We demonstrate the trapping of a conduction electron between two identical adatom impurities in a one-dimensional semiconductor quantum-dot array system (quantum wire). Bound steady states arise even when the energy of the adatom impurity is located in the continuous one-dimensional energy miniband. The steady state is a realization of the bound state in continuum (BIC) phenomenon first proposed by von Neuman and Wigner [Phys. Z. 30, 465 (1929)]. We analytically solve the dispersion equation for this localized state, which enables us to reveal the mechanism of the BIC. The appearance of the BIC state is attributed to the quantum interference between the impurities. The Van Hove singularity causes another type of bound state to form above and below the band edges, which may coexist with the BIC

Garmon, S.

Ordonez, Gonzalo

Petrosky, T.

Tanaka, S.

Digital Commons @ Butler University

Butler UniversityDigital Commons @ Butler UniversityScholarship and Professional Work - LAS College of Liberal Arts & Sciences2007Electron Trapping in a One-DimensionalSemiconductor Quantum Wire with MultipleImpuritiesS. TanakaS. GarmonGonzalo OrdonezButler University, gordonez@butler.eduT. PetroskyFollow this and additional works at: http://digitalcommons.butler.edu/facsch_papersPart of the Quantum Physics CommonsThis Article is brought to you for free and open access by the College of Liberal Arts & Sciences at Digital Commons @ Butler University. It has beenaccepted for inclusion in Scholarship and Professional Work - LAS by an authorized administrator of Digital Commons @ Butler University. For moreinformation, please contact fgaede@butler.edu.Recommended CitationTanaka, S.; Garmon, S.; Ordonez, Gonzalo; and Petrosky, T., "Electron Trapping in a One-Dimensional Semiconductor QuantumWire with Multiple Impurities" Physical Review B / (2007): -.Available at http://digitalcommons.butler.edu/facsch_papers/722Electron trapping in a one-dimensional semiconductor quantum wire with multiple impuritiesS. Tanaka,1 S. Garmon,2 G. Ordonez,3 and T. Petrosky2,*1Department of Physical Science, Osaka Prefecture University, Sakai 599-8531, Japan2Center for Complex Quantum Systems, The University of Texas at Austin, Austin, Texas 78712, USA3Physics and Astronomy Department, Butler University, 4600 Sunset Avenue, Indianapolis, Indiana 46208, USAReceived 7 September 2007; published 19 October 2007We demonstrate the trapping of a conduction electron between two identical adatom impurities in a one-dimensional semiconductor quantum-dot array system quantum wire. Bound steady states arise even whenthe energy of the adatom impurity is located in the continuous one-dimensional energy miniband. The steadystate is a realization of the bound state in continuum BIC phenomenon first proposed by von Neuman andWigner Phys. Z. 30, 465 1929. We analytically solve the dispersion equation for this localized state, whichenables us to reveal the mechanism of the BIC. The appearance of the BIC state is attributed to the quantuminterference between the impurities. The Van Hove singularity causes another type of bound state to formabove and below the band edges, which may coexist with the BIC.DOI: 10.1103/PhysRevB.76.153308 PACS numbers: 73.20.Hb, 73.21.Cd, 73.21.Fg, 78.67.LtOne-dimensional semiconductor quantum wires 1D-QWRs have been extensively investigated theoretically andexperimentally over the past two decades.1–3 Thanks to ad-vances in nanotechnology, various ways to manufacturesemiconductor 1D-QWR have been developed.4 The quan-tum confinement of electrons in these structures greatlymodifies the density of states of carriers resulting in com-pletely different electronic and optical properties from thebulk system.1–4 The 1D-QWR has also been fabricated inmetallic systems and organic molecular systems.5,6In a previous report,7 we presented the charge transferfrom an adatom localized state to the 1D conduction mini-band associated with a quantum wire or quantum-dot array.Due to a singularity in the density of electron states at eitheredge of the electronic miniband, there is a nonanalytic g4/3enhancement of the charge transfer rate when the adatomenergy is near the band edges, where g is the dimensionlesscoupling constant describing the hybridization interactionbetween the adatom localized state and the miniband. Wealso presented Fano’s absorption spectrum attributed to thetransition of an electron from a core level of the adatom tothe conduction miniband, with which we were able to explic-itly separate the irreversible Markovian exponential decayand the reversible non-Markovian power law decay contri-butions to the time evolution of the system.Here, we shall report a different phenomenon: electrontrapping between two identical adatoms in the 1D minibanddue to quantum interference. The physical situation we con-sider in this Brief Report is depicted in Fig. 1a. We find thata bound steady state appears even when the energy of theadatom impurity energy is located in the 1D miniband suchthat an electron is trapped between the two identical ada-toms. The steady state is an example of “bound states incontinuum” BIC first proposed by Von Neumann andWigner8 for oscillating attractive potentials. Since then, anumber of theoretical studies of BIC have been reported.9,10Experimental evidence of BIC has also been reported inpositive energy bound states in superlattice structures ofquantum wells with a single impurity site11 or in a singledefect stub.12 In a recent article, one of the authors presentedanother example of BIC in a double-cavity or double-dot,two-dimensional electron waveguide.10 In this Brief Report,we shall present this interesting feature of BIC in a 1D-QWRwith multiple impurities, which is physically quite differentfrom the waveguide.We consider a 1D-QWR consisting of N+1 quantum dots,where we consider a single bound state for each quantumdot, as shown in Fig. 1b. An electron can tunnel betweenthese bound states through the transfer integral −B /2, whichleads to the formation of the continuous 1D miniband alongthe wire with the width B in the limit N→.1–3 In thepresent work, we assume that the lateral extent perpendicularto the wire axis is small such that the quantum confinementin a given quantum dot is so strong that the energy separationbetween the lowest bound state and the first excited boundstate in that dot is far larger than the 1D miniband width.Therefore, we can consider electron transport along the chainwithout causing any excitation perpendicular to it.In addition, the adatom states with energy E0 are locatedat the nth and −nth sites; they are hybridized with thebound states within these dots with the coupling strength gbecause the size of the quantum dot, which usually rangesFIG. 1. a Two identical adatoms attached to a 1D quantum-dotarray and b level structures of the adatom localized states and thebound states in each quantum dot. The adatoms are located at thenth and −nth dots.PHYSICAL REVIEW B 76, 153308 20071098-0121/2007/7615/1533084 ©2007 The American Physical Society153308-1from several to hundreds of nanometers, is far larger than thesize of the adsorbates. We shall simply refer to the adatomlocalized states as the impurity states. We represent thissingle electron system by the tight binding Hamiltonian in-cluding the two impurity states hybridized with the minibandasH = E0d−n+ d−n + dn+dn −B2 l,la˜l+a˜l + gB l=−n,ndl+a˜l + a˜l+dl ,1where d−n and dn are the fermionic annihilation operators forthe electron in the impurity states, a˜l is the fermionic opera-tor representing the bound state in the lth quantum dot, andthe symbol l , l represents the sum over nearest neigh-bors. We introduce the wave number representation ak=l=−N/2N/2 a˜leikl /N+1 and impose the usual periodic boundaryconditions on the 1D-QWR that will result in a continuousenergy spectrum for the miniband in the limit N→. Forfinite N, we have a discrete wave number with kj =2j / N+1, where j is an integer that runs from −N /2 to N /2. TheHamiltonian H is then cast intoH = E0d−n+ d−n + dn+dn + j=−N/2N/2Ekjakj+ akj+gBN =−n,n j=−N/2N/2dl+akjeikjl + akj+ dle−ikjl , 2where Ek=−B cos k is the energy dispersion of the miniband,which leads to the density of states Ek	1 B2−Ek2−1/2with singularities at both edges of the miniband, Ek= ±B.One advantage of 1D-QWR is that one can change the valueof B by changing the potential barriers between the quantumdots using precise nanotechnology1,2: the value of B is esti-mated to be about 140 meV for GaAs/AlGaAs QWRwhen the adjacent distance of the quantum dots is 25 nm.1The parameter gB is the coupling constant representing thecharge transfer between the adatom and the semiconductorsurface, and it has been estimated to be less than 70 meV forphysisorbed molecules on the semiconductor surface.13The Hamiltonian may be decoupled by introducing sym-metric s state and antisymmetric states p state both for theimpurity states and the 1D miniband with mode k throughs	dn+d−n /2, p	dn−d−n /2, Sk	ak+a−k /2, andPk	ak−a−k /2. We then have H=Hs+Hp, whereHs = E0s+s + j=0N/2EkjSkj+ Skj +2gBN j=0N/2s+Skj coskjn + H.c. ,3aHp = E0p+p + j=0N/2EkjPkj+ Pkj +2igBN j=0N/2p+Pkj sinkjn − H.c. ,3bwhere H.c. stands for Hermitian conjugate. Both Hs and Hptake the form of the Friedrichs-Fano Newns-AndersonHamiltonian,16–19 which has been used extensively to de-scribe the electronic states of adsorbates on metal and semi-conductor surfaces.20 These bilinear Hamiltonians can be di-agonalized by a linear transformation.7,16 However, we donot need the explicit form of the transformation for our pur-poses in this Brief Report.The self-energies for the s and p symmetrized impuritystates are given in the limit N→ bys,pz =g2B22 −dk1 ± cos2knz + B cos k=g2B2 − 11 ± −  + 2 − 12n=g2Bi sin 	1 ± ei2n	 , 4where we have put 	z /B=−cos 	. In Eq. 4, the  sign isfor the s state while the  sign is for the p state. Care mustbe taken when choosing the appropriate branch for the ana-lytic continuation of 2−1 so that the localized state willdecay toward the future.21The solutions of the dispersion equation for the s and psymmetrized states,s,pz 	 z − E0 −s,pz = 0, 5consist of the pole in the lower-half complex plane in thesecond Riemann sheet corresponding to the unstable decay-ing state and the poles on the real axis of the first sheetcorresponding to the stable states.It is found from Eq. 4 that as z moves on the real axisfrom −B to B, the self-energies for the s and p states peri-odically vanish as expi2n	m=1  and  signs are forthe s and p states, respectively at z=Em, whereEm=−B cosm /2n	−B cos	m, with m=1,3 , . . . ,2n−1for the s state and m=2,4 , . . . ,2n−2 for the p state. As anillustration, we show the real part and imaginary part ofpE for the p states in Fig. 2 for the case of n=4, wherethe horizontal axis stands for E /B. As shown, RepE=ImpE=0 at 	m=2 /8, 4 /8, and 6 /8. The real andimaginary parts of sE for the s states are also shown inFig. 3 for the case of n=4. In this case, for 	m= /8, 3 /8,5 /8, and 7 /8, ResE=ImsE=0. It should be em-phasized that the values of Em do not depend on the couplingconstant g, that is, the position of Em is exclusively deter-mined by the energy level structures of the 1D miniband aswill be shown below.The solution of the dispersion equation Eq. 5 is deter-mined by the intersection of E−E0 and E as a function ofE. Therefore, when E0=Em, z=E0 is a stable solution of thedispersion equation in the 1D miniband; this gives a BICstate which never decays even though the energy E0 is lo-cated in the 1D miniband, irrespective of the coupling con-stant g. The number of such solutions Em increases as theseparation between the impurity sites 2n increases: n−1 isthe number of solutions for the p state and n for the s state.We shall now show that in the BIC states, the electron iscompletely trapped between the two impurity sites. We con-BRIEF REPORTS PHYSICAL REVIEW B 76, 153308 2007153308-2sider the s symmetry case the p symmetry case can be dem-onstrated in a similar manner. In terms of the site represen-tation, the symmetrized basis is given by S˜0= a˜0 and S˜l= a˜l+ a˜−l /2 for l=1,2 , . . ., where a˜n	 a˜n+0, etc., with0 the electron vacuum state. In terms of this basis, Hs isrewritten as Hs=Hin+Hout, whereHin = E0ss + l=0n−1vlS˜lS˜l+1 + H.c. + gBsS˜n + H.c. ,6aHout = l=nvlS˜lS˜l+1 + H.c. . 6bIn Eqs. 6a and 6b, v0=−B /2 and vl=−B /2 forl=1,2 , . . .. Let us first consider the eigenvalue problem ofHin instead of Hs: Hinin=E˜ inin. For the eigenstate ofHin, it can be proven thatE˜ in− E0 = gBS˜ninsin. 7It follows from Eq. 7 that if E˜ in=E0 as we have discussedin the case for the BIC, the eigenstate does not involve theS˜n component, i.e., in= ss+l=0n−1S˜lS˜lin. As a re-sult, the state in does not couple with the outer states ofS˜l l=n+1,n+2, . . .  through Hout. That is, it is completelylocalized within the two impurities; it is also an eigenstate ofthe total Hamiltonian Hs: Hsin=E˜ inin with E˜ in=E0.We show the explicit form of the eigenstates forsome cases: the BIC for the n=1, s case is written as1,s= s+2gS˜0 /1+2g2, with Hs1,s=0, andfor the n=2, p case 2,p= p+2gP˜ 0 /1+4g2, withHp2,p=0. For the n=2, s case, they are representedby ±2,s= s+2gS˜1± S˜0 /1+8g2, with Hs2,s= ± 2,s /2. On the other hand, the eigenstates of Hs witheigenvalues different from E0 necessarily involve the S˜ncomponents, such that those states couple with the outerstates and are therefore delocalized. This is consistent withthe ordinary observation that the continuum under most cir-cumstances has a destabilizing effect. However, below, wewill discuss an exception which results in a different kind ofbound state, even though E˜ in=E0 is not satisfied.The mechanism by which these steady states occur insidethe continuous energy spectrum may be understood as fol-lows. Suppose that an electron is localized at one of the twoimpurity states. Due to the resonance interaction with the 1Dminiband, the impurity state decays spontaneously into the1D miniband emitting an electron wave, as shown in Fig. 4.When the free electron wave reaches the other impurity, theelectron transitions into the other localized impurity state.This impurity state also decays into the 1D miniband due tothe resonance interaction, just as the first site. The emittedfree propagating electron wave thus goes back and forth be-tween the impurity sites, just like a ping-pong game.14 Ourresults show that one can practically manufacture such asemiconductor 1D-QWR with double impurities that theFIG. 2. The real part RepE and the imaginary part ImpEof the self-energy of the p state for n=4 as a function of real E,where the horizontal axis stands for E /B. Inside the minibandE /B=−1 to E /B=1, there are three points of Em where Em=0.In the figure, we indicate the value of 	m corresponding to the Em’s.No divergence in pE occurs at the band edges.FIG. 3. The same plot of the s state for n=4 as in Fig. 2. Insidethe 1D miniband, there are four points of Em where Em=0. Thevalue of 	m corresponding to the points of Em are indicated.ResE diverges at E /B= ±1, while ImsE does not.FIG. 4. Schematic picture of the quantum interference in BIC.The spontaneously decaying waves constructively interfere witheach other between the impurities, while they destructively interfereoutside the impurities.BRIEF REPORTS PHYSICAL REVIEW B 76, 153308 2007153308-3electron waves emitted from the impurities constructivelyinterfere with each other inside the two impurities, whilethey destructively interfere outside the impurities, as sche-matically illustrated in Fig. 4. As a result of quantum inter-ference, the BIC is localized between the two impurities.As for the s state, two stable states always appear justbelow and above the 1D miniband due to the divergence ofthe density of states at the band edges.7,15 The wave functionof this state outside the miniband is largely extended to statesrepresenting sites outside the impurities, contrary to the BIC.Despite being coupled to the outer sites, this state remainsstable, as mentioned above. It is interesting that differenttypes of steady states coexist when the BIC appears. For thep symmetry, however, the factor of sinkn in the hybridiza-tion term in Eq. 3b suppresses the divergence of the densityof states, such that the stable state outside the miniband doesnot appear when E0 is located inside the miniband.These BIC states in the 1D-QWR may be experimentallyobserved with the use of the optical absorption spectrum aswe discussed in Ref. 7. We hope to challenge the experimen-talist to observe this interesting phenomena through the ab-sorption spectrum or by other means.We thank W. Schieve, G. Sudarshan, L. E. Reichl, N.Hatano, and H. Nakamura for insightful discussions. Thismaterial is based upon work supported by the National Sci-ence Foundation under Grant No. 0611506. This work wassupported by the Grant-in-Aid for Scientific Research fromthe Ministry of Education, Science, Sports, and Culture ofJapan.*petrosky@physics.utexas.edu1 S. E. Ulloa, E. Castano, and G. Kirczenow, Phys. Rev. B 41,12350 1990.2 L. P. Kouwenhoven, F. W. J. Hekking, B. J. van Wees, C. J. P. M.Harmans, C. E. Timmering, and C. T. Foxon, Phys. Rev. Lett.65, 361 1990.3 A. Richter G. Behme, M. Suptitz, C. Lienau, T. Elsaesser, M.Ramsteiner, R. Notzel, and K. H. Ploog, Phys. Rev. Lett. 79,2145 1997.4 R. Nötzel and K. H. Ploog, Adv. Mater. Weinheim, Ger. 5, 221993; X.-L. Wang and V. Voliotis, J. Appl. Phys. 99, 1213012006.5 I. Barke et al., Solid State Commun. 142, 617 2007.6 C. Joachim, J. K. Gimzewski, and A. Aviram, Nature London408, 541 2000.7 S. Tanaka, S. Garmon, and T. Petrosky, Phys. Rev. B 73, 1153402006.8 J. von Neumann and E. Wigner, Phys. Z. 30, 465 1929.9 F. H. Stillinger and D. R. Herrick, Phys. Rev. A 11, 446 1975;L. Fonda and R. G. Newton, Ann. Phys. N.Y. 10, 490 1960;W. Vanroose, Phys. Rev. A 64, 062708 2001; K.-K. Voo andC. S. Chu, Phys. Rev. B 74, 155306 2006; S. Longhi, Eur.Phys. J. B 57, 45 2007.10 G. Ordonez, K. Na, and S. Kim, Phys. Rev. A 73, 022113 2006.11 F. Capasso, C. Sirtori, J. Faist, D. L. Sivco, S.-N. G. Chu, and A.Y. Cho, Nature London 358, 565 1992.12 P. S. Deo and A. M. Jayannavar, Phys. Rev. B 50, 11629 1994.13 O. Björneholm, A. Nilsson, A. Sandell, B. Hernnäs, and N.Mårtensson, Phys. Rev. Lett. 68, 1892 1992.14 G. Ordonez and S. Kim, Phys. Rev. A 70, 032702 2004.15 T. Petrosky, C.-O. Ting, and S. Garmon, Phys. Rev. Lett. 94,043601 2005.16 K. Friedrichs, Commun. Pure Appl. Math. 1, 361 1948.17 E. C. G. Sudarshan, 1962 Brandeis Lectures in Physics Ben-jamin, New York, 1962.18 U. Fano, Phys. Rev. 124, 1866 1961.19 P. W. Anderson, Phys. Rev. 124, 41 1961.20 G. P. Brivio and M. I. Trioni, Rev. Mod. Phys. 71, 231 1999.21 T. Petrosky, S. Tasaki, and I. Prigogine, Physica A 173, 1751991.BRIEF REPORTS PHYSICAL REVIEW B 76, 153308 2007153308-4

Electron Trapping in a One-Dimensional Semiconductor Quantum Wire with Multiple Impurities

http://digitalcommons.butler.edu/cgi/viewcontent.cgi?article=1723&context=facsch_papers

Electron Trapping in a One-Dimensional Semiconductor Quantum Wire with Multiple Impurities

Abstract

Similar works

Full text

Available Versions

Digital Commons @ Butler University