Abstract
We investigate quantum interference effects in a doubleAharonovBohm (AB) interferometer consisting of five quantum dots sandwiched between two metallic electrodes in the case of symmetric dotelectrode couplings by the use of the Green’s function equation of motion method. The analytical expression for the linear conductance at zero temperature is derived to interpret numerical results. A threepeak structure in the linear conductance spectrum may evolve into a doublepeak structure, and two Fano dips (zero conductance points) may appear in the quantum system when the energy levels of quantum dots in arms are not aligned with one another. The AB oscillation for the magnetic flux threading the doubleAB interferometer is also investigated in this paper. Our results show the period of AB oscillation can be converted from 2π to π by controlling the difference of the magnetic fluxes threading the two quantum rings.
Keywords:
AharonovBohm interferometer; Fano effects; Quantum dots; Transport propertiesNano Ideas
Thanks to rapid developments in the fabrication and selfassembly techniques, the electrical transport through nanoscale quantum systems such as a single quantum dot, multiple quantum dots, atoms or molecules coupled to metallic electrodes has been an interesting subject in recent years [16]. In the nanoscale quantum systems, the electrical transport is ballistic, while the phase coherence of the electrons is preserved. Especially, the quantum interference effects in an AB ring including a quantum dot have been reported [7]. The results showed that Fano effect with asymmetric parameters was a good probe to quantum interference effects in the nanoscale systems. The transport properties of a quantum ring consisting of two parallelcoupled quantum dots sandwiched between two metallic electrodes have been also studied theoretically and experimentally in the last few decades [722]. For example, an AB interferometer including two coupled quantum dots with each quantum dot inserted in each arm was presented, and an oscillating electric current was detected experimentally [8,9]. Such a doublequantumdot model consisting of the parallelcoupled doubledot system has been studied extensively in some previous theoretical works [1318]. When the interdot coupling is considered, a bonding molecular state and an antibonding molecular state are developed. A swap effect can be found in the quantum system by tuning the magnetic flux threading the quantum ring, which may be used in the future quantum computations [15].
Recently, the transport properties of multiparallelcoupled quantum dots have attracted considerable attention due to their potential applications and abounding physics [2331]. Zeng et al. studied the AB effects in a quantum ring consisting of four quantum dots sandwiched between two metallic electrodes, and a Fano dip is developed when the energy levels of quantum dots in two arms are mismatched [23]. Guevara et al. offered a quantum model describing multiparallelcoupled quantumdot molecule, and Fano effects in the quantum system were studied in detail [24]. More recently, Li et al. studied the electrical transport through a triplearm AB interferometer consisting of three parallel quantum dots with electronelectron interactions under an applied electric field [25].
In this work, we study the quantum interference effects in a doubleAharonovBohm interferometer consisting of five quantum dots sandwiched between two metallic electrodes as shown in Fig. 1. The left quantum ring consisting of the quantum dot 1, the quantum dot 2, quantum dot 3 and the left metallic electrode encloses the magnetic flux . The right ring consisting of the quantum dot 3, the quantum dot 4, the quantum dot 5, and the right metallic electrode encloses the magnetic flux . Two quantum rings are connected together by the quantum dot 3. For simplicity, we consider that only one energy level exists in each quantum dot, and the energy levels of all quantum dots can be tuned by the voltages applied on the quantum dots. In this work, we studied in detail the quantum interference effects in the doublequantumring structure in the case of symmetric dotelectrode couplings. The results show that the linear conductance peaks at zero temperature can be effectively tuned by using the intermediate quantum dot 3. As the energy level of the quantum dot 3 changes, a threepeak structure may evolve into a twopeak structure in the linear conductance spectrum. When the energy levels of quantum dots in arms are not aligned with one another, two Fano dips may appear in the doublequantumring device. Two Fano peaks can be effectively modified by tuning the energy level of quantum dot 3. The AB oscillation for magnetic flux is also studied in this work. Our results show that the AB oscillating behavior depends strongly on the difference between the magnetic fluxes threading the left and right rings. In addition to the general AB oscillation with 2πperiod, an AB oscillation with πperiod can be developed when the difference between the two magnetic fluxes is (2n + 1)π(n = 0, 1, 2,…).
Figure 1. Schematic plot of a doubleAharonovBohm interferometer consisting of five quantum dots sandwiched between two metallic electrodes. The left ring encloses a magnetic flux , and the right ring encloses a magnetic flux
Model and Methods
The parallelcoupled quantumdot structure that we consider is illustrated in Fig. 1. For simplicity, the interdot and intradot coulomb interactions are neglected, and the total Hamiltonian describing the quantum system is written as
where the first term H_{leads} in Eq. (1) describes the left and right electrodes in the noninteracting electron approximation
where denotes the creation(annihilation) operator for an electron with energy ε_{αk} and momentum k in electrode α. The second term in Eq. (1) describes the dynamics of the five parallelcoupled quantum dots, which can be modeled by using the following fivesite Hamiltonian
where creates (annihilates) an electron with the energy ε_{j} in the jth quantum dot, and the AB phase and the flux quantum . t_{ij} describes the interdot tunneling coupling between the dot i and dot j, for convenience, which is regarded as a real number. The third term in Eq. (1) describing the tunneling coupling between the quantum dots and the electrodes is divided into two parts
describes the tunneling coupling between the isolated quantum dots and the electrode α
and
A phase factor is attached to V_{αj} in the presence of the magnetic flux, and they can be written as , , , . In order to study the transport through the doublequantum ring, we need know the retarded (advanced) Green’s function . The retarded Green’s function is defined as
Applying the equation of motion method, we obtain a series of equations [32],
Four new Green’s functions arising from the tunneling couplings between the quantum dots and metallic electrodes appear in the above Eqs. (8–9) and Eqs. (11–12), and they are
Using the above Eqs. (8–16), the retarded Green’s function can be calculated by the following 5 × 5 matrix
where and )(j = 1, 2 for α = L; j = 4, 5 for α = R). The advanced Green’s function G^{a} is obtained by the relation . The linear conductance of the double quantum ring at zero temperature can be calculated by the Landauer Formula [32],
The linewidth matrices and are given
and
In the case of the symmetrical dotelectrode tunneling couplings , the linear conductance has the following form,
where and . From the above equation, we note that the linear conductance includes the contributions from four different electron tunneling paths (1→3→ 4, 1→3→5, 2→3→ 4;2→3→5) and interference terms among them. Equation (21) is also written as:
Results and Discussion
In this section, the dependence of the linear conductance σ on system parameters is discussed numerically and analytically. The coupling strength between the quantum dots and the metallic electrodes is taken as the energy unit. Through this paper, all energy levels in quantum dots, tunneling couplings and Fermi energy are measured by .
Without Magnetic Flux
We first study the linear transport properties of the doubleAB interferometer consisting of five parallelcoupled quantum dots with the same energy levels (ε_{j} = 0, j = 1, 2, 3, 4, 5) in the absence of the magnetic fluxes (ϕ_{L} = ϕ_{R} = 0). Figure 2a shows the linear conductance σ of the doublequantumring structure as a function of the Fermi level E_{F} in the case of symmetrical interdot tunneling couplings. The same interdot tunneling coupling strengths are chosen as t_{13} = t_{23} = t_{34} = t_{35} = t. There are total five molecular states in center fivedot quantum system, but three molecular states of which with zero energy are degenerate. Other two molecular states are located at −2t and 2t, respectively. So we find that there are three resonance peaks with σ = 2e^{2}/h appearing at 0, 2t and −2t, respectively. With t increasing, the position of the center conductance peak has no changing, while the other two conductance peaks move in the opposite direction as shown in Fig. 2a. In Fig. 2b and 2c, we display the linear conductance σ as a function of the Fermi energy E_{F} in the case of asymmetrical interdot tunneling couplings. In Fig. 2b, we assume the interdot tunneling coupling strengths as and t_{34} = t_{35}. In this case, three conductance peaks are located at 0, , and , respectively. As t_{34} decreases, the conductance peaks are suppressed and become narrower as shown in Fig. 2b. In Fig. 2c, we examine the transport properties of the doublequantumring structure by tuning t_{23} and t_{34}. Here we fix the interdot tunneling couplings t_{13} and t_{35} at . A Fano dip (zero conductance point) appears at E_{F} = 0 when t_{23} (t_{34}) is different from t_{13}(t_{35}). The behind reason is the destructive interference effects between the electron waves directly transmitting through the three coupled quantum dots in series and sidecoupled quantum dots (see the inset in Fig. 2c). As t_{23} and t_{34} decrease, a threepeak structure in the linear conductance spectrum evolves into the fourpeak structure. When t_{23} = t_{34} = 0, the quantum device becomes a three quantumdot array sandwiched between two metallic electrodes with sidecoupled quantum dots as shown in the inset of Fig. 2c. The Fano dip is located at the energy levels of the sidecoupled quantum dots (ε_{2} = ε_{4} = 0). It is noted that two peaks become closer and narrower as t_{23} and t_{34} decrease. When t_{23} = t_{34} = 0, the two conductance peaks are located at .
Figure 2. Linear conductance σ as a function of E_{F} under different interdot tunneling couplings. The energy levels of the quantum dots are chosen as ε_{1} = ε_{2} = ε_{3} = ε_{4} = ε_{5} = 0
Figure 3 displays the linear conductance σ as a function of E_{F} when the energy level of the third quantum dot ε_{3} is not aligned with those of the other four quantum dots. Five molecular states appear at , respectively. When ε_{3} = 0, three conductance peaks with σ = 2e^{2}/h are centered at 0, and , respectively. The linear conductance σ as a function of E_{F} under the different values of ε_{3} is plotted in Fig. 3a and 3b. The left two conductance peaks evolve into a single conductance peak, and the right conductance peak moves in the right direction in the case of ε_{3} > 0. We also find the height of the left conductance peak is suppressed. In the case of ε_{3} < 0, the situation is in inverse. The right two conductance peaks evolve into a single conductance peak, and the height of the conductance peak is suppressed. The left conductance peak moves in the left direction, and the height of the conductance peak has no changing. In order to explore the dependence of the linear conductance on the energy level of the intermediate quantum dot 3, we plot the linear conductance as a function of ε_{3} under several different interdot couplings in Fig. 4. With ε_{3} increasing, the linear conductance first increases and reaches the maximum value at the certain value ε_{3m}. When the interdot couplings become small, ε_{3m} moves toward the Fermi energy. Once the interdot couplings are enough small, the resonant conductance peak is almost pined at E_{F}. The transport properties of the doublequantum ring are similar with these of a single quantum dot coupled straightly to metallic electrodes.
In Fig. 5, we study the transport properties of the doublequantum ring when the quantumdot levels are not aligned with one another. The same interdot coupling strengths (t_{13} = t_{23} = t_{34} = t_{35} = t = ) are considered, and the energy levels of quantum dots parameters are chosen as , and , respectively. Since the degenerate states are opened in this case, five conductance peaks appear in the linear conductance spectrum. The linear conductance σ consists of three BreitWigner peaks and two sharp asymmetric Fano peaks. A Fano dip happens at , while the other Fano dip appears at . The numerical results may be explained by using Eq. (21). The linear conductance in the absence of magnetic flux can be written as
Figure 5. Linear conductance σ as a function of the Fermi energy E_{F} under several different values of the quantum dot 3. Other quantumdot energy levels are chosen as , , and , respectively
Using Eq. (17), we arrive at
From the above equation, we see clearly σ = 0 for or . The linear conductance spectrum has mirror symmetry around ε_{3} in the case of ε_{3} = 0. With increasing ε_{3}, the mirror symmetry is broken. The left Fano peak is suppressed, while the right Fano peak is firstly suppressed, then it is enhanced.
With Magnetic Flux
The periodic oscillation in the linear conductance for the magnetic flux is key features when the phase coherence of the electrons is preserved. In this section, we start with the study of dependence of the linear conductance on the magnetic fluxes through the doublequantum ring. The dependence of the linear conductance on the Fermi energy in the presence of the same magnetic flux through the left and right rings (ϕ_{L} = ϕ_{R} = ϕ) is shown in Fig. 6. The same interdot tunneling couplings and the same quantumdot energy levels ε_{1} = ε_{2} = ε_{3} = ε_{4} = ε_{5} = ε_{0} = 0 are considered. When the magnetic flux ϕ is presented, a Fano dip is developed when the Fermi energy is aligned with ε_{0}. With ϕ increasing, the threepeak structure disappears, while a fourpeak structure in the linear conductance spectrum appears. When the magnetic fluxes are further increased, the fourpeak structure evolves into a doublepeak structure. The results may be explained by the following expressions. After some algebra, we derive the linear conductance σ in the presence of the magnetic flux as the following analytical form
Figure 6. Linear conductance σ as a function of the Fermi energy E_{F} under several different magnetic fluxes threading the left and right rings ϕ_{L} = ϕ_{R} = ϕ
From the above equation, we see clearly that the linear conductance disappears when E_{F} = 0, and four molecular states are located around and , respectively. With ϕ increasing, the two middle conductance peaks are suppressed obviously, while the outside conductance peaks become shaper as shown in Fig. 6. When ϕ approaches π, we can obtain an approximate formula for the linear conductance
where δ_{1} and δ_{2} represent two small quantities. Equation (26) shows two narrower conductance peaks are centered at as shown in Fig. 6. When ϕ = nπ(n is odd number), the linear conductance disappears everywhere for any value of E_{F}.
Figure 7 shows the images of the linear conductance σ as functions of ϕ_{L} and the Fermi energy E_{F}. The same interdot couplings and same quantumdot levels are considered. denotes the difference between magnetic fluxes through the left quantum ring and right quantum ring. When , a general 2πperiodic AB oscillation for ϕ_{L} is obtained as shown in Fig. 7a and 7b. When , a πperiodic AB oscillation appears in the quantum system as shown in Fig. 7c. The above results can be explained as following expressions. When the magnetic fluxes through the doublequantum ring are considered, using Eq. (24), we arrive at
When , the above equation can be simplified as
Figure 7. Images of the linear conductance σ as functions of the fermi energy E_{F} and ϕ_{L} for a, b, and c, respectively. d Linear conductance σ as a function of magnetic flux ϕ_{L} for [solid (black) line], [dashed (red) line], and [dotted (blue) line], respectively. Other system parameters are chosen as in Fig. 6
So an AB oscillation with πperiod is developed as shown in Fig. 7c and 7d.
It is well known that an oscillating current in the AB interferometer has been detected experimentally [8,9]. For a single quantum ring consisting of two parallelcoupled quantum dots sandwiched between two metallic electrodes, 2πperiodic oscillation of the linear conductance σ is reported in the pervious works [13]. The linear conductance σ as a function of ϕ_{L} under the different energy levels in the quantum dot 3 or several different values of ϕ_{R} is shown in Fig. 8. The Fermi energy E_{F} is fixed at , and other system parameters are chosen as in Fig. 6. The AB oscillation for ϕ_{L} in the presence of different energy levels of the quantum dot 3 is potted in Fig. 8a. For ε_{3} = 0, a series of shaper resonant peaks appear at 2n π(n = 0, 1, 2…), and the linear conductance reaches a minimum (σ = 0) when ϕ_{L} approaches (2n + 1) π(n = 0, 1, 2…). When the energy level in quantum dot 3 is not aligned with other quantumdot levels, the single conductance peak around 2nπ splits into the two conductance peaks. With ε_{3} moving away from zero energy point, two conductance peaks move in the opposite direction. It is noted that the doublepeak structure around 2nπ disappears slowly as the magnetic flux threading the right quantum ring increases.
Figure 8. a AB oscillations for ϕ_{L} in the presence of several different energy levels of the quantum dot 3 with ϕ_{R} = 0; b AB oscillations for ϕ_{L} in the presence of several different magnetic fluxes ϕ_{R} with the fixed . Other system parameters are chosen as ε_{1} = ε_{2} = ε_{4} = ε_{5} = 0, , and
Summary
In summary, the transport properties and quantum interference effects in a doubleAB interferometer in series consisting of five quantum dots in the case of symmetric dotelectrode tunneling couplings are studied by using Green’s function equation of motion method. The energy levels of all quantum dots can be tuned by the voltages applied on the quantum dots in experiments. The linear conductance can be effectively modified by the intermediate quantum dot 3. As the energy level in the quantum dot 3 changes, a threepeak structure in the linear conductance spectrum evolves into a twopeak structure. When the quantumdot levels in arms are not aligned with one another, two Fano resonances with different Fano factors may appear in the quantum device. The AB oscillation for the magnetic flux in the doubleAB interferometer is also studied in this work. The results show that the AB oscillating behavior depends strongly on the difference between the magnetic fluxes threading the left and right quantum rings. An AB oscillation with πperiod for the magnetic flux threading the left quantum ring is developed when the difference between the two magnetic fluxes is (2n + 1)π(n = 0, 1, 2,…).
Acknowledgments
The authors thank the supports of the National Natural Science Foundation of China (NSFC) under Grant No. 10947130 and the Science Foundation of the Education Committee of Jiangsu Province under Grant No. 09KJB140001. The authors also thank the supports of the Foundations of Changshu Institute of Technology.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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