J Comput Electron (2006) 5:247–250

DOI 10.1007/s10825-006-8852-z

Scattering resonances in 1D coherent transport through acorrelated quantum dot: An application of the few-particlequantum transmitting boundary methodAndrea Bertoni · Guido Goldoni

C© Springer Science + Business Media, LLC 2006

Abstract We present a method for the calculation of the

scattering states of a N + 1th-particle coherently interact-

ing with N correlated particles confined in a nanostructure

and placed within an open domain. The method is based on a

generalization of the quantum transmitting boundary method[C. Lent and D. Kirkner, J. App. Phys. 67, 6353 (1990)].

The antisymmetry conditions of the N + 1-identical parti-

cles current-carrying state results from a proper choice of

the boundary conditions. As an example which is relevant to

coherent electronics, we apply the method to compute the ex-

act transmission functions and phases of an electron crossing

a 1D quantum dot with zero, one or two bound electrons.

Keywords Scattering states · Fano resonances · Quantum

dot · Open boundaries · 1D coherent transport

Introduction

The process of quantum scattering of a charge by a system of

bound particles has attracted much attention since the early

days of quantum mechanics. Inelastic scattering of electrons

by atoms served as a validation for theoretical approaches

beyond the single-particle approximation [1, 2]. The above

methods, especially those based on phase-shift analysis of

scattered wavefunctions, were also applied to the study of

carrier scattering by a localized impurity in bulk metals and

A. Bertoni (�)INFM-S3 National Research Center on nanoStructuresand bioSystems at Surfaces,Via Campi 213/A, 41100 Modena, Italye-mail: bertoni.andrea@unimore.it

G. GoldoniINFM-S3 National Research Center on nanoStructuresand bioSystems at Surfaces, Modena, Italy; Dipartimento diFisica, Universita di Modena e Reggio Emilia, Italy

semiconductors, thanks to the similarities of the impurity

states to atomic orbitals.

An electrostatically-defined semiconductor quantum dot

(QD) is an ideal laboratory to test and quantify quantum

coherence and correlation effects among carriers in scatter-

ing experiments, with a single electron transmitted through

or reflected by electrostatically charged QDs [3]. Indeed,

QDs posses many desirable features from this point of

view: an atomic-like structure that can be fully controlled

by electrostatically tuning the confining potential; tunable

coupling to source and drain leads; precise determination

of conductance down to the quantized transmission limit;

tunable number of electrons within the same sample; and last

but not least, transmission phases can be directly measured

by experimental setups exploiting quantum-interference. In

this context, Yacoby et al. [4] and Schuster et al. [5] recently

measured the transmission phase of an electron scattered

by a QD. This stimulated a novel interest on transmission

phases; indeed, not only these studies have a fundamental

interest, but may directly play a role in the next-generation

of quantum nanoelectronic devices and, possibly, of future

quantum computing architectures [6].

In this work we generalize the quantum transmittingboundary method (QTBM) [7], which will be briefly re-

viewed below, to the solution of the few-particle spinless

Schrodinger equation. The new method, which we termed

few-particle QTBM (FP-QTBM), allows us to include proper

open boundary conditions and simulate the scattering of one

electron by a few charges confined in a QD. Below we dis-

cuss its application to a 1D double-barrier resonant-tunneling

(DBRT) device, with an electron incoming from one lead

and a number of other electrons occupying a bound state of

the central dot (see Fig. 1). We found that, in spite of the

single dimensionality of the structure, the coupling between

the continuum of the single-particle scattering states and the

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248 J Comput Electron (2006) 5:247–250

Fig. 1 Potential profile V (x) of the simulated system (see Eq. (1)).Four bound single-particle levels χm are represented. The QD electronis initialized in the ground state and the scattered electron, arriving fromthe left lead, is given a kinetic energy of 10 meV. Two 50 meV tunnelbarriers mimic the quantum point contacts that connect the QD to theleads. The bottom potential of the QD is varied in the simulations

discrete spectrum of the bound states, gives rise to Fano reso-

nances, where the transmission phase shows an abrupt jump

of π at the zeros of the transmission coefficient.

Few-particle quantum transmitting boundarymethod and scattering resonances

The mechanism of charge transport through a QD operating

in single-electron regime (i.e., when only the scattered elec-

tron is localized in the dot) is well understood: Breit-Wigner

resonances are found in the transmission spectra, stemming

from the coupling of a quasibound state to the scattering states

in the leads. Such transmission resonances show a Lorentzian

line shape of the amplitude t = C(�/2)/(E − Eqb + i�/2),

where C is a complex constant, � the width of the resonance,

inversely proportional to the life time of the quasibound state

with energy Eqb, and E is the energy of the incoming particle.

Below we briefly discuss the FP-QTBM which allows us to

investigate scattering experiments beyond this single-particle

picture.

Let us consider the envelope function of two spinless iden-

tical particles in 1D, ψ(x1, x2), describing the stationary state

with one electron in a QD bound state (we take here the

ground state without loss of generality) χ0(x), with energy

E0, and a second electron incoming from the left boundary

(Fig. 1) as a plane wave of energy (E − E0), where E is

the total energy of the two-particle system. If one supposes

that a change in the energy of the quasibound state does not

alter significantly its life time, t can be equivalently seen

as a function of E at a fixed QD potential or as a function

of Eqb at a fixed incoming-electron energy. We chose this

second perspective considering a fixed Fermi level in the

leads and a variable QD potential, and we suppose that the

applied source-drain bias is sufficiently small that either zero

or one quasibound state lies inside the transport window. The

Hamiltonian of the system reads

H = − h2

2m

(∂2

∂x1

+ ∂2

∂x2

)+V (x1)+V (x2)+ e2

4πε|x1 − x2| ,

with m, e and ε effective mass, elementary charge and dielec-

tric constant, respectively and V (x) the given single-particle

potential (see Fig. 1). We aim at solving Hψ(x1, x2) =Eψ(x1, x2) in the square domain x1, x2 ∈ [0, L]. Due to

the Pauli exclusion principle the two-particle wave function

must be antisymmetric ψ(x1, x2) = −ψ(x2, x1). Therefore

the computational domain can be reduced to the black re-

gion in Fig. 2. Along the diagonal boundary, the condition

ψ(x, x) = 0 is imposed by the wave function antisymmetry

(this would also apply to charged bosons due to the diverging

Coulomb energy). To include the other two boundaries we

follow the QTBM. Owing to our choice of boundary condi-

tions (one particle coming from the left lead as a plane wave

and the other in the ground state of the QD) the total energy

E belongs to a continuous region of the energy spectrum.

Furthermore we limit ourself to energies E that are lower

than the double ionization energy, i.e., the incoming electron

does not posses enough energy to unbind the other particle

without being captured in the QD. This condition means that

the probability to find both electrons in the leads is zero, i.e.

ψ(x1, x2) = 0 when both x1 and x2 are outside [0, L] (regions

F in Fig. 2). Following the QTBM and taking into account

Fig. 2 Computational domain and boundaries. The Schrodinger equa-tion is numerically solved in the black triangular region. Due to thechoice of the system parameters, the wave function is zero in fourforbidden regions marked by F that represent the possibility of bothelectrons being in the leads. The two-particle wave function is incidentand reflected through boundary A while it is transmitted through bound-ary B. On the diagonal boundary, i.e. when x1 = x2, ψ(x1, x2) = 0 dueantisymmetrization. In the calculation performed we took L = 50 nm

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J Comput Electron (2006) 5:247–250 249

the antisymmetry of the wave function, the form of the wave

function in the two leads A ≡ (0 < x1 < L , x2 < 0) and B ≡(x1 > L , 0 < x2 < L) is

ψ A(x1, x2) = −[χ0(x1)eik A

0 x2 +MA∑

m=0

bAmχm(x1)e−ik A

m x2

+∞∑

m=MA+1

bAmχm(x1)ek A

m x2

](1a)

ψ B(x1, x2) =[

MB∑m=0

bBmχm(x2)e−ik B

m x1

+∞∑

m=MB+1

bBmχm(x2)ek B

m x1

], (1b)

respectively, where χm(x) is the m-th bound state, with en-

ergy Em , of the QD; k Am =

√(2m/h2)|E − Em − V (0)| and

k Bm =

√(2m/h2)|E − Em − V (L)|; bA

m and bBm are the m-th

channel reflection and transmission coefficients, respectively

(and are unknowns of the problem); MA and MB represent

the number of bound QD states that can be excited when

the incoming electron is either reflected back in lead A or

transmitted to lead B. Equations (1a) represent the general

form of a two-particle wave-function of energy E when one

particle is on a constant-potential lead and the other is bound

by the confining potential. Note that also the higher energy

channels must be included in the calculation as evanescent

waves, in perfect analogy with the QTBM, where the χm’s

represent the transverse modes of the leads.

The boundary Eqs. (1a) and (1b) are coupled to the

Schrodinger equation for the internal points and discretized

by finite difference method. The resulting linear system is

solved by means of the parallel direct sparse solver library

Pardiso [8, 9]. Explicit calculations are performed using

GaAs material parameters. Below we show results in a en-

ergy range for which the bound electron is left in the ground

state after the scattering, i.e., MA = MB = 0. In particular,

we consider a 30 nm potential well of variable negative depth,

with two 50 meV barriers, each 3 nm wide. The simulation

domain is L = 50 nm and the energy of the injected electron

is 10 meV. The leads are at zero potential. Once the linear sys-

tem has been solved, the transmission probability and phase

are easily obtained from the bB0 complex coefficients.

A generalization of the procedure described above can

be derived in order to calculate the scattering states of an

impinging electron when N electrons are bound in the QD.

In this case the first step is the computation of the correlated

N -electron eigenstates of the QD χm(x1, x2, ..., xN ); then

the (N + 1)-electron Schrodinger equation is solved together

with (N + 1) open boundary conditions, similar to Eq. (1a),

in a (N + 1)-dimensional domain [10].

Results and conclusions

Figure 3 shows the transmission probability (solid line) and

the transmission phase (dotted line) for one (A), two (B)

and three (C) particles, as a function of the QD potential.

Single-particle Breit-Wigner resonances are present in the

three cases (see, as an example, Fig. 3(D)): they appear as

large Lorentzian peaks centered around the eigenenergies

of the closed system. As the number of electrons bound in

the QD increases, the above resonances shift toward stronger

confining energies: this is due to the Coulomb blockade, orig-

inated by the larger negative charge of the carriers bound in

the QD. Note that in our approach the Coulomb blockade

effect is automatically obtained without the need of an elec-

trostatic model. The transmission phase changes smoothly

by π on each Breit-Wigner resonance, as contemplated by the

Fig. 3 Transmission spectra for one electron (A), two electrons (B)and three electrons (C). The transmission probability (solid curve) andthe transmission phase (dotted curve) are reported as a function of thebottom potential of the QD. While in the single-particle case only Breit-Wigner resonances are present, in the other cases, with electron-electroncorrelation, also Fano resonances appear. Lower graphs show two par-ticular resonances, of different types (D: Breit-Wigner, E: Fano) of thethree-electron spectrum

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250 J Comput Electron (2006) 5:247–250

single-particle approach. In addition to the above resonances,

a number of extremely sharp peaks with typical asymmetric

Fano lineshape (see, as an example, Fig. 3(E)) is present in

the transmission spectra of the two- and three-particle scat-

tering states. The transmission phase increases smoothly by

π on the resonance peak but shows an abrupt drop of π in

correspondence of the zero of the transmission probability.

This behavior of the phase, already demonstrated in the liter-

ature for single-channel single-particle transport through 2D

QDs [11,12], and experimentally measured in single-electron

transistors [13], is not obtainable for 1D single-particle sys-

tems: Fano resonance in 1D system are a genuine few-particle

phenomenon brought about by electron-electron correlation.

In conclusion, we have developed a generalization of the

QTBM which allowed us to simulate the scattering of one

electron through a nanostructure with confined carriers cou-

pled to the leads with full inclusion of the Coulomb corre-

lation. Application to scattering from a charged 1D QD has

been shown. As a final remark we note that related experi-

ments on charge transport through QDs [14,15] demonstrated

also that the transmitted current include a coherent compo-

nent also in the case of a multi-electron occupancy of the dot.

With the few-particle QTBM it should be possible to quantify

this coherent part [16], since it corresponds to the probabil-

ity that the electron is transmitted in the same channel from

which is injected.

Acknowledgments This work has been partially supported by projectsMIUR-FIRB n.RBAU01ZEML, MIUR-COFIN n.2003020984, INFMCalcolo Parallelo 2004, and MAE, Dir.Gen. Promozione CooperazioneCulturale.

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