Two-level quantum dot in the Aharonov Bohm ring. Towards understanding phase lapse *

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1 Mateials Science-Poland, Vol. 5, No. 4, 007 Two-level quantum dot in the Ahaonov Bohm ing. Towads undestanding phase lapse * P. STEFAŃSKI ** Institute of Molecula Physics of the Polish Academy of Sciences, Smoluchowskiego 17, Poznań, Poland It is shown theoetically that indiect inteaction between two quantum dot levels can geneate an anomalous phase shift fo a system of the dot placed in one of the ams of the Ahaonov Bohm ing. The inteaction between levels aises fom the non-consevation of the obital quantum numbe duing the hopping pocess of electons between the levels and leads. Such an unusual phase lapse behavio is obseved expeimentally and still lacks of pope theoetical desciption. Key wods: quantum dot; Ahaonov Bohm ing; phase shift; phase lapse 1. Intoduction The phase lapse is a phenomenon which is chaacteized by a sudden decay of the phase shift [1,] measued fo the quantum dot (QD) in Ahaonov Bohm (A B) geomety, when the gate voltage shifts the dot enegy levels with espect to chemical potential of the leads. Seveal theoetical attempts have been made (see fo example [3, 4]) to descibe this unusual featue but none of them seems to be satisfactoy. In the pesent wok the evolution of the phase shift is investigated fo a model of two-level quantum dot placed in one of the ams of Ahaonov Bohm ing. The levels have diffeent hybidization stengths to the leads; one is well coupled to the leads and conducting, and the second is shap in enegy scale and inactive in tanspot. The obital quantum numbe is not conseved while hopping pocess of electons occus between the dot and the leads. As a consequence, both the levels ae coupled to each othe via the leads. It causes a consideable deviation fom the usual electon wave phase shift behaviou which ises fom zeo to π when QD level cosses effective Femi enegy. In paticula, a lapse of the phase appeas. It coesponds well to ecent expeimental obsevations of the phase evolution fo the QD in Ahaonov * Pesented at the Confeence of the Scientific Netwok New Mateials fo Magnetoelectonics MAG-EL-MAT 007, Będlewo nea Poznań, 7 10 May 007. ** piots@ifmpan.poznan.pl

2 144 P. STEFAŃSKI Bohm geomety in the limit of small electon numbe in the dot [], when the tanspot though the dot has just been initiated.. Hamiltonian of the system We conside a system composed of a quantum dot placed in one of the ams of Ahaonov Bohm ing (see the inset to Fig. 3 fo the schematic pictue). The Hamiltonian descibing the system has the fom: H = ε c c + ε d d + + kα kα, kα, γ γ γ k, α, = L, R γ= 1, + t c d + hc.. + t c c + hc.. ~ + + γα kα, γ ql, kr, k, α, = L, R k, q, γ = 1, (1) The fist tem descibes enegy of the left and ight lead. The next tem epesents quantum dot enegy with two levels γ = 1,. The thid tem shows the hopping between the dot and electodes. The last tem descibes tunneling though the diect Ahaonov Bohm channel and the tack of the phase evolution of electon wave is kept by intoduction of the phase dependence of the hopping matix element between the iφ leads, t = te. The phase acquied by electon wave in the magnetic field pependicula to the device plane is φ = πφ/φ 0, Φ being enclosed flux by the ing and Φ 0 = hc/e is a flux quantum. 3. Calculation of the conductance The cuent is calculated stating fom time evolution of non-equilibium Geen functions unde assumption that the γ = 1 QD level is well coupled to the leads and active in electon tanspot, wheeas γ = level wave function has only a small, finite ovelap with the states in the leads but does not diectly paticipate in tanspot. It is indiectly coupled to the γ = 1 level via the leads, which has consideable effect on the phase evolution as will be shown below. In the cuent calculation fom the left lead we stat fom the time evolution of the paticle numbe N L : JL = ed NL / dt = ( ie/ ) [ NL, H], which gives the following expession: ie J = t c d t d c + t c c t c c + * + + * + L 1 L kl, 1 1L 1 kl, ql, kr, kr, ql,, k q ()

3 Two-level quantum dot in the Ahaonov Bohm ing 145 Eq. () is then witten in tems of non-equilibium lesse Geen functions < + < + G1, k (, t t α ) = i ckα, ( t ) d1 () t and Gkα, q (, t t α ) = i cq, ( t α ) ckα, () t fo t = t. These functions descibe electon popagation though QD placed in one of the ams of the Ahaonov Bohm ing and diect popagation though the second A B am, espectively. Afte taking tempoal Fouie tansfom the cuent takes the fom: e dω dω < < J = 1 1, ( ), ( ) L R t G ω t G ω π L kl + R, π ql kr (3) k k q To obtain explicit expession fo the cuent the lesse Geen functions have to be known. They ae calculated fom the equation of motion (EOM) of the equivalent time-odeed Geen functions and then Langeth continuation is pefomed to obtain lesse Geen functions [5]. The calculated expession is exact, possible appoximations ae included only in the QD etaded Geen function calculation (see Eq. (5) below). Simila steps have been pefomed to obtain the cuent fom the ight lead, J R. Symmetization of the cuent and assumption on popotionate coupling to the leads gives the expession fo the total cuent: e J = dω fl fr T h (4) whee the tansmission is expessed as: 4 + T ω = T + T R cosϕ RG ω ( ) 1L 1R 1L 1R b b b ( 1 1 ) 1 L + + R ( ) I 1 1L 1R 1L 1R ( ) ( ) 1L 1R 1 T cos ( ) b ϕ Tb G1 ω (5) The tansmission though diect am is: T b = 4 ( 1+ ) and π t ρlρr =, = π t γα γα α ρ ρ α being the spectal density of the lead α, constant and featueless and the spin index has been suppessed fo bevity. The above geneal equation fo the cuent though A B ing with the QD has been obtained fo the fist time in [6]. Conductance though the system in a steady situation fo the limit of zeo bias voltage is of the fom: dj e df ( ) ev 0 G = = d T dev h dω ω (6)

4 146 P. STEFAŃSKI 4. Retaded quantum dot Geen function To calculate the conductance though the device, the etaded dot Geen function is needed + G1 ( t t ) = iθ ( t t ) d1 ( t), d1 ( t ) We emphasize that it has to be calculated in pesence of the γ = level and in pesence of the diect channel. It is deived by the EOM appoach and can be witten in the fom of Dyson equation (in enegy domain) fo one spin diection: 0 G = G + Σ G, whee: G i cos( φ) ( 1L + 1R) (7a) L 1R 1 = ω ε i ( 1 1 ) cos( ) ( 1 1 ) + φ + + = ( 1+ ) L R L R L R 1 ( 1L R L1R ) sin( φ) + G ( 1+ ) 0 (7b) 0 and G ( ) ω has the fom simila to 0 G ( ) 1 ω, whee index 1 has been eplaced by, γγ α = π tγαtγα ρα, the dot levels ae shifted unifomly by gate voltage V g : εγ = εγ Vg. 5. Calculation of the phase shift The phase shift of the electon wave popagating though the device is calculated fom the genealized Fiedel sum ule [7] which descibes the elation between phase shift of the electon wave scatteed by an impuity and the paticle numbe pesent at the impuity site. Fo one spin diection, it has the fom of: δ = π n = π dωρ (8) imp ε F imp

5 Two-level quantum dot in the Ahaonov Bohm ing 147 whee the integation is taken up to the Femi level ε F assumed to be zeo, and the spectal density ρ = (1/π) I G1 of the impuity (in the consideed case QD imp level ε 1 ) is calculated in the pesence of the diect channel and the ε dot level. 6. Numeical esults The evolution of the phase shift fo vaious values of the level splitting defined by Δ is shown in Fig. 1, Δ > 0 means that ε is situated above ε 1 in the enegy scale. When the gate voltage inceases, both the dot levels ae shifted unifomly towads the esultant chemical potential defined in the leads. Fo a single ε 1 dot level which cosses chemical potential, the phase shift changes fom 0 to π as expected fo a esonant level model (see inset of Fig. 1). Fig. 1. Evolution of the phase shift of the electon wave tavesing the device with espect to gate voltage fo vaious spacings between QD levels ε 1 and ε = ε 1 +Δ; ε is above ε 1 in enegy scale. The values of Δ ae indicated in the figue. The bold solid line (and the inset) shows a phase shift fo a single ε 1 active dot level. The calculation has been made fo the Ahaonov Bohm phase Φ = 0 The situation changes when the influence of the ε level is also consideed. When ε appoaches the Femi level, an incease of the phase shift appeas which is followed by a dip. The incease of the phase is due to a tempoal incease of the paticle numbe of ε 1 level when ε (inteacting with ε 1 via electodes) cosses chemical potential and is being filled by electons. The anomaly then vanishes when ε is futhe shifted below the Femi level, becoming fully occupied, and its influence on ε 1 level stats to be sceened and negligible. The anomaly gets smoothe shape when the splitting between the levels inceases.

6 148 P. STEFAŃSKI Fig.. Conductance though the device calculated fo the same paametes as in Fig. 1. The bold solid line shows the conductance fo a single QD level ε 1 only The esults of conductance calculations fo the same paametes as in Fig. 1 ae depicted in Fig.. Although ε does not paticipate diectly in electon tanspot though the device, it has consideable influence on the conductance shape. The bold continuous cuve shows the conductance fo ε 1 only level. The asymmety of the conductance peak Fig. 3. Evolution of the phase shift with espect to gate voltage of the electon wave tavesing the device fo vaious spacings between QD levels ε 1 and ε = ε 1 +Δ; ε is below ε 1 in enegy scale. The values of Δ ae indicated in the figue. The inset shows the scheme of the device unde consideation

7 Two-level quantum dot in the Ahaonov Bohm ing 149 aises fom the Fano effect [8] which develops itself due to the pesence in tanspot of the diect channel apat fom esonant dot level. Inclusion of the ε level, which is coupled indiectly to conducting ε 1, causes the appeaance of shap Fano esonances, whose shapes depend on the QD level splitting. A simila featue (not shown) appeas fo Δ < 0, with Fano esonance shapes being mio eflected. It shows that the position of ε with espect to the Femi level detemines the shape of the Fano conductance peaks though ε 1 and Fano paamete q /ε. A simila situation has been encounteed in the case of a lage dot with stongly vaiable hybidization of the levels to the leads [9]. Results in Figue 3 ae calculated fo the case when the ε level lies below ε 1 in the enegy scale (Δ < 0). When the gate voltage deceases, ε 1 cosses the Femi enegy as the fist, is followed by ε cossing. It esults in a tempoal decease of paticle numbe of ε 1 ; a pat of chage fom ε 1 can be absobed into ε which becomes unoccupied when appoaching the Femi level. It esults in the appeaance of the minimum of the phase shift, also obseved expeimentally []. The inset of Fig. 3 shows the scheme of the device unde consideation. The calculated anomalies ae vey weakly sensitive to the extenal Ahaonov Bohm phase (not shown). In conclusion, we have shown that indiect coupling between the QD enegy levels and lage diffeence of thei coupling stength to the leads causes an anomaly of the phase shift, as obseved expeimentally fo the Ahaonov Bohm geomety. The shape of the anomaly depends on the QD level aangement in enegy scale. Acknowledgements The wok was suppoted in pat by the Ministy of Science and Highe Education within the eseach poject fo yeas , as pat of the Euopean Science Foundation EUROCORES Pogamme FoNE by funds fom the Ministy of Science and Highe Education and EC 6FP (contact N. ERAS-CT ), and the EC poject RTNNANO (contact N. MRTN-CT ). Refeences [1] SCHUSTER R., BUKS E., HEILBLUM M., MAHALU D., UMANSKY V., STRINKMAN H., Natue, 385 (000), 779. [] AVINUN-KALISH M., HEILBLUM M., ZARCHIN O., MAHALU D., UMANSKY V., Natue, 436 (005), 59. [3] SILVESTROV P.G., IMRY Y., Phys. Rev. Lett., 85 (000), 565. [4] SILVA A., OREG Y., GEFEN Y., Phys. Rev. B, 66 (00), [5] LANGRETH D.C., [in:] Linea and Nonlinea Tanspot in Solids, Plenum Pess, New Yok, [6] HOFSTETTER W., KÖNIG J., SCHOELLER H., Phys. Rev. Lett., 87 (001), [7] HEWSON A.C., The Kondo Poblem to Heavy Femions, Cambidge Univesity Pess, [8] FANO U., Phys. Rev., 14 (1961), [9] STEFAŃSKI P., TAGLIACOZZO A., BUŁKA B.R., Phys. Rev. Lett., 93 (004), Received 5 May 007 Revised 10 July 007