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Int. J. Nanotechnol., Vol. 4, No. 5, 2007

Spin pumping with quantum dots E.R. Mucciolo Department of Physics, University of Central Florida, P.O. Box 162385, Orlando, FL 32816-2385, USA E-mail: [email protected]

C.H. Lewenkopf* Departmento de Física Teórica, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier 524, Rio de Janeiro 20550-900, Brazil Fax: +55-21-25877296 E-mail: [email protected] *Corresponding author Abstract: The purpose of this paper is to discuss how lateral semiconductor quantum dots can be used as pumps to produce spin polarised currents, by exploring quantum phase coherence phenomena. Electronic transport in phase coherent or mesoscopic systems has been intensively studied for more than three decades. While there is a substantial understanding of the stationary regime, much less is known about phase-coherent non-equilibrium transport when pulses or ac perturbations are used to drive electrons at low temperatures and at small length scales. However, about 20 years ago Thouless proposed to drive non-dissipative currents in quantum systems by applying simultaneously two phase-locked external perturbations. The so-called adiabatic pumping mechanism has been revived in the last few years, both theoretically and experimentally, in part because of the development of lateral semiconductor quantum dots. Here we show how open dots can be used to create spin-polarised currents with little or no net charge transfer. The pure spin pump we propose is the analog of a charge battery in conventional electronics and may provide a needed circuit element for spin-based electronics. We also briefly discuss other relevant issues such as rectification and decoherence and point out possible extensions of the pumping mechanism to closed quantum dots. Keywords: nanodevices; quantum dots; spintronics; pumping; spin currents. Reference to this paper should be made as follows: Mucciolo, E.R. and Lewenkopf, C.H. (2007) ‘Spin pumping with quantum dots’, Int. J. Nanotechnol., Vol. 4, No. 5, pp.482–495. Biographical notes: Eduardo R. Mucciolo is an Associate Professor at the University of Central Florida. He graduated from MIT in 1994, was a Postdoctoral Associate at Nordita, Denmark (1994–1996), Associate Professor at the PUC-Rio de Janeiro, and Visiting Professor at Duke (2003–2004). His research area is theoretical condensed matter physics, with focused interests in electronic properties of nanostructures, spin transport and quantum computation in solid-state. Copyright © 2007 Inderscience Enterprises Ltd.

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Caio H. Lewenkopf is an Associate Professor at the Universidade do Estado does Rio de Janeiro, conducting research in theoretical condensed matter physics. He graduated at the University of Heidelberg, Germany, in 1991, was a Postdoctoral Associate at the Max-Planck-Institute in Heidelberg, Michigan State University, University of Washington, and University of São Paulo. He recently received the prestigious ‘Cientistas do Nosso Estado’ fellowship. He has authored more than 50 research papers mainly on electronic transport properties in nanodevices. His work on transport and non-equilibrium statistical mechanics has received international recognition through his appointment to the Editorial Board of Physica A.

1

Introduction

In the past few years we have seen important advances in the coherent control of micro and nanoelectronic devices. The experimental effort, driven by the quest for the implementation of quantum computation in semiconductor and superconductor devices, has increased substantially the breath and scope of the study of mesoscopic systems, in particular lateral semiconductor quantum dots [1]. In these systems, electrons within a two-dimensional gas (2DEG) are confined to small puddles by the application of gate voltages. The shape and size of these puddles can be controlled and fine-tuned by the same or additional gate voltages. Electrodes also allow one to vary the width of the point contacts that connect the electron puddle to the 2DEG. By acting on these points contacts one can operate the quantum dots in ‘open’ (at least one propagating channel allowed per point contact) or ‘closed’ (fully pinched point contacts) regimes. A great variety of stationary transport phenomena have been observed in these systems over the past 15 years [2]. Recently, a new generation of experiments has started to probe the dynamical transport properties of quantum dots. A remarkable attempt to explore phase-coherent, pulsed response of an open quantum dot was led by Switkes et al. [3]. Their motivation was the observation of the so-called adiabatic quantum pumping effect, first discussed by Thouless in the context of one-dimensional electronic systems more than 20 years ago [4]. Adiabatic quantum pumping takes place when one slowly modulates two or more external parameters of a quantum system, resulting in a net dc current without the application of any bias [5]. The effect requires phase-coherent electrons and a system well coupled to reservoirs. The generation of spin currents in semiconductor heterostructures is a topic of great interest presently [6,7]. It is believed that spin currents may find applications in in-chip quantum communication, where light propagation is not practical. The full control of the electronic spin degree of freedom in semiconductors also promises to have a large impact in the future of conventional technologies by increasing processing speed, storage capacity and functionality. While our work does not explore these issues, it does show that quantum dots are versatile enough to yield spin currents with high efficiency, albeit only at low temperatures. The remaining sections are divided as follows. In Section 2 we provide a general qualitative discussion of adiabatic quantum pumping. In Section 3 we show that pumping in the presence of a sufficiently strong magnetic field breaks spin symmetry and produces spin-polarised currents. We also estimate the magnitude of the effect under realistic

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assumptions. The detection of pure spin currents generated with a quantum dot spin pump is discussed in Section 4.

2

Adiabatic quantum pump

Let us suppose that a certain quantum system is connected to two particle reservoirs. Quantum pumping can be defined as the production of net dc currents between reservoirs by acting solely on the quantum system with ac perturbations. No bias is applied between reservoirs. For open quantum systems, when no substantial potential barriers exist between the system and the reservoirs, adiabatic pumping is achieved when the frequency of the ac perturbations is much smaller than the inverse dwell time of the electron in the quantum dot, namely, ω « 1/τd. For adiabatic pumping one needs to vary at least two independent parameters, say, X1 and X2, to induce a dc current. In Figure 1 we show schematically how a quantum pump can be implemented with a lateral quantum dot. Figure 1

A quantum dot electron pump. The dark grey elements represent electrodes and the arrows indicate electron flow through the dot-reservoir contact regions. The two gate voltages X1(t) and X2(t) act as pumping parameters, continuously deforming the dot shape. On the right-hand side: a pumping cycle in parameter space. The voltages sweep a closed area A with contour Σ in parameter space. After a cycle is completed, a net charge is transferred between the two reservoirs

The quantum pumping current described above contained spin-degenerate electrons. Therefore, the contributions coming from ‘up’ and ‘down’ spin components of the charge current had identical direction and amplitude, leading to zero net spin transport. In order to generate a net spin flow out of spin-depolarised electron sources, one needs to add a spin-symmetry breaking field to the system. There are two simple ways to do that: •

creating a Zeeman splitting in the spectrum by applying an external magnetic field, or

•

using a pump with strong spin-orbit coupling.

While the latter has been recently studied theoretically in different contexts [8,9], it remains very challenging to implement experimentally. Here we will focus on the former case [10], which has already been tested and shown to produce clear evidence of spin-polarised transport [11].

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The main idea can be understood through the scheme shown in Figure 2. Since the materials underlying reservoirs and quantum dot is the same, upon the application of an external magnetic field B, energy levels inside and outside the dot will be spin split by the Zeeman energy Ez = g*µBB/2, where g* is the effective gyromagnetic factor and µB is the Bohr magneton. Let us assume that Ez is much smaller than the Fermi energy yet sizeable in comparison to the mean level spacing inside the dot, ∆. The Zeeman splitting in the reservoirs amounts to a small shift in the wavelengths of the ‘up’ and ‘down’ electron states at the Fermi surface and nothing else. However, the effect in the quantum dot states can be much more pronounced. Since there are marked differences in the spatial distribution of eigenfunctions of states even if they are close in energy, changing the orbital content of states near the Fermi level will strongly affect the pumping current. Recall that the matrix elements of the scattering matrix fluctuate in energy for systems, which have a chaotic dynamics in the classical limit. Therefore, by having EZ > ∆ we make the ‘up’ and ‘down’ components of the pumping current close to uncorrelated for a chaotic pump. If we define charge and spin pumping currents as [12]: IC = I↑ + I↓

(1)

I S = I↑ − I↓

(2)

and

respectively, we see that while IS(B = 0) ≡ 0 due to spin degeneracy, typically we have IS(B > BC) ≠ 0, with BC being a characteristic Zeeman field related to the pumping current correlation energy, EC : BC = EC/g*µB. For weakly coupled quantum dots at very low temperatures EC = ∆. However, energy levels in open dots are broad resonances instead of sharp discrete levels. Moreover, lowest achievable temperatures with present technologies are comparable to the level spacing found in all but the smallest quantum dots. Thus, in the general case, EC = max{kB, T, ∆, ħγ}, where γ is the electron escape rate (i.e., the inverse dwell time). This is scale is not too large: For a ballistic GaAs quantum dot with a 1 µm in linear size and one open propagating channel per lead, with electron densities of 1011 cm2 (that are standard for in high-quality GaAs wafers), one usually finds ∆ = πħγ ≈ 1 µeV, leading to BC ≈ 1 T at temperatures below 100 mK. In practice, one should avoid using a magnetic field perpendicular to the 2DEG underlying a lateral quantum dot. This is because even at fields of about 1 T there is already a significant reduction on the sensitivity of the wave functions to external perturbations (such as shape distortions) due to the formation of Landau states. This in turn reduces the dependence of the scattering matrix elements on parametric driving and, consequently, decreases the pumping current amplitude. Thus, a parallel magnetic field that only couples significantly to the electron spin and leaves the orbital motion unaltered is a more sensible choice for producing spin-polarised currents. The drawback is that this choice limits the spin polarisation to only one direction.

486 Figure 2

E.R. Mucciolo and C.H. Lewenkopf Schematic view of the states involved in the quantum pumping current through a dot in the presence of a Zeeman energy splitting EZ of the order of the mean level spacing ∆. The solid arrows indicate spin orientation. The Zeeman splitting does not significantly modify the orbital nature of the states in the reservoirs found near the Fermi level. However, if a and b are states with distinct wave function content, the splitting can make I↑ ≠ I↓ (for colours see online version)

We have argued that phase coherence combined with wave function sensitivity to parametric changes make ‘up’ and ‘down’ spin components of an adiabatic pumping current nearly independent when even a moderate magnetic field is applied. This effect can be explored to produce dc spin transport with zero net charge transfer, the so-called pure spin current. The mechanism is illustrated in Figure 3. The idea here is again based on the large sensitivity of confined quantum states to parametric changes when the underlying electronic motion is classically chaotic. If a third tuning parameter, X3 is provided besides the other two used to drive the system adiabatically, X1 and X2, such that H = H(X1, X2, X3), one can try to search for a realisation of the Hamiltonian when ‘up’ and ‘down’ spin components of the pumping current have the same amplitude but opposite directions. This point is denoted in Figure 3 by XPS. Notice that IS ≠ 0 while IC = 0 at this point. Figure 3

Schematic plot of dc pumping currents as functions of a quantum dot tuning parameter X. The plots in (a) and (b) show the spin up and spin down components of the pumping current, I↑,↓, the total charge IC, and total spin currents IS when no parallel magnetic field is applied, B|| = 0. Notice that in this case IS = 0. In plots (c) and (d) B|| ≠ 0, making I↑ ≠ I↓ and IS ≠ 0, in general. There are values of the tuning parameter, such as XPS, when a finite spin current occurs without net charge transport (IC = 0) (for colours see online version)

Spin pumping with quantum dots

3

487

Theory

When no spin-symmetric breaking field exists, both ‘up’ and ‘down’ spin components of the pumping current are identical. For an irregularly shaped quantum dot, as the amplitude of the external magnetic field is increased past the characteristic field BC the two spin components become uncorrelated. Since the ensemble averaged value of the dc pumping current is zero in the absence of bias, 〈I↑〉 = 〈I↓〉 = 0, we can write that  I 2 , B = 0 corr↑↓ ( B ) ≡ I ↑ I ↓ =  ↑ 0, B  BC .

(3)

For intermediate values of the magnetic field, the correlation interpolates monotonically between the two limiting values. The full spin polarisation of the pumping current only occurs at those special configurations where I↑ = I↓ exactly, regardless to how large the magnetic field is (provided it is nonzero). For all other configurations, the polarisation will be smaller, random, and dependent on the magnetic field. In order to quantify the typical amplitude of the spin current in comparison to the charge current, we introduce the quantity rpol = 〈 I S2 〉 / 〈 I C2 〉 . Both charge and spin currents can be written in terms of the

correlation function corr↑↓(B). It is then straightforward to show that rpol ( B ) ≡

corr↑↓ (0) − corr↑↓ ( B)

0, B = 0 = corr↑↓ (0) + corr↑↓ ( B) 1, B >> BC.

(4)

This means that a substantial spin polarisation may be achieved for sufficiently large magnetic fields, as we have anticipated using qualitative arguments in Section 2. Therefore, it is important to be able to quantify this effect, as well as the characteristic dependence of the spin current amplitude on temperature and on the number of propagating channel in the leads. The non-trivial aspect of evaluating corr↑↓(B) is that one needs to find a formulation where information about the energetics inside the dot can be incorporated. In other words, one has to be able to track down how wave functions and energy levels depend not only on the driving parameters but also on how scattering matrix elements corresponding to different spin orientations become progressively uncorrelated as the Zeeman splitting widens. Thus, a microscopic Hamiltonian formulation is unavoidable and the calculation cannot be performed within the static, random scattering matrix approach used by Brouwer [5]. Luckily, however, there are at least two ways to carry out the analytical calculation of corr↑↓(B). Moreover, it is also possible to complement the analytical calculations with numerical simulations when the former becomes too involved. One of us authored the first to study this problem [10] where it was used a suitably adaptation of the formalism developed by Vavilov et al. for the spinless charge pumping [13] variance. Their approach used the non-equilibrium Keldysh technique to relate the instantaneous pumping current to the real-time scattering matrix of the dot using. Being very general and applicable beyond the adiabatic approximation, this formulation is rather sophisticated. In what follows we show that it is possible to obtain the pumped spin

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current in the adiabatic regime in simple way, using the Keldysh formalism and the equations of motion method. Before going into the details of the calculation, it is important to highlight the assumptions used in the model. Vavilov and coworkers computed the pumping current assuming that: •

the electron eigenstates of a non-interacting dot could be described the unitary ensemble of Gaussian random matrix

•

that the leads contained many propagating channels.

These assumptions are well justified for the case of open dots with sufficiently complicated geometry, large contacts, and in the presence of a time-reversal symmetric breaking field. In practice, the latter condition can be enforced by letting the external magnetic field have a weak (tens of mT) perpendicular component. Following [13], let us divide the total Hamiltonian of the total system into three parts, H total = H dot + H leads + H dot-leads .

(5)

For the dot Hamiltonian we have H dot =

M



∑ ∑  Hnm + δ nm

n , m =1 σ =±1

σ EZ 

+

anσ amσ 2 

(6)

where am+σ (and amσ) are creation (and annihilation) electron operators defined over a single-particle basis of size M and {Hmn} denote matrix elements of the orbital contribution to the electron energy on that basis. The spin-dependent, diagonal term accounts for the Zeeman energy. Notice that H = H(t). For the leads Hamiltonian we have H leads =

N



∑ ∑ ∑  Eα (k ) +

α =1 k σ =±1

σ EZ 

+

cα kσ cα kσ 2 

(7)

where cα+kσ (and cakσ) are creation and annihilation electron operators in the leads, the index α runs over all B = NR + NL propagating channels in the right and left leads, and Eα(k) is the electron energy dispersion relation in the channel α. Finally, for the dot-leads coupling Hamiltonian, we have M

N

H dot-leads = ∑ ∑ ∑

∑ (Wnα cα+kσ a nσ + H.c.)

(8)

n =1 α =1 k σ =±1

where Wnα are coupling matrix elements related to the overlap between dot and lead single-particle wave functions at the contact regions. These matrix elements are assumed energy and spin independent, as the electronic density of states ρR(L) at the left and right leads.

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489

The pumped current can be exactly written in terms of the dot Green’s function [14] J Lσ (t ) = −

2e dω Γ  Im tr  L Gσ< (t , t ) + ∫ fσ (ω )∫ dt ′e − iω (t ′ − t ) / = Γ L Gσr (t , t ′ )  π = 2 2  

(9)

where [Γ L ]nm = 2πρL ∑ α ∈L Wα nWα*m and fσ(ω) is the Fermi function with the chemical potential shifted by the Zeeman energy. The Green’s functions are defined as standard: < + the lesser one reads Gmn ,σ (t , t ′ ) = i 〈 d mσ (t ′ ) d nσ (t ) 〉 , whereas the retarded one is r + Gmn ,σ (t , t ′ ) = −iΘ(t − t ′ )〈{d mσ (t ), d nσ (t ′ )}〉. The Green’s functions are determined by the

Dyson equation:      G (t , t ′ ) = g (t , t ′ ) + ∫ dt1dt2 G (t , t1 )Σ (t1 , t2 ) g (t2 , t ′ )

(10)

in matrix notation   Ar A=  0

A<   Aa 

(11)

 for the bare ğ, the full Green’s function Ğ, and the self-energy Σ . The problem becomes very amenable due to the simple structure of the bare Green’s function, whose retarded component Fourier transform is gr(ω,t) = (ω – H(t) + i0)–1. In addition, the self-energy can be written in a simple closed form as ∑ mn,σ (t , t ′) = ∑α Wα m,σ g α kσ (t − t ′)Wα n,σ . The latter bare Green’s function refers to electrons propagating in the leads, and gαr kσ (ω ) = (ω − ε α kσ + i 0) −1. We proceed analytically by exploring the fact that the time scales coming into play are very different. Following the strategy proposed in [15,16], we write the time dependent Green’s functions as   G (t , t ′ ) = G (t − t ′, t ) (12)

where t = ((t + t ′ )) / 2 is associated with slow variations. Writing the Dyson equation in terms of the new time variables, using the explicit form of g, expanding g (t − t ′, t + δ t ) = g (t − t ′, t ) + δ t ∂g (t − t ′, t ) ∂ t and taking the Fourier transform with respect to the fast time variations, t – t′, we obtain    G0 (ω , t ) = [ g −1 (ω , t ) + Σ (ω )]−1 (13) and



    i=  ∂ 2 g (ω , t ) i= ∂ 2 G0 (ω , t )   G1 (ω , t ) = − G0 (ω , t )Σ (ω ) + Σ (ω ) g (ω , t ) 2 ∂ω∂ t 2 ∂ω∂ t 



(14)

where G ≈ G0 + G1. That is, we approximate Ğ by a sum of its equilibrium part Ğ0 and the first order adiabatic correction due to the time dependent driving Ğ1. Since the equilibrium Green’s function obeys the fluctuation-dissipation theorem, the above equations allow us to evaluate Ğ0 + Ğ1 in a closed analytic form. The current can be cast as

490

E.R. Mucciolo and C.H. Lewenkopf  ∂G r (ω , t ) r a   ∂f  Σ Gσ 0 (ω , t )  J Lσ (t ) = −e∫ dω  − σ  Im Tr  Γ L σ 0  ∂ω  ∂t    ∂fσ  I (ω , t ). ≡ − e ∫ dω  −  ∂ω  Lσ

(15)

We recall that the time-dependence of the Hamiltonian H = H(X1, X2) comes from the functions X1(t) and X2(t). Thus, I Lσ (ω , t ) =

dX 1 dX ALσ ;1 + 2 ALσ ;2 dt dt

(16)

where ALσ ;i (ω , X 1 , X 2 ) =

 ∂S  1 Im Trα ∈L  σ Sσ+  2π  ∂X i 

(17)

and Sσ (ω , X 1, X 2 ) = I − 2π iWGσr 0 (ω , X 1, X 2 )W + is the standard resonance scattering matrix [17], whose elements are the amplitudes of the electrons scattered by the quantum dot. Notice that equation (17) recovers exactly the result obtained by Büttiker et al. [18] by very different means. We are left to evaluate a time integral to compute the spin transmitted during a pumping cycle. By using Green’s theorem it is possible to convert this time integral into a an area integral over parameter space, namely 〈 J Lσ 〉cycle = −

 ∂Sσ ∂Sσ+   ∂ fσ  − d ω d d Im Tr X X 1 2 L α ∈  ∂ X ∂X  π ∫  ∂ω  ∫A  2 1

eΩ

(18)

that is the spin analogue for the adiabatic charge pumping expression derived in [5]. The surface integral in equation (19) suggests a geometric nature of quantum pumping in the adiabatic limit [19]. This aspect of the problem has been nicely explained by several authors [20–22]. Our approach, based on non-equilibrium Green’s functions is slightly more general, since it allows addressing heating and dissipation – key for proposing a device – in a direct manner. We have now all necessary elements to compute corr↑↓(B). One possibility is to try to model a given quantum dot the most accurate way possible. As a second step one would try to optimise its area and shape to produce a large pure spin current. This is a costly calculation. Therefore, we pursue a different strategy here. We first use a statistical model to obtain a figure of merit to support further experimental and theoretical. The dot single-particle Hamiltonian is separated into static and time-dependent or driving terms, H(t ) = H0 + λ1 (t ) X 1 + λ2 (t ) X 2 .

(19)

The static or unperturbed term H0 is taken to be a member of the Gaussian unitary ensemble with variance 〈|[H0]mn|2〉 = M∆2π2, where M » 1 is the matrix rank and ∆ is the level space near the band centre and where the Fermi level is located. The scalar functions λ1(t) and λ2(t) modulate the amplitude of the time-dependent perturbation. We choose, as it is usually done, a pair of phase-locked harmonic functions:

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491

λ1(t) = cos(Ωt) and λ2(t) = cos(Ωt + ϕ). It turns out that for sufficiently large dots, the

diagonal parts of X1 and X2 are strongly suppressed and these matrices can be taken traceless. The strength of the perturbation is then fully characterised by only three numbers, namely Tr( X 12 ) , Tr( X 22 ) and Tr(X1X2). These quantities can be related to the so-called ‘velocity’ correlator [23], which measures how energy levels in the quantum dot respond to a linear, static perturbation of the form λ1X1 + λ2X2: 2Tr( X i X j ) M

2

=

∂ε a ∂ε a ∂ε a − ∂λ i ∂ λ j ∂λ i

∂ε a ∂λ j

(20)

where {εa(λ1, λ2)} are the energy eigenvalues of the isolated quantum dot. This velocity correlator can be measured, thus providing information about the traces. After presenting theses elements, it becomes clear that the calculation of corr↑↓(B) involves an average over 4-point S-matrix elements, or 4-point Green’s functions. This can be done either by a diagrammatic technique, suitable for N » 1, or by numerical simulations. In the adiabatic bilinear regime, the leading term for the spin current correlator in inverse powers of N becomes relatively compact, namely ∞

corr↑↓ ( B ) =

e 2 Ω 2 gτ d det(C ) sin 2 ϕ −τ / τ d ∫ dτ e 2π 0 2

  kT τ / ђ cos(τ EZ / 2ђ) ×   sinh(π kT τ / ђ) 

 τ   1 + τ  d

(21)

where g = NRNL/N is the dot dimensionless conductance, τd = 2πħ/N∆ is the dwell time, and Cij =

π M 2∆

Tr( X i X j ).

(22)

It is important to remark that equation (21) can also be derived using a semiclassical, trace formula representation of the scattering matrix. In that case, the validity of the semiclassical formulation is guaranteed by the large number of channels in the leads and the presumed chaotic electronic motion inside the dot. Details of the semiclassical calculation can be found in [24]. In Figure 4 we show the resulting polarisation ratio as a function of magnetic field for different temperatures and escape rates.

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E.R. Mucciolo and C.H. Lewenkopf

Figure 4

4

The dependence of the Zeeman splitting energy on the relative spin polarisation for (a) different temperatures and (b) for different number of channels in the leads (for colours see online version)

Results and discussion

Once a spin-polarised current is generated through the mechanism presented in Section 3, the question that naturally arises is how to detect or measure the spin polarisation. One could imagine using ferromagnetic leads to spin filter the current, like polarisers are used to filter out any component of a light beam. However, adding ferromagnetic leads is not yet a realistic option for lateral quantum dot setups. A more appropriate and readily available option is to use quantum point contacts (QPCs) where the width of the constriction can be controlled by gate voltages [25]. A schematic view of such device connected to a spin pump is shown in Figure 5. Figure 5

(a) Schematic illustration of a quantum dot spin pump connected to a spin filtering point contact. The arrows indicate the direction of the current and (b) linear conductance of the point contact as a function of the constriction width in the absence and in the presence of a parallel magnetic field (for colours see online version)

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The detection works as follows. At sufficiently low temperatures and in the absence of a spin-splitting magnetic field, the conductance in the QPC is spin degenerate and presents steps at even values of the conductance quantum e2/h. When the spin-splitting field is turned on, the degeneracy is broken and intermediate steps appear at odd values of e2/h. In this case, if one operates the QPC in the range where just one propagating channel is allowed, only the ‘up’ spin component of the current will be transmitted and the ‘down’ component will be filtered out. When the quantum pump is set to work at a configuration where I↑ = –I↓ as to have a zero net flow of charge, the QPC will block I↓, resulting in IC ≠ 0 through the constriction a nonzero voltage drop across it (VQPC ≠ 0). However, if the QPC conductance is brought to the second plateau at 2e2/h, both spin components are allowed to flow, making IC = 0 and VQPC = 0 as well. Thus, by monitoring the voltage across the QPC the number of propagating channels goes from odd to even, it is possible to detect the spin current. When the pump works away from the pure spin configuration, I↑ and I↓ do not fully compensate each other and the charge current never drops to zero when the number of propagating channels in the QPC is even. This scheme was implemented experimentally by Watson et al. [11]. In their setup a moderate perpendicular magnetic field was used to focus the current flow into the QPC (the cyclotron radius was considerably larger than the linear size of the effective well inside the dot, thus Landau level quantisation was not sufficiently strong to impair parametric pumping.). Their results show clear evidence that spin polarisation is achieved by parametrically driving the quantum dot in the presence of a parallel magnetic field. The spin current observed corresponded to tens of ħ per cycle at a frequency of 10 MHz. While this seems to confirm at least qualitatively our proposal, a quantitative description requires taking into account rectification [26,24] and dephasing [27,24] effects. A complete test would also require collecting enough statistics to compare the statistics of the spin current to the theoretical predictions based on random matrix theory, as discussed in Section 3. This has yet to be done.

5

Conclusions

In this paper we have argued that adiabatic pumping in open, lateral quantum dots can be used effectively to generate spin-polarised currents. The method is based on the phase coherence and the sensitivity of wave functions and energy levels inside the dot to changes in the shape of the confining potential. We also presented a new derivation of the pumping current, based on the non-equilibrium Green’s functions technique, which opens the possibility of estimating the dissipation and optimising the pumping parameters. Recently, this spin pump proposal passed its first experimental test. We hope that more groups will become interested in this subject.

Acknowledgements We thank C. Chamon, C. Marcus, and M. Martínez-Mares for the fruitful collaboration on this subject. This work was supported by CNPq, FAPERJ, and the Instituto do Milênio de Nanotecnologia (CNPq).

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References and Notes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

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