Heat generation by spin-polarized current in a quantum-dot spin battery

Physics Letters A 379 (2015) 613–618

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Physics Letters A www.elsevier.com/locate/pla

Heat generation by spin-polarized current in a quantum-dot spin battery Feng Chi a,b,∗ , Lian-Liang Sun c , Jun Zheng b , Yu Guo b a b c

School of Physical Science and Technology, Inner Mongolia University, Huhehaote 010023, China College of Engineering, Bohai University, Jinzhou 121013, China College of Science, North China University of Technology, Beijing 100041, China

a r t i c l e

i n f o

Article history: Received 17 October 2014 Received in revised form 10 December 2014 Accepted 13 December 2014 Available online 17 December 2014 Communicated by R. Wu Keywords: Quantum dot Heat generation Spin battery

a b s t r a c t We study the heat generation by spin-polarized current due to the electron–phonon coupling in a singlelevel quantum-dot, which is connected to an external either symmetric or asymmetric dipolar spin battery. We find that the heat generation depends sensitively on the configuration of the spin battery’s chemical potentials. In the case of the dot coupled to symmetric spin battery, there is remarkable negative differential of the heat generation, which disappears if the spin battery is asymmetric. We also find that the magnitude of the heat generation in asymmetric spin battery is much larger than that in symmetric one despite of the reduced intensity of the electric current. Moreover, the heat generation is insensitive to the system’s temperature, and develops peaks when the dot-lead coupling strength or the intradot Coulomb interaction approaches the phonon’s energy quanta. © 2014 Elsevier B.V. All rights reserved.

1. Introduction In recent years, the advance in nano-fabrication techniques has made it possible to have electrons transported through continuously miniaturized systems. Compared with traditional macroscopic devices, submicro- and nano-scale ones may run faster and become cheaper. Meanwhile, many researchers are being devoted to the investigation of how the heat is generated, transported, converted, and stored in nanodevices [1]. Such scientific studies will lead to the understanding of the laws of the heat generation for designing energy conservation electric components. It is known that the most important causes of heating in nanoscale systems are the inelastic electron–electron (e–e) and electron–phonon (e–p) scatterings [2–6], which are the same as those in bulk materials. Theoretically, Sun and Xie have studied the heat generation by electric current in a quantum-dot (QD) system by the nonequilibrium Keldysh Green’s function technique [7]. In their work, the heat originates from the e–p interaction, through which the energy associated with the electric current in the dot is transferred to the phonon bath in the form of heat [7]. Many subsequent works concerning the QD structure indicated that the Joule law applies to the bulk system breaks down, and various interesting properties unique to nanodevices were predicted [8–17].

*

Corresponding author. Tel.: +86 0416 3400202. E-mail address: [email protected] (F. Chi).

http://dx.doi.org/10.1016/j.physleta.2014.12.019 0375-9601/© 2014 Elsevier B.V. All rights reserved.

The QD or QD molecular are man-made quasi-zero-dimensional structures in solid, in which electrons are confined in all three directions and take fully quantized energy levels [18]. As compared to the one- or two-dimensional heterostructures [19], parameters in QD structures including the electron number on the dot, the energy level, and the dot-lead coupling strength can be precisely adjusted by gate voltages. Since the electron spin state of an isolated electron in QD is a natural two-level system, it has been proposed as a promising candidate for a quantum bit [20], which is the basic requirement for solid-state quantum information processing. QD structures have also become one of the most important objects in spintronics [21,22], whose central theme is the active manipulation of spin degrees of freedom in solid-state systems. Many effective methods have been developed to generate and control spinpolarized current in semiconductor spintronics, including electrical [22–26], optical [27–30] and even thermal techniques [31–34]. The spin polarization of the current p = ( J ↑ − J ↓ )/( J ↑ + J ↓ ) can reach as high as 100% or even infinite. Here 100% polarized spin means that there is only one spin component current that can transport through the system. Whereas infinite spin polarization implicates the achievement of a pure spin current driven out from a “spin battery” or “spin cell”. Several schemes have been put forward for the realization of spin battery, such as by using a ferromagnetic resonance process or a rotating external magnetic field [35–37]; all-optical injection of pure spin current based on the quantum interference of two-color laser fields [38,39]; and thermospin effect [31–34]. Pure spin current in spin battery has recently been

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F. Chi et al. / Physics Letters A 379 (2015) 613–618 †

Fig. 1. (Color online.) Schematic plot of the spin battery based on a single level quantum dot: (a) symmetric dipolar spin battery; (b) asymmetric dipolar spin battery. The dot level locates at the electron–hole symmetric point (εd = −U /2) which is outside of the transport window, and the other level due to the existence of intradot Coulomb interaction is also above the left and right chemical potentials.

where dσ (dσ ) creates (annihilates) an electron with energy ε0 and intradot Coulomb interaction U 0 . The quantity λ is the coupling constant between the dot electron and located phonon mode.  † The last term H T = k,β,σ ( T kβ σ ckβ σ dσ + H .c ) describes the conduction electron hopping between the QD and the leads with T kβ σ being the tunneling matrix element. Here we do not consider the ferromagnetism of the leads, and then T kβ σ and εkβ σ are actually independent on electron spin. Following previous works, we make a canonical transformation to eliminate the electron–phonon  ˜ = Xˆ H Xˆ † with Xˆ = exp[(λ/ωq )2 σ (aq† − coupling terms, i.e., H †

aq )dσ dσ ]. The transformed Hamiltonian reads [7,8,46]

˜ = H



εkβ σ ck†β σ ckβ σ +

σ

k,β,σ

electrically generated in a micro-wide quantum wire defined by electrostatic gates on top of a GaAs/AlGaAs heterostructure [25,26]. Some theoretical proposals for spin battery [40–43] or 100% [44] spin-polarized current were also suggested. Until now, most of the above-mentioned theoretical works about the heat generation or local heating have mainly focused on the structure composed of a QD coupled to normal metal [7–14,16], superconductor [15], or ferromagnetic [17] leads. The role of the electron spin has received less attention [17]. A recent work concerning Seebeck spin tunneling effect in a ferromagnet– oxide–silicon tunnel junction has shown that the heat generation is significantly influenced by the spin-dependent electric band structure in the electron reservoir [45]. This is very different from the heat generation in the usual charge-based electric devices [7]. The properties of the heat generation in the above experimental work were explained by a rough free-electron elastic tunneling model, and the energy-dependence of the spin polarization, i.e., the spindependent band structure, is introduced as a phenomenological parameter [45]. The roles of the e–e and e–p interactions were neglected in their work [45]. In the present paper we propose to couple a QD to an external spin battery, in which the chemical potential depends on electron spin (see Fig. 1). Our following results will show that the heat generation is very sensitive to the configuration of the spin battery’s chemical potentials. In a symmetric dipolar spin battery illustrated by Fig. 1(a), the magnitude of the heat generation may decrease with increasing spin bias, showing the negative differential of the heat generation (NDHG). This effect disappears if the spin battery is made of an asymmetric one as shown in Fig. 1(b). The magnitude of the heat generation in asymmetric spin battery is much larger than that in symmetric one due to the enhanced electron average occupation number blockaded on the dot. By tuning the dot level, coupling strength between the dot and the electron reservoirs, and the intradot e–e interaction, one can effectively adjust the magnitude of the heat generation.

Our QD spin battery system shown in Fig. 1 is described by the Hamiltonian of H = H leads + H ph + H dot + H T , where H leads = k,σ ,β∈ L , R

εkβ σ ck†β σ ckβ σ is the Hamiltonian for electrons in the †

left (L) and right (R) leads, and ckβ σ (ckβ σ ) is the conduction electron creation (annihilation) operator with wave vector k, electron energy εkβ σ , and spin σ . The second term stands for the



†

†

single-phonon mode H ph = q ωq aq aq , where aq (aq ) is the creation (annihilation) operator of a phonon having frequency ωq . In the following we consider the dot is coupled to a single phonon mode, and then the summation over q is omitted. The third term is the electron Hamiltonian on the QD [7,8],

H dot =

 σ





ε0 + λ aq† + aq d†σ dσ + U 0d†↑ d↑d†↓ d↓ ,

(1)

† T˜ kβ σ ckβ σ dσ

εd d†σ dσ + U d†↑ d↑d†↓ d↓

 + H .c ,

(2)

k,β,σ

where the renormalized dot level and the intradot Coulomb interaction are respectively given by εd = ε0 − λ2 /ωq and U = U 0 − 2λ2 /ωq . The tunneling matrix element is also renormalized to T˜ kβ σ = T kβ σ X , where the phonon operator is X = exp[−(λ/ωq )2 × †

(aq − aq )]. The operator X can be replaced by its expectation value  X  = exp[−(λ/ωq )2 ( N ph + 1/2)], where N ph = 1/[exp(ωq /k B T ph ) − 1] being the phonon distribution function with T ph the temperature of the phonon bath. Since in the above unitary transforma†

†

tion, the operators aq aq and dσ dσ remain unchanged, and then the heat flow between the electrons on the dot and the phonon †

bath Q σ (t ) = ωq daq (t )aq (t )/dt  [2,7,12] can be calculated from the transformed Hamiltonian in Eq. (2). In the absence of any timedependent fields, the Fourier transform of the heat generation can be obtained as [7,8],

Q = Re



ωq λq2

σ

dω  < ¯) G˜ σ (ω)G˜ > σ (ω

2π

  ˜ a ¯ ) + G˜ rσ (ω)G˜ > ¯) , − 2N ph G˜ > σ (ω)G σ (ω σ (ω

(3)

¯ = ω − ωq . The dot single-electron Green’s functions where ω r ,a,<,> G˜ σ (ω) are the Fourier transforms of G˜ rσ,a,<,> (t ) defined in terms of Hamiltonian (2), i.e., the retarded (advanced) one † r (a) G˜ σ (t ) = ∓i θ(∓t ){dσ (t ), dσ (0)}, the lesser Green’s function

˜> G˜ < σ (t ) = i dσ (0)dσ (t ), and the greater Green’s function G σ (t ) = † r −i dσ (t )dσ (0) [47,48]. Based on Hamiltonian (2), G˜ σ (ω) can be easily calculated by the equation of motion method as follows [8, 47,48], †

G˜ rσ (ω) =

2. Model and method



+





1 r −1

g˜ σ

(ω) + i (Γ˜L + Γ˜R )/2

,

(4)

where the dressed Green’s function without coupling to the leads is g˜ σr (ω) = [ω − ε˜ d − U˜ (1 − nσ¯ )]/[(ω − ε˜ d )(ω − ε˜ d − U˜ )]. Other Green’s functions can be determined accordingly [48], <(>) <(>) G˜ aσ (ω) = [G˜ rσ (ω)]∗ , G˜ σ (ω) = G˜ rσ (ω)Σ˜ σ G˜ aσ (ω), in which the lesser and greater self-energies are respectively given by Σ˜ < = i [Γ˜L f L σ (ω) + Γ˜R f R σ (ω)], and Σ˜ σ> = −i {Γ˜L [1 − f L σ (ω)] + Γ˜R [1 − f R σ (ω)]} [47]. The Fermi distribution function is f L ( R )σ (ω) = 1/{exp[ω − μ L ( R )σ ]/k B T e + 1} [40–43], where μβ σ and T e are the spin-dependent chemical potential in lead β and the temperature of the electrons, respectively. The average occupation number nσ needs

to be self-consistently calculated by the equation of nσ = Im dω G˜ < σ (ω)/2π . The line-width function in the above equations is Γ˜β = 2π | T˜ kβ |2 ρβ and ρβ being the local density of states in lead β .

F. Chi et al. / Physics Letters A 379 (2015) 613–618

Fig. 2. (Color online.) (a) Pure spin current in unit of e ωq /2π and (b) heat generation in unit of λ2 ωq /2π as functions of the spin bias V s and renormalized dot level εd for a symmetric dipolar spin battery. The adopted parameters are Γ0 = T e = T ph = 0.1ωq , U = 3ωq and λ = 0.6ωq .

The spin-dependent electric current can be calculated by the Keldysh nonequilibrium Green’s function formalism,

and its standard form is obtained as [46–48], J σ = −(e /h) dω[ f L σ (ω) − f R σ (ω)] T σ (ω), where the transmission function is given by,

T σ (ω) = 2

ΓL Γ R Im G rσ (ω). ΓL + Γ R

(5)

The line-width function in the above equation is Γβ = Γ˜β × exp [(λ/ωq )2 (2N ph + 1)], and the single-electron Green’s functions r ,a,<,> Gσ (ω) in the above equation are the Fourier transforms of r ,a,<,> Gσ (t ) defined in terms of the un-transformed Hamiltonian. Based on the above canonical transformation, G rσ (ω) is related to G˜ rσ (ω) by [7,8] r

G σ (ω) =

n =∞



L n G˜ rσ (ω − nωq )

n=−∞

+



 1 < G˜ σ (ω − nωq ) − G˜ < σ (ω + nωq ) , 2

(6)

where L n in the equation is given by L n = exp[− g (2N ph + 1)] ×

exp(nωq /2k B T ) I n [2g 2N ph ( N ph + 1)], with I n (x) the modified nth Bessel function. 3. Numerical results In the following numerical investigations, we set the phonon energy ωq = 1 as the energy unit (h¯ = 1), and assume symmetric barriers with Γ˜L = Γ˜R = Γ0 . We also assume that the QD and the phonon bath are held at the same temperature, i.e., T ph = T e . Figs. 2(a) and (b) show the pure spin current J s = h¯ ( J ↑ − J ↓ )/2 and the heat generation Q = Q ↑ + Q ↓ when the dot is coupled to a symmetric dipolar spin battery whose chemical potentials are related to the spin bias via V s = μ L ↑ = −μ L ↓ = −μ R ↑ = μ R ↓ , which

615

is shown in Fig. 1(a). It should be noted that in such a structure, equal number of spin-up and spin-down electrons flow in opposite directions, and then the total charge current J e = J ↑ + J ↓ is zero. Fig. 2(a) shows that the behaviors of the spin current are analogous to those of the charge current in the system composed of a QD coupled to normal metals [8,46]. When the dot level is at the electron–hole symmetric point, i.e., εd = −U /2, the spin current remains almost at zero for small spin bias, and then starts to increase until the spin bias is large enough to ensure εd to enter into the transport window. In the regime of dot level εd > −U /2, the magnitude of the spin current increases when the renormalized dot level εd enters into the transport window of the spin bias, and then develops into a plateau. The spin current remains at this plateau until the other level εd + U enters into the spin bias window and achieves another plateau. Meanwhile, there are also some phonon-induced sub-levels, at which the spin current increases due to the phonon-assisted tunneling processes [8,46]. We emphasize here that this phonon-assisted futures are in fact hardly distinguishable by the subtle changes of the colors. One can see it more clearly in Fig. 3(a). In the case of dot level εd < −U /2, the level of εd + U enters into the transport window of the spin bias first followed by the level of εd with increasing spin bias. As compared to Fig. 2(a), it is found that the heat generation in Fig. 2(b) has no direct relationship with respective to the spin current. This implicates that the usual Joule law for the heat generation in bulk macroscopic system breaks down now. Generally, the heat generation is independent on the direction of the electron flow, and then finite heat generation emerges despite of zero charge current. Different from the spin current, the heat generation has a delay of ωq /2 with respective to the spin bias as compared to the spin current. The reason is that the heat generation originates from the process of phonon emitting when an electron at state of ω jumping to an empty state of ω − ωq [7,8]. If the spin bias is small, the electrons cannot absorb enough energy to emit phonons and then the heat generation is zero. This delay effect demonstrates that the energy is carried by the phonons in a quantized form. It should be indicated that here the heat generation has a delay of ωq /2 with respect to the spin bias. In Refs. [7] and [8], however, the delay is ωq . This difference is caused by the different relationships between the chemical potentials and the spin and charge biases. For a fixed dot level (taking εd = 0 for an example), the heat generation starts to increase as V s > ωq /2, and then reaches a plateau until the level of εd + U enters into the spin bias transport window, i.e., V s = εd + U . With further increasing spin bias, the magnitude of the heat generation decreases in the regime of εd + U < V s < εd + U + ωq , exhibiting the NDHG. For V s > εd + U + ωq , the heat generation approaches to another plateau, whose magnitude is smaller than that in the spin bias regime of εd + ωq < V s < εd + U . The NDHG phenomenon arises from the phonon absorbtion processes when the electrons jump from the level of ˜d + U˜ to ˜d + U˜ + h¯ ωq [8,18]. For the present chosen parameters, we find that the magnitude of the heat generation in the regime of high spin bias is smaller than the maximum value in the spin bias regime of εd + ωq < V s < εd + U . This is quite different from the case of heat generation by charge current in Refs. [7,8]. This effect may be attributed to the reduced electron occupation number when the dot is connected to a symmetric spin battery. The spin current and the heat generation varying as functions of the spin bias for different values of the system temperature are illustrated in Figs. 3(a) and 3(b), respectively. In the low temperature regime, the two plateaus and the phonon-induced sub-steps in the current are clearly visible. With raising temperature, the cures of both J s − V s and Q − V s become more smooth, with unchanged qualitative behaviors. Here one can also see that unlike the current, the derivative of the curve of the heat generation with

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F. Chi et al. / Physics Letters A 379 (2015) 613–618

Fig. 3. (Color online.) (a) Spin current and (b) heat generation as functions of the spin bias V s for εd = 0 and different values of the temperature. (c) and (d) are individually for heat generation as a function of Γ0 and U with different values of the dot level. The spin bias and the temperature in (c) and (d) are fixed as V s = 2ωq and T = 0.1ωq . The intradot Coulomb interaction is set as U = 3ωq in (c), and Γ0 in (d) is 0.1ωq .

respective to the bias d Q /dV s shows no phonon side peaks [7,8]. In Fig. 3(c) we show the dependence of the heat generation on the line-width function Γ0 for different values of dot level. The heat generation first increases with increasing Γ0 , reaching a maximum and then decreases with further increased line-width function. The enhanced heat generation by increasing Γ0 can be attributed to the broadening of the dot level. In the case of T e = T ph , the heat generation can be simplified to the following form [7,11],

Q =

 ωq λ2  dω ¯ ) T σ (ω) T σ (ω¯ ), f L R σ (ω) f L R σ (ω ˜ ˜ 2π σ ΓL Γ R

(7)

¯ ) are individually f L σ (ω) − f R σ (ω) where f L R σ (ω) and f L R σ (ω ¯ ) − f R σ (ω¯ ). Obviously, larger line-width function will and f L σ (ω make the transmission spectrum wider and enlarge the overlaps of ¯ ) T σ (ω¯ ). In this way, the heat generation f L R σ (ω) T σ (ω) and f L R σ (ω is enhanced. With further increasing Γ0 , however, the time of an electron spent in the QD will be shortened and then the heat generation will be decreased. Fig. 3(d) shows that the heat generation has a huge peak at U = ωq . This peak is induced by the resonant phonon emitting process happening between the two real electron levels εd and εd + U [8]. In Fig. 4 we present the contour plot of the heat generation and the current versus the dot level εd and the spin bias V s in the system composed a QD coupled to asymmetric spin battery. In such a system, the chemical potentials of the left lead is spin dependent (V s = μ L ↑ = −μ L ↓ ), whereas the right lead is made of normal metal (μ R ↑ = μ R ↓ = 0) [42–44]. In the dot level range of 0 < εd < V s , the spin-up current J ↑ is always positive and has a broad resonance. The reason is that the energy range between μ L ↑ and μ R is the spin-up transport window, through which spin-up electrons tunnel from the left lead to the right one via the dot [44].

Fig. 4. (Color online.) (a) Total current J = J ↑ + J ↓ in unit of e ωq /h and (b) heat generation in unit of λ2 ωq /2π as functions of the spin bias V s and renormalized dot level εd for an asymmetric dipolar spin battery. For the cases of εd > 0 and εd < −U , the current is nearly 100% spin-polarized. Otherwise, both the spin-up and spin-down currents are nearly zero. The other parameters are as in Fig. 2.

F. Chi et al. / Physics Letters A 379 (2015) 613–618

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spin orientations [44], and the phonon absorb processes also cannot happen. Similar to the cases of symmetric dipolar spin battery in Fig. 3, the heat generation in asymmetric dipolar spin battery is also enhanced for moderate line-width function (Fig. 5(a)), and develops a large peak when the intradot Coulomb e–e interaction equals to the phonon frequency. The reason is just the same as that in Figs. 3(c) and (d). 4. Conclusion In conclusion, heat generation by both pure spin current and 100% spin-polarized current is investigated in a system composed of a QD coupled to two types of dipolar spin battery. We find that the heat generation in the asymmetric dipolar spin battery is much larger than that in the symmetric one due to the enhanced electron occupation number blockaded on the dot. There is negative differential of the heat generation phenomenon originated from the resonant phonon absorbtion processes in symmetric dipolar spin battery system. This effect disappears if the spin battery is asymmetric due to the special electron occupation configurations in the high and deep dot level cases. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 61274101). Fig. 5. (Color online.) Heat generation as a function of Γ0 (a) and U (b) with different values of the dot level for an asymmetric spin battery. The spin bias and the temperature in (a) and (b) are set to be V s = 4ωq and T = 0.1ωq . The other parameters are as in Fig. 3.

Meanwhile, the spin-down current J ↓ in this energy regime approaches to zero because the spin-down energy level is pushed up to εd + U and is empty, resulting in zero J ↓ . It means that a pure spin-up current is generated. The intensity of the current increases with increasing spin bias. Lowering the dot level by external gate voltage to the energy region of −U < εd < 0, both J ↑ and J ↓ are nearly zero. This is because now a spin-up electron is injected from the left lead due to μ L ↑ > εd , and can hardly tunnel out of the dot to the right lead as μ R > εd . In this way a spin-up electron is blockaded on the dot, and spin-up current is nearly zero. Meanwhile, the spin-down energy level εd + U is always kept out of its transport window (εd + U > μ R , μ L ↓ ), and is empty. When the dot level is lowered to −( V s + U ) < εd < −U , the spin-down energy level enters into its transport window resulting in finite J ↓ and zero J ↑ [44]. If the dot level is further lowered, both εd and εd + U are occupied by electrons of opposite directions, and then the current is zero. Note that in this asymmetric spin battery, the current’s intensity is roughly half of that in symmetric one because there is only one spin component current. The magnitude of the heat generation in Fig. 4(b), however, is greatly enhanced. We argue that this enhancement originates from the increased electron occupation number on the dot [44], which can emit more phonons. Comparing Fig. 2(b) with Fig. 4(b), one can find that the NDHG effect disappears in the latter case. The reason can be explained as follows: For εd > 0 in asymmetric dipolar spin battery, the dot level εd is the transport channel of spin-up electron as μ L ↑ > εd > μ L ↓ when the spin bias is applied. Even if the level εd + U enters into the transport window, it cannot carry any electrons since εd + U > μ R , μ L ↓ . Spin-up electron also cannot take εd + U due to the Pauli exclusion principle, i.e., εd and εd + U cannot be simultaneously occupied by electrons of the same spin orientation. Consequently, resonant phonon absorbtion due to the electron jumping between εd and εd + U cannot occur and the NDHG disappears. For −( V s + U ) < εd < −U , the level of εd and εd + U are all occupied by different electrons of different

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