Chen Hao

Duan Wenhui

Gu Bing-Lin

Kawazoe Yoshiyuki

Wu Jian

Physical Review Letters

Wu, Jian

Gu, Bing-Lin

Chen, Hao

Duan, Wenhui

Kawazoe, Yoshiyuki

Institutional Repositories DataBase (IRDB)

Resonant Tunneling in an Aharonov-Bohm Ring with a Quantum Dot

Tohoku University Repository (TOUR) / 東北大学機関リポジトリ

Resonant Tunneling in an Aharonov-Bohm Ringwith a Quantum Dot著者 Wu Jian, Gu Bing-Lin, Chen Hao, Duan Wenhui,Kawazoe Yoshiyukijournal orpublication titlePhysical Review Lettersvolume 80number 9page range 1952-1955year 1998URL http://hdl.handle.net/10097/53166doi: 10.1103/PhysRevLett.80.1952VOLUME 80, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 2 MARCH 1998d int andaturesures1952Resonant Tunneling in an Aharonov-Bohm Ring with a Quantum DotJian Wu,1,2 Bing-Lin Gu,1,2 Hao Chen,3 Wenhui Duan,2 and Yoshiyuki Kawazoe41Center for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China2Department of Physics, Tsinghua University, Beijing 100084, People’s Republic of China3Department of Physics, Fudan University, Shanghai 200433, People’s Republic of China4Institute for Materials Research, Tohoku University, Sendai 980-77, Japan(Received 14 May 1997)We investigate the resonant tunneling in an Aharonov-Bohm ring with a quantum dot embeddeone of its arms. A complete transmission mechanism, considering both the resonance of the dothe interference effect, is presented. As a consequence, the experimental results, the phase fefor conductance peaks, are well explained by a one-dimensional noninteracting model. New featconcerning temperature effects are predicted. [S0031-9007(98)05304-6]PACS numbers: 73.40.Gk, 72.20.– i, 73.20.Dxndn-eembyaseyce,er-ee.ededurhenghesectinThe first measurement of the transmission phase throua mesoscopic ring was reported by Yacobyet al. in 1995[1]. The authors devised a modified Aharonov-Bohm (ABring with a quantum dot embedded in one of its arm(see Fig. 1). Two distinguishing features were observedthis experiment: first, the phase of AB oscillations changabruptly when the conductance of the AB ring reachespeak; second, the AB oscillations at consecutive condutance peaks are in phase. Since then, many attempts [2were made to explain those two features. One year lathe same group went on to measure the phase and matude of the reflection coefficient through a quantum dot ba new interferometry device [6]. While in a more recenexperiment, they studied a modified interferometry systeand observed similar features [7]. These experimentsof fundamental importance since they directly reveal thphase of electron transport, which is usually lost in mesurements of transmission or reflection probabilities, amight provide additional information about a mesoscopsystem. It has been shown theoretically that there shobe a phase change ofp for every conductance peak. However, the second feature that the AB oscillations at consetive conductance peaks are in phase is not well understoyet [7].It is the purpose of this Letter to present a completransmission mechanism for such a system. This mecnism reveals the functions of both the dot and the ring. Tsystem, a mesoscopic ring with a quantum dot embeddin one of its arms, can be divided into two parts: Thsmall dot and its big complementary partner (CP) whicis attached by two ideal leads. These two parts formloop—an interferometry system. The states of the sytem can be divided into two sets: the states correspondto the bound states of an isolated dot, and the preliminastates of the CP from which the electrons tunnel to the dThe isolated CP can be considered as a quantum line wtwo stubs, which will cause small oscillations on the coductance, at both ends. The transmission character issically dominated by the CP when there is no resonance0031-9007y98y80(9)y1952(4)$15.00gh)sinesac-–5]ter,gni-ytmareea-ndiculd-cu-odteha-heedehas-ingryot.ithn-ba-int e transmission through the dot. However, while a boustate of the dot is close to the Fermi level, resonant tuneling occurs and the interference effect will appear. Wfound that the resonant tunneling through the whole systcan be observed only when the phase shift introducedthe resonant state of the dot matches the transmission phof the CP. In other words, if the phase shift introduced bthe resonance of the dot causes constructive interferenthe resonant peak of the conductance will appear. Othwise, the resonance of the dot will not contribute to thtotal transmission because of the destructive interferencIn order to explain it more explicitly, we model thesystem by a one-dimensional noninteracting ring attachby two ideal leads from both sides. As Yeyatie al. did[2], we use a quantum well (dot potentialeD) with twobarriers (barrier heightseB) to simulate the dot. Then, theeffective electrostatic potential on the ring is parametrizby the quantitieseD , eB, and e0 (potential outside thedot), which is schematically represented in Fig. 1. Oaim is to investigate how the states of the dot affect tresonant tunneling of the system. We use a tight-bindi(TB) representation of the electron states. To study tmultienergy-level effects of the dot, a number of siteare assumed to be present in the well region. The effof the magnetic fluxf is included by introducing thePeierls phase factor of the hopping integralt. Then theFIG. 1. Geometry of an Aharonov-Bohm ring with a quantumdot embedded in one of its arms (left) and the on-site energythe tight-binding model (right).© 1998 The American Physical SocietyVOLUME 80, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 2 MARCH 1998thefon.ri-herotesHamiltonian of theN sites ring can be written asH ­NXj­1sej j jl k jj 2 eiwj jl k j 1 1j2 e2iwj j 1 1l k jjd , (1)where the TB hopping integral,t ­ h̄2y2mpa2, is taken tobe unity as the energy unit, andjN 1 1l is identical withj1l. As shown in Fig. 1, the on-site energyej is equalto the effective electrostatic potential on the lattice sitIn this Letter,e0 ­ 0 and eB ­ 3 if not specified. Thephase factor is given byw ­ 2pfyNf0, wheref0 is theelementary flux quantumf0 ­ hcye.The conductance of the system is calculated by usethe real space Green’s function (RGF) and temperatuKubo formula [8–10]. Here, only the Green’s functionG of the isolated sample, ring with a dot, is required ithe calculation [10]. Moreover, the local density of state(LDOS) can be conveniently obtained from the RGF byLDOS ­ 21pIm Gjj , (2)whereGjj is the diagonal matrix element of the RGF [9]Thus, the density of states (DOS) within the dot, and ththe energy levels of the dot, can be obtained in sequenNow, let us focus on the two features of the experime[1]. In Fig. 2, we plotted the conductanceg and the phaseb of AB oscillations versus dot potentialeD at zero mag-netic flux. Hereb is defined to be the phase introduceby the magnetic flux at which the conductance has a mamum, and the temperatureT is taken to be zero. We ob-serve the abrupt phase change and in-phase AB oscillatiat consecutive conductance peaks. Meanwhile, it is ainteresting to note that there is another abrupt phase chaof p between two peaks, just similar to the results metioned in Ref. [7]. This should be observed, otherwise wwould not have observed the in-phase peaks. In orderFIG. 2. Conductance (in units ofe2yh) as a function of thedot potentialeD for different magnetic flux. The marks givethe calculated values of the conductance. The dashedwithout marks represents the phase (in units of2p) of the ABoscillations.e.ofrens.ence.ntdxi-onslsongen-etolinereveal the nature of this phase change, we calculatedDOS within the dot near the Fermi level and the LDOS othe dot at the Fermi level in the new phase change regiThe results are given in Fig. 3. For the sake of compason, the DOS within the dot near the Fermi level and tFIG. 3. (a) The relative DOS in an antiphase region (fodifferent dot potentialeD) as a function of energy. (b) Therelative DOS in the adjacent in-phase region (for different dpotentialeD) as a function of energy. (c) The relative LDOSat the Fermi level at the site positions. The calculated valuare given as marks. The lines are given only as a guide.1953VOLUME 80, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 2 MARCH 19982).ottoantdu--ithheuson.eof-yngoeelalfh-hre-opli-nel-eas-ile,LDOS of the dot at the Fermi level for an adjacent condutance peak are also given. By tracing the movement of tenergy levels with increasing dot potentialeD, we find alsoa bound state of the dot near the Fermi level [Fig. 3(a)]the new phase change region. This is the same as the cfor the adjacent conductance peak [Fig. 3(b)]. And thtwo bound states, the state near the Fermi level in the nphase change region and the state contributing to thejacent conductance peak, correspond to the adjacent stin an isolated dot. They are shifted to the Fermi level bthe well potentialeD one by one. As shown in Fig. 3(c),they are antiphase, which is also a direct result of the quatum theory. If we notice the existence of the AB ring, ainterferometer, it can easily be understood that the adtional phase change ofp in this resonant state will causedestructive interference between the two arms of the rinand then will not contribute to the conductance. How athe things going on if we introduce another phase chanof p? Motivated by this thought, the conductanceg ver-sus dot potentialeD at half flux quantum is also calcu-lated, and the result is given in Fig. 2. It can be seen thnew peaks appear exactly in the new phase change regand there are no peaks in the old region. This impliethat the resonance of the dot does introduce phase shbetween two arms and results in interference. This ressuggests an interesting experiment, which would check oconjecture.Another interesting feature of our result is the finitwidth of the phase change region. Though previouauthors alleged that there should be a sharp jumpzero width [2], a finite width was actually observed inthe experiment [1]. In Ref. [2], Yeyati and Büttikerassumed that the conductance (as a function of flux) canpresented by a uniform functional form. However, it habeen pointed out that the resonance of the system canbe described by the concept of a “phase shift” [4] in thphase change region. What happens instead is thatconductance changes its functional form, just like the lineconductance in Ref. [4]. Obviously, the magnetic fielwill cause the shift of the energy levels. The phase chanreaches a value ofp only after the resonant state of the dohas passed through the Fermi level at all magnetic fieldTherefore, the effect of the flux on the energy levels resuin the finite width of the phase change region. The scaof the transition is determined by the energy differencetwo adjacent states of the CP, but not the difference of tbound states of the dot.What is the transmission mechanism of such a systeThe curve of the conductance resonant oscillations versthe plunger gate voltage on the dot showed that tresonance is related to the resonant states of the dotBut in this interferometer system, the coherence effecaused by the ring cannot be ignored. The conductanpeaks appear only when the phase shift introduced byresonant state of the dot matches the transmission phasthe CP. Otherwise, the destructive interference will resu1954c-heinaseeewad-atesyn-ndi-g,regeation,siftsulturesofbesnotetheardgets.ltsleofhem?ushe[7].ctcethee ofltin the decrease of the conductance (as observed in Fig.In addition, the maximum conductance at a definite dpotential is given as the function of the dot potentialeD inFig. 4. It can be seen that once the additional phase dueth magnetic flux causes the phase matching, the resonstates of the dot will contribute to the conductance, anresult in conductance peaks.To complete our study, the temperature effects are calclated and shown in Fig. 5. It is well known that high temperatures are disadvantageous to quantum effects. Wincreasing temperatures the quantum oscillations of tresonant tunneling decrease gradually. What surprisedis the appearance of new peaks in the antiphase regiHow can it occur? The answer is in Fig. 3. There arsome energy levels which are close to the resonant statethe dot. By tracing these energy levels in different well potentialeD , we found that these levels are affected little bthe eD in most cases. In other words, these states beloto the tunneling states of the CP, which will contribute tthe conductance. By comparing Figs. 3(a) with 3(b), wfound five tunneling states of the CP near the Fermi levwhose relative DOS is small in normal case. The originpositions of states are, respectively, atE ­ 20.27, 20.15,20.02, 0.10, and0.22. Meanwhile, the resonant state othe dot, whose relative DOS is much larger than the oters, moves from the Fermi level to higher energies witincreasing well potentialeD. When the resonant state ofthe dot passes through a tunneling state to an in-phasegion, the latter will not be affected. In other words, the twstates will form accidental degenerate states and the amtude of the DOS will remain unchanged. However, whethe resonant state of the dot is passing through a tunning state [for example, the state at20.04 , E , 20.01 inFig. 3(a), and the state at0.10 , E , 0.15 in Fig. 3(b)]to the antiphase region, it will not be degenerate with thlatter. (For convenience, we note this tunneling statestateP in the following context.) It appears that the resonant state and tunneling state repel each other. MeanwhFIG. 4. The maximum conductance (in units ofe2yh) at adefinite dot potentialeD as the function of the dot potential.The phase of the AB oscillations is replotted contrastively.VOLUME 80, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 2 MARCH 1998antets.eea-aonbyct.e-ofg-eey.FIG. 5. Conductance (in units ofe2yh) as a function of thedot potentialeD for different temperatures at zero magnetifield. The marks give the calculated values of the conductanThe conductance forT ­ 0 at half flux quantum is given as adotted line for comparison.the amplitude of the DOS for the tunneling statePincreases greatly. This is very similar to the caseelectrons at Bloch zone boundaries. Actually, thephenomena are caused by the same reason—periodicMoreover, the boundary conditionjN 1 1l ­ j1l in thering is much more restricted than the one for a latticThe repelling effect can be understood by the degenerperturbation theory. In this case, the phase shiftthe resonant state of the dot is also favorable to ttransmission state of the CP, from which the electrotunnel to the dot to form the tunneling stateP. As aresult, the tunneling stateP is enhanced greatly. This isjust the case that the first order perturbation is nonzein the degenerate perturbation cases. Consequently,two states are separated away. As shown in Fig. 3(when the resonant state of dot is shifted to the Ferlevel, the tunneling stateP is repelled away from theFermi level. With increasingeD , the resonant state isshifted away, and the tunneling stateP is shifted back.Therefore, at finite temperatures, the contribution by ttunneling stateP leads to an increase of the conductancwith increasing eD. Simultaneously, as the resonanstate of the dot is shifted away from the tunneling stacce.ofseity.e.atebyhensrothea),miheetteP, the conductance decreases due to decaying resoneffect. The formation of new peaks are determined by thcompetition between the tunneling and resonant effecWith increasing temperature, the contributions from thtunneling stateP become dominant and consequently thnew peaks appear.In conclusion, we have presented a complete mechnism for the interference phenomena in an AB ring withquantum dot embedded in one of its arms. The electrtransmission through such a system is characterizedboth the resonance of the dot and the interference effeThe experimental results are well explained by a ondimensional noninteracting resonant tunneling modelelectron transmission. An interesting experiment is sugested to support our viewpoint. A new phenomenon, thtemperature effect in such systems, is predicted.We are grateful for the useful discussion with LeChang. This work is supported by the High TechnologResearch and Development Program of China.[1] A. Yacoby, M. Heiblum, D. Mahalu, and H. Shtrikman,Phys. Rev. Lett.74, 4047 (1995).[2] A. L. Yeyati and M. Büttiker, Phys. Rev. B52, 14 360(1995).[3] G. Hackenbroich and H. A. Weidenmüller, Phys. RevLett. 76, 110 (1996).[4] C. Bruder, R. Fazio, and H. Schceller, Phys. Rev. Lett.76,114 (1996).[5] P. S. Deo and A. M. Jayannavar, Mod. Phys. Lett. B10,787 (1996).[6] E. Buks, R. Schuster, M. Heiblum, D. Mahalu,V. Umsndky, and H. Shtrikman, Phys. Rev. Lett.77, 4664(1996).[7] R. Schuster, E. Buks, M. Heiblum, D. Mahalu,V. Umsndky, and H. Shtrikman, Nature (London)385,417 (1997).[8] H. Chen, J. Wu, Z. Q. Li, and Y. Kawazoe, Phys. Rev. B55, 1578 (1997).[9] H. Chen, J. Wu, Z. Q. Li, and Y. Kawazoe (to bepublished).[10] A. Aldea, P. Gartner, and I. Corcotoi, Phys. Rev. B45,14 122 (1992).1955

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Resonant Tunneling in an Aharonov-Bohm Ring with a Quantum Dot

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