (ω)GAB↓↓(ω)
(ω − εB↓− Σ (B,B) ↓↓ (ω))GBB↓↓(ω) − γBAGAB↓↓(ω) − Σ (A,B) ↓↓ (ω)GAB↓↓(ω) = 1 (28) . Now that the equations have been defined we are going to need to construct
to consider the diagonal compositions according to equations 25, 26, 27 and
28. Recall that, ˆHGR=1 this comes in handy in the form,
ˆ HR
AA↑↑ HˆRAA↑↓ HˆRAB↑↑ HˆRAB↑↓
ˆ
HR
AA↓↑ HˆRAA↓↓ HˆRAB↓↑ HˆRAB↓↓
ˆ HR BA↑↑ HˆRBA↑↓ HˆRBB↑↑ HˆRBB↑↓ ˆ HR BA↓↑ HˆRBA↓↓ HˆRBB↓↑ HˆRBB↓↓ GR
AA↑↑ GRAA↑↓ GRAB↑↑ GRAB↑↓
GRAA↓↑ GRAA↓↓ GRAB↓↑ GRAB↓↓
GRBA↑↑ GRBA↑↓ GRBB↑↑ GRBB↑↓ GR BA↓↑ GRBA↓↓ GRBB↓↑ GRBB↓↓ =14x4.
We can associate the ˆHmnσσ0 components of the above identity relation
with those of 25, 26, 27 and 28. A simple staring contest will reveal the following matrix, ˆ Hmnσσ0 = (ω − εA↑− Σ (A,A) ↑↑ ) 0 −γAB− Σ (B,A) ↑↑ (ω) 0 0 (ω − εA↓− Σ (A,A) ↓↓ ) 0 −γAB− Σ (B,A) ↓↓ (ω) −γBA− Σ (A,B) ↑↑ (ω) 0 (ω − εB↑− Σ (B,B) ↑↑ ) 0 0 −γBA− Σ (A,B) ↓↓ (ω) 0 (ω − εB↓− Σ (B,B) ↓↓ ) . (29)
We can simplify the diagonal elements to express the self-energy terms in terms of the line width broadening parameter Γ which is the imaginary
com-ponet of the Σ. We therefore write, Σ(m,n)σσ (ω) = −2iΓσ which corresponds
to the interdot interaction with the junction. As the intradot tunneling is a
scalar value γAB = γBA, so we can write them simply as tc which was used
in the main text. The full matrix is then significantly simplified to read, ˆ Hmnσσ0 = (ω − εA↑+i2Γ11↑(ω)) 0 −tc+i2Γ12↑(ω) 0 0 (ω − εA↓+i2Γ11↓(ω)) 0 −tc+2iΓ12↓(ω) −tc+i2Γ21↑(ω) 0 (ω − εB↑+2iΓ22↑(ω)) 0 0 −tc+i2Γ21↓(ω) 0 (ω − εB↓+2iΓ22↓(ω)) . (30)
Notice that the off-digonal portions of the matrix 30 contains Γσ(ω). These
are non-zero due to the possiblity of transport through either QD. Therefore multiple pathways are present and interference phenomena may be observed.
Compacting the spin components to reduce cluttering the Hamiltonian is simply, ˆ Hmnσσ0 = (ω − εAσ+ 2iΓ11σ(ω)) −tc+2iΓ12σ(ω) −tc+2iΓ21σ(ω) (ω − εAσ+2iΓ22σ(ω)) (31) Inverting the matrix 31 will finally give us the Green’s function,
ˆ H−1mnσσ0 = GRmnσσ0 = 1 Ω (ω − εAσ+2iΓ22σ(ω)) −tc+ 2iΓ21σ(ω) −tc+ 2iΓ12σ(ω) (ω − εAσ+2iΓ11σ(ω)) (32)
where, Ω = (ω − ε1σ+ i 2Γ11σ(ω))(ω − ε2σ+ i 2Γ22σ(ω)) − (−tc+ i 2Γ12σ(ω))(−tc+ i 2Γ21σ(ω)) (33)
The coupling between the central device and the environment is given by the level-width functions which are by definition imaginary. We will not prove anything here but it is enough to consider it as the representation of the flow of particles in and out of the central region, we simply give the definition here,
Γαijσ(ω) = 2πρασ(ε) Tijα
2
δ(ω − ε) (34)
where Tijα is again the tunneling from the leads to the central device and
ρα
σ(ε) is again the density of states in the lead α for dot level spin σ.
More generally the dot-lead coupling can be seen as the spin-dependent hybridization of the states in the central device due to the continuum of the leads.
We can realize the different possible pathways through the central region by changing the appearance of Γ which allows us to tune the relative position of the QD to the left and right lead making the one site the more favorable path for the current to tunnel through.
tc
L R
(a) The Parallel configuration is
given by γ2 = γ1.
tc
L R
(b) The serial configuration is
given by γ2 = 0.
Figure 3: Schematics of the DQD system coupled to left and right leads.
Writing Γ = ΓL+ ΓR we associate the left and right leads to both the
3b we would like to create a general Γ matrix that represents both case in
figure 3. To do this we allow γ2 to approach γ1, where γ1 = p × tc for
some p ∈ [0, 1]. The serial configuration is associated with a diagonal Γ, with zero contributions from the non-diagonal elements, while the parallel configuration has a finite non-diagonal elements in Γ. We therefore write the
following ΓR11σ = ΓL22σ = γ1 similarly ΓR22σ = ΓL11σ = γ2. The non-diagonal
matrix elements are then ΓL
21σ = ΓL12σ = ΓR21σ = ΓR12σ =
√
γ1γ2. We can then
write the Γ matrix as,
Γασ = Γα 11σ pΓα11σΓα22σ pΓα 11σΓα22σ Γα11σ Density of states
The Density of state provides useful insight into any quantum system, we
write the non equilibrium density as ρ(ε) = −1
π Im T r[G
>
ii(ε) − G<ii(ε)].
Fur-thermore we consider the retarded Green’s function given in equation 32, we extract the relevant information by letting the denominator Ω = 0.
0 = (ω − ε1σ+ i 2Γ11σ(ω))(ω − ε2σ+ i 2Γ22σ(ω)) − (−tc+ i 2Γ12σ(ω))(−tc+ i 2Γ21σ(ω))
= ω2− ω(ε1σ− iΓ11σ+ ε2σ− iΓ22σ)) + (ε1σ− iΓ11σ)(ε2σ− iΓ22σ)
− (tc− iΓ12σ)(tc− iΓ21σ)
The roots are then given by,
ω± = 1 2(ε1σ+ ε2σ− i(Γ11σ+ Γ22σ) ± r (∆ε − i 2[Γ11σ − Γ22σ]) 2+ 4(t c− i 2Γ12σ)(tc− i 2Γ21σ) (35)
where it is convenient for later use to define ∆ε = ε1σ−ε2σ. Let ε1σ = ε2σ = ε0
and Γ11σ = Γ22σ = Γ0. We can therefore rewrite the above as,
ω±= ε0− i 2Γ0 ± r (tc− i 2Γ12σ)(tc− i 2Γ21σ) (36)
Recall that we defined the non-diagonal elements of Γ as Γ12σ = Γ21σ =
√ γ1γ2
we may therefore write (tc− iΓ12σ)(tc− iΓ21σ) = (tc− iΓ12σ)2.
ω± = ε0−
i
2Γ0± (tc−
i
2Γ12σ) (37)
If we for simplicity look at the serial configuration shown in figure 3b, we can define the Green’s function for this system by defining the Γ matrix appropriately. As seen in the figure the leads do not symmetrically couple to
the Central device, we therefore write ΓR
22σ = ΓL11σ = γ2 = 0. The off-diagonal
elements are then ΓL
ijσ = ΓRijσ =
√
γ1γ2 = 0 where i 6= j. Setting the energy
levels of the quantum dots as degenerate i.e. ∆ε = 0 at some energy ε0 under
a symmetric level-width broadening ΓR
11σ = ΓL22σ = Γ0 for some coupling Γ0.
The center of the distributions of the two quantum dots will differ from the
eigenvalues for the isolated level by equation 37, i.e, ω± = ε0± tc.
Figure 4 represents the density of state of the DQD system as it was
described in figure 3. For ∆ε = 0, γ1 = tc and γ2 = 0 as shown in the left
figure the distributions are identical represented by the superposition of two
Lorentzian separated by the inter dot coupling 2tc. If we approach the limit
of the parallel coupled system γ2 = 0.25γ1 we begin to see a narrowing of one
distribution and the widening of the other. For γ2 = 0.75γ1 which is shown
on the right, one of the states approaches a δ-function.
− 10 − 5 0 5 100.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0 2 4 6 8 10 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 − 10 − 5 0 5 10 0.000 0.002 0.004 0.006 0.008 0.010
Figure 4: Density of states ρ(ε) as a function of the Fermi energy for
degen-erate QD levels ∆ε = 0 and tc= γ1 with varying γ2
The serial and parallel configurations are compared in figure 5, where the electron distribution was initially distributed across two Lorentzian in the
serial system we find that one state is darkened and the other broadened as a result. This indicates that the darkened states becomes localized.
− 10 − 5 0 5 10 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025
Figure 5: Density of states ρ(ε) as a function of the Fermi energy for
4
Probability Current
We first consider the transmission as a function of the Fermi energy of the leads, figure 6 gives us our first results in the behaviour of the current passing through the central region.
We begin looking at the simplest case where the QDs are positioned
such that their level energies coincide i.e. ε1σ = ε2σ hence the difference
∆ε = 0. We allow the inter dot coupling strength to equal the level-width
broadening from the leads tc = γ1. The first figure in the upper left corner
denotes the serial configuration where γ2 = 0 that is characterised by two
equal resonance indicating that the probability of the current passing the two dots are equal. This is clear from the two super positioned Lorentzian. As
γ2 increases approaching γ2 = γ1 one of the resonances begins to diminish
resulting in the well known Fano resonance, characterised by the line shape
seen at γ2 = 0.25γ1 and γ2 = 0.5γ1. When γ2 = γ1 the Fano line shape
vanishes completely and we are left with a broadened Lorentzian indicating that the current is only associated with the single QD analogous to an isolated state.
0.0 0.2 0.4 0.6 0.8 1.0 − 30 − 20 − 10 0 10 20 30 0.0 0.2 0.4 0.6 0.8 1.0 − 30 − 20 − 10 0 10 20 30
Figure 6: The transmission as a function of the Fermi energy, for ∆ε = 0,
tc= γ1 whilst varying γ2
If we now consider how a finite difference in the energy levels i.e. ε1σ6= ε2σ
of the QDs effects the transmission for the parallel coupled system. We distance ourselves from the symmetric system that we have discussed so far
and allow the energy levels to be separated by tc, i.e. kε2σ− ε1σk = tc.
We again consider equation 33 and rewrite the last term as,
(tc− i 2Γ12σ)(tc− i 2Γ12σ) = |tc| 2− 1 4Γ12σΓ21σ − i 2(Γ12σ+ Γ21σ) = r (|tc|2 − 1 4Γ12σΓ21σ) 2+1 4(Γ12σ+ Γ21σ) 2e−i arctan 1 2(Γ12σ +Γ21σ ) |tc|2−14Γ12σ Γ21σ (38)
We can now extract a phase factor from equation 38 which will associate the quantum interference effects with the inter dot coupling and the lead coupling. We only consider the denominator,
|tc|2− 1 4Γ12σΓ21σ = |tc| 2 −1 4γ1γ2 =⇒|tc| ≥ √ γ1γ2 2 (39) The inequality |tc| ≥ 12 √
γ1γ2 allows for some prediction when setting the
pa-rameters that result in the desired resonance. The Fano resonance appears clearly when the inequality is satisfied and diminishes as
√ γ1γ2
2 → |tc|.
In figure 7 the darkened state appears as a Fano line shape convoluted with a Lorentzian located at the conducting state. The dashed line indicates the position of the Lorentzian before symmetry breakage of the QD energy levels.
Figure 7: The transmission as a function of the Fermi energy for the parallel
coupled system (γ2 = γ1 = tc), where ∆ε = tc.
The antiresonance is found at ε2σ= tc, similar to the case of ∆ε = 0 and
ε0 = 0 the transmission is characterized by a convolution of a Lorentzian
indicating cotunnelling of electrons through this the state of energy ε1σ and
a Fano line shape at the darkened state of energy ε2σ. This is made more
clear when considering the local DOS illustrated in figure 8 of the QDs,
given by ρi(ε) = −1πIm(G>ii(ε) − G
<
ii(ε)) for i = 1, 2. The contributions to
the total DOS is separated into local DOS ρi corresponding to the individual
QDs, where the full line denotes the density distribution ρ1 at the QD of
energy ε1σ and the dashed line corresponds to the distribution ρ2 at the QD
Figure 8: The local DOS as a function of the Fermi energy for the parallel
coupled system (γ2 = γ1 = tc), where ∆ε = tc.
It is clear that the distributions retain characters from both QDs at dif-ferent energy intervals. In connection to the formation of the Fano lineshape shown in figure 7, at the energy range associated with the lineshape we find
that ρ1 and ρ2 also displays similar characteristics, i.e. the formation of a δ
function in combination with a ”dip” at zero distribution. Analogously the energy range associated with the dominate transport channel has a wider
distribution from both ρ1 and ρ2. Effectively the DOS is enough to predict
the behaviour of the transmission due to the width of the distribution func-tion being inversely proporfunc-tional to the occupafunc-tion lifetime. It is also clear that the convention of bonding and antibonding is quite convenient as we
5
Conclusion
In this work we have studied the transmission at zero temperature of a lead-DQD-lead system where the QDs are connected to the leads in serial as well as in a parallel configuration. We found that that the transmission was composed of a convolution of a Lorentzian centered at the bonding energy and a Fano lineshape appearing at the energy of the anti-bonding level. We found that the formation of the Fano peak narrows as we approach the
limit of a symmetrical system until it completely vanishes when γ1 = γ2 for
∆ε1σ = ε2σ. In the parallel configuration the anti-bonding states has no
occupancy as shown by the density of state of system where the peak at the anti-bonding energy goes towards the limit of a δ -function until disappearing completely. The transmission pathway is dominated by the bonding state which is shown by the widening of the Lorentzian. We can interpret the widening of the Lorentzian as the increase of the lifetime of the corresponding QD. The corresponding density of states becomes more localized which is characteristic for a localized antibonding state.
References
[1] George Arfken. Mathematical Methods for Physicists. Academic Press, Inc., San Diego, third edition, 1985.
[2] D. D. Awschalom, D. Loss, and N. Samarth, editors. Semiconductor
Spintronics and Quantum Computation. Springer Berlin Heidelberg,
2002.
[3] K. Cahill. Physical Mathematics. Cambridge University Press, 2013. [4] AM Calle, M Pacheco, GB Martins, VM Apel, GA Lara, and PA
Orel-lana. Fano–andreev effect in a t-shape double quantum dot in the kondo regime. Journal of Physics: Condensed Matter, 29(13):135301, 2017. [5] ML Ladron De Guevara, F Claro, and Pedro A Orellana. Ghost fano
resonance in a double quantum dot molecule attached to leads. Physical Review B, 67(19):195335, 2003.
[6] U Fano. Effects of configuration interaction on intensities and phase shifts 1961 phys. Rev, 124:1866.
[7] S. Hassani. Mathematical Physics: A Modern Introduction to Its Foun-dations. Springer International Publishing, 2013.
[8] Antti-Pekka Jauho, Ned S. Wingreen, and Yigal Meir. Time-dependent transport in interacting and noninteracting resonant-tunneling systems. Phys. Rev. B, 50:5528–5544, Aug 1994.
[9] Kicheon Kang and Sam Young Cho. Tunable molecular resonances of double quantum dots embedded in an aharonov-bohm interferometer. arXiv preprint cond-mat/0210009, 2002.
[10] G. D. Mahan. Many Particle Physics, Third Edition. Plenum, New York, 2000.
[11] PA Orellana, F Dominguez-Adame, I G´omez, and ML Ladron De
Gue-vara. Transport through a quantum wire with a side quantum-dot array. Physical Review B, 67(8):085321, 2003.
[12] Pedro A Orellana, ML Ladr´on de Guevara, and F Claro. Controlling
fano and dicke effects via a magnetic flux in a two-site anderson model. Physical Review B, 70(23):233315, 2004.
[13] Gianluca Stefanucci and Robert van Leeuwen. Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction. Cambridge University Press, 2013.
[14] Yusuke Takazawa, Yoshiki Imai, and Norio Kawakami. Electron trans-port through t-shaped double-dots system. Journal of the Physical So-ciety of Japan, 71(9):2234–2239, 2002.
