Bound states in the continuum and spin filter in quantum-dot molecules

Physica B 455 (2014) 66–70

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Physica B journal homepage: www.elsevier.com/locate/physb

Bound states in the continuum and spin filter in quantum-dot molecules J.P. Ramos a, P.A. Orellana b,n a b

Departamento de Física, Universidad Católica del Norte, Casilla 1280, Antofagasta, Chile Departamento de Física, Universidad Técnica Federico Santa María, Vicuña Mackenna 3939, Santiago, Chile

art ic l e i nf o

a b s t r a c t

Available online 30 July 2014

In this paper we study the formation of bound states in the continuum in a quantum dot molecule coupled to leads and their potential application in spintronics. Based on the combination of bound states in the continuum and Fano effect, we propose a new design of a spin-dependent polarizer. By lifting the spin degeneracy of the carriers in the quantum dots by means of a magnetic field the system can be used as a spin-polarized device. A detailed analysis of the spin-dependent conductance and spin polarization as a function of the applied magnetic field and gate voltages is carried out. & 2014 Elsevier B.V. All rights reserved.

Keywords: Quantum dot Spin filter Bound states in the continuum

1. Introduction Spintronics is a wide field of research that exploits the spin degree of freedom of the electrons [1]. In spintronics, the precise control of spin currents leads to new functionalities and higher efficiencies than conventional electronics. In the last few years, there have been a renewed interest in the design and use of nanostructures as spintronic devices in which the spin degree of freedom is used as the basis for their operation [2–6]. In order to generate and detect spin polarized currents, these devices exploit the natural connection between the conductance and the quantum-mechanical transmission. The quantum dots are suitable systems to be used as spintronic devices. They are artificial nanostructures in which electrons can be confined even in all space dimensions [7,8], resulting in charge and energy quantization. Both features present in real atomic systems, so useful analogies between real and artificial atomic systems have been exploited recently. By enforcing this analogy, effects like Fano [9] and Dicke [10] have also been found to be present in quantum dot configurations. In this context, Song et al. [4] described how a spin filter may be achieved in open quantum dot systems by exploiting the Fano resonances occurring in their transmission characteristics. In a quantum dot in which the spin degeneracy of a carrier has been lifted, they showed that the Fano effect might be used as an effective means to generate spin polarization of transmitted carriers. The electrical detection of the resulting spin polarization should then be possible.

n

Corresponding author. Tel.: þ 56224326751. E-mail address: [email protected] (P.A. Orellana).

http://dx.doi.org/10.1016/j.physb.2014.07.047 0921-4526/& 2014 Elsevier B.V. All rights reserved.

In a previous work [11], we have shown that in a side-coupled double quantum dot system the transmission shows a large peak-tovalley ratio. The difference in energy between the resonances and antiresonances can be controlled by adjusting the difference between the energy levels of the two quantum dots by gate voltages. The above behavior in the transmission is a result of the quantum interference of electrons through the two different resonant states of the quantum dots, coupled to common leads. This phenomenon is analogous to the Dicke effect in quantum optics, which takes place in the spontaneous emission of two closely lying atoms radiating a photon into the same environment. In the electronic case, however, the decay rates (level broadening) are produced by the indirect coupling of QDs through leads [12,13], giving rise to a fast (superradiant) and a slow (subradiant) mode. Also, Vallejo et al. [14] have shown that the coexistence of a bound state in continuum (BIC) and Fano antiresonance can be exploited to result in polarizations close to 100% in wide regions in the space of used parameters. The existence of BICs was predicted at the dawn of quantum mechanics by von Neumann and Wigner [15] for certain spatially oscillating attractive potentials, for a one-particle Schödinger equation. Much later, Stillinger and Herrick [16] generalized von Neumann's work by analyzing a two-electron problem, they found that BICs were formed despite the interaction between electrons. BICs have also shown to be present in the electronic transport in meso- and nanostructures [17–21]. Several mechanisms of formation of BICs in open quantum dots (QDs) have been reported in the literature. The simplest one is based on the symmetry of the systems and, as a consequence, in the difference of parity between the QD eigenstates and the continuum spectrum [22]. Recently, González-Santander et al. extended the notion of BICs to the domain of time-dependent potentials [23].

J.P. Ramos, P.A. Orellana / Physica B 455 (2014) 66–70

67

In the present work, we study the formation of BICs in a system formed by a quantum dot coupled to two leads with a side attached molecule and their possible application in spintronics. We show that the combination of BICs and Fano effect can be used to design an efficient spin filter. By tuning the positions of BICs and Fano antiresonances with a magnetic field and gate voltages, this system can be used as an efficient spin filter even for small values of magnetic fields.

2. Model The system is formed by a quantum dot coupled to two leads with a side attached molecule, as is shown schematically in Fig. 1 (upper panel). The full system is modeled by a generalized non-interacting Anderson Hamiltonian, namely H ¼ H l þH m þ H in , where H l represents the Hamiltonian of the leads, H m represents the Hamiltonian of the embedded quantum dot with a side attached quantum dot molecule and H in represents the interaction between the leads and the embedded quantum dot. We consider each quantum dot with a single energy level εi;σ ði ¼ 1; …; 4Þ. Then, Hl ¼

εkα c†kα ;σ ckα ;σ ;

∑

ð1Þ

σ ;α ¼ L;R 3

†

∑ εi;σ di;σ di;σ  t

Hm ¼

i ¼ 0;σ

†

∑ ðd1;σ dj;σ þ h:c:Þ

j ¼ 2;3;σ

†

 ∑νðd0;σ d1;σ þ h:c:Þ; σ

H in ¼

∑

σ ;α ¼ L;R

†

ðV α d0;σ ckα ;σ þh:c:Þ;

t2 t2  ε  ε2;σ ε  ε3;σ

:

ð5Þ

By using Dyson's equation G ¼ g 0 þ g 0 H in G, where G is Green's function of the system and g0 is Green's function of the effective quantum dot, Green's function of the effective quantum dot coupled to two leads is obtained 1 ; G0;σ ðεÞ ¼ ε  ε~ 0;σ  Σ l ðεÞ where Σ l ðεÞ is the self-energy due to leads.

ðε  ε~ 0;σ Þ2 þ Γ

2

;

ð7Þ

where Γ ¼  iΣ l , is the effective coupling between the effective quantum dot with the leads. The linear conductance G is obtained by Landauer's formula at zero temperature, Gσ ðεf Þ ¼ ðe2 =hÞτσ ðε ¼ εf Þ [25]. Then, the linear conductance of the system is given by e2 Γ : h ðεf  ε0;σ  Σ m ðεf ÞÞ2 þ Γ 2 2

The resonances in the linear conductance are obtained by the equation

where ε0;σ  ε0 þ εz σ zσσ , and Σ m;σ ðεÞ is the self-energy due to the presence of the side attached molecule given by

ε  ε1;σ 

Γ2

ð3Þ

ð4Þ

ν2

τσ ðεÞ ¼

G σ ðεf Þ ¼

†

Σ m;σ ðεÞ ¼

Once we obtain Green's function G0;σ , we can evaluate the spin dependent transmission probability τσ by using the Fisher–Lee relation [24]

ð2Þ

where di;σ ðdi;σ Þ is the electron creation (annihilation) operator in the ith QD with spin σ ðσ ¼ ↓; ↑Þ, the c†kα ðckα Þ is the electron creation (annihilation) operator in the lead α (α ¼L,R) with momentum k and spin σ; while V α is the tunneling coupling between the α lead and the embedded quantum dot. Besides, ν is the coupling between the embedded quantum dot and the quantum dot molecule and t is the interdot coupling in the side attached quantum dot molecule. We consider a system in equilibrium, at zero temperature with εi;σ ¼ εi þ σεz (i ¼0,1,2,3) where εi is the energy level of the isolated i quantum dot, εz ¼ g μB B is the Zeeman energy due to the presence of the magnetic field B, with g being the electron Landé factor and μB being the Bohr magneton. By using a re-normalization process based on the Dyson equation for Green's functions (see the appendix), we obtained an effective model for the artificial molecule. Fig. 1 (lower panel) displays the effective model which consists in a single quantum dot coupled to two leads with an effective energy ε~ 0;σ ,

ε~ 0;σ ¼ ε0;σ þ Σ m;σ ðεÞ;

Fig. 1. Upper panel: quantum dot coupled to two leads with a side attached molecule. Lower panel: effective model, derived by a decimation process.

εf  ε0;σ  Σ m;σ ðεf Þ ¼ 0;

ð8Þ

ð9Þ

while the zeros of the conductance coincide with the poles of the self-energy Σ m;σ ðεf Þ, which correspond to the spectrum of the side attached molecule. In order to study the BICs, we setting the energy levels of the side attached molecule as, ε1;σ ¼ ε0;σ , ε2;σ ¼ ε0;σ þ Δ, ε3;σ ¼ ε0;σ  Δ. On one hand, the resonances are obtained from the eigenvalues of Eq. (2) qffiffiffi 2 2 2 2 2ðε þ ð10aÞ ξ; 7 ;σ  ε0;σ Þ ¼ Δ þ ν þ 2t  2 2 2 2ðε  7 ;σ  ε0;σ Þ ¼ Δ þ ν þ 2t þ 2

qffiffiffi ξ;

ð10bÞ

where ξ ¼ ðΔ  ν Þ þ 4t 2 ðΔ þ ν2 Þ þ 4t 4 . On the other hand, the antiresonances are obtained from the spectrum of the side attached molecule, which is given by 2

2 2

2

ω1;σ ¼ ε0;σ ;

ð11aÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ω2;σ ¼ ε0;σ þ 2t 2 þ Δ2 ;

ð11bÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω3;σ ¼ ε0;σ  2t 2 þ Δ2 :

ð11cÞ

We can rewritten the self-energy Σ m;σ ðεÞ in terms of the spectrum of the side attached molecule as following:

Σ m;σ ðεÞ ¼ Σ m1;σ ðεÞ þ Σ m2;σ ðεÞ þ Σ m3;σ ðεÞ; Σ mα;σ ðεÞ ¼

ð6Þ

ν21 ¼

ν2α ðα ¼ 1; 2; 3Þ; ε  ωα ; σ

Δ2 ν2 ; Δ2 þ 2t 2

ð12aÞ ð12bÞ

ð12cÞ

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J.P. Ramos, P.A. Orellana / Physica B 455 (2014) 66–70

ν22 ¼ ν23 ¼

t 2 ν2

Δ þ 2t 2

2

:

ð12dÞ

In order to quantify the degree of spin polarization, we introduce the weighted spin polarization as [4] Pσ ¼

jτ↑  τ↓ j τσ : jτ↑ þ τ↓ j

ð13Þ

Notice that this definition takes into account not only the relative fraction of one of the spins, but also the contribution of those spins to the electric current. In other words, we will require that not only the first term of the right-hand side of Eq. (13) to be of the order of unity, but also the transmission probability τσ . 3. Results In what follows all energies will be given in units of the parameter Γ and we setting t ¼ ν ¼ Γ =2 and ε0 ¼ 0. In first place, we show results for the linear conductance in the absence of magnetic field. Fig. 2 displays the linear conductance versus the Fermi energy for Δ ¼ 0 (solid black line), Δ ¼ 0:1Γ (dotted red line) and Δ ¼ Γ (dashed blue line). For Δ ¼ 0 the conductance shows two Fano antiresonances and three resonances. The Fano antiresonances arise from the destructive interference between the continuum state of the embedded quantum dot and the localized states in the side-attached molecule [26]. From Eqs. (12a–c) we can see that when Δ ¼0, the molecular state with energy ω1;σ ¼ ε0;σ decouples from the continuum and therefore does not contribute to the conductance (Σ m1;σ ðεÞ ¼ 0, ν1 ¼ 0). This state becomes a BIC. This state arises from the hybridization of the states of the quantum dots 2 and 3 through the quantum dot 1 and has no projection on the continuum, as it is shown schematically in Fig. 4. Besides, we see from Eq. (10) that two resonances collapse at εf ¼ 0. When Δ is turned on, this BIC becomes a quasi-BIC. Now, this state contributes with an extra-Fano antiresonance to the conductance. Besides, the conductance displays four resonances due to the lifting of the degeneracy of the resonances. For instance, for Δ ¼ 0:1Γ a sharp antiresonance appears at εf ¼ 0 due to the quasiBIC and the resonance around εf ¼ 0 splits into two resonances. This split grows as Δ increases (Δ ¼ Γ ) as we can see in Fig. 3. The separation between resonances grows and the width of the lateral Fano line shapes decreases with Δ. In fact, Fig. 5 displays as ν2α

evolves with Δ. On one hand, ν21 (solid black line) vanishes at Δ ¼0 and grows as Δ increases and saturates for Δ b Γ . On the other hand, ν22;3 (red and blue dashed lines) decreases almost quadratically with Δ and tends to zero for the unphysical limit Δ-1. In this limit, two of the molecular states become BICs. However we can see from Figs. 3 and 5, for approximately that for Δ  3Γ these two states become quasi-BICs. In the presence of a magnetic field, the spin degeneracy is broken and the conductance becomes spin dependent. Fig. 6 displays the spin dependent conductance as a function of the Fermi energy for two values of magnetic field, εz ¼ 0:1Γ (upper panel) and εz ¼ 0:05Γ (lower panel) with Δ ¼ 0:2Γ . By adjusting Δ and magnetic field, it is possible to control the position and width of the resonances and antiresonances in the conductance. This property can be used in order to produce a high spin-polarization, turning a resonance for one spin with the antiresonance for the opposite spin. In fact, Fig. 7 displays the spin-polarization for two values of the magnetic field, εz ¼ 0:1Γ (lower panel) and εz ¼ 0:05Γ (upper panel) for fixed Δ ¼ 0:1Γ . We can observe that the system shows optimal spinpolarization in a wide energy range. In order to show the robustness of the high spin-polarization under changes of the parameters of the system, Fig. 8 displays a contour plot with the weighted spin-polarization in the space of the parameters Δ and Zeeman energy εz for a fixed Fermi energy (εf ¼  0:1Γ ). This demonstrates that the spin-polarization

Fig. 3. Linear conductance as a function of the Fermi energy for Δ ¼ 2Γ (solid black line), Δ ¼ 3 Γ (dotted red line) and Δ ¼ 4Γ (dashed blue line), in the absence of magnetic field. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 2. Linear conductance as a function of the Fermi energy for Δ ¼ 0 (solid black line), Δ ¼0.1 Γ (dotted red line) and Δ ¼ Γ (dashed blue line), in the absence of magnetic field. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 4. Schematic view of the molecular states with the embedded quantum dot, for Δ a 0 (upper panel) and Δ ¼ 0 (lower panel).

J.P. Ramos, P.A. Orellana / Physica B 455 (2014) 66–70

Fig. 5. Square effective coupling ðν2α Þ as a function of Δ. ν21 (solid black line) and ν22;3 (red and blue dashed lines). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

69

Fig. 7. Weighted spin-polarization for σ ¼ ↓ (dashed red line) and for σ ¼ ↑ (solid black line), for Δ ¼ 0:1Γ and Zeeman energy εz ¼ 0:1Γ (lower panel) and εz ¼ 0:05Γ (upper panel). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 8. Weighted polarization at the energy εf ¼  0:1Γ in the space of the parameters Δ and B.

the inter-dot coupling and the on-site Coulomb interaction, different regimes arise. For t=U≫1, the resonances and antiresonances would split into two, separated by the on-site Coulomb energy, while for t=U≪1, the resonances and antiresonances would occur in pairs. Moreover, it was demonstrated that the Fano effect is robust against e–e interaction even in the Kondo regime [29,30]. Besides, recently Žitko et al. [31] found in parallel double- quantum-dot systems that the BICs are robust even if electron–electron interaction is taken into account. Fig. 6. Spin dependent conductance as a function of the Fermi energy, σ ¼ ↓ (dashed red line) and for σ ¼ ↑ (solid black line). The upper panel corresponds to εz ¼ 0:1Γ and the lower panel to εz ¼ 0:05Γ. Here we have same considerations than (3), but Δ ¼ 0:35Γ. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

filtering capabilities of our device can be made optimum (  100%) within a wide range of parameters. We expect that the above results remain valid even if the electron–electron interaction is taken into account. In fact, in embedded coupled QD, the main effect of the electron–electron interaction is to shift and to split the resonance positions [27,28]. This occurs because the on-site Coulomb repulsion energy U introduces a renormalization of the site energies. In analogy with coupled QD, we expect that, depending on the relation between

4. Summary In summary, we have studied the formation of bound states in the continuum in a side attached quantum dot molecule dot. We show that the bound states in the continuum are formed by the internal interference in the quantum dot molecule. Besides we propose a novel spin filter device based on the combination of bound states in the continuum and Fano effect. Assuming perfectly coherent transport, we have demonstrated that the spin-polarization filtering capabilities of our device can be made optimum ( 100%) within a wide range of parameters (Fermi energy and applied magnetic field). It is important to emphasize that the proposed system can exhibit high spin polarization even for very weak magnetic fields.

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Appendix A

References

In this section, we show the re-normalization process used to obtain the effective model shown in Fig. 1 (lower panel). By using Dyson's equations for Green's functions we can obtain a set of equation for the model indicated in Fig. 1. In the first place, by decimating the external quantum dots (2 and 3) into the central quantum dot (1), we obtain the following equations for Green's functions,

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

G11 ¼ g 1 þ g 1 tG21 þ g 1 tG31

ð14aÞ

G21 ¼ g 2 tG11

ð14bÞ

G31 ¼ g 3 tG11 ;

ð14cÞ

so we have G11 G11 ¼

1  g~ 1 ; t2 t2 ε  ε1   ε  ε2 ε  ε3

ð15Þ

we note that

ε~ 1  ε1  t 2 ½ðε  ε2 Þ  1 þ ðε  ε3 Þ  1 ;

ð16Þ

and

Σ m ðεÞ ¼

ν : ε  ε~ 1 2

ð17Þ

Now we decimate the normalized lateral part with the embedded QD in a continuum (QD0), so we have

[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

G00 ¼ g 0 þ g 0 νG~ 10

ð18aÞ

[30]

G~ 10 ¼ g~ 1 νG00 :

ð18bÞ

[31]

F. Pulizzi, Spintronics, Nat. Mater. 11 (2012) 367. S. Datta, B. Das, Appl. Phys. Lett. 56 (1990) 665. J.A. Folk, R.M. Potok, C.M. Marcus, V. Umansky, Science 299 (2003) 679. J.F. Song, Y. Ochiai, J.P. Bird, Appl. Phys. Lett. 82 (2003) 4561. F. Mireles, G. Kirczenow, Phys. Rev. B 64 (2001) 024426. F. Mireles, S.E. Ulloa, F. Rojas, E. Cota, Appl. Phys. Lett. 88 (2006) 093118. L. Jacak, P. Hawrylak, A. Wójs, Quantum Dots, Springer, Berlin, 1998. T. Chakraborty, Quantum Dots, Elsevier Science, North-Holland, 1999. U. Fano, Phys. Rev. 124 (1961) 1866. R.H. Dicke, Phys. Rev. 89 (1953) 472473. P.A. Orellana, F. Domínguez-Adame, Phys. Status Solidi A 203 (2005) 1178. P.A. Orellana, M.L. Ladrón de Guevara, F. Claro, Phys. Rev. B 70 (2004) 233315. P. Trocha, J. Barnas, J. Nanosci. Nanotechnol. 10 (2010) 2489. M.L. Vallejo, M.L. Ladrón de Guevara, P.A. Orellana, Phys. Lett. A 374 (2010) 4028. J. von Neumann, E. Wigner, Phys. Z 30 (1929) 465. F.H. Stillinger, D.R. Herrick, Phys. Rev. A 11 (1975) 446. R.L. Schul, H.W. Wyld, D.G. Ravenhall, Phys. Rev. B 41 (1990) 12760. J.U. Nöckel, Phys. Rev. B 46 (1992) 15348. I. Rotter, A.F. Sadreev, Phys. Rev. E 71 (2005) 046204. M.L. Ladrón de Guevara, P.A. Orellana, Phys. Rev. B 71 (2006) 205303. J.W. González, M. Pacheco, L. Rosales, P.A. Orellana, Europhys. Lett. 91 (2010) 66001. C.J. Texier, Phys. A: Math. Gen. 35 (2002) 3389. C. González-Santander, P.A. Orellana, F. Domínguez-Adame, Europhys. Lett. 102 (2013) 17012. D.S. Fisher, P.A. Lee, Phys. Rev. B 23 (1981) 12. R. Landauer, J. Phys.: Condens. Matter 1 (1989) 8099. K. Kobayashi, H. Aikawa, A. Sano, S. Katsumoto, Y. Iye, Phys. Rev. B 70 (2004) 035319. T. Aono, K. Takahashi, New J. Phys. 9 (2007) 114. H. Lu, R. Lü, B.-F. Zhu, Phys. Rev. B 71 (2005) 235320. A.C. Johnson, C.M. Marcus, M.P. Hanson, A.C. Gossard, Phys. Rev. Lett. 93 (2004) 06803. M. Sato, H. Aikawa, K. Kobayashi, S. Katsumoto, Y. Iye, Phys. Rev. Lett. 95 (2005) 066801. R. Žitko, J. Mravlje, K. Haule, Phys. Rev. Lett. 108 (2012) 066602.