Effects of Interference and Inelastic Scattering in Aharonov-Bohm Ring with Quantum Dot

Interference effects are studied in an Aharonov-Bohm ring with a quantum dot within a one-dimensional model with effects of phase-breaking scattering. When a dot level crosses the Fermi level, the phase of the reflection coefficient changes over a wide range, inducing a resonance in various regions of the ring. Such reflection effects compete with those of resonant transmission through the dot. The peak asymmetry is determined by a combination of these effects and therefore exhibits a complicated behavior even as a function of the phase coherence length.


§1. Introduction
In an Aharonov-Bohm (AB) ring with a quantum dot in an arm, a conductance peak due to resonant tunneling can exhibit an asymmetric lineshape due to interference of waves passing through an arm without a dot and those through an arm with a dot.−4) The purpose of this paper is to demonstrate, with the use of a simple one-dimensional model (1D), the fact that the asymmetric peak is a result of complicated resonant interferences inherent in such systems rather than a simple Fano interference between waves passing through different arms.
An AB ring with a quantum dot may be modeled as illustrated in Fig. 1(a).Let t d be the transmission coefficient through the lower arm containing a dot.It is approximately written as where V Rn and V nL are the matrix elements of transitions from the dot state n with energy E n to the right-going states and from the incident to the dot state, respectively, and F n and Γ n are the energy shift and the broadening, respectively, of the dot level due to coupling with states outside of the dot.This can be written as t d = α/( +i) with = (E−E n −F n )/Γ n and α ∝ V Rn V nL /Γ n .By adding the transmission coefficient t 0 of the upper arm, we have the total transmission probability of the AB ring with a (complex) Fano asymmetry parameter q = i + (α/t 0 ).Except in the case of pure imaginary q, the conductance exhibits an asymmetric lineshape depending on the sign of the real part of q.Because V Rn V nL changes its sign depending on the parity of dot states and the parity changes alternately (more generally by Friedel's sum rule), the direction of the asymmetry becomes opposite between adjacent dot levels.Furthermore, AB magnetic flux changes the relative phase between α and t 0 and consequently the phase of q with period given by the magnetic-flux quantum φ 0 = ch/e.Therefore, the phase of the AB oscillation is expected to change by π between adjacent dot levels.−7) A surprising finding was that the phase becomes the same between adjacent peaks, i.e., phase persistence.−15) It was suggested, for example, that between neighboring peaks there is a hidden electron charging event that does not cause conductance peaks. 9)The possible disappearance of some peaks due to an interference inside the AB ring 10) and the vanishing of the transmission coefficient occurring in a 1D model due to Fano-type interference 15) were suggested.Since the observation of a clear asymmetric peak, 1) 1D model calculations were carried out further.For example, an alternating direction change of the asymmetry was obtained again. 16,17)Several attempts at studying the effects of dephasing were also made. 18)−28) The coexistence of a small number of strongly coupled states and many weakly coupled states in the dot with finite width was shown to be responsible for the observed phase persistence.There are various strange features which remain to be understood, however.For example, the asymmetry of the peaks changes continuously but in a complicated manner as a function of a control voltage changing the effective path length of an arm of the ring [see Fig. 1(a)].Furthermore, all the peaks have the same asymmetry depending on Submitted to Journal of Physical Society of Japan the control gate.The origins of these peculiar behaviors remain to be understood because explicit calculations can only be performed for a limited set of parameters in realistic systems.
In this work, we use a 1D model with the effects of inelastic scattering disturbing the phase coherence and demonstrate that the asymmetric lineshape is a result of complicated resonances in various regions of an AB ring with a dot and therefore is not determined by a simple Fano-type interference of waves passing through two arms.The organization of the paper is as follows: In §2, a one-dimensional model with inelastic scattering is introduced.In §3, some examples of explicit numerical results are presented with emphasis on the roles of resonance in different regions of the ring.The effects of the presence of several channels as in actual systems are discussed in §4.A summary is given in §5.§2.Model Aharonov-Bohm Ring We consider a one-dimensional chain of atoms denoted by j = 1, . . ., N with site energy u j +2t, nearestneighbor hopping integral −t, and magnetic flux φ.−35) To each lattice point j, a one-dimensional chain consisting of sites l = 1, 2, . . . is attached and the chain is connected to a reservoir, with the site l = 0 being the same as the site j of the ring.The nearest-neighbor hopping integral and the site energy of the chain are denoted by −t and ε , respectively.The lattice constant of the chain is denoted by a .
The equation of motion of the chain is given by Its eigenenergy is written as ε (k ) = ε −2t cos k a with wave vector k which is chosen as k > 0 in the following.The outgoing (+) and incoming (−) waves satisfy the relation respectively.
The equation of motion at the site j of the ring is given by We decompose c j 1 into the outgoing and incoming parts, i.e., Each part can be written as where we have decomposed the amplitude into with + and − denoting outgoing and incoming waves, respectively, in the chain.Therefore, we have Thus, the equation of motion at site j becomes (2.9) A chain connected to a source electrode S is attached to the site m and another connected to a drain electrode D is attached to the site n.We shall confine ourselves to the case n = m.The hopping integral in these source and drain chains is chosen as t, the same as that in the ring, the site energy is chosen as 2t, and the lattice constant as a.We have (2.10) At the site m, the equation of motion is given by where m represents the amplitude of the wave coming from the source chain.The equation at the site n can be obtained by replacing m with n.
Let H be the effective Hamiltonian defined by and the corresponding Green's function (2.14)Then, we have The amplitude of the outgoing waves is given by (2.17) We define the velocities as v = |∂ε (k )/h∂k | and v = |∂ε(k)/h∂k|.Then, the transmission coefficients are given by t DS = G nm 2it sin(ka), (2.18) ) ) and the reflection coefficients are given by The transmission and reflection probabilities are given by the absolute square of these quantities, i.e., T DS = |t DS | 2 , etc. Let I S be the current flowing into the system from the source electrode with voltage V S , I D that from the drain electrode with voltage V D , and I j that from the reservoir j with voltage V j .We define further ) and ) ) ) Then, we have (2.29) The current flowing into and out of each reservoir should vanish under the condition of flux conservation, i.e., I 2 = 0. Therefore, we have giving Because of the current conservation, we should have which give This symmetry relation should be satisfied in any system and can be used for a check of numerical accuracy.
In the following, we shall choose a = a without loss of generality.Furthermore, we shall choose k a = π/2.Then, the diagonal site energy due to a chain connected to a reservoir is given by a pure imaginary number, i.e., The strength of the inelastic scattering is characterized by the phase coherence length L φ defined by with v F the Fermi velocity.We shall consider the case of small ka for which the lattice model can simulate a continuous system.The energy is expanded as with m * = h2 /2ta 2 .We choose the Fermi energy E F for the energy scale and the Fermi wavelength λ F for the length scale.They are related to each other through (2.38) The control gate in the upper arm and the quantum dot in the lower arm are modeled by a potential illustrated in Fig. 2. As for the control gate, the potential is given by , where V C is the potential height, L B is the length of the region where the potential is applied, and x C is the center of the barrier potential.The dot potential is symmetric around the dot position x D .In the right hand side, it is given by , where L D is the dot length, W is the barrier height, and V G is the gate voltage controlled externally.We shall confine ourselves to the case of an AB ring with the same upper-arm length L/2 and lowerarm length L/2.Further, the control gate is fixed at the center of the upper arm.The dot position is placed first at the center of the lower arm and can be shifted to the left side by Δ.The actual AB ring is formed by a gate structure fabricated on top of a twodimensional system and therefore both control potential and dot potential are slowly varying in comparison with the Fermi wavelength λ F .In the following, we choose and L/λ F = 10.5.For this L B the reflection due to the control gate is negligible until the barrier height V C exceeds the Fermi level.For the lattice constant, we choose λ F /a = 20.Details of the results are dependent on these parameters, but qualitative behaviors remain the same.In particular, the ring size L/λ F = 10.5 is smaller than that of experiments, but the results periodically vary when the length of each part is changed with period of λ F as long as the phase coherence length is much larger.
At the T junction where the source wire and the AB ring joins, the transmission coefficient t and reflection coefficient r can be calculated as (2.42) These satisfy the flux conservation condition |r| 2 +2|t| 2 = 1.For the present choice of the parameter, we have |r| ≈ 0.753, arg(r) ≈ 0.854 × π, |t| ≈ 0.465, and arg(t) ≈ 0.254×π, showing that the total transmission probability and the reflection probability are comparable to each other.Therefore, about half of the wave incident from the source is reflected back and the remaining half is transmitted into two arms with equal probability.We can define an effective amplitude in the upper arm by (2.43) Similarly, the effective amplitude in the lower left arm and lower right arm can also be defined.In the limit of weak inelastic scattering, this amplitude is proportional to the electron density in the region corresponding to the incident wave.§3.Numerical Results Figure 3 shows some examples of the calculated conductance for several values of the control voltage V C /E F in the absence of an AB flux (φ = 0) and L φ /L = 100 for which inelastic scattering is negligible.With the increase in the gate voltage, a quasi-bound level localized in a dot crosses the Fermi level, where the conductance exhibits an asymmetric lineshape.The level corresponding to the largest gate voltage, i.e., V G /E F ∼ 0.9 denoted by (1), is associated with the lowest level, that at V G /E F ∼ 0.7 to the first excited level denoted by (2), etc.The conductance when dot levels are away from the Fermi level exhibits a peak around V C /E F ≈ 0.45 as a function of V C .This resonance behavior becomes clear if we take an average over V G as shown by the dotted line.Furthermore, the resonance manifests itself in the change in the asymmetry direction of the conductance peak as a function of V C .This asymmetry change will be discussed later near the end of this section and we shall first concentrate our attention to the resonance in the upper arm.
Figure 4 shows the effective amplitude in the upper arm, lower left arm, and lower right arm as a function of V C , when the dot is pinched off.The amplitude in the upper arm shows a clear Lorentzian lineshape due to a resonance in the upper arm and is well proportional to the conductance given by the dotted line in Fig. 3.The amplitudes in the lower left and right arms show a slight asymmetric increase corresponding to the resonance in the upper arm, but effects are not so important.The effective path length of the upper arm can be calculated using a WKB approximation for the potential of the control gate and adding the contribution of the phase shift of the reflection r at the T junction.The resulting path length is roughly given by L upper arm eff /λ F ≈ 5−0.46× (V C /E F − 0.43) giving a resonance at V C /E F ≈ 0.43 in agreement with the numerical result.
That the resonance is actually taking place in the upper arm can also be understood from the effects of inelastic scattering.Figure 5 shows the conductance at the resonance for various values of the phase coherence length L φ .When a dot level crosses the Fermi level, a resonant tunneling through the dot or a resonant reflection by the dot disturbs the resonance in the upper arm and therefore the conductance exhibits a dip.With the increase in inelastic-scattering intensity or with the decrease in L φ , this strong interference effect is suppressed and the lineshape approaches a symmetric Lorentzian.That the conductance peak becomes nearly symmetric for L φ /L = 0.5 is consistent with the fact that the resonance takes place in the upper arm with a long path length.A careful examination of the results shows that the direction of the asymmetry changes between L φ /L = 5 and 2 for peaks (2) and (4).
Figure 6 shows similar results for V C /E F = 0.5.The asymmetry direction changes between adjacent dot peaks when L φ is sufficiently large.With the decrease in L φ , the peaks corresponding to the lowest and second excited dot levels, denoted by ( 1) and (3) in the figure, change their lineshape simply from asymmetric to symmetric.However, peaks (2) and ( 4) exhibit first a change in the asymmetry direction between L φ /L = 2 and 1, and then turn into a symmetric Lorentzian with the further decrease in L φ .This behavior clearly shows that the asymmetric lineshape is not a result of simple interferences of waves passing through the upper arm and those through the dot level in the lower arm.
Figure 7 shows the conductance at V C /E F = 0 for several values of the shift Δ in the dot position when the effects of inelastic scattering are unimportant.The conductance when dot levels are away from the Fermi level exhibits a peak at around Δ/λ F = 0.14.This is a result of resonance in the upper arm induced by a resonance in the lower left arm.Figure 8 shows the effective amplitudes as functions of Δ for V C /E F = 0 when the dot is depleted.The amplitude in the lower left arm exhibits a symmetric Lorentzian lineshape at around Δ/λ F = 0.13.The strong enhancement of the amplitude in the upper arm is induced by this resonance.The conductance shown in Fig. 7 when dot levels are away from the Fermi level as a function of Δ is highly proportional to this amplitude in the upper arm.
When dot levels are away from the Fermi level, most of the electron wave is reflected by the dot.Therefore, the system can be represented by the model with the left T junction with a wire with length d and the right T junction with a one wire with length L lower −d as illustrated in the inset of Fig. 8, where L lower is the effective path length of the lower arm.Then, the transmission t L and reflection coefficient r L of the left T junction are given by where r and t are given by eq.(2.42).This relation between r L and t L and the flux conservation condition |r L | 2 + |t L | 2 = 1 lead to the conclusion that r L lies on the circle with center at −1/2 and radius 1/2.Thus, the reflection coefficient r L executes a rotation along the circle and its phase changes between π/2 and 3π/2, when e 2ikd changes its phase by 2π as a function of d.A total reflection r L = −1 occurs at a certain value of d.
The phase of the reflection coefficient r R at the right T junction also changes as a function of d.
Because the phase of r L (as well as that of r R ) directly influences the resonance condition of the upper arm, the upper arm becomes resonant when Δ (or d) is swept near the resonance in the lower left arm.Furthermore, a total reflection at the T junction, i.e., r L = −1, occurs also near the resonance in the lower arm.In fact, the peak of the amplitude in the upper arm around Δ/λ F ≈ 0.14 corresponds to this induced resonance in the upper arm and the vanishing amplitude around Δ/λ F ≈ 0.08 corresponds to the induced total reflection at the left T junction.
Figure 9 shows the effects of inelastic scattering for Δ/λ F = 0.14 and V C /E F = 0.When a dot level crosses the Fermi level, a resonant tunneling through the dot or a resonant reflection by the dot disturbs the resonance in the lower left arm and therefore the conductance exhibits a dip.With the increase in inelastic-scattering intensity or with the decrease in the phase coherence length, this strong interference effect is suppressed and the lineshape approaches a symmetric Lorentzian.This behavior is qualitatively the same as in Fig. 5.However, the ap-proach to the symmetric lineshape is slower because the resonance occurs in the lower left arm with short length.
The most noteworthy feature of Fig. 7 is that for Δ/λ F = 0.13, 0.12, and 0.11 the conductance exhibits a peak with the same asymmetry direction.Figure 10 shows the conductance at Δ/λ F = 0.14 for different values of V C .With the increase in V C , the asymmetry of all the dot peaks remains the same until V C /E F reaches ∼ 0.43 where the resonance condition is satisfied in the upper arm, and then changes its direction at the same time.This change in the asymmetry direction across this value of V C is essentially the same as the feature noted in Fig. 3.
Figure 11 shows the conductance in the presence of a magnetic flux.When an asymmetric peak appears owing to simple interference of waves passing through the upper arm and the lower arm, the direction of the asymmetry is expected to become opposite in the presence of the half flux quantum φ = φ 0 /2.In the present case, however, the asymmetry direction does not change at all with the flux.This clearly shows that a resonant reflection by a dot level plays dominant roles here.
When the Fermi level is close to a dot level E 0 , the reflection coefficient r D of the dot can be approximately written as where Γ is the broadening of the dot level.This r D lies on the circle with the center at −1/2 and a radius of 1/2 and therefore its phase changes between π/2 and 3π/2 when E 0 crosses E F .The corresponding r L and t L become Obviously, the phase change of r D near the resonance of the dot level can approximately be absorbed into the change in the effective arm length d.This indicates that the dot-level crossing is almost equivalent to a change in the arm length d for a wide range as far as the effects of the resonant reflection are concerned.When the Fermi level crosses the dot level, the resulting rapid change in the phase of r D causes a resonance in the lower left arm and then a resonance in the upper arm also, leading to a resonant transmission through the upper arm.It is possible to reproduce the qualitative behavior of the conductance peaks shown in Fig. 10, i.e., that all the peaks have the same asymmetry direction and the direction changes across V C corresponding to the resonance in the upper arm, within a model including the effects of resonant reflection at each dot level.
Because a resonance reflection by a dot level causes adjacent conductance peaks to have the same asymmetry direction, a resonant transmission through a dot level is necessary to cause the alternate asymmetry change as in Figs. 3 and 6 for sufficiently large L φ .This can be confirmed by the change in the asymmetry direction of all the peaks in the presence of the AB flux φ/φ 0 = 1/2 and across the value of V C corresponding to the resonance in the upper arm shown in Fig. 3.
In Fig. 5, the lineshape of ( 1) and ( 3) already becomes symmetric for L φ /L = 2.This shows that for levels (1) and (3) the transmitted wave through the dot has a constructive interference with the wave in the upper arm.This further shows that the wave through the dot has a destructive interference for levels (2) and (4), because of the parity change of the dot levels.The effects of the dot-level crossing are much weaker for levels (1) and (3) than for (2) and (4).Therefore, when the resonantly transmitted wave through the dot has a constructive interference with the wave in the upper arm, the effects of resonant reflection and transmission tend to cancel each other, whereas they become stronger when the wave through the dot has a destructive interference.This seems to be valid for most of the results obtained, although reasons remain unclear.
In Fig. 3, conductance peak (2) becomes smaller than (1) for the control voltage V C /E F = 0.55 and 0.6.This behavior differers from that for smaller values of V C /E F .The phase of the wave passing through the upper arm changes rapidly and over a wide range when V C is swept through the resonance in the upper arm.As a result, the wave transmitted through a dot level now has a constructive interference for level (2) instead of level (1) for V C /E F = 0.55 and 0.6.Similar behavior can be seen in Fig. 7.
Some resonance usually occurs in the whole part of the AB ring whenever a dot level crosses the Fermi level either by resonant reflection, transmission, or their combinations.The asymmetric lineshape is the result of such resonance due to complicated interferences.The effects of resonant reflection by a dot level are equally as important as those of resonant transmission and the degree of the competition between reflection and transmission can change as a function of the phase coherence.This is the origin of the change in the asymmetry direction when the phase coherence length is reduced, as obtained in Figs. 5 and 6. §4.Discussion Usually, actual AB rings with a quantum dot have several conducting channels in the wire region.In the adiabatic limit, where the confinement potential varies slowly on the scale of the Fermi wavelength, each onedimensional channel has its own effective potential and mixing between different channels is neglected.Therefore, dot levels are classified by associated 1D channels in the wire region.Because the lowest 1D channel has the largest kinetic energy in the wire direction and the lowest effective barrier height, only the lowest 1D channel and the associated dot levels contribute to the resonance reflection and transmission.−28) In an actual quantum dot, such a classification is slightly violated by mixing due to the confinement potential or by possible disorder.−28) Because of the large charging energy (so far neglected completely), these weakly coupled dot levels contribute to Coulomb peaks that are almost equally spaced as a function of the gate voltage.
−28) It is clear that resonant reflections and transmissions have only a small influence on the wave function in the ring for these levels.The present 1D model applies to strongly coupled levels instead.When they happen to have the same asymmetry as obtained here, every conductance peak including that associated with a weakly coupled state has the same asymmetry as obtained in ref. 26.
−4) Actual observation the same asymmetry for most of dot levels by a slight change in the configuration, preferably with a narrower effective width and at low temperatures giving sufficiently long phase coherence length, is highly expected, because it provides a direct evidence of complicated interferences occurring in actual Aharonov-Bohm rings.§5.

Summary and Conclusion
The roles of interference effects have been demonstrated in an Aharonov-Bohm ring with a quantum dot within a one-dimensional model including the effects of inelastic scattering breaking the phase coherence.It has been shown that resonances appearing in different regions of the ring play dominant roles in the behavior of the conductance peak associated with dot levels.In fact, when a dot level crosses the Fermi level, the phase of the reflection coefficient changes over a wide range, inducing a resonance in the left or lower right arm and then also giving rise to a resonance in the upper arm.Such reflection effects compete with those of a resonant transmission through a dot level.The asymmetry of the peak is determined by a combination of these effects and therefore exhibits a complicated behavior even as a function of the phase coherence length.V C /E F = 0.When a resonance occurs in the lower left arm, the amplitude in the upper arm also is strongly enhanced.The inset shows an effective model when the transmission through the dot is not allowed.

Fig. 1 Fig. 2
Fig. 1 Schematic illustration of AB ring with quantum dot.(a) Model corresponding to double slit.The transmission coefficient of the upper arm is denoted by t 0 and that of the lower arm by t d .When the dot level E n crosses the Fermi level, t d exhibits a

Fig. 3
Fig. 3 Calculated conductance as function of gate voltage V G for several values of control voltage V C .The dotted lines show the conductance averaged over V G as a function of V C .It takes a maximum at V C /E F ≈ 0.44.The dot levels are labeled as (1), (2), (3), and (4) from the bottom.

Fig. 6
Fig. 6 Calculated conductance for different values of L φ at V C /E F = 0.5 slightly away from resonance in upper arm.Δ/λ F = 0.The asymmetry of the peaks change with the decrease in L φ .This change is clear particularly for the first and third excited dot levels, i.e., (2) and (4).

Fig. 7
Fig. 7 Calculated conductance as function of V G for several values of shift Δ of dot position.The conductance when dot levels are away from the Fermi level exhibits a peak around Δ/E F ≈ 0.14 owing to resonance in lower left arm.

Fig. 8
Fig. 8 Integrated squared amplitude of wave function as function of shift in dot position.V G /E F = 1.V C /E F = 0.When a resonance occurs in the lower left arm, the amplitude in the upper arm also is strongly enhanced.The inset shows an effective model when the transmission through the dot is not allowed.

Fig. 9
Fig. 9 Calculated conductance for different values of L φ at strong resonance in lower left arm.V C /E F = 0. Δ/λ F = 0.14.The conductance peaks exhibit an asymmetric lineshape even when L φ ∼ L/4, demonstrating a resonance in a short region.

Fig. 10 Fig. 11
Fig. 10 Calculated conductance as function of V G for different values of V C when strong resonance occurs in lower left arm.Δ/λ F = 0.14.All the peaks have the same asymmetry direction and their direction changes simultaneously with V C .Fig. 11 Calculated conductance as function of gate voltage in presence of AB flux at resonance in lower left arm.V C /E F = 0.6.Δ/λ F = 0.14.The asymmetry direction is independent of the AB flux.