# Quantum interference effect in electron tunneling through a quantum-dot-ring spin valve

- Jing-Min Ma
^{1}, - Jia Zhao
^{1}, - Kai-Cheng Zhang
^{1}, - Ya-Jing Peng
^{1}and - Feng Chi
^{1}Email author

**6**:265

**DOI: **10.1186/1556-276X-6-265

© Ma et al; licensee Springer. 2011

**Received: **12 September 2010

**Accepted: **28 March 2011

**Published: **28 March 2011

## Abstract

Spin-dependent transport through a quantum-dot (QD) ring coupled to ferromagnetic leads with noncollinear magnetizations is studied theoretically. Tunneling current, current spin polarization and tunnel magnetoresistance (TMR) as functions of the bias voltage and the direct coupling strength between the two leads are analyzed by the nonequilibrium Green's function technique. It is shown that the magnitudes of these quantities are sensitive to the relative angle between the leads' magnetic moments and the quantum interference effect originated from the inter-lead coupling. We pay particular attention on the Coulomb blockade regime and find the relative current magnitudes of different magnetization angles can be reversed by tuning the inter-lead coupling strength, resulting in sign change of the TMR. For large enough inter-lead coupling strength, the current spin polarizations for parallel and antiparallel magnetic configurations will approach to unit and zero, respectively.

PACS numbers:

## Introduction

Manipulation of electron spin degree of freedom is one of the most frequently studied subjects in modern solid state physics, for both its fundamental physics and its attractive potential applications [1, 2]. Spintronics devices based on the giant magnetoresistance effect in magnetic multi-layers such as magnetic field sensor and magnetic hard disk read heads have been used as commercial products, and have greatly influenced current electronic industry. Due to the rapid development of nanotechnology, recent much attention has been paid on the spin injection and tunnel magnetoresistance (TMR) effect in tunnel junctions made of semiconductor spacers sandwiched between ferromagnetic leads[3]. Moreover, semiconductor spacers of InAs quantum dot (QD), which has controllable size and energy spectrum, has been inserted in between nickel or cobalt leads[4–6]. In such a device, the spin polarization of the current injected from the ferromagnetic leads and the TMR can be effectively tuned by a gate nearby the QD, and opens new possible applications. Its new characteristics, for example, anomalies of the TMR caused by the intradot Coulomb repulsion energy in the QD, were analyzed in subsequent theoretical work based on the nonequilibrium Green's function method [7].

The TMR is a crucial physical quantity measuring the change in system's transport properties when the angle *φ* between magnetic moments of the leads rotate from 0 (parallel alignment) to arbitrary value (or to *φ* in collinear magnetic moments case). Much recent work has been devoted to such an effect in QD coupled to ferromagnetic leads with either collinear[4–13] or noncollinear[14–16] configurations. It was found that the electrically tunable QD energy spectrum and the Coulomb blockade effect dominate both the magnitude and the signs of the TMR[4–16].

On the other hand, there has been increasing concern about spin manipulation via quantum interference effect in a ring-type or multi-path mesoscopic system, mainly relying on the spin-dependent phase originated from the spin-orbit interaction existed in electron transport channels[17–20]. Many recent experimental and theoretical studies indicated that the current spin polarization based on the spin-orbital interaction can reach as high as 100%[21–23] or infinite[24–29]. Meanwhile, large spin accumulation on the dots was realized by adjusting external electrical field or gate voltages to tune the spin-orbit interaction strength (or equivalently the spin-dependent phase factor)[27–30]. Furthermore, there has already been much very recent work about spin-dependent transport in a QD-ring connected to collinear magnetic leads[31–34]. Much richer physical phenomena, such as interference-induced TMR enhancement, suppression or sign change, were found and analyzed[31–34].

## Model and Method

where
is the creation (annihilation) operator of the electrons with momentum *k*, spin-*σ* and energy *ε*
_{
kβσ
} in the *β* th lead (*β* = *L, R*);
creates (annihilates) an electron in the QD with spin *σ* and energy *ε*
_{
d
} ; *t*
_{
βd
} and *t*
_{
LR
} describe the dot-lead and inter-lead tunneling coupling, respectively; *U* is the intradot Coulomb repulsion energy. *φ* denotes the angle between the magnetic moments of the leads, which changes from 0 (parallel alignment) to *π* (antiparallel alignment).

*β*is calculated from the time evolution of the occupation number , and can be written in terms of the Green's functions as[20, 30]

where the Keldysh Green's function *G*(*ε*) is the Fourier transform of *G*(*t* - *t'*) defined as
,
. In our present case, it is convenient to write the Green's function as a 6 *×* 6 matrix in the representation of (|*L* ↑〉, |*R* ↓〉, |*d* ↑〉, |*L* ↓〉, |*R* ↓〉, |*d* ↓〉). Thus the lesser Green's function *G*
^{
<
} (*ε*) and the associated retarded (advanced) Green's function *G*
^{
r(a)}(*ε*) can be calculated from the Keldysh and the Dayson equations, respectively. Detail calculation process is similar to that in some previous works [20, 30], and we do not give them here for the sake of compactness. Finally, the ferromagnetism of the leads is considered by the spin dependence of the leads' density of states *ρ*
_{
βσ
}. Explicitly, we introduce a spin-polarization parameter for lead *β* of *P*
_{
β
} = (*ρ*
_{
β↑}- *ρ*
_{
β↓
})/(*ρ*
_{
β↑}+ *ρ*
_{
β↓}), or equivalently, *ρ*
_{
β↑(↓)}= *ρ*
_{
β
}(1 ± *P*
_{
β
}), with *ρ*
_{
β
}being the spin-independent density of states of lead *β* .

## Result and Discussion

In the following numerical calculations, we choose the intradot Coulomb interaction *U* = 1 as the energy unit and fix *ρ*
_{
L
} = *ρ*
_{
R
} *= ρ*
_{0} = 1, *t*
_{
Ld
} = *t*
_{
Rd
} = 0.04. Then the line-width function in the case of *p*
_{
L
} *= p*
_{
R
} *=* 0 is Γ _{
β
} ≡ 2π*ρ*
_{
β
} |*t*
_{
βd
} |^{2} ≈ 0.01, which is accessible in a typical QD[41–43]. The bias voltage *V* is related to the left and the right leads' chemical potentials as *eV* = *μ*
_{
L
} - *μ*
_{
R
} , and *μ*
_{
R
} is set to be zero throughout the paper.

*J*=

*J*

_{↑}+

*J*↓, where

*J*

_{ σ }= (

*J*

_{ Lσ }-

*J*

_{ Rσ })/2 is the symmetrized current for spin-

*σ*, current spin polarization

*p*= (

*J*

_{↑}

*- J*↓)

*/*(

*J*

_{↑}+

*J*↓), and TMR=[

*J*(

*φ =*0) -

*J*(

*φ)*]/

*J*(

*φ)*are shown in Figure 2 for selected values of the angle

*φ*. In the absence of inter-lead coupling (

*t*

_{ LR }= 0), the electric current in Figure 2(a) shows typical step configuration due to the Coulomb blockade effect. The current step emerged in the negative bias region occurs when the dot level

*ε*

_{ d }is aligned to the Fermi level of the right lead (

*μ*

_{ R }= 0). Now electrons tunnel from the right lead via the dot to the left lead because

*μ*

_{ L }=

*eV < εε*

_{ d }= 0. The dot can be occupied by a single electron with either spin-up or spin-down orientation, which prevents double occupation on

*ε*

_{ d }due to the Pauli exclusion principle. Since the other transport channel

*ε*

_{ d }+

*U*is out of the bias window, the current keeps as a constant in the bias regime of

*eV*<

*ε*

_{ d }= 0. In the positive bias regime of

*ε*

_{ d }<

*eV*<

*ε*

_{ d }+

*U*a single electron transport sequentially from the left lead through the dot to the right lead, inducing another current step. The step at higher bias voltage corresponds to the case when

*ε*

_{ d }+

*U*crosses the Fermi level. Now the dot may be doubly occupied, and no step will emerge regardless of the increasing of the bias voltage.

When the relative angle between the leads' magnetic moments *φ*. rotates from 0 to *π*, a monotonous suppression of the electric current appears, which is known as the typical spin valve effect. The suppression of the current can be attributed to the increased spin accumulation on the QD[14–16]. Since the line-width functions of different spin orientations are continuously tuned by the angle variation, a certain spin component electron with smaller tunneling rate will be accumulated on the dot, and furthermore prevents other tunnel processes. As shown in Figure 2(b), the current spin polarizations in the bias ranges of *eV* < *ε*
_{
d
} and *eV* > *ε*
_{
d
} + *U* are constant and monotonously suppressed by the increase of the angle, which changes the spin-up and spin-down line-width functions. In the Coulomb blockade region of *ε*
_{
d
} < *eV* < *ε*
_{
d
} + *U*, the difference between the current spin polarizations of different values of *φ* is greatly decreased, which is resulted from the Pauli exclusion principle. The current spin polarizations also have small dips and peaks respectively near *eV* = *ε*
_{
d
} and *eV* = *ε*
_{
d
} + *U*, where new transport channel opens. The most prominent characteristic of the TMR in Figure 2(c) is that its magnitude in the Coulomb blockade region depends much sensitively on the angle than those in other bias ranges. The deepness of the TMR valleys are shallowed with the increasing of the angle. Meanwhile, dips emerge when the Fermi level crosses *ε*
_{
d
} and *ε*
_{
d
} + *U*. In the antiparallel configuration (*φ* = *π*), the magnitude of the TMR is larger than those in other bias voltage ranges.

When the inter-lead coupling is turned on as shown in Figure 2(d)-(f), both the studied quantities are influenced. Since the bridge between the leads serves as an electron transport channel with continuous energy spectrum, the system electric current increases with in-creasing bias voltage [Figure 2(d)]. For the present weak inter-lead coupling case of *t*
_{
LR
} < *t*
_{
βd
} , the transportation through the QD is the dominant channel with distinguishable Coulomb blockade effect. The current spin polarizations for different angles in the voltage ranges out of the Coulomb blockade one now change with the bias voltage value, but their relative magnitudes somewhat keep constant. The difference between the current spin polarization magnitude of different angle is enlarged by the interference effect brought about by the inter-lead coupling. Comparing Figure 2(f) with 2(c), the behavior of TMR is less influenced by the bridge between the leads in the present case.

*t*

_{ LR }= 0.01 and the angle

*φ*=

*π/*2, i.e., the magnetic moments of the leads are perpendicular to each other, to examine the bias dependence of these quantities for different values of leads' polarization

*P*

_{ L }=

*P*

_{ R }=

*P*. The electric currents in the bias voltage ranges of

*eV*<

*ε*

_{ d }and

*eV*>

*ε*

_{ d }+

*U*. are monotonously suppressed with the increase of

*P*[Figure 3(a)]. This is because the spin accumulation on the dot in these bias ranges is enlarged by the increase of the leads' spin polarization. In the Coulomb blockade region, however, current magnitudes of different

*P*are identical. The reason is that in this region the spin accumulation induced by the Pauli exclusion principle, which was previously discussed, plays a decisive role compared with that brought about by the leads' spin polarization. As is expected, the current spin polarization is increased with increasing

*P*, which is shown in Figure 3(b). The magnitude of the TMR in Figure 3(c) increases with increasing

*P*. For the half-metallic leads (

*P*

_{ L }=

*P*

_{ R }=

*P*= 1), the magnitude of the TMR is much larger than those of usual ferromagnetic leads (

*P*

_{ β }< 1). All these results are similar to those of a single dot case[14–16].

*t*

_{ LR }influence these quantities. In Figure 4 we show their characteristics each as a function of

*t*

_{ LR }with fixed bias voltage

*eV*=

*U*and

*ε*

_{ d }= 0.5, which means that we are focusing on the Coulomb blockade region. It is shown in Figure 4(a) that in the case of weak inter-lead coupling, typical spin valve effect holds true, i.e., the current magnitude is decreased with increasing

*φ*as was shown in Figure 2(a) and 2(d) (see the Coulomb blockade region in them). With the increase of

*t*

_{ LR }, reverse spin valve effect is found, in other words, current magnitudes of larger angles become larger than those of smaller angles. This phenomenon can be understood by examining the spin-dependent line-width function. The basic reason is that in this Coulomb blockade region, the relative magnitudes of the currents through the QD of different angle will keep unchanged regardless of the values of

*t*

_{ LR }(see Figure 2). But the current through the bridge between the leads, which is directly proportional to the inter-lead line-width function , will be drastically varied by the angle. In the parallel configuration, for example, spin-up inter-lead line-width function is larger than the spin-down one since

*ρ*

_{ L↑}=

*ρ*

_{ R↑}=

*ρ*

_{0}(1 +

*P*

_{ β }) and

*ρ*

_{ L↓}=

*ρ*

_{ R↓}=

*ρ*

_{0}(1 -

*P*

_{ β }). So the current polarization will increase with increasing

*t*

_{ LR }as shown by the solid curve in Figure 4(b). As the polarization of the leads is fixed, both spin-up and spin-down line-width functions will be enhanced with increasing

*t*

_{ LR }, resulting in increased total current as shown in Figure 4(a). For the antiparallel case (

*φ*=

*π*), the current magnitude will also be enhanced for the same reason. But the current spin polarization is irrelevant to the tunnel process through the bridge since

*ρ*

_{ L↑}=

*ρ*

_{ R↓}=

*ρ*

_{0}(1 +

*P*

_{ β }) and

*ρ*

_{ L↓}=

*ρ*

_{ R↑}=

*ρ*

_{0}(1 -

*P*

_{ β }). The inter-lead line-width functions of both spin components are equal . The current spin polarization is mainly determined by the transport process through the QD. From the above discussion we also know that the current magnitude of the parallel configuration through the bridge is larger than that of the antiparallel alignment. With the increase of

*t*

_{ LR }, current through the bridge play a dominant role as compared with that through the dot, and the reverse spin valve effect may emerge accordingly. For the case of 0 <

*φ < π*, the behavior of the current can also be understood with the help of the above discussions. Due to the reverse spin valve effect, the TMR in Figure 4(c) is reduced with increasing

*t*

_{ LR }, and becomes negative for high enough inter-lead coupling strength.

## Conclusion

We have studied the characteristics of tunneling current, current spin polarization and TMR in a quantum-dot-ring with noncollinearly polarized magnetic leads. It is found that the characteristics of these quantities can be well tuned by the relative angle between the leads' magnetic moments. Especially in the Coulomb blockade and strong inter-lead coupling strength range, the currents of larger angles are larger than those of smaller ones. This phenomenon is quite different from the usual spin-valve effect, of which the current is monotonously suppressed by the increase of the angle. The TMR in this range can be suppressed even to negative, and the current spin polarizations of parallel and antiparallel configurations individually approach to unit and zero, which can then serve as a effective spin filter even for usual ferromagnetic leads with 0 < *P*
_{
β
} < 1.

## Declarations

### Acknowledgements

This work was supported by the Education Department of Liaoning Province under Grants No. 2009A031 and 2009R01. Chi acknowledge support from SKLSM under Grant No. CHJG200901.

## Authors’ Affiliations

## References

- Prinz GA: Magnetoelectronics.
*Science*1998, 282: 1660. 10.1126/science.282.5394.1660View ArticleGoogle Scholar - Wolf SA, Awschalom DD, Buhrman RA, Daughton JM, von Molnér S, Roukes ML, Chtchelka-nova AY, Treger DM: Spintronics: A Spin-Based Electronics Vision for the Future.
*Science*2001, 294: 1488. 10.1126/science.1065389View ArticleGoogle Scholar - Jacak L, Hawrylak P, Wójs A:
*Quantum dots*. New York: Springer-Verlag; 1998.View ArticleGoogle Scholar - Hamaya K, Masubuchi S, Kawamura M, Machida T, Jung M, Shibata K, Hirakawa K, Taniyama T, Ishida S, Arakawa Y: Spin transport through a single self-assembled InAs quantum dot with ferromagnetic leads.
*Appl Phys Lett*2007, 90: 053108. 10.1063/1.2435957View ArticleGoogle Scholar - Hamaya K, Kitabatake M, Shibata K, Jung M, Kawamura M, Machida T, Ishida S, Arakawa Y: Electric-field control of tunneling magnetoresistance effect in a Ni/InAs/Ni quantum-dot spin valve.
*Appl Phys Lett*2007, 91: 022107. 10.1063/1.2759264View ArticleGoogle Scholar - Hamaya K, Kitabatake M, Shibata K, Jung M, Kawamura M, Ishida S, Taniyama T, Hirakawa K, Arakawa Y, Machida T: Oscillatory changes in the tunneling magnetoresistance effect in semiconductor quantum-dot spin valves.
*Phys Rev B*2008, 77: 081302(R). 10.1103/PhysRevB.77.081302View ArticleGoogle Scholar - Stefański P: Tunneling magnetoresistance anomalies in a Coulomb blockaded quantum dot.
*Phys Rev B*2009, 79: 085312.View ArticleGoogle Scholar - Bulka BR: Current and power spectrum in a magnetic tunnel device with an atomic-size spacer.
*Phys Rev B*2000, 62: 1186. 10.1103/PhysRevB.62.1186View ArticleGoogle Scholar - Rudziński W, Barnaś J: Tunnel magnetoresistance in ferromagnetic junctions: Tunneling through a single discrete level.
*Phys Rev B*2001, 64: 085318.View ArticleGoogle Scholar - Cottet A, Belzig W, Bruder C: Positive Cross Correlations in a Three-Terminal Quantum Dot with Ferromagnetic Contacts.
*Phys Rev Lett*2004, 92: 206801. 10.1103/PhysRevLett.92.206801View ArticleGoogle Scholar - Weymann I, König J, Martinek J, Barnaś J, Schön G: Metallic Si(111)-(7 × 7)-reconstruction: A surface close to a Mott-Hubbard metal-insulator transition.
*Phys Rev B*2005, 72: 115314. 10.1103/PhysRevB.72.115314View ArticleGoogle Scholar - Misiorny M, Weymann I, Barnaś J: Spin effects in transport through single-molecule magnets in the sequential and cotunneling regimes.
*Phys Rev B*2009, 79: 224420. 10.1103/PhysRevB.79.224420View ArticleGoogle Scholar - Weymann I, Barnaś J: Kondo effect in a quantum dot coupled to ferromagnetic leads and side-coupled to a nonmagnetic reservoir.
*Phys Rev B*2010, 81: 035331. 10.1103/PhysRevB.81.035331View ArticleGoogle Scholar - König J, Martinek J: Interaction-Driven Spin Precession in Quantum-Dot Spin Valves.
*Phys Rev Lett*2003, 90: 166602.View ArticleGoogle Scholar - Braun M, König J, Martinek J: Theory of transport through quantum-dot spin valves in the weak-coupling regime.
*Phys Rev B*2004, 70: 195345. 10.1103/PhysRevB.70.195345View ArticleGoogle Scholar - Rudziński W, Barnaś J, Świrkowicz R, Wilczyński M: Spin effects in electron tunneling through a quantum dot coupled to noncollinearly polarized ferromagnetic leads.
*Phys Rev B*2005, 71: 205307.View ArticleGoogle Scholar - Li SS, Xia JB: Spin-orbit splitting of a hydrogenic donor impurity in GaAs/GaAlAs quantum wells.
*Appl Phys Lett*2008, 92: 022102. 10.1063/1.2833692View ArticleGoogle Scholar - Li SS, Xia JB: Electronic structures of N quantum dot molecule.
*Appl Phys Lett*2007, 91: 092119. 10.1063/1.2778541View ArticleGoogle Scholar - Sun QF, Wang J, Guo H: Quantum transport theory for nanostructures with Rashba spin-orbital interaction.
*Phys Rev B*2005, 71: 165310. 10.1103/PhysRevB.71.165310View ArticleGoogle Scholar - Sun QF, Xie XC: Spontaneous spin-polarized current in a nonuniform Rashba inter-action system.
*Phys Rev B*2005, 71: 155321. 10.1103/PhysRevB.71.155321View ArticleGoogle Scholar - Chi F, Li SS: Spin-polarized transport through an Aharonov-Bohm interferometer with Rashba spin-orbit interaction.
*J Appl Phys*2006, 100: 113703. 10.1063/1.2365379View ArticleGoogle Scholar - Chi F, Yuan XQ, Zheng J: Double Rashba Quantum Dots Ring as a Spin Filter.
*Nanoscale Res Lett*2008, 3: 343. 10.1007/s11671-008-9163-zView ArticleGoogle Scholar - Chi F, Zheng J: Spin separation via a three-terminal AharonovCBohm interferometers.
*Appl Phys Lett*2008, 92: 062106. 10.1063/1.2857471View ArticleGoogle Scholar - Xing YX, Sun QF, Wang J: Nature of spin Hall effect in a finite ballistic two-dimensional system with Rashba and Dresselhaus spin-orbit interaction.
*Phys Rev B*2006, 73: 205339. 10.1103/PhysRevB.73.205339View ArticleGoogle Scholar - Xing YX, Sun QF, Wang J: Symmetry and transport property of spin current induced spin-Hall effect.
*Phys Rev B*2007, 75: 075324. 10.1103/PhysRevB.75.075324View ArticleGoogle Scholar - Xing YX, Sun QF, Wang J: Influence of dephasing on the quantum Hall effect and the spin Hall effect.
*Phys Rev B*2008, 77: 115346. 10.1103/PhysRevB.77.115346View ArticleGoogle Scholar - Lü HF, Guo Y: Pumped pure spin current and shot noise spectra in a two-level Rashba dot.
*Appl Phys Lett*2008, 92: 062109.View ArticleGoogle Scholar - Chi F, Zheng J, Sun LL: Spin-polarized current and spin accumulation in a three-terminal two quantum dots ring.
*Appl Phys Lett*2008, 92: 172104. 10.1063/1.2918843View ArticleGoogle Scholar - Chi F, Zheng J, Sun LL: Spin accumulation and pure spin current in a three-terminal quantum dot ring with Rashba spin-orbit effect.
*J Appl Phys*2008, 104: 043707. 10.1063/1.2973339View ArticleGoogle Scholar - Sun QF, Xie XC: Bias-controllable intrinsic spin polarization in a quantum dot: Proposed scheme based on spin-orbit interaction.
*Phys Rev B*2006, 73: 235301. 10.1103/PhysRevB.73.235301View ArticleGoogle Scholar - Trocha P, Barnaś J: Quantum interference and Coulomb correlation effects in spin-polarized transport through two coupled quantum dots.
*Phys Rev B*2007, 76: 165432. 10.1103/PhysRevB.76.165432View ArticleGoogle Scholar - Weymann I: Effects of different geometries on the conductance, shot noise, and tunnel magnetoresistance of double quantum dots.
*Phys Rev B*2008, 78: 045310. 10.1103/PhysRevB.78.045310View ArticleGoogle Scholar - Chi F, Zeng H, Yuan XQ: Flux-dependent tunnel magnetoresistance in parallel-coupled double quantum dots.
*Superlatt Microstruct*2009, 46: 523. 10.1016/j.spmi.2009.04.002View ArticleGoogle Scholar - Trocha P, Weymann I, Barnaś J: Negative tunnel magnetoresistance and differential conductance in transport through double quantum dots.
*Phys Rev B*2009, 80: 165333. 10.1103/PhysRevB.80.165333View ArticleGoogle Scholar - Chen JC, Chang AM, Melloch MR: Transition between Quantum States in a Parallel-Coupled Double Quantum Dot.
*Phys Rev Lett*2004, 92: 176801. 10.1103/PhysRevLett.92.176801View ArticleGoogle Scholar - Wang ZhM:
*Self-Assembled Quantum Dots*. New York: Springer; 2008.View ArticleGoogle Scholar - Wang ZhM, Holmes K, Mazur YI, Ramsey KA, Salamo GJ: Self-organization of quantum-dot pairs by high-temperature droplet epitaxy.
*Nanoscale Res Lett*2006, 1: 57. 10.1007/s11671-006-9002-zView ArticleGoogle Scholar - Strom NW, Wang ZhM, Lee JH, AbuWaar ZY, Mazur YuI, Salamo GJ: Self-assembled InAs quantum dot formation on GaAs ring-like nanostructure templates.
*Nanoscale Res Lett*2007, 2: 112. 10.1007/s11671-007-9040-1View ArticleGoogle Scholar - Lee JH, Wang ZhM, Strm NW, Mazur YI, Salamo GJ: InGaAs quantum dot molecules around self-assembled GaAs nanomound templates.
*Appl Phys Lett*2006, 89: 202101. 10.1063/1.2388049View ArticleGoogle Scholar - Hankea M, Schmidbauer M, Grigoriev D, Stäfer P, Köhler R, Metzger TH, Wang ZhM, Mazur YuI, Jalamo G: Zero-strain GaAs quantum dot molecules as investigated by x-ray diffuse scattering.
*Appl Phys Lett*2006, 89: 053116. 10.1063/1.2240114View ArticleGoogle Scholar - Hanson R, Kouwenhoven LP, Petta JR, Tarucha S, Vandersypen LMK: Spins in few-electron quantum dots.
*Rev Mod Phys*2007, 79: 1217. 10.1103/RevModPhys.79.1217View ArticleGoogle Scholar - Li SS, Abliz A, Yang FH, Niu ZC, Feng SL, Xia JB: Electron and hole transport through quantum dots.
*J Appl Phys*2002, 92: 6662. 10.1063/1.1518754View ArticleGoogle Scholar - Li SS, Abliz A, Yang FH, Niu ZC, Feng SL, Xia JB: Electron transport through coupled quantum dots.
*J Appl Phys*2003, 94: 5402. 10.1063/1.1609648View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.