PHYSICAL REVIEW B 15 AUGUST 1998-IVOLUME 58, NUMBER 7

Phase evolution of the transmission coefficient in an Aharonov-Bohm ring with Fano resonance

Chang-Mo Ryu and Sam Young ChoDepartment of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea

~Received 14 January 1998!

Phase of the transmission coefficient for a mesoscopic loop with Fano resonance has been investigated byusing the scattering matrix method, for a model of an Aharonov-Bohm ring with a double barrier coupled to at-stub resonator in one arm. It is found that the phase varies rapidly byp on the shoulder of a resonance peak.In addition, it is found that there is a sudden phase jump byp near the tail of a resonance peak. Each resonancepeak is found to return to the same phase via two processes; in one process, the phase shift byp occurs rathersmoothly through the excitation of theh/2e oscillation near the resonance peak, and in the other process, thesame amount of phase shift occurs very abruptly across the zero transmission of the resonator. These phasebehaviors show some resemblance to the ones found in the recent phase measurement of a quantum dot viadouble-slit interference experiment.@S0163-1829~98!02331-5#

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I. INTRODUCTION

Quantum transport in mesoscopic systems has drawnsiderable interests in the last decade.1–4 Recently, Yacobyet al.5 reported an interesting experiment, which seemedhave fundamental importance. In an experiment employan Aharonov-Bohm~AB! ring with a quantum dot embeddein one arm, they observed the conductance behavior confiing that the quantum dot supports a phase coherent transsion, in spite of the fact that a quantum dot retains maelectrons. In addition, the phases of conductance measurdifferent positions on a single Coulomb peak revealed sprising results that the phase changed abruptly byp betweenthe two shoulders of the peak, and that successive pwere in the same phases. These unexpected results, depfrom the predictions based on the existing theory, spurfurther theoretical investigation.6–9,11 The mechanism of thephase shift that apparently comes from the interplay betwthe AB ring and the quantum dot, however, is still not copletely understood.

Levy Yeyati and Bu¨ttiker6 and Yacobyet al.9 showed thatthe phase rigidity seen in the experiment comes fromOnsager relation based on the time-reversal symmetryconservation of current in the two-terminal measuremenconductance. Furthermore, in a subsequent study, Yacet al. reported an experimental result showing that under ctain conditionsh/2e oscillations dominate the conductance9

A simple analysis based on a one-dimensional double-baresonant tunneling~DBRT! model also supported the possbility for this phase shift viah/2e oscillations.10

The first experiment of Yacobyet al. was set up with atwo-terminal configuration, and thus the phase measuremwas limited to either 0 orp. Very recently, Schusteret al.reported an improved measurement using a four-termprobe.11 In the four-terminal configuration, magnitude anthe phase of the transmission coefficient can be measdirectly. Schusteret al. carried out the measurement ofphase for a quantum dot in the Coulomb blockade regiand could confirm again that the phase behavior in the renance peak is described by a single electron model. Talso observed another striking phenomenon that the p

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rise of p within a peak was followed by a sharp drop of thphase byp near the tail. The sharp drop of the phase wfound to occur in accordance with the zero transmissionthe quantum dot,tQD .

As in the case of the analysis of Yacobyet al., the quan-tum dot is often simulated by the simple Fabry-Pe´rot modelof a one-dimensional DBRT. This model can explain wthe phase behavior near a resonance peak, but away fromresonance some conflicts arise. Firstly, the identical phashown on the successive resonances could not be explain the single electron picture. Moreover, when the phasehavior is compared with the recent experiment of Schuset al., the transmission amplitude through a double-barrresonator becomes

tdb5tatbeiu

12r br b8ei2u , ~1!

where ta,b and r a,b are the transmission and reflection amplitudes across the barriersa and b from left to right, withthe prime denoting the amplitudes of electrons from rightleft. As is well known,tdb can never be zero, which does ncompletely agree with the experimentally observationvanishingtQD under certain condition.

This seems to indicate that certain important physics mbe missing in the simple one-dimensional Fabry-Pe´rot modelof the quantum dot, especially in the presence of a magnfield. That is, the one-dimensional Fabry-Pe´rot model maynot fully represent the quantum dot. In fact, it was shownsome recent theoretical studies that the resonance of thetron transmission through the quantum dot becomes Ftype when the dot is treated as non-one-dimensional one.12 Inthe case when the lateral effect is included the transmisthrough the quantum dot could become a Fano type. Inpresence of Fano resonance, the transmission coefficiendrop sharply to zero, whereas it can never be zero forsimple DBRT.13,14 In the former case, zero-pole pairs afound in the complex energy plane whereas in the latter conly the poles appear. Therefore, it seems worthwhile

3572 © 1998 The American Physical Society

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study the phase behavior of electron transmission throthe Fano resonance in comparison with the simple DBmodel.

Motivated by the above interests in the phase behaviothe quantum transmission through the Aharonov-Bohm riwe, therefore, investigate the phase evolution of the tramission coefficient of the AB ring with a modified doubbarrier to have a Fano resonance. The Fano resonanincorporated by thet stub. In a double barrier and at-stubstructure, Fano resonance appears. This is manifested istudy of Porodet al.14 As shown in the pioneering study bPrice,13 a Fano resonance can occur only for a weacoupledt stub in a simple structure,13 but in the case of acomplicated t structure such as the one coupled withdouble barrier or the Aharonov-Bohm ring, strong couplito the t resonator can also give rise to the Faresonance.14,15 The investigation of the system of thAharonv-Bohm ring coupled with a stub was initiatedButtiker for the purpose of the study of the capacitaneffect.16 In our model, we neglect the charging effect athough it may be important. We just assume it to giveenergy gap, as in Ref. 10. In the present analysis, we emthe scattering matrix method.2,3,17

Main results we found are that the phase shift betwtwo resonant peaks occurs through two generic processeone process, the phase shift byp occurs rather smoothlythrough the excitation of theh/2e oscillation near the resonance peak, and in the other process, the same amouphase shift occurs very abruptly across the zero transmisof the resonator. The first process seems to be similar toone found in the simple Fabry-Pe´rot model of DBRT, but thesecond one seems to be the unique one for the Fano m

II. PHASE OF THE TRANSMISSION COEFFICIENTIN AN AHARONOV-BOHM RING WITH

AN ATTACHED RESONATOR

The schematic diagram of the AB ring with an attachresonator is shown in Fig. 1. The incoming and outgowaves at each junction are related to each other by the stering matrices described according to the form givenRefs. 18 and 19.

FIG. 1. Schematic representation of an Aharonov-Bohm rwith a double barrier and at-stub resonator in one of the armthreaded by a magnetic flux. The inset shows thet junction at theresonator.

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At the left junction the scattering matrix becomes

F S2

L12

L22G5F 2c Ae Ae

Ae a b

Ae b aGF S1

L11

L21G . ~2!

At the right junction it is described by the same form. For tcircular sections, the matrix becomes

F L11

L21

R11

R21

G5F r 18 0 t18 0

0 r 28 0 t28

t1 0 r 1 0

0 t2 0 r 1

GF L12

L22

R12

R22

G . ~3!

The scattering matrix through a barrier can be written as

@Sb#5F iA12b Ab

Ab iA12bG . ~4!

The transmission probability through a barrier is decidby the square of barrier tunneling parameterb. That is, if bis 1, there is no barrier, and ifb is zero, the transmissionvanishes.

The scattering matrices at the upper branch coupledstub in the Ballistic limit become

FL11

JL1 G5F iA12beikl /2 Abei ~kl2 f M !/4

Abei ~kl1 f M !/4 iA12bGFL1

2

JL2 G , ~5!

F JR1

R11G5F iA12b Abei ~kl2 f M !/4

Abei ~kl1 f M !/4 iA12beikl /2 GF JR2

R12G , ~6!

FJL2

JR2G5F ~2112i tan kls!

21 S 11i

2cot klsD 21

S 11i

2cot klsD 21

~2112i tan kls!21G

3FJL1

JR1G . ~7!

The total circumference of the ring is denoted byl , thelength of thet stub byl s , and the dimensionless Fermi wavvector byk. f M represents the magnetic flux. In the preseanalysis,k was normalized byl .

After some algebra, we can obtain the transmission aplitude T for the AB ring as a function of the transmissioand reflection amplitudest1 ,t2 andr 1 ,r 2. Here, 1 and 2 de-note the indices in the upper and lower arms, respective

T5AeS H1t11H2t2

E1

H18r 11H28r 2

E8D , ~8!

with

E[~AA82b2t1t282B2B28t1t18!~AA82b2t18t22B1B18t2t28!

2b2~B28t181B1t28!~B2t181B18t28!t1t2 ,

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H1[AeA@C28~AA82b2t18t22B1B18t2t28!

1bC18~B28t181B1t28!t2#,

H18[Ae@~AA82b2t1t282B1B18t2t28!~B2C28t11bC18t2!

1b~B2t11B18t2!~bC28t11B1C18t2!t28#,

and

A[~12ar1!~12ar2!2b2r 1r 2 ,

B1,2[a1~b22a2!r 1,2,

C1,2[11~b2a!r 1,2.

A8, B1,28 , and C1,28 are obtained fromA, B1,2, and C1,2 byrepacingr 1,2 with r 1,28 . E8 can be obtained fromE by ex-changing the primed amplitudes with the unprimed one.H2

and H28 are the same asH1 and H18 , respectively, if theindices of the amplitudes are interchanged. Heree is thecoupling constant, which we assume to be constant at ejunction.

Figure 2~a! showsuTu2, the transmission coefficient asfunction of the length of the resonator for the barrier tunning parameterb50.3. The normalized Fermi wavelengthtaken ask55p/ l . The calculated transmission coefficieshows periodicity with respect to the length of the resonaIn Fig. 2~a!, three cycles aroundl s / l 51 are plotted. Theresonance peaks are asymmetric. The degree of asymmeshown to vary according to the strength of the tunnelparameter and the magnitude of magnetic fields. Figure~b!shows the conductance oscillation atl s / l 50.98, 1.015,1.026, 1.032, and 1.038, marked byA, B, C, D, and E in

FIG. 2. ~a! Transmission coefficient as a function of the resontor length.~b! Transmission coefficient as a function of the manetic field at the positions marked in~a!.

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Fig. 2~a!, respectively. To manifest the effect of the phashift, the magnitude of the oscillation has been normalizedthe plot at about the same level. Oscillation changes theity from A to B by p and again byp from C to E. We seethat the parity of the phase changes rapidly on both sidethe peak. AtD, appearance of double frequency oscillatican be seen. The position of the phase jump betweenA andB is shown to be independent of variation of the magnituof the magnetic flux and the strength of the barrier tunneli

The phase behavior of the transmission when there isbarrier is shown to somewhat resemble the case of the dobarrier. This seems to indicate that when the Fano resonais present the phase behavior may be dominated by thisfect.

In a recent study,9 Yacoby et al. reported an experimendemonstrating that the phase jump at a resonance peacompanies theh/2e oscillation. We see a similar behavioappearing atD. When the value ofl s / l is increased fromCto D, we see that new troughs occur ath/2e periods. It wasfound that asl s / l comes closer toE, the newly developedtroughs become deepened~this cannot be clearly seen on thnormalized scale!, while the original troughs located ath/eperiods become flattened, thus, theh/e oscillation is againrecovered. This process gives rise to the phase shift bpfrom C to E. Similar process of phase shift via the excitatioof h/2e harmonic component was noted in the experimenresult of the quantum dot@Fig. 2~a! of Ref. 9#. Yacobyet al.tried to explain this behavior of phase shift based onDBRT model alone@Fig. 5~c! of Ref. 10#, but our studyindicates that the Fano model can also give rise to such pshift by p via h/2e oscillation.

It is noted that acrossl s / l 51 the phase of the transmission abruptly changes byp @betweenA andB in Fig. 2~b!# atthe tail of the peak. Schusteret al.11 reported very recentlythat a phase rise ofp in a peak was followed by a sharp droby p near the tail of each peak. They also pointed out tthe sharp drop of the phase occurred when the transmisamplitude of the quantum dottQD vanished while the totatransmission amplitude still remained finite. We find a simlar behavior, as shown in Fig. 3. The length of the resonain our model corresponds to the plunger gate voltage inexperiment. We see that atl s / l 51 the transmission of theresonatorutJu vanishes and the phase drops sharply, as in

-

FIG. 3. utJu2 ~solid line! and u(tJ) ~dotted line! for a doublebarrier with at-stub resonator as a function ofl s / l . Note the sharpdrop of the phase atl s / l 51, whereutJu250. Transmission coeffi-cient uTu2 changes the parity of the phase atl s / l 51, as shown inFig. 2~b! (A andB).

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case oftQD . In the case of a symmetrically resonantt stub,the scattering matrix for the strongly coupled resonator~withthe coupling constant49 ) becomes constant,14 and thus, in theballistic limit, the transmission amplitudetJ for a doublebarrier with at stub attached becomes

tJ52b~211exp@2ikl s# !

2~42b12iA12b!1~423b22iA12b!exp@2ikl s#.

~9!The value oftJ becomes zero whenk5np/ l s with n an

integer. Figure 3 shows the coefficientutJu2 and the phaseuof tJ for a double barrier with at stub. Although the shape outJu2 is not Lorenzian, the general behaviors ofutJu2 andu(tJ) for the resonator seem to resemble the variationutQDu2 andu(tQD) measured by Schusteret al. for the quan-tum dot. The slow rise of the phase around the peak ofutJu2and sudden drop of the phase whenutJu250 seem to re-semble the phase behaviors shown in experiment@Figs. 2~c!and 3~c! in Ref. 11#.

We also note that atl s / l 51 the transmission coefficienuTu2 is constant and does not vary with the magnetic fluxsimilar behavior was noted in the case of the ring attacwith a simple t stub, for the variation of the incidenenergy.20 The constant transmission is associated withfact that electrons are totally reflected at the junction ofresonator. From Eq.~3!, tJ becomes zero whenk5np/ l s , inwhich case electrons cannot pass through the upper arm.phase acquired by the electrons moving along the loweralone does not cause any interference, and thus the transion coefficient becomes constant, being independent of

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magnetic field. The sharp phase shift atl s / l 51 comes fromthe fact thattJ ~and thus the total transmission amplitud!changes sign crossing this zero transmission point.

III. CONCLUSION

In conclusion, we have studied the phase shift oftransmission coefficient for the Aharonov-Bohm ring wiFano resonance, using a model of a ring connected to DBcoupled to at-stub resonator. As a result, we found that theare two generic processes of the phase shift; one, viaexcitation ofh/2e oscillation and the other, crossing the zetransmission of the resonator. In the simple Fabry-Pe´rot-typemodel of DBRT, only the first process is found, but in thFano resonance model, there appear two different proceInterestingly, the phase behavior of the transmission coecient for a double barrier with a lateral effect simulatedthe t stub seems to coincide with the ones observed inrecent interference experiments of the AB ring embeddwith a quantum dot. From this study, we conclude thatphase behavior of an Aharonov-Bohm ring with Fano renance is quite different from that of the simple Fabry-Pe´rotmodel of DBRT, and that Fano resonance might play soimportant roles in the transmission through the quantumembedded in the AB ring.

ACKNOWLEDGMENTS

This work was supported by Pohang University of Sence & Technology/BSRI special funds and KOSEF. Wappreciate the discussions and encouragement given byfessor C. K. Kim.

nd

. B

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