Phase lapses in scattering through multi-electron quantum dots: Mean-field and few-particle regimes
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Phase lapses in scattering through multi-electron quantum dots:Mean-field and few-particle regimes
Andrea Bertoni ∗ CNR-INFM National Research Center on nanoStructures and bioSystems at Surfaces ( S ),Via Campi 213/A, 41100 Modena, Italy Guido Goldoni
CNR-INFM National Research Center on nanoStructures and bioSystems at Surfaces ( S ),Via Campi 213/A, 41100 Modena, Italy andDipartimento di Fisica, Universit`a di Modena e Reggio Emilia, 41100 Modena, Italy Abstract
We show that the observed evolution of the transmission phase through multi-electron quantumdots with more than ∼
10 electrons, which shows a universal (i.e., independent of N ) as yetunexplained behavior, is consistent with an electrostatic model, where electron-electron interactionis described by a mean-field approach. Moreover, we perform exact calculations for an open1D quantum dot and show that carrier correlations may give rise to a non-universal (i.e., N -dependent) behavior of the transmission phase, ensuing from Fano resonances, which is consistentwith experiments with a few ( N <
10) carriers. Our results suggest that in the universal regimethe coherent transmission takes place through a single level while in the few-particle regime thecorrelated scattering state is determined by the number of bound particles.
PACS numbers: 73.63.Kv, 03.65.Nk, 72.10.-d 73.23.Hk . INTRODUCTION Among the experiments that exploit the coherent dynamics of carriers, the one performedin 1997 by Schuster et al. , in which the transmission phase of an electron scattered througha quantum dot (QD) was measured, constitute an ideal test on the validity of differenttheoretical models for the inclusion of electron-electron interaction. In fact, the ability tomodel coherent carrier transport experiments in low-dimensional semiconductor systems isessential for designing possible future devices for coherent electronics or quantum computing.In the experiments of Refs. 1 and 2, two paths are electrostatically defined in a high-mobilityAlGaAs 2DEG, within a multi-terminal setup that allows to overcome the phase-rigidityconstraint of a two-terminal one. Two narrowings along one of the paths define a QD whichis operated in the Coulomb blockade regime (a different set of experiments, performed in theKondo regime, presents another peculiar phase behavior ). The transmission phase acrossthe QD is measured by an electron interferometry technique in which electrons are emittedat a given energy from a quantum point contact at one end of the two-path system: Whenthe energy corresponds to a quasi-bound level (QBL) of the QD, a transmission resonanceoccurs. The depth of the QD confining potential V d is tuned by charging a nearby “plunger”gate and the transmittance, together with the corresponding phase, is obtained as a functionof V d .The process of electron scattering through the QD has been modeled by means of a num-ber of different approaches, ranging from multi-particle few-sites to lattice and Hubbard model Hamiltonians . Still, none of the proposed approaches has been able to fully repro-duce the main feature of the measured transmission phase θ , namely, the recurring behaviorfound in the many-particle regime of the QD, where θ smoothly changes by π on each trans-mission peak of the N -electron system, and then abruptly drops to the initial value in eachvalley between the N and N +1 resonances, this leading to in-phase transmission resonances.This is called the universal behavior since it does not depend on the charge status of theQD. While the change of the phase at each resonance is well described by the Breit-Wignermodel, the nature of the phase drops remains substantially unexplained.Recently, an enhanced version of the electron interferometer system , allowing for theprecise control of the number of electrons inside the QD down to zero, has been used tomeasure the coherent transmission amplitude for small N . The results show that when only2 few electrons ( N <
6) are bound into the QD, the universal behavior of the phase is lost,and the phase drop occurs only for certain values of N . Furthermore, it was confirmed thatthe measured phase evolution is indeed related to the N -electron dot and not to the largertwo-path device.The aim of the present paper is to show that the universal behavior of the phase (large N ) is consistent with an electrostatic approximation, where the electron-electron interactionbetween the scattered carrier and the bound ones is included as a mean Coulomb field.This is done in Sec. II, where the transmission probability and phase are computed fora 2D potential representing the QD (attached to source and drain leads) plus a “large”number of bound electrons. Furthermore, in Sec. III we show that an exact few-particlecalculation performed on an effective 1D model of the system leads to the appearance of bothBreit-Wigner and Fano resonances , with continuous and discontinuous phase evolution,respectively, consistent with the experimental findings in the small N regime. Finally, inSec. IV, we draw our conclusions. II. MEAN-FIELD APPROACH: RECURRING PHASE DROPS
Let us resume the expected phase evolution for a single electron crossing an empty QD.We do so for a specific 2D potential V s [Figs. 1(a) and 1(b)] which mimics the one generatedby the surface metallic gates in the 2DEG of the devices of Refs. 1 and 2. Along thepropagation direction [Fig. 1(b)], we take two smoothed barriers (with a maximum heightof 10 meV and a maximum width of 10 nm) that connect a 60 nm flat negative region whichmimics the QD potential which is tuned in the simulations; along the transverse direction,we consider a harmonic confinement with ~ ω =30 meV. In the setup used in Refs. 1 and 2, nobias is applied between the QD source and drain leads since the coherent electron traversingthe dot is emitted by a quantum point contact at one end of the two-path system (notincluded in our simulations). Accordingly, we keep the Fermi energy of the two leads at zeropotential and fix the energy of the incoming electron. Material parameters for GaAs havebeen used. The open-boundary single-particle 2D Schr¨odinger equation has been solved byusing the quantum transmitting boundary method in a finite-difference scheme. Figure 1(c)shows the transmission probability and phase as a function of the QD potential. As the QDpotential is varied, the incoming electron comes into resonance with higher single-particle3
100 -80 -60 -40 -20 0 dot potential (meV) |t| θ position (nm) -50510 V ( m e V ) (c)(b)(a) FIG. 1: (color online). The adopted 2D potential profile (a) consisting of a harmonic potentialwith level spacing of 30 meV in the transverse direction and a double barrier along the propagationdirection (b). Since no bias is applied the Fermi levels in the source and the drain coincide, and aretaken to be zero. Single-particle transmission probability and phase (c) are shown, as a functionof the QD potential. The energy of the incoming carriers is 1 meV.
QBLs. At each resonance peak, the phase increases by π in agreement with the Breit-Wignermodel, while it is substantially constant in the low-transmission valleys.We show next that when the mean Coulomb field of electrons that populate the QD istaken into account, the behavior of the transmission phase shows the observed drops. Inour model, the maximum of a transmission peak corresponds to the alignment of the energyof the scattered electron with the energy of a QBL of the mean-field potential. When theenergy bottom of the QD is further lowered and the alignment is lost, the transmissionprobability decreases until the QBL becomes a genuine bound state, i.e., its energy fallsbelow the Fermi energy, and it is occupied by an additional electron. The new mean-fieldpotential has the QBL of the previous resonance shifted by the addition energy and, aftera further lowering of the QD potential, it produces another resonance. This phenomenon,that is essentially a Coulomb blockade effect, is repeated each time a carrier is added to theQD. As the mean fields produced by N or N + 1 electrons are very similar in the large N regime, the QBL that generates the resonances and the corresponding transmission phaseis always the same at each peak, with an abrupt drop each time a new electron occupies abound state of the QD.We now apply our model to a QD with the structure potential V s of Figs. 1(a) and1(b). In order to estimate the QD electrostatic potential we first solve the closed-boundarySchr¨odinger equation then add the field generated by an electron in the ground state ψ ,4amely V ( x, y ) = e πǫ Z dx ′ dy ′ | ψ ( x ′ , y ′ ) | e − r/λ D r (1)with r = p ( x − x ′ ) + ( y − y ′ ) + ( d/ and where d = 1 nm represents the thickness ofthe 2DEG and λ D = 30 nm is the Debye length . The Fermi levels of the source and drainleads are fixed, i.e., we neglect the effect of the charge inside the QD on the leads. Wecompute the ground state of the new potential V s + V and we repeat the whole procedureuntil we reach a number N of bound particles for which the potential V s + V + · · · + V N hasan unbound (positive energy) ground state. Then we compute the 2D scattering state foran incoming electron with the boundary conditions already described for the single-particlecalculation. For simplicity the bound states are calculated in a finite domain by solvingthe closed-boundary Schr¨odinger equation. This leads to a shift in the energy of the boundstates that has no effect on the qualitative results of the present work, i.e., the phase dropsbetween the transmission resonances.We show two sets of calculations in Fig. 2. In the top panel (a), the system parametersare chosen as in Fig. 1 in order to obtain a clear resolution of the resonances, althoughthey do not correspond to the experiments in Refs. 1 and 2. For the chosen parameters,the transmission occurs through the fourth excited QBL. All resonances, corresponding todifferent N , are in phase and this trend continues as the potential of the QD deepens, i.e., forlarger numbers of bound electrons. Note that, although the effect of the charging of the QDis essentially classical, the transmitted electron must be obviously modeled in a quantumapproach in order to obtain the transmission phase.In Fig. 2(b), we consider a structure with parameters closer to those of the experimentalconditions in Ref. 2; in particular, the confinement potential is much weaker and the energyof the incoming electron smaller than in Fig. 2(a) (see caption), leading to less definedresonances. The transmission phase evolution is similar to the previous case, in spite of thefact that differences between the two calculations are not only quantitative, showing thatthe results obtained are robust against the details of the calculation and of the system. Inparticular, (1) due to the low energy of the incoming carrier, the transmission takes placethrough the ground QBL rather than an excited state; (2) the two lowest QBLs are, for N >
4, quasi-degenerate. The latter effect is due to charge accumulation in the center ofthe QD, away from the barriers, inducing a double-well-like profile along the propagation5
140 -120 -100 -80 dot potential (meV) |t| θ -8 -7 -6 -5 -4 -3 dot potential (meV) |t| θ FIG. 2: (color online). (a) Transmission probability and phase for an electron scattered by thepotential corresponding to the sum of the structure potential described in Fig. 1 and the meanfield generated by N bound electrons. The numbers indicate the value of N at each transmissionpeak: as the QD potential V d decreases N increases. (b) Transmission probability and phase areshown for a system similar to the one of Fig. 1 whose parameters are tuned in order to matchthe energy levels of Ref. 2, namely: kinetic energy of the scattered electron ≈ µ eV, chargingenergy ≈ ≈ µ eV, and difference between the first twosingle-particle QBLs ≈ µ eV. The above values are obtained with a 100 nm well and two 50 µ eVbarriers 4 nm wide in the longitudinal direction and a harmonic confinement with ~ ω = 1 meV inthe transverse direction. direction. In this regime, the transmission peaks corresponding to the two lowest QBLsmerge and, for each N , a single transmission resonance is found that, being originated bytwo quasi-degenerate states, is characterized by a phase change of 2 π . However, since thetrapping of an additional electron in a localized state takes place just after the transmissionmaximum, the resulting phase evolution spans only a range of π . A further effect of thecharge accumulation in the center of the QD is the decrease of the maximum value of the6ransmission probability on the resonances. The above trend is clear in the left part ofFig. 2(b). We note that our simulations are performed at zero temperature and with anexact energy of the incoming carriers, this leading to the steep transitions in the transmissionprobability of Fig. 2(b). Such steepness is not expected in experiments due to the uncertaintyof the incoming carriers’ kinetic energy and the temperature dependence of bound levels’occupancy. In the simulations based on the mean-field approach, the universal behavior, i.e.the phase drops occurring between successive resonances, persists down to N = 0, in contrastwith experiments of Ref. 2 where phase drops may or may not occur for N <
6. It should benoted, however, that the phase drops are a necessary consequence of the electrostatic modelemployed for the coupling between the bound and incoming electrons: Such a mean-fieldpicture is expected to break down at small N . Indeed, we show in the following section thatthe inclusion of carrier correlation may give rise to an N -dependent phase evolution.To conclude our mean-field analysis we discuss the similarities between the results hithertopresented and the ones obtained in Ref. 8, also including electron-electron interaction in amean-field approximation. In the above work, the lead-dot-lead system is modeled with across-bar geometry and the transmission amplitude is obtained by using the non-equilibriumGreen function approach and a Hubbard Hamiltonian. The recurring phase drops are foundat zeros of the transmission and persist when the electron-electron interaction is turned off.While the first effect agrees with our simulations, we find no drops in the non-interacting case.The difference can be explained by the different models adopted for the dot: a 2D double-barrier structure in our case and a 1D bar orthogonal to the lead-to-lead direction in Ref. 8.This is confirmed by the further agreement between our Fig. 1(c) and a non-interactingsimulation for a double-barrier 1D structure reported in the above work. There, the universalbehavior of the transmission phase seems induced by the cross-bar configuration, with asingle site between the two leads, regardless of the Coulomb interaction. III. FEW-PARTICLE APPROACH: BREIT-WIGNER AND FANO RESO-NANCES
In order to obtain numerically the transmission coefficient of a fully correlated systemthe calculation must be able to solve the few-particle problem exactly in an open domain,a difficult task for a general 2D potential. We therefore chose to simulate the dynamics for7wo electrons in a strictly 1D quantum wire with the same profile of Fig. 2(b); the lateralextension of the wire is, however, taken into account by an effective Coulomb potential V C ( x ) = e / [4 πǫ ( x + d )], where the Coulomb singularity is smoothed by a cutoff d = 1 nm .Then, we solve exactly the few-particle open-boundary Schr¨odinger equation in the realspace, using a generalization of the quantum transmitting boundary method mentionedabove, whose general derivation is detailed elsewhere . In the following, we describe itfor a 1D spinless system.Let us consider a region of length L , with a single-particle potential V ( x ), constantoutside that region (leads): V ( x ) = V (0) if x < V ( x ) = V ( L ) if x > L . Although themethod is valid for the general case we consider here V (0) = V ( L ) = 0 for simplicity. Letus take ( N −
1) interacting identical particles bound by V ( x ) in its ( N − χ ( x , ..., x N − ). The m -th excited eigenstates of ( N −
1) interacting particles will bedenoted by χ m .Our aim is to find the correlated scattering state of N -particles ψ ( x , . . . , x N ) that hasthe following form when the n -th (with n ≤ N ) particle is localized in the left lead, i.e.,when x n < ψ ( x , ..., x n , ..., x N ) | x n < = ( − n (cid:20) χ ( x , ..., x n − , x n +1 , ..., x N ) e ik l x n ++ M l X m =0 b
10 -8 -6 -4 -2 0 dot potential (meV) |t| θ -5.61 -5.60 -5.59 dot potential (meV) −π0π θ -9.22 -9.21 -9.20 -9.19 dot potential (meV) -5 |t| (a)(b) (c) FIG. 3: (color online). (a) Correlated transmission probability (bottom) and phase (top) for anelectron scattered by the potential described in Fig. 2 caption when a second electron is bound inthe QD. Three Breit-Wigner and two narrow Fano resonances are present. (b) and(c) Details of thetransmission spectrum, with transmission probability in logarithmic scale, showing the asymmetricFano resonances and the corresponding phase jump of π . similar to the ones already seen in Fig. 1. The two remaining resonances, shown in detailin Figs. 3(b) and 3(c) (note the logarithmic scale for the transmission amplitude), are verynarrow (few µ eV) and present a typical asymmetric Fano line shape . They are a signatureof electron-electron correlation and, from their small width, we deduce that the effect of theCoulomb potential is very limited in our model. Nevertheless, the distinctive behavior ofthe transmission phase is clearly visible in the upper plots of Figs. 3(b) and 3(c). An abruptphase jump of π takes place near the resonance, where the transmission probability vanishes.This shows that the origin of the phase jumps detected in the few-particle experiments mayreside in correlation-induced Fano resonances. A similar behavior, with the presence of bothBreit-Wigner and Fano resonances, showing continuous and discontinuous phase evolutionsrespectively, is found in the simulation of the correlated three-electron scattering state (notshown here). In the latter case, the ratio of Fano resonances is larger and keeps increasing10ith the number of particles. IV. CONCLUSIONS
In summary, we showed that in the few-particle correlated regime, both Breit-Wignerand Fano resonances are found, while in the mean-field regime, a single type of resonanceis present, which is repeated for each number of bound electrons, this leading to a phasejump of π each time a new electron enters the QD. While these results are consistent withexperiments, the microscopic nature of the recurring phase drops in the latter regime remainsunclear. In fact, they are well reproduced by the first of our approaches in which no quantumcorrelation is present between the bound electrons and the scattered one. On the other handdifferent models that take into account the fully-correlated dynamics of the carrier suchas our few-particle calculations cannot be applied to the many-particle regime. The aboveconsiderations suggest that in the latter regime, the coherent component of the transmittedelectron wave function (i.e., the component that does not get entangled with the QD andwhose phase is detected by the interferometer) behaves as if the QD, together with the boundelectrons, was a static electric field, being unable to discriminate between two different valuesof N . On the other hand, when N is small, the transmission phase is able to provide apartial information on the number of electrons confined in the QD through the characterof the transmission resonance and the possible phase lapse. The transition regime betweenthe many- and few-particle conditions needs further analysis since it can clarify theconnections between the two opposite approaches used in the present work.We finally note that the Fano resonances found in the 1D two-particle scattering states area genuine effect of carrier-carrier correlation, in accordance to the original concept developedin Ref. 20. A more general definition is often adopted, ascribing the Fano line shape of thetransmittance to the interference of two alternative real-space pathways. In fact, Fanoresonances have been obtained previously by means of 2D , multi-channel , and two-path single-particle calculations. In those cases, however, the ratio between the numberof Fano and Breit-Wigner resonances is not expected to vary when varying the confinementenergy of the QD, while the number of correlation-induced Fano resonances becomes shortlydominant when moving from a low- N to a high- N condition in the framework of a fullfew-particle modeling. 11pon completion of this work we learned about a recent work by Karrasch et al. wherethe π lapses are also ascribed to (anti)resonances of Fano type. Acknowledgments
We are pleased to thank M. Heiblum, E. Molinari and M. Rontani for fruitful discussions.We acknowledge financial support by MIUR-FIRB no. RBAU01ZEML, EC Marie Curie IEFNANO-CORR, and INFM-Cineca Iniziativa Calcolo Parallelo 2006.12EFERENCES ∗ e-mail: [email protected]; group web-page: R. Schuster, E. Buks, M. Heiblum, D. Mahalu, V. Umansky, and H. Shtrikman, Nature ,417 (1997). M. Avinun-Kalish, M. Heiblum, O. Zarchin, D. Mahalu, and V. Umansky, Nature , 529(2005). A. L. Yeyati and M. Buttiker, Phys. Rev. B , R14360 (1995). Y. Ji, M. Heiblum, D. Sprinzak, D. Mahalu, and H. Shtrikman, Science , 779 (2000). P. G. Silvestrov and Y. Imry, Phys. Rev. Lett. , 106602 (2003). Y. Oreg and Y. Gefen, Phys. Rev. B , 13726 (1997). A. L. Yeyati and M. Buttiker, Phys. Rev. B , 7307 (2000). H. Q. Xu and B.-Y. Gu, J. Phys.: Condens. Matter , 3599 (2001). P. G. Silvestrov and Y. Imry, Phys. Rev. Lett. , 2565 (2000). For a rewiev see G. Hackenbroich, Phys. Rep. , 463 (2001). J. U. Nockel and A. D. Stone, Phys. Rev. B , 17415 (1994). C. S. Lent and D. J. Kirkner, J. Appl. Phys. , 6353 (1990). Although the width and energies of the resonances vary with d and λ D , we found that the quali-tative behavior of the transmission amplitude is not affected by the choiche of those parameters. In addition, due to the very small energy of the incoming electron which is comparable to thenumerical error in the determination of the energy of the QBLs, the energy where an additionalelectron gets trapped into the QD is fixed in the middle of the first two QBLs rather than fromcomparison to the Fermi energy. While in our case the phase drops happen as the occupancy of the dot changes by one unitand do not correspond necessarily to a zero of the transmission, in Ref. 8, the drops alwayscoincide with a transmission zero. This is due to the constraint of an integer occupancy in ourelectrostatic model. M. M. Fogler, Phys. Rev. Lett. , 56405 (2005). A. Bertoni and G. Goldoni, J. Comp. Electron. , 177 (2006). A. Bertoni and G. Goldoni (2007), (unpublished). The approach described replicate the “quantum transmitting boundary method” of Ref. 12 andthereafter applied to the modeling of many micro- and nano-electronic systems. In fact, in theoriginal formulation, the problem was the solution of a single-particle 2D Schr¨odinger equationwith open boundaries, and its extension to three (and conceptually even more) dimensions wasstraightforward. By considering that the equation for a single particle in N dimensions and thatfor N particles in one dimension have the same form, it is possible to take profit of the techniquedeveloped for the former in order to solve the latter. U. Fano, Phys. Rev. , 1866 (1961). T. Taniguchi and M. B¨uttiker, J. Phys.: Condens. Matter , 13814 (1999). M. Rontani, Phys. Rev. Lett. , 76801 (2006). E. Tekman and P. F. Bagwell, Phys. Rev. B , 2553 (1993). E. R. Racec and U. Wulf, Phys. Rev. B , 115318 (2001). O. Entin-Wohlman, A. Aharony, Y. Imry, and Y. Levinson, J. Low Temp. Phys. , 1251(2002). A. Fuhrer, P. Brusheim, T. Ihn, M. Sigrist, K. Ensslin, W. Wegscheider, and M. Bichler, Phys.Rev. B , 205326 (2006). C. Karrasch, T. Hecht, A. Weichselbaum, Y. Oreg, J. von Delft, , and V. Meden, Phys. Rev.Lett. , 186802 (2007)., 186802 (2007).