Electron-correlation driven capture and release in double quantum dots
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Electron-correlation driven capture and release in double quantum dots
Federico M. Pont ∗ Facultad de Matem´atica, Astronom´ıa y F´ısica, Universidad Nacional de C´ordoba,and IFEG-CONICET, Ciudad Universitaria, X5000HUA, C´ordoba,Argentina and Theoretische Chemie, Physikalisch-Chemisches Institut,Im Neuenheimer Feld 229, 69120 Heidelberg, Germany
Annika Bande † Institute of Methods for Material Development and Joint Ultrafast Dynamics Lab in Solutions and at Interfaces (JULiq),Helmholtz-Zentrum Berlin f¨ur Materialien und Energie,Albert-Einstein-Str. 15, 12489 Berlin, Germany andTheoretische Chemie, Physikalisch-Chemisches Institut,Im Neuenheimer Feld 229, 69120 Heidelberg, Germany
Lorenz S. Cederbaum ‡ Theoretische Chemie, Physikalisch-Chemisches Institut,Im Neuenheimer Feld 229, 69120 Heidelberg, Germany (Dated: October 24, 2018)We recently predicted that the interatomic Coulombic electron capture (ICEC) process, a long-range electron correlation driven capture process, is achievable in gated double quantum dots(DQDs). In ICEC an incoming electron is captured by one QD and the excess energy is usedto remove an electron from the neighboring QD. In this work we present systematic full three-dimensional electron dynamics calculations in quasi-one dimensional model potentials that allow fora detailed understanding of the connection between the DQD geometry and the reaction probabilityfor the ICEC process. We derive an effective one-dimensional approach and show that its resultscompare very well with those obtained using the full three-dimensional calculations. This approachsubstantially reduces the computation times. The investigation of the electronic structure for vari-ous DQD geometries for which the ICEC process can take place clarify the origin of its remarkablyhigh probability in the presence of two-electron resonances.
PACS numbers: 73.21.La, 73.63.Kv, 34.80.Gs, 31.70.Hq
I. INTRODUCTION
The technical ability of producing nanosized materi-als lead among other achievements to the discovery - andnowadays the technological application - of semiconduc-tor (SC) QDs. In these structures some typical featuresof SC bulk material are prevailed and married to typi-cal atomic properties emerging from the energy levelquantization in the QDs, motivating their name: ar-tificial atoms. DQDs can either be coupled (artificialmolecules ) or uncoupled. The latter arrangement weconsider here for the investigation of an energy transferprocess between QDs.The electron confinement achieved through differentQD geometries (disc shaped, spherical, wires, double lay-ered, etc.) presents an interesting variety of electronicproperties that are, however, similar for various kindsof QDs. Epitaxially-grown self-assembled QDs are mostcommonly disc or pyramidally shaped InGaAs islandsonto a GaAs substrate fed through a wetting layer byfree electrons from the substrate. Vertical stackingof layers allows to obtain a nanostructure of verticallyarranged DQDs.
In electrostatically defined QDs, a two-dimensionalelectron gas is created between two semiconductors withdifferent gaps. The gas can carry free electrons which can be further confined using charged metallic gates to definethe regions of one, two or more QDs. In the last years theadvances in nanowire fabrication allowed the construc-tion of QDs inside long nanowires using interlaced layerof different semiconductors. Colloidal nanocrystals cannowadays be constructed small enough to observe quan-tization of the electronic levels. They have attracted a lotof attention in the past few years as materials in modernthird generation solar cells.
In all theses QD struc-tures the manipulation of the electronic levels of the QDsis straightforward. Particularly, manipulation of levelswith different spin quantum numbers by magnetic or elec-tric fields is possible. This allows the study and charac-terization of transitions between them, whichare an appealing and desirable property in the field ofquantum information.Many experimental techniques are employed in cur-rent research to measure the properties of QDs. Theelectrical current through QDs can be obtained by trans-port spectroscopy. Transport on electrostatically de-fined QDs, nanowire based QD structures, and nan-otube defined QDs is widely used to determine thelevel structure inside the QDs. Another important fieldof research in various nanostructures is carrier relax-ation dynamics within excitons after an optical excita-tion. Pump-probe schemes with time resolution in theorder of ten of picoseconds can resolve processes suchas electron-phonon interactions, multiple excitongeneration, Auger relaxation also far-IR relaxationand relaxation into defects, impurities especially at sur-faces. The characteristics can be measured by photolu-minescence spectroscopy and complementary pho-tocurrent measurements can give information on the non-radiative decay time and energy of the excitons or intra-conduction band excited states. In the specific case ofDQDs, the transitions and tunneling dynamics of elec-trons of vertically coupled QDs were studied and inter-dot phonon-relaxation processes were detected betweenthe QDs. P to S orbital electron relaxation via elec-tron correlation has also been demonstrated in uncou-pled n -doped DQDs and after electric pulse excita-tion. In this case the relaxation in one QD occurs viaenergy transfer and emission of an electron in a neigh-boring QD in a process called intermolecular Coulombicdecay (ICD).
In the present work we focus on the less intensivelystudied capture dynamics of free electrons into n-dopedDQDs mediated solely by long-range electron correla-tion. In general the most important electron cap-ture mechanism is via emission of longitudinal opticalphonons, that has been studied before in single anddouble QDs. It has been analyzed theoretically in singleQDs along with electron collisions and emission.
Inour previous work we showed for the first time that elec-tron capture can as well be mediated efficiently by long-range electron correlation in the interatomic Coulom-bic electron capture (ICEC) in DQDs. The process wasnamed after the one originally predicted to be operativein atoms and molecules. In atoms the electron cap-ture by one atom occurs while another electron is emittedfrom an atom into its environment. In DQDs the elec-tron capture by one QD leads to an emission of electronsfrom neighboring QDs with controlled energy propertiesthat can be tuned by changing the geometric DQD pa-rameters. We postulated ICEC for n-doped DQDs em-bedded in nanowires (Fig. 1) using an effective mass ap-proximation (EMA) based model potential in which weperformed numerically exact electron dynamics calcula-tions. The relaxation dynamics of an excitonic electronin undoped materials can be described within the samemodel provided that the hole relaxation to the band edgehas been faster than that of the electron. We showed already that the probability for ICEC isnon-negligible and can be greatly enhanced in the pres-ence of two-electron resonance states that are capable ofundergoing fast ICD-related energy transfer. Here, wesystematically add other DQD configurations to thosestudied before and analyze how and for which energies inthe different configurations ICEC in the general and theresonance case becomes most effective.The paper is organized as follows: First we presentsome general considerations on the ICEC process (II), in-troduce our model and the DQD electronic structure (III)followed by the electron dynamics methods used (IV) and the results (V). Since numerically exact computa-tions in the full six-dimensional Hilbert space are verytime consuming, we additionally include an effective two-dimensional description of the nanowires and compare tothe full dimensional results (V B 4). The discussion ofthe results using realistic semiconductor parameters aregiven in (VI) followed by the conclusions (VII). II. CONDITIONS FOR ICEC IN DQDS
In this work we consider a system of two fully corre-lated electrons and two QDs which we call the left andright QD and which are described by two different modelpotentials (see Fig. 2). For the time being consider a leftpotential well that supports only a single one-electronlevel L with energy E L and a right one with one single-electron level R with energy E R such that E L = E R .The tunneling and hybridization between L and R inthe DQD is vanishingly small due to the long interdotdistance R of the considered system. The ICEC processoccurs as depicted in Fig. 2 where an electron is initiallybound to the right QD and another electron with mo-mentum p i is coming in from the left side of the DQD.The incoming electron can then be captured into the L ground state of the left QD while the electron on theright is emitted from the R ground state of the rightQD. Energy conservation dictates that the total energyof the system E T (in) E T = ε i + E R (1)(out) E T = ε f + E L , (2)is conserved and the kinetic energy acquired by theoutgoing electron can be expressed as ε f − ε i = ∆ E (3)with the corresponding momentum p f = q p i + 2 m ∗ ∆ E (4)where ε i,f = p i,f / m ∗ , ∆ E = E R − E L and m ∗ is theelectron effective mass in atomic units. As one can noticefrom Eq. (4) the emitted electron can have a higher ora lower momentum than the initial electron, dependingon the relation between the bound-state energies E R and E L . However, for negative values of ∆ E the ICECchannel is closed if the incoming electron energy is lowerthan | ∆ E | (see Eq. (4)). Note also that since ∆ E is theenergy acquired by the outgoing electron, then − ∆ E isconversely the energy gain/loss suffered by the DQD. III. MODEL
The motion of two electrons inside a nanostructuredsemiconductor can be accurately described using a few-electron effective mass model potential in which elec-tron dynamics calculations are feasible. This approach Figure 1. (Color online) Schematic view of two experimentalsetups to achieve the electron confinement inside a nanowire.In panel (a) a 3D confinement is obtained using a layeredsemiconductor structure, in (b) the nanowire is built of a sin-gle semiconductor material and the barriers are obtained byelectrostatic depletion (areas indicated with green shading).The depletion is achieved by setting different electrostatic po-tential energies in the metallic gates below the wire. offers then straightforward observability of how electroncorrelation can lead to ICEC in general two-site systemswhere electron correlation between moieties plays a fun-damental role as well as in the specific case of a QD. Weadopt here the model for the DQD used previously tostudy the dynamics of ICEC and ICD.
Thedots are represented by two Gaussian wells aligned in z direction. In x and y direction we assume a strong har-monic confinement which could be attributed either todepleting gates or to the actual structure of the semi-conductor. Besides the full three-dimensional calcula-tions we also considered a simpler one-dimensional modelthat uses an effective electron-electron interaction to takethe wire shape of the system in x and y direction im-plicitly into account. In this one-dimensional effectivemodel electron dynamics calculations are much more ef-ficient because only the z coordinates of the electrons areevolved in time. Figure 2. (Color online) Schematic view of the interatomicCoulombic electron capture for a double quantum dot. Theeffective mass approximation is used to describe the quantumdots as two potential wells. The capture of the incomingelectron by the left dot (dashed green state) is mediated byits correlation with the electron initially bound to the rightdot (full green state). While the electron is captured in theleft dot, the electron on the right is excited into the continuumand becomes an outgoing electron.
A. Hamiltonian
The two-electron effective mass Hamiltonian for thesystem is H ( r , r ) = h ( r ) + h ( r ) + 1 ε r | r − r | (5)where ε r is the relative dielectric permittivity and h ( r i ) = − m ∗ ∇ i + V c ( x i , y i ) + V l ( z i ) (6)is a one-electron Hamiltonian in which V c ( x i , y i ) = 12 m ∗ ω ( x i + y i ) (7) V l ( z i ) = − V L e − b L ( z i + R/ − V R e − b R ( z i − R/ (8)are the transversal confinement and longitudinal openpotentials, respectively. m ∗ is the effective mass, R isthe distance between the QDs and b L,R are the sizes ofthe left and right QD while V L,R their depths. Perform-ing the scaling r i → ε r m ∗ r i of the electronic coordinatesone can obtain the scaling relationships of the Hamil-tonian parameters shown in Tab. I. Clearly, we can usethe effective mass and the relative permittivity equal toone and rescale the parameters afterwards to obtain theenergies and distances for a specific semiconductor.Due to the comparably strong confinement ( ω = 1 . > V L,R ) the excited states relevant to this study areonly in z direction. We will correspondingly have a levelstructure L n ( R n ), n = 0 , , . . . in the left (right) QD withenergies E L n ( E R n ). The orbital symmetry is simply thatof a symmetric well: L corresponds to an S-symmetryaround the left dot, L to a P-symmetry and so on. Parameter Scaled value H (or E ) m ∗ ε r Hm ∗ ε r ω ωR ε r m ∗ R ( b L , b R ) ε r m ∗ ( b L , b R )( V L , V R ) ε r m ∗ ( V L , V R )Table I. Scaling of the Hamiltonian and parameters under thetransformation r i → ε r m ∗ r i . B. Effective one-dimensional approach
As mentioned in Sec. III A the system under consider-ation has a strong lateral confinement. It is then possibleto construct an effective one-dimensional Hamiltonian using the wave function separation ansatzΨ( r , r ) = ψ ( z , z ) φ ( x , y ) φ ( x , y ) , (9)where φ are two-dimensional single-electron groundstate functions and ψ ( z , z ) is the longitudinal effec-tive wave function. Since essentially the same results areobtained for singlet and triplet states, we chose tripletsymmetry throughout our study. Ψ has the proper sym-metry under exchange of electrons given by the longitu-dinal wave function ˆΠ ↔ ψ ( z , z ) = − ψ ( z , z ). Theone-dimensional Hamiltonian can be deduced from theanalysis of the expectation value of the full Hamiltonianwith the product wave function of Eq. (9) h Ψ | H | Ψ i = 2 ω − X i =1 , (cid:28) ψ (cid:12)(cid:12)(cid:12)(cid:12) m ∗ i ∂ ∂z i + V long ( z i ) (cid:12)(cid:12)(cid:12)(cid:12) ψ (cid:29) +1 ε r (cid:28) Ψ (cid:12)(cid:12)(cid:12)(cid:12) | r − r | (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:29) . (10)The last term can be explicitly written in the form D Ψ (cid:12)(cid:12)(cid:12) | r − r | (cid:12)(cid:12)(cid:12) Ψ E = Z Z | Ψ( r , r ) | | r − r | d r d r = Z Z | ψ ( z , z ) | V eff ( z ) dz dz , (11)with the squared longitudinal wave function and the ef-fective z -potential V eff ( z ) = r π l e ζ (1 − erf ( ζ )) , (12)which depends on z = | z − z | , the variable remainingafter integrating over the x and y coordinates.The size of the two-dimensional ground state wavefunction is given by l = p h φ | x | φ i = p /m ∗ ω and ζ = z / √ l is the distance z between the electrons in terms of the confinement size l . The asymptotic behaviorof V eff ( z ) exhibits a Coulombic decay behavior at largeelectron separation. However, at small distances betweenthe electrons this effective potential does not diverge at z = z which is beneficial for numerical treatments: V eff ( z ) (cid:12)(cid:12)(cid:12) z →∞ −→ z − l ( z ) ! (13) V eff ( z ) | z → −→ l (cid:18)r π − z l + · · · (cid:19) (14)The validity of the effective potential in different con-finement regimes was studied in [41] for double QDs as afunction of the distance R between QDs. From Eq. (13)we see that l/z defines the correction order of the effec-tive interaction at large distances. If we take the distancebetween the dots R as a measure of the closest distancethat electrons will be from each other, then z /l ≈ R/l .We realize then from Eq. (13) that in the regime studiedin this work ( l ≈ R ≈ z = z scales as 1 /l (seeEq. (14)) indicating that in truly narrow confinements( l →
0) there is less room for the electrons to avoid thedivergence of the Coulomb interaction.
IV. COMPUTATIONAL DETAILS
The dynamical evolution of the system was obtained bysolving the time-dependent electronic Schr¨odinger equa-tion employing the multiconfiguration time-dependentHartree (MCTDH) approach.
The triplet wave func-tion Ψ( r , r , t ) = n X i,j A ij ( t ) ϕ i ( r , t ) ϕ j ( r , t ) , (15)was expanded in time-dependent single particle functions ϕ i ( r , t ) (SPFs) and coefficients A ij ( t ) that fulfill the anti-symmetry condition A ij ( t ) = − A ji ( t ) for all times. TheDirac-Frenkel variational principle (cid:28) δ Ψ (cid:12)(cid:12)(cid:12)(cid:12) H − i ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:29) = 0 (16)was used to obtain the equations of motion for the coef-ficients and SPFs.They were efficiently solved using a constant mean fieldapproach as implemented in the MCTDH-Heidelbergpackage. The convergence of numerical results wasensured by monitoring the population of the least popu-lated SPF. This is reasonable because the SPFs are adap-tive in time and are optimized to describe Ψ( r , r , t )with the least possible number of SPFs.The multimode SPFs ϕ i ( r q , t ) were in turn expandedin one-dimensional time-dependent SPFs for each of theCartesian coordinates ( x, y, z ) as ϕ i ( r q , t ) = X lmn C ( q ) lmn ( t ) χ ( x ) l ( x q , t ) χ ( y ) m ( y q , t ) χ ( z ) n ( z q , t ) . (17)These one-dimensional SPFs χ l are expanded on aDVR-grid (discrete variable representation). We choseharmonic oscillator DVRs for the x and y , and a sineDVR for the z coordinate as listed in Tab. II.In the full 3D calculations the Coulomb potential wasregularized as 1 /r → / p r + a with a = 0 .
01 toprevent divergences at r = r , and then transformedinto sums of products using the POTFIT algorithm ofMCTDH.A quadratic complex absorbing potential (CAP) wasplaced at the position ± z cap along the z coordinate toabsorb the outgoing electron before it reaches the end ofthe DVR grid. The CAP obeys W ± = − iη ( z ∓ z cap ) Θ( z ∓ z cap ) (18)where η is the CAP strength and Θ is the Heavysidestep function. The absorption prevents the unphysicalreflection of outgoing electrons at the grid boundaries.The absorption of the WP is also used to analyze theenergy distribution of the outgoing WP. The quantitythat we want to compute is the reaction probability (RP)for ICEC which corresponds to the scattering matrix ele-ment | S L ,R ( E T ) | which is the probability that an elec-tron impinging from the left on the DQD possessing anelectron bound at R leads to emission of an electron tothe right leaving behind a DQD with an electron boundto L .The computation of the matrix element was performedby using the expression for the stationary scatteringeigenfunctions in terms of the initial wave packet W P i in order to obtain the amount of emitted density fromthe wave packet absorbed by the CAP. The energy dis-tribution | ∆ W P i ( E T ) | of the incoming WP i is used tonormalize the Fourier transform of the absorbed density g L ( τ ) to obtain the reaction probability (RP). We ex-plicitly computed RP ( E T )100 = | S L ,R ( E T ) | = 2Re R ∞ g L ( τ ) e iE T t/ ~ d τπ | ∆ W P i ( E T ) | (19)where g L ( τ ) = Z ∞ D Ψ( t ) (cid:12)(cid:12)(cid:12) P (1) L W (2)+ P (1) L (cid:12)(cid:12)(cid:12) Ψ( t + τ ) E d t + Z ∞ D Ψ( t ) (cid:12)(cid:12)(cid:12) P (2) L W (1)+ P (2) L (cid:12)(cid:12)(cid:12) Ψ( t + τ ) E d t = 2 Z ∞ D Ψ( t ) (cid:12)(cid:12)(cid:12) P (1) L W (2)+ P (1) L (cid:12)(cid:12)(cid:12) Ψ( t + τ ) E d t (20) and∆ W P i ( E T ) = s m ∗ πp R Z ∞−∞ f W P i ( z ) e ip R z d z (21)where the function f W P i ( z ) is a Gaussian wave packetwith a spatial width ∆ x W P i . ∆ W P i ( E T ) is the en-ergy distribution of the incoming WP i peaked around ε W P i and given by the appropriate Fourier trans-form which uses the incoming momentum p R = p m ∗ ( E T − E R ) ≡ p i . g L ( τ ) is the absorbed electronic density by the rightCAP while another electron is bound in the L state.The projectors P ( q ) L acting on electron q specify whichelectron is in the L state, and the sum over both possibleconfigurations gives the total absorbed density. Note thatthis quantity explicitly correlates both events, emissionand capture, and thus gives only the ICEC contributionof the total emitted density. The scattering matrix inEq. (19) corresponds to the R initial state because theinitial wave functionΨ(0) = [ f W P i ( z ) φ R ( z ) − f W P i ( z ) φ R ( z )] × φ ( x , y ) φ ( x , y ) (22)represents a bound electron at R plus an incoming elec-tron both in the ground state of the confinement poten-tial.The RP is a wave-packet independent quantity in theenergy range of the size of the energy width of the in-coming wavepacket WP i (see Eq. (19)). At each energy,the RP gives the relative amount (in %) of the electrondensity that would be emitted in the calculation with amonoenergetic electron at that energy. The absorptionof WP i by the CAP outside the DQD economizes thecomputation time needed to obtain the RP. V. RESULTS
In this section we analyze the electronic structure (Sec.V.A) and the dynamics of the electrons (Sec. V.B) inthe DQD relevant for ICEC. We compare a number ofdifferent configurations that can be classified accordingto the general setups of the QD model potentials shownin Fig. 3. In setup A only the right QD with a singleone-electron state R is present. The only purpose ofinvestigating this setup is to prove that, for the incomingelectron energies considered in this work, no transmissionto the right is possible when the left QD is not present.The configurations belonging to setup B have one leftand one right QD and each dot has a single one-electronstate, L and R , respectively. In these cases ICEC isallowed and occurs as visualized in Fig. 2. Finally,setup C comprises configurations where the left QD hasan excited one-electron state L in addition to the L ground state allowing for the intermediate state | L R i to be formed. Since electrons located in the left and ABC
Figure 3. The three QD model potential setups studied in thiswork. Setup A is briefly analyzed and used only to clarify thatno transmission to the right is possible without a left QD. Insetup B each QD has one bound one-electron state, L and R ,respectively, and B is used to show how ICEC works in doubleQDs. In setup C the left QD has an additional one-electronexcited state L . In such a configuration the energy of thetwo-electron resonance | L R i can be tuned to substantiallyincrease the ICEC reaction probability. right QD are interacting with each other through thelong-range Coulomb interaction pushing the state intothe continuum, this state turns out to be a two-electron resonance. We will show that under certain conditionsthis resonance leads to a remarkable increase of the ICECprobability. A. Electronic Structure
As a first step in our analysis we want to study theelectronic structure of the DQD embedded in the wire.As explained in Sec. III the two-electron states can benamed after the one-electron states of the DQD. Theconfinement part of the wave function is described bythe lowest energy harmonic oscillator wave functions in x and y both with frequency ω and effective mass m ∗ and we therefore concentrate only on the z wave functionanalysis in what follows.The potential energy curves and the wave functionsof the states for two of the configurations used in thedynamical calculations are shown in Fig. 4. The concreteconfiguration of setup B in Fig. 4(a) has two bound one-electron states L and R . It is clearly visible that bothstates are localized in the respective QDs and that thereis no hybridization of the states. Two characteristics ofthis configuration make this possible. One is the distance R between the QDs, which is large compared to their size,and the other is the asymmetry of the DQD which leadsto different energies for the left and right QDs. -1.00-0.75-0.50-0.250.000.250.500.751.00 V l [ a u ] , ψ ( z ) V L = 0.80, b L = 1.0E L = -0.377 E R = -0.246V R = 0.60, b R = 1.0V L = 0.71, b L = 0.3R = 10V R = 0.60, b R = 1.0R = 10E R = -0.246E L = -0.441E L = -0.048 (a) -30 -20 -10 0 10 20 30 z [au] -1.00-0.75-0.50-0.250.000.250.500.751.00 V l [ a u ] , ψ ( z ) (b) Figure 4. (Color online) The potential V l ( z ) (black dashedlines) and its bound states for two different configurations.(a) The DQD potential binds only two one-electron stateswith wave functions ψ R (green squares) and ψ L (blue cir-cles). The energy levels of these states are marked by dashedlines and the respective binding energies are indicated. (b)The left well is shallower and wider than in (a) and bindsone additional excited one-electron state with a p-type wavefunction ψ L (red crosses). The configuration shown in Fig. 4(b) is a representa-tive of setup C. It shows a wider and shallower left QDwhich allows for an excited one-electron state L . We seethat the binding energy E L is much smaller than E L and E R and the wave function ψ L is therefore moreextended than ψ L and ψ R .We set the origin of the energy scale to 2 ω through-out the study. It amounts to the energy contributed byboth electrons in the ground state of the transversal con-finement potential V c (Eq. (7)). With this choice thebound (unbound) states of the longitudinal potential ofthe DQD have negative (positive) energies. B. Dynamical calculations and results
By employing electron dynamics calculations we caninvestigate what happens when an electron coming fromthe left side approaches the DQD where one electron isinitially bound and how, if at all, ICEC occurs. We startwith the simplest case of setup A (Sec. V B 1) where onlythe right QD is present and then move on to differentconfigurations of setup B (Sec. V B 2) and C (Sec. V B 3).All examples were computed using both the 1D model(Sec. III B) and the full 3D Hamiltonian (Sec. III A) fortriplet symmetry. In all cases we chose the energy of theincoming wave packet (WP i ) such that it is to low toionize the electron initially bound to the R state, evenif the full energy width of the WP i is considered.
1. One single QD
The initial state of the two-electron systems is an in-coming free electron from the left and a bound one in theright QD. A similar setup was studied before, however,for a different energy regime of the incoming electron inwhich two-electron ionization was allowed. The param-eters V R = 0 . b R = 1 . E R = − . i ) is an energy normalizedGaussian peaked around ε W P i = 0 .
056 a.u. The packethas a spatial width ∆ x W P i = 10 . ε W P i ≈ .
033 a.u. which is not enough to ion-ize the bound electron by the incoming one. Moreover,excitation to higher states in the transversal directionsare energetically forbidden for these parameters.The dynamics of the full 3D scattering process calcu-lated according to the method described in Sec. IV isvisualized in Fig. 5(a) by the longitudinal electronic den-sity ρ ( z, t ) = Z d r ′ Z d x Z d y | Ψ( r , r ′ , t ) | (23)as a function of z and t . The incoming electron is com-pletely reflected starting at about t =3 a.u. while theother electron remains bound in the right QD. The samecalculation was made using the one-dimensional modeldescribed in Sec. III B and is shown in Fig. 5(b) for com-parison. The evolution is in both cases very similar, onlythe population P of the lowest populated SPF (which isa measure of the convergence as explained in Sec. IV) isdifferent (but however small) in each case giving a valueof P = 1 × − for the simplified model and P = 1 × − for the full calculation. For long times ( t ≈
25 a.u.) thetotal density ρ ( z, t ) in the system decreases to zero. Thereason for this unphysical behavior is the CAP absorbingthe continuum electron. This effect has no impact on theobserved results, because the reflection process is alreadycompleted within a much shorter time of about 10 a.u. (a)(a)(b)3D1D Figure 5. (Color online) Evolution of the electronic densityEq.(17) for a QD of setup A using the full three-dimensionalHamiltonian (a) and (b) the one-dimensional model of sec-tion III B. The incoming wave packet (WP i ) approaches fromthe left to the QD located at z = 5 a.u. (right dot) whichis initially occupied by one electron ( R state). The incom-ing packet is initially located at z = −
125 a.u. with energy ε WP i = 0 .
056 a.u. and has a spatial width ∆ x WP i = 10 . ε WP i = 0 .
033 a.u. The parametersused for the MCTDH simulations are given in Tab. II. Notethat the energy covered by the
W P i is to low to remove theelectron in the right QD ( ε i + ∆ ε WP i < | E R | ). Since theleft QD is missing, no emission to the right is observed.
2. ICEC in a double quantum dot
We now focus on configurations of setup B where weadded the left QD at a distance R = 10 . ρ ( z, t ) of fourconfigurations is shown in each left panel of Fig. 6 (a)-(d). The right QD and the incoming wave packet WP i are the same in all four configurations with V R = 0 . b R = 1 . Table II. Parameters used in the MCTDH calculations. The discrete variable representation (DVR) types correspond toharmonic oscillator (HO) and sine DVR (SIN). x y z
DVR type HO HO SINDVR points/Primitive Basis 5 5 431Range / a.u. ( − . , .
02) ( − . , .
02) ( − . , . x ) / a.u. 1 .
01 1 .
01 1 . x, y, z combined) z cap - - 168 . (a) (c)(b) (d) Figure 6. (Color online) Evolution of the electronic density (left panels) and the obtained ICEC reaction probabilities (RP)(right panels) for setup B of Fig. 3. The incoming wave packet (WP i ) approaches the DQD centered at z = 0 a.u. from the leftwhich is initially occupied by an electron in the right QD ( R state). ε i = 0 .
130 a.u. and the parameters of the right potentialare the same in all four cases. The depth of the left dot V L is varied as indicated for each case: (a) V L = 0 . V L = 0 . V L = 0 . V L = 0 .
725 a.u. The emission of the electron seen to the right is from R and takes place through ICEC(compare Fig. 2). The emitted packet acquires momentum in all cases according to the energy conservation (Eq. (3)) and isfaster than the incoming packet. This is clearly visible from the slope of the outgoing density which is smaller than that ofthe incoming density. The reaction probabilities shown in the right panels exhibit a peaked energy distribution centered at thevalues ε ( peak ) i (depicted as dashed vertical lines computed by Eq. (24) and listed in Tab. III). ε i = 0 .
130 a.u., ∆ x W P i = 10 a.u., ∆ ε W P i = 0 .
051 a.u.The left QD is characterized by b L = 1 . V L = 0 . , . , . , .
725 a.u. The correspondingenergies E L and ∆ E = E R − E L are given in Tab. III.Electron emission to the right is clearly visible in allfour cases. The flatter slope of the final wave packet(WP f ) trajectory traveling to the right indicates thatthe emitted electron has higher momentum than the in-coming electron. According to Eq. (3) the final energy ofthe outgoing electron represented by p f calculated fromEq. (4) (see Tab. III) decreases when the depth V L de-creases. The RP gives a quantitative measure of ICECand can be computed using Eq. (19). Descriptively, it isthe probability of capturing an electron in the left QDwhile simultaneously emitting an electron to the rightfrom the right QD. The RP as a function of the incom- Table III. The parameters used in the four configurations forsetup B discussed in the text and in Fig. 6, and the resultingcomputed energies, final momenta p f , and positions ε ( peak ) i ofthe peak values of the reaction probability (RP). All valuesare given in a.u. V L E L ∆ E p f ε ( peak ) i . − . . .
722 0 . . − . . .
698 0 . . − . . .
673 0 . . − . . .
648 0 . ing electron energy ε i is shown in each right panel ofFig. 6 (a)-(d). The energy range covered in the RP plotsis determined by the peak ε W P i with the energy width∆ ε W P i of the incoming wave packet. It is possible toobtain reliable results from one simulation within the en-ergy range ε W P i ± ε W P i , which is used for the RPplots.At this point we would like to discuss more the mean-ing of the RP. The values given in the plots for ICECare exactly the amount of the total electron density inpercent that would be ejected from R to the right andcorrespondingly the increase of the population of L , ifthe electron incoming from the left was mono-energeticwith energy ε i . On the other hand, a mono-energeticelectron implies an infinitely wide WP i (∆ x W P i → ∞ ),which cannot be realized numerically on our finite DVRgrid. In our calculations we take a rather broad incomingwavepacket and by employing Eq. (19) we can computethe RP.Let us analyze the results for RP shown in Fig. 6.They clearly show that ICEC is no at all constant oreven monotonic in the covered energy range. On thecontrary, it is seen that ICEC is very selective in energy.This is a non-trivial result considering that the ICECchannel into L is open for all incoming electron energies(Eq. (4)). The peak of the RP has its origin in the factthat the total energy E T (see Eqs. (1) and (2)) is the rel-evant energy in a scattering process. The RP shows amarked increase in the probability when the total energy E T matches the energy gained by the DQD ( − ∆ E ) inthe ICEC process in which the emitted electron takes anenergy ∆ E . Using Eq. (1) we obtain the value of ε i atwhich the peak of the RP is located, ε ( peak ) i = − E R − ∆ E. (24)The values obtained for ε ( peak ) i are given in Tab. IIIand depicted with vertical dashed lines in the RP plotsof Fig. 6. We see that the RP peaks obtained from thedynamics fit exactly the predicted values using Eq. (24).The RP values for the configurations of setup B all re-vealed probabilities below 1%.
3. Capture in the presence of a two-electron resonance
The physics of the capture is complicated in the pres-ence of an increased number of bound states of the QDs.In general, several capture and decay channels will beopen before and after the capture and the physics of res-onance states comes into play. We analyze the probablymost simple extension to the DQDs described in the pre-vious sections (setups A and B) by including one extraexcited state in the left QD (setup C).Accordingly, we modify the potential well of the leftQD by choosing b L = 0 . b L = 1 . V L . Thisdependence is shown in Fig. 7 for the three-dimensionalmodel. Due to the Coulomb interaction the DQD accom-modates a two-electron resonance which derives from the L [a. u.] -0.6-0.5-0.4-0.3-0.2-0.10 E [ a . u . ] -0.65-0.55-0.45-0.35-0.25-0.15-0.05 Γ [ a . u . ] R L L L R - ∆Ε Figure 7. (Color online) Width (top panel) and energies (bot-tom panel) of the | L R i two-electron resonance playing arelevant role in enhancing the ICEC probability in setup Cas function of the depth V L of the left QD. The energy andwidth of the resonance obtained in the one-dimensional modelof section III B are shown with brown crosses and those ob-tained for the full three-dimensional system with black dots.Both sets of results are very similar. Shown are also the en-ergies of all the single-electron states computed for the fullthree-dimensional system. The energies of the L , R , and L states of the DQD are depicted as dashed green, red, andblack lines, respectively, while the value of the energy differ-ence − ∆ E = E L − E R is indicated by a dash-dotted orangeline. one-electron states L and R . The | L R i resonance en-ergy and decay rate (inverse lifetime) are shown as blackdots in Fig. 7. Decay rates in QDs can be computed us-ing different methods. We follow here the approachemployed in in which the resonance state | L R i is pre-pared by imaginary time propagation followed by the realtime evolution to find its total decay rate.The capture process occurs in the presence of the res-onance as indicated in Fig. 8 so that different electroncapture scenarios can be imagined.As before in setup B, electron capture into the L statewith simultaneous release of the other electron from the R state is one possible pathway (direct ICEC). More-over, if the energy of the resonance is above the thresh-old, the incoming electron can be captured into the two-electron resonance state | L R i . After this it decaysthrough a process called interatomic Coulombic decay(ICD), that means by deexcitation of the elec-tron in the left QD ( | L i → | L i ). The released en-ergy is used to emit the electron from the right QD( | R i → e − ). We denote this pathway as the resonancechannel and the process as resonance-enhanced ICEC.After being populated by the incoming electron, the res-onance can also decay by emitting elastically the electronto the left. This decay resembles that of a shape reso-nance: e − + | R i → | L R i → | R i + e − . This decay isof course only possible when the resonance energy E L R is higher than E R , a situation that was not usually ful-0 Figure 8. (Color online) Schematic view of interatomicCoulombic electron capture in a model potential for a dou-ble QD in the presence of a two-electron resonance | L R i (dashed red lines). The incoming electron can be capturedinto | L R i (middle panel) because the resonance energy liesabove the threshold. Then, the resonance decays by ICD(middle to bottom panel), a process in which the excited elec-tron of the left QD decays from | L i to the | L i state whiletransferring the excess energy to the electron in the right QDwhich is emitted to the continuum. filled in the systems where ICD was investigated earlier.For completeness we mention that the incoming electronenergy is sufficiently low so that direct electron captureinto the L state is energetically forbidden for all casesconsidered here.The time evolution of the electron density ρ ( z, t ) hasbeen calculated for different left well depths V L =0.65,0.67, 0.71, and 0.74 a.u. (Fig. 9, left panels). Comparingwith the results for setup B (Fig. 6) a clear difference isobserved for the density emitted from z = 0 to the right.In setups C a continuous decay with an exponential timeconstant is visible while an almost instantaneous electronemission takes place for setups B . This indicates that themechanisms involved in the capture and emission pro-cesses are different for both setups. It is also noteworthythat the emitted electronic density to the left becomesmore complex in case C showing clear signatures of in-terference with the incoming WP i . The electron emittedelastically to the left is responsible for these interference effects.The results obtained for ICEC in section V B 2 showthat the ICEC probability is highest if the total energy E T matches the negative of the energy difference ∆ E .It is, therefore, worthwhile to study the behavior of theICEC probability in relation to the value of ∆ E in thepresence of a resonance. Fig. 7 shows that the resonanceenergy crosses − ∆ E around the value V L = 0 .
70 a.u. Wepreviously addressed the configuration with V L = 0 . E L R = − ∆ E . In this case, the coincidence of the RPpeak and the resonance energy lead to an extraordinaryincrease of the ICEC probability. The presence of theresonance enables an extra channel that can be tuned tocooperatively augment the emission. The RP for this andthree other V L values belonging to configurations aboveand below the mentioned crossing point are shown in theright panels of Fig. 9. The incoming WP i also depictedin Fig. 9 is different for each of the configurations becausethe RP region of interest changes with the resonance en-ergy. Nevertheless, the energy range shown is the samein the four plots.We observe that for V L = 0 .
65 and 0 .
67 a.u. the RPdevelops one large peak with a shoulder indicating a sec-ond peak. These two peaks correspond to the direct andthe resonance-enhanced ICEC channels of the scatteringprocess. The vertical lines depicted in the correspondingpanels of Fig. 9 stand for the energy of the resonance andof the ICEC peak computed from Eq. (24). The maximaof the RP are seen to be slightly displaced from theselines. In this sense the simple picture of independent res-onance and direct ICEC peaks is not strictly valid anda correction taking the interaction between them intoaccount is needed in order to obtain the correct peak po-sitions. It should also be clear that both channels mayinterfere. It is noteworthy that the RPs now take on val-ues of 10 and 16 %, respectively, which are substantiallyhigher than in the case of setup B where only the directICEC channel is operative.The choice of V L = 0 .
71 a.u. in panel (c) provides anextraordinary increase of the capture and emission prob-ability. This probability of 22 % indicates that the directand resonance ICEC pathways coherently contribute tothe same channel | R i + e − . The peak height stronglydepends on whether the values of E res and − ∆ E (de-picted in Fig. 9 and listed in Tab. IV) coincide. We seein Fig. 9 for case (d) where V L is slightly enhanced thatthe peak height, now about 5 %, is again smaller than incase (c). Clearly, the increase of the ICEC probability incase (c) derives from the concurrence of both processes.The total width of the RP peak for case (c) is very nar-row and given by the inverse lifetime of the resonance,as opposed to the other cases where a wider RP withmore than one peak is obtained. This narrowness can beutilized to design an energy selective device. In case (c) the emitted electron density reaches the gridboundary before the resonant emission from the DQD hasterminated. This has no effect on the RP values as we1
Table IV. Depth V L of the left QD, resonance and ICEC peakvalues in a.u. for the setup C cases. V L E res − ∆ E ε ( peak ) i Γ ( × − Γ (RP)( × − . − . ± . − .
148 0 . ± ± . − . ± . − .
164 0 . ± ± . − . ± . − .
194 0 . ± ± . − . ± . − .
218 0 . ± ± find when using longer grids where the full emission ispossible before reaching the absorbing boundary. This isdemonstrated explicitly in the following section.
4. ICEC in the one-dimensional effective model
In addition to the results given by the full three-dimensional simulations we performed computations us-ing the one-dimensional model described in section III B.These calculations are much less time consuming and alsoallow to use much larger grids.The result for configuration (a) of Setup B is shownin Fig. 10 demonstrating that the RP is structurallyand quantitatively similar to that of the full three-dimensional computation. Without showing the picturewe note that also the evolution of the electron densityin the one-dimensional effective model is very similar tothat of Fig. 6 for the full three-dimensional computation.Since the computation times are considerably reducedfor the one-dimensional model, we can perform the sim-ulations on much longer grids than those used for the 3Dcalculations. Now, we can address numerically the ques-tion whether the RP obtained from Eq. (19) reproducesthe population of the L state via ICEC computed byemploying incoming mono-energetic electrons. The ini-tial wave packet WP i can now be chosen to be spatiallywider with ∆ x = 20 a.u., with a reduced dispersion inenergy ∆ ε W P i ≈ . L state over time cannow be computed for a selected value of the energy ε i ofthe incoming electron. This determines the RP at thatenergy. Clearly, we need to repeat the simulation usingdifferent incoming energies in order to construct a fullRP curve. An example of an RP curve constructed inthis manner is depicted in Fig. 11. We observe that themaxima of the L populations follow closely the values ofthe RPs obtained from the flux determined via Eq. (19),even though the energy distributions ∆ ε W P i of the WP i sused to describe mono-chromatic incoming electrons arenot extremely narrow as they should be. If they wereinfinitely narrow, then we would expect both RP resultsto coincide.The RP does not change if we use different WPs. Wecan demonstrate this by using an energetically narrowwave packet with ∆ ε W P i ≈ . ε W P i ≈ . P L or P R of theone-electron states L and R of the left and right QDscomputed using an energetically narrow WP and a longgrid. For setup B this estimate works well as we did usea narrow WP. For setup C, however, we used in the full3D calculations an energetically wide WP and a rathershort grid and one cannot expect the above mentionedestimate to produce realistic results. Indeed, our cal-culations of these populations and of the norm N ( t ) ofthe wave packet show that these quantities decay due toabsorption into the boundaries of the grid before the es-timate takes on the correct value. This is mainly becausethe WP used is very wide. This raises the question onwhy is then the RP computed employing Eq. (19) notaffected by the grid size as is demonstrated in Fig. 12.The answer is that this equation keeps collecting the fluxon the boundary as long as the population P R on theright QD decreases and that of the left QD, P L , increases(see Eqs. (19-20). Clearly, absorption on the boundariesdoes not affect the RP of ICEC when computed via theseequations. In other words, the RP is very robust againstabsorption and this also explains the insensitivity of theresults to the size of the grid and width of the wave packetas found above.The results show that the overall density evolution isvery similar and the 1D model provides very good resultsfor the RP in both setups B and C. Moreover, sometimesit is only possible to perform one-dimensional computa-tions using grids long enough to show the complete ICECprocess. This assertion strongly supports the use of one-dimensional effective models when ε i is low and thus isnot able to produce excitations in the lateral confine-ment. The one-dimensional model is a very useful tool ifthe RPs of many different configurations needs to be an-alyzed, because it allows to quickly identify the relevantenergy range and shape of the RPs.2 (a) (c)(b) (d) Figure 9. (Color online) Evolution of the electronic density and obtained reaction probability (RP) for setup C of Fig. 3. Theincoming wave packet (WP i ) approaches from the left to the DQD centered at z = 0 a.u. which is initially occupied by anelectron in the right QD ( R state). The left dot binds two states L and L and the depth of the left dot V L is varied as: (a) V L = 0 . V L = 0 . V L = 0 . V L = 0 . R takes placethrough the process shown in Fig. 8. In the right panels the energy of the two electron resonance ( L R ) and that of the directICEC peak are indicated with vertical lines (green continuous and black dashed, respectively) and tabulated in Tab. IV. For thesimulations we used different WP i (red dash-dotted line), because the resonance and direct ICEC peak energies vary and thenthe relevant region of the RP is different for each case. The RP (black solid line) shows two distinguishable contributions to theenergy distribution in case (a): one from the resonance state (peak, 0.060 a.u.) and the other from direct ICEC (shoulder, 0.085a.u.). For case (b) both contributions have nearly the same energy and the emission increased markedly near the resonanceenergy. In case (c) the matching of both energies (resonance and direct) gives a huge enhancement of the emission with anarrow energy distribution, which corresponds to the width of the resonance (see table IV). The enhancement is lost in case(d) where the energy mismatch between the resonance and direct ICEC is enough to destroy the correlation of the processes. ε [au]0.10.20.30.40.5 R P t o L [ % ] Figure 10. (Color online) Comparison of the ICEC reac-tion probability (RP) for configuration (a) of setup B (seeFig. 6) obtained using the one-dimensional effective modelof Sec. III B and the result of the full 3D computation. TheRP of the full calculation (black) compares very well to theone-dimensional result (orange dashed). The vertical blackdashed line indicates the value of ε ( peak ) i given by Eq. (24). VI. DISCUSSION
We demonstrated that ICEC is operative and in somecases a very effective electron capture mechanism inDQDs. In the previous sections we have shown howa simple full-dimensional model can be constructed todescribe the process. Nevertheless, our model includesonly electron correlation to mediate electron capture, al-though other capture mechanisms are likely to be as effec-tive as ICEC. Therefore we stick to an estimation on theimportance of ICEC with respect to other processes. Aswe will show, the capture times for ICEC are in the sameorder or even faster than other common mechanisms.The capture rate into QDs is the commonly used quan-tity to characterize the efficiency of an electron captureprocess and it depends strongly on the amount of timeit takes for the capture to be completed, i. e. a fastercapture leads to a greater efficiency. The importance ofICEC is then determined by comparing the time it takesICEC to complete capture compared to the electron cap-ture times reported for other processes available in thesystem.
To estimate the speed of ICEC we transfer the param-eters of our model to realistic semiconductor structuresusing the effective mass conversion of Table I. It is appli-cable to gate defined DQDs with quasi-one dimensional3 ε [au]0.20.40.60.8 R P t o L [ % ] au] P opu l a ti on o f L [ % ] Figure 11. (Color online) Upper panels: Comparison of twomethods to determine the reaction probability RP for case(a) of setup B. The one-dimensional model was employed.The left panel shows the RP obtained by the flux anal-ysis using Eq. (19) with a single energetically wide WP i , ε WP i = 0 . , ∆ ε WP i = 0 . x WP i = 10 . L population of ener-getically narrower WP i ’s, ∆ x WP i = 20 .
0, ∆ ε WP i ≈ . L as a function of time for three different WP i ’s ε WP i = 0 . . .
150 a.u. (green middle, red top and blue bottom line,respectively) is shown in the right panel. The correspondingWP i ’s are shown in the left panel (dashed lines, green on theleft, red in the middle and blue in the right). To obtain thebrown curve in the left panel each maximum of the L pop-ulation was assigned to the respective ε WP i in the left paneland values were interpolated. Lower panels: Comparison ofthe reaction probabilities (RPs) obtained from different WPs.The RP from the wide WP i shown in the upper panels (blacksolid line) is compared with the RP obtained from the en-ergetically narrower WP i s, ε WP i = 0 . . .
150 a.u.,∆ ε WP i ≈ . x WP i = 20 . i s areshown with dashed lines. geometry or to QDs embedded in nanowires, so wecompare ICEC times with those obtained for other cap-ture processes in these systems. Table V shows the en-ergies and sizes for different materials in setup B, case(a) and Table VI those for setup C, case (c). The en-ergies obtained are well in the range of intraband levelspacings of QDs in nanowires and of intrashell levelsin self-assembled QDs. ε [au]102030 R P t o L [ % ] z = 270 a.u. z = 600 a.u.z = 960 a.u.z = 270 a.u. (3D) Figure 12. (Color online) Comparison of the ICEC reac-tion probability (RP) for configuration (c) of setup C (seeFig. 9) obtained using the one-dimensional effective modelof Sec. III B and the result of the full 3D computation. TheRP of the full calculation (black crosses) compares very well tothe one-dimensional results (circles) even for very large grids.The vertical black dashed line indicates the value of ε ( peak ) i given by Eq. (24) and the vertical full green line the | L L i resonance energy. Let us first analyze setup B. The time window shownin Fig. 6 is about T = 1400 a.u. and by transformingto SC materials of Table V we obtain T GaAs = 77 . T InP = 71 . T AlN = 6 . T InAs = 267 . t RPICEC = 28 au, and the times in differentmaterials are accordingly: t GaAsICEC = 1 . t InPICEC = 13 . t AlNICEC = 0 .
12 and t InAsICEC = 5 .
45 ps. We stress that thistime estimation only makes sense because we obtained aresonant behavior, rather than a non zero contributionfor all energy values.The surprisingly short time scale it takes ICEC tooccur makes ICEC a promising mechanism competitivewith other capture processes. It is faster than the re-ported capture times of 100 ps for free carriers in bulkGaAs into InAs/GaAs QDs in single layer samples mea-sured at room temperature .The time scale of ICEC obtained for the different ge-ometries always gives shorter times for smaller sizes of theDQD. This fact stresses the importance of confinementfor the process to be competitive. It can be connectedto previous studies on ICD in molecular dimers, wherethe length scale of about 0 . For the setup C case (c) the time window shown inFig. 9 is of T = 2700 a.u. and transforming it to the semi-conductor materials of Table V we obtain T GaAs = 150 . T InP = 137 . T AlN = 11 . T InAs = 515 . Table V. Realistic values of the parameters in different semi-conductors for geometry (a) in setup B. The energies are givenin meV and the lengths in nm. Effective masses and dielectricconstants taken from [56 and 57].Parameter GaAs InP AlN InAs R .
94 89 .
89 11 .
24 286 . √ b R/L l V L V R E L -4.47 -4.87 -56.78 -1.30 E R -2.92 -3.18 -37.10 -0.85Table VI. Realistic values of the parameters in different semi-conductors for geometry (c) in setup C. The energies are givenin meV and the lengths in nm. Effective masses and dielectricconstants taken from [56 and 57].Parameter GaAs InP AlN InAs R .
94 89 .
89 11 .
24 286 . √ b L √ b R l V L V R E L -5.23 -5.69 -66.40 -1.52 E R -2.92 -3.18 -37.10 -0.85 E L − E R can in this setup estimate the duration of the emissionusing the lifetime of the involved resonance | L R i . Wehave that for case (c) τ = 256 . τ GaAs = 14 . τ InP = 13 . τ AlN = 1 . τ InAs = 49 .
04 ps. From theobserved values of the GaAs energy spacings and elec-tron energies in the range of < R resonance in ICEC seems to be competitive withrelaxation via phonons. The times for ICEC are, how-ever, faster than reported intraband decay times due toacoustic phonon emission for InGaAs/GaAs QDs of 100ps. Our work is focused on strongly laterally confinedstructures, such as nanowires, and is thus suitable forthe use of a one-dimensional effective potential. In allcases and setups treated here both the full and one-dimensional descriptions provided almost identical quali-tative and quantitative results. The main result obtainedfrom this comparison for the cases studied in this work isthat the physics in the strongly laterally confined modelcan be correctly described using the effective potentialwhen the characteristic lateral energies are about twice or more than those of the QDs.
VII. CONCLUSIONS
Ultrafast electron capture in single QDs is an exten-sively studied topic nowadays due to its relevancein the development of a wide variety of technological ap-plications.
As shown here, electron capture via theICEC processes, in which the neighboring QD in a DQDis getting ionized, is particularly fast and can play a sig-nificant role in the dynamics contributing to the energytransfer between QDs. The ICEC mechanisms in DQDscould, in principle, be exploited to be implemented in de-vices which generate a nearly monochromatic low energyelectron in a given direction.The implementation of DQDs in nanowires using mate-rials with long carrier lifetimes such as InP should befavorable for ICEC. The rate at which the electron cap-ture occurs varies with material and radius of the wire.Reported times for carrier trapping cover a large rangefrom fast values of 10 ps for GaAs and 160 ps for ZnO to very slow ones such as 1 ns for InP nanowires. Usingwires with long carrier trapping times are favorable forICEC to be active.The process is driven by long-range Coulomb inter-actions, so we expect ICEC to be also applicable toother QDs geometries like, e.g. , self-assembled verticallystacked dots.
We have derived an effective one-dimensional approachthat correctly describes the dynamics and RPs of allthe cases we have considered. This approach reducesconsiderably the computational efforts and also demon-strates, by comparison with full 3D computations, thatthe physics involved is described correctly by a one-dimensional model as long as the characteristic confine-ment energy is about twice or more than that of the QD.The calculations presented were performed for thesame distance R between the dots. Since long-range cor-relation is involved in ICEC a rather pertinent questionis how the reaction probability changes with R . The an-swer has been partially given in the first publications onICEC in atoms and molecules (see Ref. 38) and for therelated ICD decay (see Refs. 25 and 26). The ICEC crosssection has an asymptotic 1 /R decay with the distance,according to previous theoretical estimates for atoms andmolecules. However, there are important contributionsnot considered in the asymptotic formulas leading to1 /R which are due to orbital overlap (see, Ref. 25 forICD in QDs and Ref. 60 for molecules). These contribu-tions can lead in some cases to a much faster ICD pro-cess. Furthermore, the quasi-one dimensional geometryof the dots considered here has a clear influence on ICD(Ref. 25) and probably also on ICEC. The calculationsare rather cumbersome and at the moment there is noexhaustive analysis of this kind for ICEC, but it will bedone in the future.5 VIII. ACKNOWLEDGMENTS
F. M. P. acknowledges financial support by DeutscherAkademischer Austauschdienst (DAAD) and ConsejoNacional de Investigaciones Cient´ıficas y T´ecnicas (CON- ICET) and A. B. by Heidelberg University (Olympia-Morata fellowship) as well as Volkswagen foundation(Freigeist fellowship). L.S.C. and A.B. thank theDeutsche Forschungsgemeinschaft (DFG) for financialsupport. ∗ [email protected] † [email protected] ‡ [email protected] A. AlAhmadi,
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