Effect of spin-flip scattering on the electron transport through double quantum dots

Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Effect of spin-flip scattering on the electron transport through double quantum dots Fu-Bin Yang a,b,n, Rui Huang c, Yan Cheng d,nn a

Department of Physics, College of Computer, Civil Aviation Flight University of China, Guanghan 618307, China Department of Mathematics, Sichuan Normal University Chengdu College, Chengdu 611745, China c College of Sciences, Southwest Petroleum University, Chengdu 610500, China d Institute of Atomic and Molecular Physics, College of Physical Science and Technology, Sichuan University, Chengdu 610065, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 19 January 2015 Received in revised form 8 February 2015 Accepted 9 February 2015

We systematically investigate the electron transport through double quantum dots (DQD) with particular emphasis on the spin-flip scattering of an electron in the DQD. By means of the slave-boson mean-field approximation, we calculate the linear conductance and the transmission in the Kondo regime at zero temperature. The obtained results show that both the linear conductance and transmission probability are quite sensitive to the spin-flip strength when the DQD structure is changed among the serial, parallel and T-shaped. It is suggested that such a theoretical model can be used to study the physical phenomenon related to the spin manipulation transport. & 2015 Elsevier B.V. All rights reserved.

Keywords: Quantum transport Kondo effect Spin-flip Quantum dot

1. Introduction Recently, many efforts have been made to investigate theoretically coupled quantum dots (QDs) [1,2]. With respect to coupled QDs, the wave functions of each quantum dot state can penetrate into the tunneling barrier and overlap each other. As well as being a model for studying the physics of strongly correlated electrons, the coupled QDs are utilized in controllable quantum coherent systems for spintronic [3] and quantum information processing devices [4]. In real spin systems, it is difficult to manipulate the spin-up state and the spin-down state individually. In contrast, since the coupled DQD system is separated, it is much easier to control the spin freedom of each dot. As a result, the physical phenomenon related to the spin-flip which is induced by the spin– orbit electrons have suggested that it can potentially offer a gate controllable approach to manipulate the spin [5,6]. Researchers have detected the spin-flip process in a single proton, a first step toward precision measurements of the antiproton's spin magnetic moment [7]. The observed photoelectron spin-flip process in Bi2Se3 also enables a precision measurement of the spin detection [8]. All of which indicate the importance of the intrinsic spin-flip effect in the mesecopical systems. n

Corresponding author at: Department of Physics, College of Computer, Civil Aviation Flight University of China, Guanghan 618307, China. nn Corresponding author. E-mail addresses: [email protected] (F.-B. Yang), [email protected] (Y. Cheng).

Due to that the electrons in the QD can flip its spin, a very sharp Kondo peak emerges in the density of states (DOS) of the QD. The parameters in the coupled QDs can be modulated experimentally in a continuous and reproducible manner, offering an appropriate platform to study the Kondo problem [9]. In particular, the observation of the Kondo effect in strongly correlated systems has opened a path for the investigation of the spin-flip effect, which stimulates further experimental [10] and theoretical studies [11]. When electron–electron correlations due to the Kondo effect are affected by such spinflip effect, the transport properties exhibit remarkable properties [12]. There have been a number of theoretical works in the DQD systems. For example, it was pointed out that the spin-flip shows a splitting effect on the Kondo resonance in the single quantum dot system [13], whereas such effects do not appear in the DQD system. The T-shaped DQD is another prototype for the special arrangement of the DQD which provides an additional path of electron propagation, but the detailed analysis about the spin-flip effect has not been done. Thus it is interesting to systematically study how the spin-flip interference together with the Kondo effect affects characteristic transport properties in a variety of DQD systems, including the serial DQD, parallel DQD, and T-shaped DQD systems.

2. The model and method In this work, we mainly focus on the effect of spin-flip scattering on the electron transport through a DQD system, which is

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F.-B. Yang et al. / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

HDQD =

∑ εmσ f m+σ fmσ

+

Γ

Γ

m, σ

tc ∑ (f + b1b2+f2σ + h. c. ) N σ 1σ

+

r ∑ bm bm+ (f m+σ fmσ¯ + h. c. ) N m, σ

+

∑ λ m ⎜⎜∑ f m+σ fmσ

⎛

Γ

Γ

Γ =Γ

Γ

Γ

m

⎝

σ

⎞ + bm+ bm − 1⎟⎟ ⎠

(2)

The last term with the Lagrange multiplier λ m is introduced so as to incorporate the constraint imposed on the slave + particles∑σ =↑↓ f m+σ fmσ + bm bm = 1. In the numerical calculations, we replace the boson operator by their expectation values + + bm (bm ) → < bm > = b˜m , which results in the renormalized quan2 ˜ tities Vmασ = Vmασ b˜m ,t˜c = tc b˜1b˜ 2, r˜m = rb˜ and ε˜mσ = εmσ + λ m in the m

Fig. 1. A schematic illustration of DQD system connected by the interdot tunneling α strength tc. Γmm (α = L, R; m = 1, 2) represents the resonance width due to transfer between the m-th dot and the α-th lead. By changing the ratio of tunneling amplitudes, we can change the system among the serial DQD (x ¼0), parallel DQD and T-shaped DQD (y¼ 0).

shown schematically in Fig.1. The Hamiltonian of the original d levels of the two dots can be described as follows:

H = HDQD + Hα + HT ,

(1)

where

HDQD =

∑

εmσ dm+σ dmσ

m, σ

+ =

r N

+ Undmσ ndmσ¯

t + c N

∑

(d1+σ d2σ

+ h. c . )

σ

∑ (dm+σ dmσ¯ + h. c. )Hα

∑ εkασ ck+ασckασHT 1 N

2 b˜ m − i∑ σ

1 < ∫ 4dπε Gmm , σ (ε) = 2

2 λ m b˜ m = i∑ σ

(3)

< ∫ 4dπε (ε − ε˜mσ ) Gmm , σ (ε)

(4)

Eq. (3) represents the constraint imposed on the slave particles, while Eq. (4) is obtained from the stationary condition that the boson field is time independent at the mean-field level. From the equation of motion of the operator fmσ [15], we have the explicit matrix form of the Green function:

m, σ

kα, σ

=

slave-boson mean-field approximation. The mean-field values of b˜m , λ m are determined by minimization of the free energy due to the Hamiltonian of the system. We can derive the set of selfconsistent equations according to the equation of motion method for the nonequilibrium Keldysh Green functions [15]:

∑

(Vmασck+ασdm+σ + h. c . )

k α , m, σ

+ Here HDQD is the Hamiltonian of the DQD and dm σ (dmσ ) is the creation (annihilation) operator of the electron in the QDs with m ¼1,2. tc describes the interdot coupling strength and r is the spin-flip strength that may cause the spin rotation of an electron in the QDs. Since the spin quantization axes in the electrodes are fixed by the internal magnetization of the magnets, an electron is in a superposition of spin-up (spin-down) states as it tunnels into (out of) the dot. As a result, the physical phenomenon related to the spin-flip may be realized in a DQD system. Hα describes the non-interacting leads with ck+ασ (ckασ ) the creation (annihilation) operator of an electron in the lead. HT denotes the tunneling between the DQD and the lead α (α = L, R). The tunneling between the lead and the QDs can be rewritten by an effective strength α ⁎ Γmm σ = π ∑k, α, σ Vmασ Vmασ δ (ε − εkασ ) . We first interpolate the serial L R DQD and parallel DQD by continuously changing x = Γ22 σ = Γ11σ L R while keeping Γ0 = Γ11σ = Γ22σ as the unity. For example, at x ¼0 the model is reduced to the serial DQD and x ¼1 the parallel DQD. We L R next change y = Γ22 with the resonance width σ = Γ22σ L R Γ0 = Γ11σ + Γ11σ fixed as unity. At y¼0, the system is equal to the T-shaped DQD and y¼ 1 the parallel DQD. In the following discussions, the intradot Coulomb interaction U (U- 1) on each dot is assumed to be sufficiently large, so that the double occupancy is forbidden. An alternative way to represent the infinite Coulomb interaction is the conventional slaveboson mean field approximation [14], where the creation (anni+ hilation) operator of electrons in the dots, dm σ (dmσ ) is replaced by + + dmσ → f mσ bm . bm (fmσ ) is the slave-boson (pseudo-fermion) annihilation operator for an empty (singly occupied) state. We can thus model the DQD Hamiltonian as follows:

[Gσr ]−1 = ⎛ ε − ε˜1σ + iΓ˜11σ − (t˜c − iΓ˜21σ ) ⎞ −r˜1 0 ⎜ ⎟ ˜ ˜ ˜ ⎜ − (tc − iΓ12σ ) ε − ε˜2σ + iΓ22σ ⎟ −r˜2 0 ⎜ ⎟ ε − ε˜1σ + iΓ˜11σ¯ − (t˜c − iΓ˜21σ¯ ) ⎟ −r˜1 0 ⎜ ⎜ ⎟ −r˜2 − (t˜c − iΓ˜12σ¯ ) ε − ε˜2σ¯ + iΓ˜22σ¯ ⎠ (5) 0 ⎝ ⎛∑ Γ˜ α f 11σ α ⎜ ⎜ α α ⎜∑ Γ˜12 σ fα ⎜ α r⎜ < G = Gσ ⎜ 0 ⎜ ⎜ 0 ⎜⎜ ⎝

∑ Γ˜21α σ fα

0

α

∑ Γ˜22α σ fα

0

α

0

∑ Γ˜11α σ¯ fα

0

∑ Γ˜12α σ¯ fα

α

α

⎞ ⎟ ⎟ ⎟ 0 ⎟ ⎟ Gσa α ˜ ∑ Γ21σ¯ fα ⎟ ⎟ α ⎟ α ∑ Γ˜22σ¯ fα ⎟⎟ ⎠ α 0

(6)

α α ˜ ˜L ˜R with Γ˜mnσ = b˜m b˜n Γmn σ and Γmnσ = Γmnσ + Γmnσ (n = 1, 2). By the Keldysh non-equilibrium Green's function (NEGF) method, we can further derive the Landauer current formula of this system 2e I = h ∑σ ∫ (fL (ε) − fR (ε)) Tσ (ε) dε,(6)

where fL (R) (ε) is the Fermi distribution function of the left (Right) lead and Tσ (ε) is the transmission probability per spin given by Tσ (ε) = Tr [Γ LGσr Γ RGσa ].

3. Numerical results In this section, we will present the numerical results of linear conductance and transmission probability for the DQD systems with the serial, parallel, and T-shaped geometries. For simplicity, we assume the energy levels for the spin up and spin down are α α degenerate (Γ˜mnσ = Γ˜mnσ¯ ). We deal with the symmetric dots in the

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Fig. 2. Linear conductance as a function of the spin-flip strength r and x when we change the system from serial DQD to parallel DQD.

Kondo regime with εmσ = − 3.5Γ0 and the bandwidth of the leads is taken as D = 60Γ0 . 3.1. Linear conductance when the system changes from serial DQD to parallel DQD The linear conductance as a function of spin-flip strength r for several values of x is exhibited in Fig. 2(a). It decreases as the increasing spin-flip strength when we consider the serial DQD

(x ¼0). However, it reduces to a minimum at r¼ 0.3 then ascends to a maximum at r ¼1.5 when x ¼0.5. However, it changes apparently when the system is close to the parallel DQD (x ¼0.8). The conductance first increases then decreases, which is obviously different from the former cases. Fig. 2(b) shows the linear conductance as a function of x for several values of the spin-flip strength r. Starting from the serial DQD (x¼ 0), the conductance monotonically decreases for r ¼0 [16], whereas it exhibits a minimum structure around x ¼0.3 and then a maximum around x ¼0.8 for

ε

ε/Γ0

ε/Γ0

Fig. 3. Transmission probability T (ε) for different x under different spin-flip strength.

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r ¼1, or a maximum around x ¼0.7 then decreases for r ¼2. It shows a reverse process with the spin-flip transformation from Fig. 2(b), even a suppressed effect for a large r. Analogous effect occurs when we choose different values of tc. Therefore, we see that apart from the detailed dependence, the characteristic behavior of the conductance in Fig. 2 is due to the Kondo effect modified by the spin-flip strength, which is also clearly seen in the transmission probability shown below. 3.2. Transmission probability when the system changes from serial DQD to parallel DQD We next show the explicit r-dependence transmission probability for different x. We first focus on the transmission probability in the serial DQD (x¼ 0), whereas it reverses with the increasing spin-flip strength. As shown in Fig. 3(a), the transmission probability around the Fermi energy reverses from a resonance to dip. The larger the spin-flip strength is, the lower the dip structure is. The resonance peak in the transmission finally splits into two peaks when r ¼ 2. We can clearly see the reversed DOS of the two dots with the increasing r from the inset of Fig. 3(a). Namely, the Kondo peak reverses around the Fermi energy in the presence of the intradot spin-flip scattering. For relatively small values of r (0 ≤ r ≤ 1), the DOS in the Fermi energy changes from a Kondo dip to a Kondo peak. Due to the coexistence of the Kondo state and the intradot spin-flips, the positions of the peaks in the DOS are renormalized by Kondo correlations. For large values of r (1 ≤ r ≤ 2), the appearance of a splitting occurs. All of which are consistent with the decreasing conductance in Fig. 2(a). The increase of r does not only increase the renormalized tunneling t˜c , but also enhance the renormalized energy ε˜m of the two dots. Due to the coupling between the dot levels and leads, the bonding state and antibonding state of the two dots have to be changed by the enhanced renormalization energy. One is wider, which is associated with the strongly coupled level, and the other narrower band is related to the weakly coupled level, which will also reduce the Kondo resonance in the Fermi energy and enlarge the splitting of the double peaks in the DOS. We wish to mention some similarities and differences between the present model and that in Ref. [13], whereas the original Kondo peak splits into two peaks appear at the positions of ε = ± r , respectively, and the original peak at ε = 0. For the serial DQD system, the interference effect due to the two leads does not exist because each dot is coupled with the single lead. However, some obvious differences appear when the DQD system changes from serial to parallel. The coupling between the dots and the leads increases because of the enhancement of x, which results in the obvious Fano shape in the transmission probability. As x increases, one of the resonances, which is composed of the “bonding” Kondo state of the DQD, becomes sharp around the Fermi energy, while the other “antibonding” Kondo state is broadened above the Fermi level. Such have already been discussed in Refs. [17,18], whereas the Fano peaks (dips) are caused by the interference effect between the two dots. However, the spin-flip effect makes the transmission Fano peaks (dips) reverse to Fano dips (peaks) when r≠0. For example, the transmission peak splits around the Fermi energy when x ¼0.2, and the Fano dip separates far away from the Fermi energy as the increasing r. Furthermore, such splitting is more obvious when x ¼0.8 or x ¼0.95. The reasons can be attributed to the effect of r on the renormalized energy ε˜m of the two dots. If the spin-flip process on the quantum dots is involved, the amplitude of conductance is small. This effect is caused by the correlation between the dots and the ferromagnetic electrodes. As the increasing spinflip strength, the renormalized energy first reduces then increases, which causes the Fano dip (resonance) far away from the Fermi

Fig. 4. Linear conductance as a function of the spin-flip strength r when we change the system from parallel DQD to T-shaped DQD.

energy. The obtained results are totally different from that in the serial DQD. We have also observed the linear conductance first decreases then increases when r = 1 and vice versa when r = 2 in Fig. 2(b). 3.3. Linear conductance when the system changes from parallel DQD to T-shaped DQD To clearly understand the spin-flip process on the Kondo resonance of the DQD system, we also discuss the transport characters when the system is changed continuously from the parallel DQD to the T-shaped DQD. From Fig. 4, the linear conductance increases substantially when we consider the T-shaped DQD (y¼0). If we choose different values of y ( ¼0.2, 0.7, 0.8), analogous interference effects occur, which is opposite to the serial case. The reasons can be attributed to the Kondo interference effect between the two dots, namely, the coexistence of a direct tunneling without a side-dot and an indirect tunneling via the side-dot results in an asymmetric transmission probability. In the T-shaped models, there exists the interference effect due to the T-shaped geometry, which is essential to control transport properties. The increase of r does not directly enhance charge fluctuations of the dot-1, but mainly increases the renormalized tunneling t˜c because electron correlations between two dots get somewhat weaker. This merely enlarge the splitting of the double peaks in the DOS of dot-1, and thus the conductance is still very small, as seen in the region of r ≤ 0.1. However, as r increases, the enhanced t˜c finally causes charge fluctuations of the dot-1, and then increases the linear conductance. All of which will be discussed in the transmission probability in the next section. 3.4. Transmission probability when the system changes from parallel DQD to T-shaped DQD Fig. 5 shows the transmission probability when the system changes from T-shaped DQD to parallel DQD. In the T-shaped DQD (y¼0 Fig. 5(a)), the increase of r makes the transmission dip reverse to a peak, however, the peak does not reverse to a dip as r increases continuously. At the same time, the double peaks of the transmission probability does not change too much if we choose y¼0.2 (Fig. 5(b)). They exhibit the same transport properties as that in the T-shaped DQD when we increase the spin-flip strength. Not only has the transmission dip reversed to a peak, but also the departure of the transmission peak around the Fermi energy. Such situation has been discussed in the previous subsection when the system is close to the parallel DQD (x¼1). However, the

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a

5

ε

b

c

ε

d

ε (Γ ·

ε (Γ ·

Fig. 5. Transmission probability T (ε) for different y under different spin-flip strength.

transmission peak also lowers down by the increasing r, which is definitely linked to the reversed current transportation. These exotic phenomena are caused by the spin-flip which will open one more path in the tunneling of an electron out of the QDs. For example, an electron with a spin-up can tunnel into the QD, but due to the spinflip, it can tunnel out of the QD as an electron with spin-down. The spin-flip strength enhances the renormalized tunneling t˜c and the renormalized energy of dot-1, but it weakens the renormalized energy of dot-2. The DOS of the two dots only reverses in the Fermi energy, which is different from the DOS transformation in the serial DQD. In the T-shaped geometry (y¼0), the Kondo resonance of the dot-1 gradually becomes symmetric with a sharp dip structure at ε = 0, whereas the DOS of the dot-2 develops a single Kondo peak located at the same position as the dip structure in the DOS of the dot-1 [19]. However, when the system is close to the parallel geometry (y → 1), the two resonances are composed of the bonding and anti-bonding Kondo states (Figs. 5(c) and (d)) (y¼0.5, 0.7). As y decreases, the double-peak structure of the Kondo resonances gradually changes its properties between the two limits mentioned above. The spin-flip strength increases the renormalized tunneling t˜c but reduces the renormalized energy of the two dots. Such effects will suppress the Fano shape of the transmission probability which is shown in Fig. 5(d).

Kondo resonance to Kondo dip in the Fermi energy when we choose the serial DQD, while the linear conductance first increases then decreases and the transmission probability splits into two Fano structures by keeping the Fano dip far away from the Fermi energy when we discuss the parallel DQD. Last, the linear conductance linearly increases as the increasing spin-flip and the transmission dip reverses to a peak when we discuss the T-shaped DQD. It has been found that the electron transport shows the characteristic properties due to the spin-flip strength, which may serve as a basis for the application of spin-manipulation devices. Recent experimental [20] results have also shown that spin-polarized conductance quantization can be probed by the electron-nuclear spin-flip scattering, which will open a new way to electrically manipulate the coherence of local nuclear spins in semiconductor nanostructures.

Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant nos. 11174214 and 11204192) and the NSAF (Grant nos. U1430117 and U1230201), and the Research Project of Education Department in Sichuan Province of China (Grant no. 15ZB0457).

4. Conclusions In summary, we have discussed the effect of spin-flip scattering on the electron transport among the serial, parallel and T-shaped double quantum dots. Due to the impact of spin-flip, the linear conductance decreases and the transmission probability undergoes

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