Physics Letters A 377 (2013) 817–821
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Physics Letters A www.elsevier.com/locate/pla
Quantum pump in a system with both Rashba and Dresselhaus spin–orbit couplings Yun-Chang Xiao a , Wei-Yin Deng b , Wen-Ji Deng b , Rui Zhu b , Rui-Qiang Wang a,∗ a b
Laboratory of Quantum Information Technology, ICMP and SPTE, South China Normal University, Guangzhou 510006, China Department of Physics, South China University of Technology, Guangzhou 510641, China
a r t i c l e
i n f o
Article history: Received 27 November 2012 Received in revised form 22 January 2013 Accepted 29 January 2013 Available online 31 January 2013 Communicated by R. Wu Keywords: Quantum pump Spin–orbit coupling Spin-dependent transport
a b s t r a c t We investigate the adiabatic quantum pump phenomena in a semiconductor with Rashba and Dresselhaus spin–orbit couplings (SOCs). Although it is driven by applying spin-independent potentials, the system can pump out spin-dependent currents, i.e., generate nonzero charge and spin currents at the same time. The SOC can modulate both the magnitude and the direction of currents, exhibiting an oscillating behavior. Moreover, it is shown that the spin current has different sensitivities to two types of the SOC. These results provide an alternative method to adjust pumped current and might be helpful for designing spin pumping devices. © 2013 Elsevier B.V. All rights reserved.
1. Introduction In recent decades, quantum pump mechanisms have been extensively studied since it was originally proposed by Thouless [1]. A cyclic adiabatic change of two or more systemic parameters can induce a pumped current even in the absence of any external bias voltage. The pumped charge per cycle is determined by the area enclosed in parameter space during the cyclic evolution. Variation of systemic parameters is realized generally by applying time-dependent potentials to a double-barrier junction [2–4]. Essentially, quantum pumping results from the interference effect between different photon-assisted coherent paths. Numerous works have made great efforts to the charge pumping in mesoscopic systems, such as bulk semiconductors [5,6], quantum dots [7–12], mesoscopic rings [13], nanowires [14], graphenes [15], and quasicrystals [16]. With the development of spintronics, much attention is paid to the ideas of combining the adiabatic pumping and spin-dependent transport. Various spin pumps or spin batteries [17] have been put forward, acting as an alternative proposal to create the spin current. In general, these setups are established with help of ferromagnetic materials. An appealing approach to generate the spin current is based on the spin–orbit couplings (SOCs), which is an intrinsic property of a two-dimensional electron gas in semiconductor heterostructures.
*
Corresponding author. E-mail address:
[email protected] (R.-Q. Wang).
0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.01.041
A remarkable advantage of the SOC is its electric tunability by gate voltage without using ferromagnetic materials or external magnetic fields. In the presence of Rashba SOC, Governle et al. [18] reported the production of spin currents. Li et al. [19] analyzed the transport property of quantum spin pumps in the adiabatic and nonadiabatic cases and found that it is possible to pump the pure spin currents by modulating the SOC. As we are known, the SOC is involved in two species, Rashba-type and Dresselhaus-type. Previous studies for dc-driven systems [20–24] revealed that this two SOCs can play different roles in spin-dependent electron transports. Naturally, it is interesting to know how the two SOC species affect the pumped spin current driven by ac voltages. Lin et al. [25] recently studied spin pump in a quantum channel with the Rashba–Dresselhaus SOCs (R–D SOCs), where the pumping functionality is achieved by means of one ac gate voltage that modulates the Rashba constant sinusoidally. They interestingly found that the spin-flip Dresselhaus interaction modifies the threshold of the spin current to switch polarization. In this Letter, we consider the same model as Ref. [25] but drive the R–D SOCs semiconductor system by two time-dependent gate voltages, respectively, exerted at two interfaces between the semiconductor and leads. The Floquet scattering theorem [3,26] developed to treat the quantum mechanical pumping in mesoscopic conductors is applied. It is shown that the SOC greatly influences the pumped charge current and completely determines the appearance of spin current. Both the direction and the magnitude of spin current can be modulated by the SOCs as well as their strength ratio.
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Y.-C. Xiao et al. / Physics Letters A 377 (2013) 817–821
and
ψnR (x)
↑ ikn x 1
= tn e
where kn =
0
↓ ikn x 0
+ tn e
1
for x > d,
(4)
2m ( h¯ 2
ε + nh¯ ω) denotes the wave vectors for the elec-
tron with n photon absorbed (n < 0) or emitted (n > 0). In the middle area (0 < x < d), the n-resolved electron wave function is [23] Fig. 1. Schematic plot of the quantum pump transport in an SFET. The left (L) and right (R) are normal leads and the middle region is the semiconductor with the R–D SOC. The interfaces are subjected to the time-dependent driven potentials u L ( R ) (t ).
+ − ψnS (x) = An+ e ikn x + B n+ e −ikn x |+ − + + An− e ikn x + B n− e −ikn x |−
(5)
with
The Letter is organized as follows. In Section 2, a model is introduced and theoretical formalisms are given correspondingly. Section 3 gives the numerical results for pumped currents, and Section 4 presents a brief conclusion of this work.
1 0 θso θso + sin , |+ = cos 2 2 0 1 1 0 θso θso |− = sin − cos ,
2. Model and formulations
Consider a spin-field-effect transistor (SFET) model, which is schematically sketched in Fig. 1. The system consists of a central region with spin–orbit coupling connected to two semi-infinite leads (L and R) where SOC is absent. In order to generate the quantum pump effect, two time-dependent potentials (or gate voltages) are applied to two connected interfaces. For simplicity, let us restrict our discussion to a quasi-one-dimensional limit, where the width of the transverse confining potential well is assumed small enough, so that the intersubband mixing can be negligible [25,27]. In the one-band effective-mass approximation, the model system can be described by the following Hamiltonian [18,21,23,25,27,28]
H (x, t ) =
p 2x 2m
−
h¯ kso m
σso p x + u (x, t ),
(1)
where p x and m are the momentum operator and the effective mass of an electron, respectively. Here, we consider both the Rashba and Dresselhaus couplings, which are combined into the second term in Eq. (1) and only appears in the central region. The spin operator is σso = σx sin θso − σz cos θso where sin θso =
kβ /kso and cos θso = kα /kso with kso =
k2α + k2β . kα (kβ ) labels the strength of the Rashba (Dresselhaus) SOC. At the positions of x = 0 and x = d, the time-dependent potentials are modeled as u (x, t ) = u L (t )δ(x) + u R (t )δ(x − d) with u j (t ) = u sj + udj cos(ωt + φ j ) ( j = L , R ). u sj is a static potential and udj is a driving potential varying in time with frequency ω and initial phase φ j . In order to determine the transport properties through this oscillating double-barrier potential, we employ the standard Floquet function method [3], in which the solution Ψε (x, t ) of Schrödinger equation with respect to H (x, t ) is expressed as a superposition of the Floquet eigenstate ψn (x), i.e.,
Ψε (x, t ) =
i (ε + nh¯ ω)t . ψn (x) exp − h¯ n=−∞ ∞
e ikn x 1 2
for x < 0,
1
↑
+ rn e −ikn x
1 0
↓
+ rn e −ikn x
1
⎛
⎞
⎛ ⎞
A n+ 1 ⎜ B n+ ⎟ √ ⎜1⎟ U n ⎝ − ⎠ − 2τ δ0,n ⎝ ⎠ 0 An 0 B n− ⎛ + ⎞ ⎛ + ⎞ A n +1 A n −1
⎜ B+ ⎟ ⎜ B+ ⎟ n +1 ⎟ ⎜ n −1 ⎟ , I = D n +1 ⎜ + n − 1 − ⎝ ⎠ ⎝ − ⎠ A n +1
B n−+1
A n −1
(6)
B n−−1
where U n , D n+1 , and I n−1 are the matrices given in Appendix A. When an electron travels from the right to the left leads, the reflection amplitude rn σ and the transmission one tn σ can be obtained by repeating the same procedure. With these amplitudes (rnσ , tnσ , rn σ , tn σ ), in the space of ( L ↑, L ↓, R ↑, R ↓) one can construct a 4 × 4 Floquet scattering matrix [3], from which the spindependent pumped current flowing through the left lead is obtained as [33]
I σL =
e ω σ 2 σ 2 n tn − tn .
2π
(7)
n
In the following calculations, the pumped charge current is as ↑ ↓ usual defined as I cL = I L + I L and the pumped spin current is ↓
I sL = I L − I L . 3. Numerical results and discussions
Here ε is the energy of injected electron, n is the channel number of photon propagating mode, and h¯ ω represents photon energy quantum. In the following, we develop the original Floquet theorem [3] and extend it to include the spin-dependent transport. When an unpolarized electron accompanied by the nth propagating mode transfers from the left to the right leads, its wave function generally is expressed as [21,23,29]
ψnL (x) = δ0,n √
2
0
and kn± = τ kn2 + k2so ± kso where τ = ms /mlead characterizes the ratio of effective masses ms in SOC region to mlead in leads. For each propagating mode, the wave functions ψnL , ψnR and ψnS in three different regions are connected by the Griffith boundary conditions [30–32]. As a result, the reflection amplitude rnσ and the transmission one tnσ in Eqs. (3) and (4) can be obtained through parameters A n± and B n± , which are in turn determined by a set of recursive equations
↑
(2)
2
0
1 (3)
Throughout this Letter, we take ms = 0.036me for typical InAsbased semiconductors [34] and mlead = me for the normal leads, where me is the mass of the free electron. Set the semiconductor length d = 0.3 μm, the energy quantum h¯ ω = 6.0 meV, and the incoming energy of electrons ε = 0.6 eV [4,9]. For simplicity, we only consider symmetric applied potentials u sL = u sR = u s = 0.4 meV and udL = udR = ud . The controllable parameters are u d , the phase difference φ = φ R − φ L , and the SOC strength. We plot the charge current I cL as a function of the phase difference φ for different SOC strengths in Fig. 2, where a nonzero
Y.-C. Xiao et al. / Physics Letters A 377 (2013) 817–821
Fig. 2. Pumped charge current versus the phase difference φ for different SOC strengths, kso = 0, 1, 3, and 5 in unit of k0 = 108 m−1 . The ac driven potentials is ud = 6 meV.
819
rent switches from a positive maximum to a negative one quickly. This is of importance as this provides an alternative method to adjust both the magnitude and direction of the pump current, instead of the phase difference φ as shown in Fig. 2. One can understand these as follows. Quantum pumping is a consequence of the interference of energetically different photon-assisted paths made by an oscillating scatterer, which is absent for the dc-driven case [23]. A propagating electron picks the coherent phase not only from the driving phase φ but also from the spatial phase due to the spin– orbit procession. This point can be seen in Eq. (5) where the phase factor kn± prominently includes the contribution of kso . As a result, this additional spatial phase leads to the change of sign in the generation of pumped spin current. However, as kso becomes large enough, the current quickly drops to a zero value and then remains it. In this region, the SOC mechanism dominates and so the ac driving potential is difficult to break the symmetry of the system kept by the SOC. In addition, we notice that the pumped current is also very sensitive to the ac driving potentials. Increasing ud can enhance the maximum in the positive and negative directions but hardly vary the sign of the pumped current. The reason is that kso also modulates the transmission amplitude by entering γn± and then U n . Physically, larger ud means that the energy level oscillates around the static potential u s with larger amplitude in which more photon-assisted channels are opened. In the discussions above, although the SOC effect plays an important role in the pump effect, the charge current is determined mainly by the total strength kso , regardless of θso , i.e., the ratio of the SOC strengths of Rashba kα to Dresselhaus kβ . Previously, it is revealed that this two types of the SOC impact considerably the spin-dependent transports in a different role [20–23]. It is interesting to know how the SOC affects the spin-resolved pump effect. ↑(↓) The pumped spin-dependent current I L , the charge current I cL , and the spin current I sL are plotted in Fig. 4 versus the phase dif↑
Fig. 3. Pumped currents versus the SOC strength for different ac driving potentials, ud = 3, 4, 5 meV. The phase difference is φ = 0.5π .
pumped current is observable. The pumped current oscillates alternatively between the positive and the negative maxima, showing an antisymmetric behavior with respect to φ = π . The appearance of current platforms makes the I cL –φ relation no longer obey simply sinusoidal function, even for vanishing kso . The similar results were also illustrated in a double-barrier oscillating structure without SOC as addressed by Wei [5]. They found that a strong pump driving potential can lead to the zero-value current platform, whose width is proportional to Fermi energy. Compared to this, the introduction of the SOC changes nonmonotonically the platform lineshape both in the width and the peak height, i.e., with increase of kso the current peak and the platform width increase first and then shrink. Intriguingly, it is noticed that the stronger SOC, e.g., kso = 5k0 , can lead to the change of current sign. This is unique as dc-driven conductances or transmission coefficients never change its sign with the increase in kso even though they also present an oscillating behavior [23]. In order to more clearly see the role of the SOC mechanism in pump effect, we in Fig. 3 depict the current I cL as a function of the SOC strength. Obviously, a nonmonotonic dependence of the pumped current on kso arises. As kso increases, the pumped cur-
↓
ference. In Fig. 4(a) I L and I L oscillate and behave in the same manner as the charge current in Fig. 4(b). They always keep the same phase but have different magnitudes. As a result, an observable spin current appears in spite of smaller value compared to the pumped charge current, as shown in Fig. 4(b). With variation of φ , the spin current changes its flowing direction alternatively. This originates from the electrons with different spins experiencing different spin scattering due to the splitting of kn± in the presence of spin–orbit interactions. In Fig. 5, the pumped spin currents are plotted as a function of the total SOC kso for different θso . The increase of θso means that the Dresselhaus component kβ is enhanced. From the figures one can see that the spin current is sensitive to kso . Between two peaks there is a sharp dip where a larger negative spin current is presented. If controlling the kso by gate voltage, one can manipulate the pumped spin current easily not only in its magnitude but also in its flowing direction. Comparing the curves with different θso , it is interesting to find that increasing the Dresselhaus component (i.e., large θso ) of the SOC can strengthen the maximum of the pumped spin current, more favorable for the formation of spin current. To a certain extent, in turn, the measurement of spin current may be applied to reveal which type of the SOCs is dominant in an unknown material. In addition, the increase in the strength of ac driving potentials [compare Figs. 5(a) and (b)] can make the pumped spin current enhance pronouncedly in positive and negative directions. 4. Conclusions In summary, we have studied the generation of spin-dependent pumped current in an SFET structure subjected to two timedependent driving potentials. The standard Floquet scattering
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Y.-C. Xiao et al. / Physics Letters A 377 (2013) 817–821
↑(↓)
Fig. 4. (a) Pumped spin-dependent current I L kso = 5k0 with 3kα = 4kβ and u d = 6 meV.
and (b) pumped charge current I cL and pumped spin current I sL versus the phase difference φ . The chosen parameters are
Fig. 5. Pumped spin current versus the SOC strength for different θso . The phase difference is chosen as φ = 0.5π , the ac driven potential is (a) u d = 4 meV and (b) u d = 8 meV.
method is employed, which allows one to analyze the timedependent electron transports in photon-assisted processes. We focus especially on the SOC effect on the quantum pump and find that the spin current as well as the charge current can be pumped out. The spin–orbit interactions not only modify their magnitudes considerably but also inverse their directions, exhibiting an oscillating behavior. This provides an alternative method to adjust both the magnitude and direction of the pumped current, instead of the phase difference. Interestingly, it is shown that two types of the SOC play different roles in generation of the spin current but play the same role in generation of the charge current. As expected, the pump effect is sensitive to the driving potential strength.
Acknowledgements
We would like to acknowledge W. Luo, D.W. Zhang, Q.H. Zhong, and S.L. Zhu for their interesting and helpful discussions. This work was supported by the Program for New Century Excellent Talents in University (Grant No. NCET-10-0090), by the NSF-China (Grant
No. 11174088), and by the Guangdong Provincial Natural Science Foundation (Grant No. S2012010010681), and SIRTPC. Appendix A Substitute Eqs. (3), (4) and (5) into the boundary conditions, we obtain the solving Eq. (6). Where U n , D n+1 , I n−1 are the matrices in according propagating modes, they are introduced below
⎛ ⎜
Un = ⎝
p γn+
qγn+
qγn+
+d
−d
p γn− e ikn
− ikn+ d
qγn e
− p γn− −qγn−
− p γn+ qγn− e ikn
− ikn− d
−
p γn−
+
− p γn+ e −ikn d −qγn+ e −ikn d
− p γn e
⎛
⎞
−qγn−
+ −ikn− d
−qγn e
⎟ ⎠,
p γn e
− phdn e −i φ −qhdn e −i φ − phdn e −i φ ⎜ −qhdn e−iφ phdn e−iφ −qhdn e−iφ
D n+1 = 2ie i φ R ⎜ ⎝
phdn e qhdn e
+
ik d n +1
+
ik d n +1
qhdn e
−
ik d n +1
− phdn e
−
ik d n +1
(8)
+ −ikn+ d
phdn e qhdn e
−ikn−+1 d
−ikn−+1 d
−qhdn e −i φ phdn qhdn e
e −i φ
−ikn++1 d
− phdn e
−ikn++1 d
⎞ ⎟ ⎟, ⎠ (9)
Y.-C. Xiao et al. / Physics Letters A 377 (2013) 817–821
⎛ I n−1 = 2ie
−i φ R
− phdn e i φ −qhdn e i φ
⎜ ⎜ + ⎝ phdn eikn−1 d + d ik qhdn e n−1
−qhdn e i φ phdn e
i φ
phdn e
i φ
−ik− d phdn e n−1 − ik −ik− d d − phdn e n−1 qhdn e n−1
qhdn e
−
− phdn e i φ
ik d n −1
⎞
−qhdn e i φ −qhdn e i φ qhdn e
−ikn+−1 d
− phdn e
−ikn+−1 d
⎟ ⎟, ⎠
(10) τ k2 u s τ k2 u where p = cos θ2so , q = sin θ2so . We set hsn = 2kF ε , hdn = 2kF εd as n n
the potential related parameters, and in corresponding definitions √
γn± =
τ kn2 +k2so kn
821
± (τ + 2ih sn ).
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