Spin polarized current generation from quantum dots without magnetic fields

An unpolarized charge current passing through a chaotic quantum dot with spin-orbit coupling can produce a spin polarized exit current without magnetic fields or ferromagnets. We use random matrix theory to estimate the typical spin polarization as a function of the number of channels in each lead in the limit of large spin-orbit coupling. We find rms spin polarizations up to 45% with one input channel and two output channels. Finite temperature and dephasing both suppress the effect, and we include dephasing effects using a new variation of the third lead model. If there is only one channel in the output lead, no spin polarization can be produced, but we show that dephasing lifts this restriction.


I. INTRODUCTION
The generation and control of spin polarized currents, in particular without magnetic fields or ferromagnets, is a major focus of recent experimental and theoretical work.This includes the spin Hall effect, which produces spin currents transverse to an electric field in a two-dimensional electron system (2DES) with spin-orbit coupling, with spin accumulation at the edges. 1 Similarly, the magnetoelectric effect 2,3,4 produces a steady state spin accumulation when an electric field is applied to a 2DES with spin-orbit coupling.The accumulation can be uniform 5,6 in the case of uniform Rashba spinorbit coupling 7 or at the edges of a channel in either the Rashba model 8,9,10 or with spin-orbit coupling induced by lateral confinement. 11,12Experiments have observed current induced spin polarization in n-type 3D samples 13 and in 2D hole systems 14,15 with spin polarization estimated to be up to 10%. 15Further work suggests a spin polarized current can be produced by a quantum point contact (QPC) with spin-orbit coupling, 16,17 in a carbon nanotube, 18 in a ballistic ratchet, 19 in a torsional oscillator, 20 or in vertical transport through a quantum well. 21ere we show that generating a polarized current from an unpolarized current is a generic property of scattering through a mesoscopic system with spin-orbit coupling.We propose using many-electron quantum dots (outside the Coulomb blockade regime) with spin-orbit coupling to produce partially spin polarized currents without magnetic fields or ferromagnets.Due to the complicated boundary conditions of the quantum dot, we do not solve for the spin polarization in terms of any particular spinorbit coupling model, geometry, and contact configuration.We estimate the effect for a ballistic system in the limit of strong spin-orbit coupling by performing a random matrix theory (RMT) calculation for the spin polarization, allowing consideration of realistic quantum dot devices robust to details of shape and contact placement.Finely tuned systems should be able to exceed these polarizations, but these results provide a useful benchmark for whether a particular tuned system is better than a FIG. 1: A tuned model quantum dot with N = 1 channels in the left lead and M = 2 channels in the right lead.A skew scatterer sends all ẑ spins to the top channel and all −ẑ spins to the bottom channel.The shaded area in the bottom channel has Rashba spin-orbit coupling of precisely the strength to rotate a down spin at the Fermi energy to an up spin, thus producing a spin polarized exit current from any input current, while respecting time reversal symmetry.
generic chaotic one.We use a density matrix formalism throughout, which allows us to develop straightforwardly a spin-conserving dephasing probe, using a new variant of the third-lead technique for accounting for dephasing.Dephasing and finite temperature both reduce the expected polarization.Without dephasing, we find that if there is only one outgoing channel then no spin polarization is possible, which was first shown by Zhai and Xu. 22nterestingly, with dephasing, spin polarization can be produced with only one outgoing channel.The case of polarized input currents will be discussed elsewhere. 23nalogous calculations have been performed by Bardarson, Adagideli, and Jacquod in a four-terminal geometry, to study the transverse spin current produced by an applied charge current. 24

II. SETUP AND SYMMETRY RESTRICTIONS
We consider non-interacting electrons in a quantum dot with two attached leads connected to large reservoirs.For any electron current entering from the leads, we can describe the output state in the leads in terms of the Smatrix of the dot, including any tunnel barriers between the leads and the dot.We assume negligible spin-orbit coupling in the leads and consider the lead on the left (right) to have N (M ) spin-degenerate channels at least partially open at the Fermi energy, and let K = N + M .As usual, the channel wavefunctions are normalized so all channels have the same flux.The S-matrix S is a 2K × 2K unitary matrix of complex numbers.For spin 1/2 particles with spin-orbit coupling, however, it is convenient to consider S to be a K × K matrix of quaternions.We give a brief introduction to quaternions and explain why they are easier to work with.
We choose a representation of the quaternions such that a quaternion q is a 2 × 2 matrix of complex numbers q = q (0) 1 1 2 + i 3 µ=1 q (µ) σ µ , where σ µ are the Pauli matrices and q (µ) ∈ C. We define three conjugates of q: complex conjugate q * = q (0) * 1 1 2 +i q (µ) * σ µ , quaternion dual q R = q (0) 1 1 2 − i q (µ) σ µ , and Hermitian conjugate By convention, the trace of the quaternion matrix ii .Given a K × K quaternion matrix Q, we can associate it with a 2K × 2K complex matrix A in the obvious way.Note then that Q † is equivalent to A † , the usual Hermitian conjugate of a complex matrix, but Q * is not equivalent to A * .Note further that the trace convention implies that trA = 2trQ.
The quaternion representation is convenient, as the time reversal operation for a scattering matrix can simply be written as S → S R . 25The S-matrix of a system with time reversal symmetry (TRS) is self-dual.
If w in (w out ) is the K × K quaternion density matrix of the incoming (outgoing) current, w out = Sw in S † .The density matrix describing the unpolarized incoherent combination of all N incoming channels is That is, w in = P L /2N where P L is the projection onto the channels of the left lead.We choose trw in = 1/2, due to the trace convention.The Landauer-Büttiker formula gives the conductance in terms of the S-matrix. 26We write the K × K quaternion S-matrix as r t ′ t r ′ with r (r ′ ) being the N × N (M × M ) reflection matrix and t (t ′ ) the M × N (N × M ) transmission matrix.Then we write the Landauer-Büttiker formula in units of 2e 2 /h as = tr(P R SP L S † ), = 2N tr(P R Sw in S † ), = 2N tr(P R w out ), where P R is the projection onto the channels of the right lead.Since w in is normalized to represent one input particle entering the system, g = 2tr(P R w out ) is the probability for that particle to exit through the right lead.The conductance is N times this probability, so we call g the conductance per channel in the left lead.
Similarly, we define a vector spin conductance 22 (i.e., exit spin current divided by voltage) G s in units of e/2π as = 2N tr( σP R w out ). Then is the spin conductance per channel in the left lead.Hence, g s µ is the µ-component of the spin polarization of the exit current times the probability of exiting into the right lead.Thus, the spin polarization of the current in the right lead is p = g s /g, with |p| ≤ 1.
We can, of course, construct g, g s , and p using only the S-matrix and not the density matrices w in and w out .The density matrix approach, however, gives the flexibility to consider arbitrarily correlated states of incoming current and also to look for arbitrary correlations in the outgoing current. 23We will also use it to straightforwardly derive a method of accounting for non-magnetic dephasing in a device with spin-orbit coupling.To complete the translation to the standard notation of conductances, we consider sending up-or down-polarized electrons into a sample and collecting either up-or down-polarized electrons, giving a conductance matrix 27 with the total charge conductance being G = G ↑↑ +G ↑↓ + G ↓↑ + G ↓↓ .G σ,σ ′ is the conductance for an input current of spin σ ′ and an exit current of spin σ, for σ, σ ′ =↑, ↓.We translate the quaternion representation into the standard notation by noting that the up-polarized incoherent input current has input density matrix w in ↑ = 1+σ3 2N P L .The output density matrix is w out ↑ = Sw in ↑ S † and the portion representing the output in the right lead is t 1+σ3 2N t † .The Landauer-Büttiker formula gives, in units of 2e 2 /h, Similarly, from which we see that G = tr(tt † ), which is the usual Landauer-Büttiker formula. 26hough there are several spin-orbit coupled systems that demonstrate spin polarization from unpolarized input, in many cases the effect is subtle. 16,17,19,28Here we give an idealized thought experiment showing that an unpolarized input current can produce a spin polarized output current.Consider a system with N = 1 and M = 2, as illustrated in Fig. 1.All input electrons are incident on a perfect skew scatterer which sends spins quantized in the +z direction into exit channel 1 and spins quantized in the −z direction into exit channel 2. Exit channel 2 has a region with Rashba spin-orbit coupling 7 which is precisely of the strength and length necessary to rotate −z spins to +z.Thus, all spins incident from the left lead exit with their spins up, and the system respects TRS, since skew scattering and Rashba spin-orbit interaction are each time reversal symmetric.
We illustrate by constructing S explicitly.We can express the scattering matrix for this thought experiment (up to an overall phase) in the 6 × 6 and 3 × 3 representations as where θ ∈ [0, 2π) and r and t have been determined by the above description, while the rest of the matrix is given by TRS and unitarity.The unpolarized input quaternion density matrix is , giving so Eq. 4 gives g s = p = ẑ, as stated above.We now prove that having at least two channels in the outgoing lead is essential.That is, for a dot with TRS and K channels in attached leads, if an unpolarized equally weighted incoherent current is sent into N = K − 1 of the channels, then the spin polarization in the remaining channel must be zero.This result has been shown before, 22 but the quaternion formalism with density matrices makes it particularly transparent, so we include the proof here.
We start with where P K is the projection onto the K th channel.The quaternion scattering matrix satisfies S = S R since TRS is unbroken, and Note that S = S R implies both S † = S * and S ii ∈ C for i = 1 . . .K.
Using Eq. 4, the spin conductance is g s µ = 2i[w out KK ] (µ) .In particular, if w out KK has no quaternion part, then g s µ = 0. We have and S KK S * KK is real, so w out KK ∈ R and g s = 0.This proof applies with channels that are fully open or have tunnel barriers, as it requires only that the S-matrix satisfy TRS and unitarity, which are unchanged by tunnel barriers.
We note further that if K > 2 then 1) the reflected current in any of the K − 1 input channels can be spin polarized, and 2) if the input current goes through less than K − 1 channels, then the remaining channels can have a spin polarization, as shown in the example of Fig. 1.

III. RANDOM MATRIX THEORY
We estimate the expected spin polarization in realistic situations by using random scattering matrix theory.We assume that the mean dwell time τ d of particles in the dot is much greater than the Heisenberg time τ H = 2π /∆, where ∆ = 2π 2 /mA is the mean orbital level spacing, m is the effective mass, and A is the area of the dot.We further assume the strong spin-orbit limit, where the spin-orbit time τ so is much less than τ d .For a chaotic quantum dot, τ d = mA/ K, where K is the number of fully open orbital channels attached to the dot, so for a sufficiently large A, even a material with "weak" spin-orbit coupling will be in the strong spin-orbit limit.The crossover from weak to strong spin-orbit coupling in chaotic quantum dots has been studied in the K ≫ 1 limit in the context of adiabatic spin pumping. 29or dots with strong spin-orbit coupling, we assume that the S-matrix is chosen from the uniform distribution of unitary matrices subject to TRS, called the circular symplectic ensemble (CSE). 25,30We find the root mean square (rms) magnitude of the spin conductance on averaging over the CSE, which gives the typical spin conductance magnitude to be expected from chaotic devices.Such an averaging can be realized in practice by small alterations of the dot shape. 31,32y symmetry, g s µ = 0 for µ = 1, 2, 3. Using Eq. 4, we evaluate (g s ) 2 = 4 tr(σ µ P R Sw in S † )tr(σ µ P R Sw in S † ) , (16)   where we sum over µ.
We use the technique for averaging over the CSE described by Brouwer and Beenakker in Section V of Ref. 33.We need just two generic averages, which we will use repeatedly.The first is of the form F 1 (S) = tr(ASBS † ) , where A and B are constant K ×K quaternion matrices and the average is taken over S chosen from the CSE of K × K quaternion self-dual matrices.Then 33 The second average we need is where A, B, C, D are constant K × K quaternion matrices and AB = AD = CB = CD = 0. We find 33 where Λ = K(2K − 1)(2K − 3).Using Eq. 18, we find where we used tr(σ µ P R ) = 0, tr(P 2 R ) = M , trw in = 1/2, and tr (w in ) 2 = 1/4N .Note that when M = 1, (g s ) 2 = 0, consistent with the general symmetry.
If we are interested in the mean square polarization of the exit current, p 2 = (g s ) 2 g −2 , we can approximate it by (g s ) 2 / g 2 .This approximate form is useful for analytical progress and will be compared to numerical results.Using Eq. 17, which, combined with Equation 19, gives We study the approximation (g s ) 2 g −2 ≈ (g s ) 2 / g 2 numerically.
We choose a 2K × 2K complex Hermitian matrix from the Gaussian unitary ensemble 25 and find the unitary matrix U which diagonalizes it.We multiply columns of U by random phases, map U into a K × K matrix of quaternions, and construct unitary self-dual S by setting S = U U R , giving S chosen from the CSE. 34igure 2 shows the numerical and analytical results, which agree quantitatively for g 2 and (g s ) 2 and qualitatively for p 2 .The largest percentage disagreement for p 2 is 7%.

IV. DEPHASING
We add dephasing to this setup using the dephasing voltage probe technique. 35,36,37We add a fictitious voltage probe drawing no current with N φ = 2π /∆τ φ fully open orbital channels, where τ φ is the dephasing time.We extend this model to preserve the spin of the reinjected electrons.
In contrast with previous work, we explicitly model reinjection of electrons from the voltage lead by modifying w in to include incoherent reinjection from the dephasing lead.The reinjection matches the total charge/spin current absorbed by the dephasing lead, but distributes FIG.2: (color online) Numerical (symbols) and analytical (lines) results for normalized mean conductance g , rms spin conductance g s , and rms spin polarization p of current exiting a chaotic quantum dot with N (M ) channels in the entrance (exit) lead.An average over 60000 S-matrices from the CSE was performed for each data point.The lines are from Eqs. 19-21.the charge/spin current evenly between the channels, and thus models dephasing processes that preserve electron spin.First, consider η µ = tr(σ µ P φ Sw in 0 S † ), where µ = 0, 1, 2, 3, σ 0 = 1 1 2 , P φ is the projection operator onto the dephasing lead's channels, and w in 0 is the input density matrix.Then 2η 0 is the probability for a particle to enter the dephasing lead, and 2 η is the spin conductance into the dephasing lead, which is proportional to the spin current into the dephasing lead.
We reinject from the dephasing lead with where we sum over repeated index µ.We set c 1 µ = η µ /N φ , which ensures the reinjected charge/spin current equals the absorbed charge/spin current.Some of this reinjected current reflects back into the dephasing lead, so it must be reinjected again.We define a 4 × 4 complex matrix Θ µν = tr(σ ν P φ Sσ µ P φ S † ), which gives the charge/spin current in the dephasing lead due to this reinjection.
FIG. 3: (color online) Numerical (symbols) and analytical (lines) results for normalized mean conductance g , rms spin conductance g s , and rms spin polarization p of current exiting a chaotic quantum dot with N = 1 channel in the entrance lead.M and N φ are the numbers of channels in the exit and dephasing leads, respectively.An average over 60000 Smatrices from the CSE was performed for each data point.The lines are from Eqs. 27 and 28.
Defining w in = w in 0 + w in φ , this procedure gives where we sum over repeated indices µ, ν = 0, 1, 2, 3.This result holds for any input current, not just the unpolarized incoherent w in 0 discussed here.We approximate w in φ by replacing Θ µν with its average in Eq. 23, similar to Eq. 21.Using Eq. 17, where We further replace tr(σ µ P φ Sw in S † ) by its average, which gives This turns out to be the same result as if we had chosen c 1 µ = 0 in Eq. 22 for µ = 1, 2, 3.That is, in the approximation of Eqs.24-25, if we have total spin decay in the dephasing lead, then Eq. 26 is unchanged.Note that Eq. 26 satisfies unitarity only on average; the total probability of exiting either through the right or left lead equals 1 only on average.In this approximation, we use Eq.18 to find where Note that even if there is only one outgoing channel, M = 1, the spin conductance is predicted to be nonzero due to dephasing.The dephasing induced spin conductance is present for N φ > 1, as shown in numerical simulations in Fig. 3 and is not an artifact of Eq. 26.In the case of N φ = M = 1, an exact treatment shows that g s = 0, contrary to Eq. 27, even with arbitrary tunnel barriers between the leads and the sample.As shown in Fig. 3, Eq.27 works well for M > 1 or N φ > 1.
We can modify this model to have N φ dephasing leads each with one channel, each separately reinjecting the same charge/spin that it absorbs.In this model, too, a nonzero g s can be produced for N φ > 1 (not shown).
Brouwer and Beenakker modified the third-lead dephasing model to make dephasing uniform in phase space by placing a tunnel barrier with transparency Γ between the dephasing lead and the dot, with Γ → 0 and N φ → ∞ while maintaining ΓN φ = 2π /∆τ φ . 38The S-matrix is then not drawn from the CSE, and simple analytical results in the spin-orbit coupled system are challenging.We study this model numerically and find that for fixed τ φ , it gives qualitatively similar results to the simpler model described above; in particular it also gives a nonzero spin current when M = 1 (not shown).Without a microscopic model of dephasing, it is possible that this dephasing induced spin current with M = 1 is an artifact of third lead dephasing models, but all three variants of third lead dephasing discussed here show this effect, so dephasing gives a loophole for producing spin currents even when M = 1.
Returning to the single dephasing lead with Γ = 1, we estimate p 2 as above, where we modify g to include the dephasing lead.Using Eqs. 17 and 26, this gives We estimate p 2 ≈ (g s ) 2 / g 2 , using Eqs.27 and 28.Comparison of these approximations to numerical evaluations is shown in Fig. 3. Again we find that the numerical and analytical results agree qualitatively, except when N φ = M = 1.

V. FINITE TEMPERATURE
If the temperature T > ∆, the polarization will be further suppressed by electrons of different energy feeling uncorrelated scattering matrices.This effectively increases the number of orbital channels, which decreases the residual polarization.We consider unpolarized incoherent flux from the left lead at temperature T .Adapting Datta, 39 we take w in (ǫ) = − ∂f ∂ǫ 1 2N

0M
, where f (ǫ) is the Fermi distribution.If the scattering matrix for particles of energy ǫ is S(ǫ), then w out (ǫ) = S(ǫ)w in (ǫ)S † (ǫ).We approximate S(ǫ) as correlated only within energy intervals of scale ∆ (see Ref. 40 for an equivalent treatment).That is, we take and which are valid only for T ≫ ∆, which is often true for chaotic quantum dots.For T ≈ ∆, S(ǫ)S † (ǫ) can be calculated using the random Hamiltonian method. 41 need an average over a new function, where AB = AD = CB = CD = 0 and f ′ = ∂f /∂ǫ.We evaluate the average of h with the K × K quaternion matrix S(ǫ) chosen from the CSE along with Eq. 30, giving, Using Eq. 31 in place of Eq. 18, we evaluate (g s ) 2 as above, which simply multiplies Eq. 19 by ∆ 6T .Also, g is unaffected by temperature, so Eq.21 is also multiplied by ∆/6T .

VI. DISCUSSION
This spin polarization should be able to be produced and detected experimentally.Even quantum dots in ntype GaAs/AlGaAs heterostructures have been observed to have sufficiently strong spin-orbit coupling to approach the RMT symplectic limit. 31,32If the spin-orbit coupling is not strong enough for the S-matrices of the dot to be drawn from the CSE, the spin polarization predicted here will be reduced but should still be present.In a given material with fixed spin-orbit coupling strength, a sufficiently large quantum dot will be well described by the CSE, with a possible increase in dephasing rate as the dot size increases.
At zero temperature, our discussion has assumed that all electrons passing through the dot see the same Smatrix, which is valid when the applied potential difference is less than the mean level spacing ∆.Since ∆ ∝ A −1 , as the dot area is increased to approach the strong spin-orbit limit, the window of voltages where these results apply shrinks.The effects predicted in this paper are most likely to be observable in a material with inherently strong spin-orbit coupling, such as p-type III/V heterostructures.Note that (g s µ ) 2 = (g s ) 2 /3 and p 2 µ = p 2 /3, so if a measurement technique or application is only sensitive to spin polarization along a particular axis, then the rms predictions for the µcomponent of the polarization and spin conductance are only √ 3 times smaller than the results stated above.We have shown that quantum dots with spin-orbit coupling can generate spin polarized currents without magnetic fields or ferromagnets, except with only one outgoing channel and TRS, when such a device cannot produce a spin current.Mesoscopic fluctuations can be large enough to give appreciable spin currents in devices with a small number of propagating channels.Even if the spin-orbit coupling is weak, a sufficiently large device will show these effects.