The shot noise in the electrical current through a ballistic chaotic quantum
dot with N-channel point contacts is suppressed for N --> infinity, because of
the transition from stochastic scattering of quantum wave packets to
deterministic dynamics of classical trajectories. The dynamics of the electron
spin remains quantum mechanical in this transition, and can affect the
electrical current via spin-orbit interaction. We explain how the role of the
channel number N in determining the shot noise is taken over by the ratio
l_{so}/lambda_F of spin precession length l_{so} and Fermi wave length
lambda_F, and present computer simulations in a two-dimensional billiard
geometry (Lyapunov exponent alpha, mean dwell time tau_{dwell}, point contact
width W) to demonstrate the scaling (lambda_F/l_{so})^{1/alpha tau_{dwell}} of
the shot noise in the regime lambda_F << l_{so} << W.Comment: 4 pages, 3 figure

Bardarson, J. H.

Beenakker, C. W. J.

Ossipov, A.

Titov, M.

Tworzydlo, J.

English

arXiv

PACS. 73.50.Td – Noise processes and phenomena

A. Ossipov

J. H. Bardarson

M. Titov

C. W. J. Beenakker

CiteSeerX

How spin-orbit interaction can cause electronic shot noise

The shot noise in the electrical current through a ballistic chaotic quantumdot with N-channel point contacts is suppressed for N -> ~, because of the transition from stochastic scattering of quantum wave packets to deterministic dynamics of classical trajectories. The dynamics of the electron spin remains quantum mechanical in this transition, and can affect the electrical current via spin-orbit interaction

Bardarson, Jens Hjörleifur

Beenakker, Carlo W. J.

KOPS - The Institutional Repository of the University of Konstanz

Bardarson, J.H.

Beenakker, C.W.J.

NARCIS 

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Leiden University Scholary Publications

Europhys. Lett., 76 (1), pp. 115–120 (2006)DOI: 10.1209/epl/i2006-10228-0EUROPHYSICS LETTERS 1 October 2006How spin-orbit interaction can cause electronic shot noiseA. Ossipov1, J. H. Bardarson1, J. Tworzydlo2, M. Titov3and C. W. J. Beenakker11 Instituut-Lorentz, Universiteit Leiden - P.O. Box 95062300 RA Leiden, The Netherlands2 Institute of Theoretical Physics, Warsaw UniversityHoża 69, 00-681 Warsaw, Poland3 Department of Physics, Konstanz University - D-78457 Konstanz, Germanyreceived 12 June 2006; accepted in final form 7 August 2006published online 30 August 2006PACS. 73.50.Td – Noise processes and phenomena.PACS. 05.40.Ca – Noise.PACS. 05.45.Mt – Quantum chaos; semiclassical methods.Abstract. – The shot noise in the electrical current through a ballistic chaotic quantumdot with N -channel point contacts is suppressed for N → ∞, because of the transition fromstochastic scattering of quantum wave packets to deterministic dynamics of classical trajecto-ries. The dynamics of the electron spin remains quantum mechanical in this transition, and canaffect the electrical current via spin-orbit interaction. We explain how the role of the channelnumber N in determining the shot noise is taken over by the ratio lso/λF of spin precessionlength lso and Fermi wavelength λF , and present computer simulations in a two-dimensionalbilliard geometry (Lyapunov exponent α, mean dwell time τdwell, point contact width W ) todemonstrate the scaling ∝ (λF /lso)1/ατdwell of the shot noise in the regime λF  lso  W .Electrical conduction is not much affected typically by the presence or absence of spin-orbit interaction. A familiar example [1–4], the crossover from weak localization to weak anti-localization with increasing spin-orbit interaction, amounts to a relatively small correctionto the classical conductance, of the order of the conductance quantum e2/h. The relativesmallness reflects the fact that the spin-orbit interaction energy Eso is much smaller than theFermi energy EF , basically because Eso is a relativistic correction to the kinetic energy [5].In this paper we identify an effect of spin-orbit interaction on the electrical current that hasa quantum mechanical origin (like weak anti-localization), but which is an order-of-magnitudeeffect rather than a correction. The effect is the appearance of shot noise in a ballistic chaoticquantum dot with a large number N of modes in the point contacts. According to recenttheory [6–8] and experiment [9], the shot noise without spin-orbit interaction is suppressedexponentially ∝ exp[−τE/τdwell] when the Ehrenfest time τE  α−1 lnN becomes greater thanthe mean dwell time τdwell of an electron in the quantum dot. (The coefficient α is the Lya-punov exponent of the classical chaotic dynamics.) The suppression occurs because electronsfollow classical deterministic trajectories up to τE (in accord with Ehrenfest’s theorem, hencethe name “Ehrenfest time”). If τE > τdwell an electron wave packet entering the quantum dotis either fully transmitted or fully reflected, so no shot noise appears [10].c© EDP SciencesArticle published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2006-10228-0116 EUROPHYSICS LETTERSWLL- -+Fig. 1 – Splitting of trajectories by spin-orbit interaction in an electron billiard. (The dotted arrowsindicate the spin bands, with ± helicities.) The splitting produces shot noise if not all trajectoriescan exit through the same opening.The electron spin of ± 12 h̄ remains quantum mechanical in the limit N → ∞. In thepresence of spin-orbit interaction the quantum mechanical uncertainty in the spin of theelectron is transferred to the position, causing a breakdown of the deterministic classicaldynamics and hence causing shot noise. The mechanism for the spin-orbit-interaction-inducedshot noise is illustrated in fig. 1. The key ingredient is the splitting of a trajectory uponreflection with a hard boundary [11].Whether a boundary is “hard” or “soft” depends on the relative magnitude of the pen-etration depth ξ into the boundary and the spin-orbit precession length lso = hvF /Eso λFEF /Eso. A soft boundary has ξ  lso, so the spin evolves adiabatically during the re-flection process [11, 12] and the electron remains in the same spin band, without splittingof the trajectory. In the opposite regime ξ  lso of a hard boundary the spin is scatteredinto the two spin bands by the reflection process. The energy splitting Eso of the spin bandsat the Fermi level amounts to a difference δp⊥  Eso/vF of the component of the momen-tum perpendicular to the boundary, and hence to a splitting of the trajectories by an angleδφso  δp⊥/pF  λF /lso. (A precise calculation of the splitting, which depends on the angleof incidence, will be given later.)Because of the chaotic dynamics, the angular opening δφso(t)  (λF /lso)eαt of a pair ofsplit trajectories increases exponentially with time t—until they leave the dot through one ofthe two point contacts after a time T . The splitting will not prevent the trajectories to exittogether through the same point contact if δφso(T ) < W/L, with W the width of the pointcontact and L the diameter of the (two-dimensional) quantum dot. The timeTso = α−1 ln(Wlso/LλF ) (1)at which δφso(Tso) = W/L is an upper bound for deterministic noiseless dynamics due tospin-orbit scattering.A. Ossipov et al.: How spin-orbit interaction can cause etc. 117Dwell times shorter than Tso may yet contribute to the shot noise as a result of diffraction atthe point contact, which introduces an angular spread δφpc  1/N  λF /W in the scatteringstates. The timeTpc = α−1 ln(WN/L) (2)at which this angular spread has expanded to W/L is an upper bound for deterministicnoiseless dynamics due to diffraction at the point contact [7]. The smallest of the two timesTso and Tpc is the Ehrenfest time of this problem,τE = α−1 ln[(W/L)min(N, lso/λF )], (3)separating deterministic noiseless dynamics from stochastic noisy dynamics. (By definition,τE ≡ 0 if the argument of the logarithm is < 1.) Since the distribution of dwell timesP (T ) ∝ exp[−T/τdwell] is exponential, a fraction∫ ∞τEP (T ) dt = exp[−τE/τdwell] of the elec-trons entering the quantum dot contributes to the shot noise.Following this line of argument we estimate the Fano factor F (ratio of noise power andmean current) as [6] F = 14 exp[−τE/τdwell], henceF =14(λFLlsoW)1/ατdwellifλFLW, ξ < lso < W. (4)The upper bound on lso indicates when diffraction at the point contact takes over as thedominant source of shot noise, while the two lower bounds indicate when full shot noise hasbeen reached (Fano factor 1/4) and when the softness of the boundary (penetration depth ξ)prevents trajectory splitting by spin-orbit interaction.Equation (4) should be contrasted with the known result in the absence of spin-orbitinteraction [6, 7]:F =14(LNW)1/ατdwellifλFLW< W < lso. (5)Clearly, the role of the channel number N in determining the shot noise is taken over by theratio lso/λF once lso becomes smaller than W .We support our central result (4) with computer simulations, based on the semiclassicaltheory of refs. [13–15]. In the limit λF → 0 at fixed lso, L,W a description of the electrondynamics in terms of classical trajectories is appropriate. For the spin-orbit interaction wetake the Rashba HamiltonianHRashba = (Eso/2pF )(pyσx − pxσy), (6)with Pauli matrices σx and σy. The two spin bands correspond to eigenstates of the spincomponent perpendicular to the direction of motion p̂ in the x-y plane (dotted arrows infig. 1). The ± helicity of the spin direction n̂ is defined by n̂ × p̂ = ±ẑ. The correspondingwave vectors arek± =√k2F + k2so ∓ kso, (7)with kso = Eso/2vF h̄ = π/lso.We consider the stadium-shaped billiard shown in fig. 1 with hard-wall confinement (ξ →0). Since λF  L we can neglect the curvature of the boundary when calculating the splittingof the trajectories by spin-orbit interaction [11]. The two reflection angles χ± ∈ (0, π/2) of the118 EUROPHYSICS LETTERSsplit trajectory, measured relative to the inward-pointing normal, are related by conservationof the momentum component parallel to the boundary,k+ sinχ+ = k− sinχ−. (8)An incident trajectory of− helicity is not split near grazing incidence, if χ− > arcsin(k+/k−)≈π/2− 2√kso/kF. Away from grazing incidence the probability Rσσ′ = |rσσ′ |2 for an electronincident with helicity σ′ at an angle χσ′ to be reflected with helicity σ at an angle χσ isdetermined by the 2× 2 unitary reflection matrixr =(r++ r+−r−+ r−−), (9a)r++ =eiχ+ − e−iχ−e−iχ+ + e−iχ−, r−− =eiχ− − e−iχ+e−iχ+ + e−iχ−, (9b)r+− = −2√cosχ+ cosχ−e−iχ+ + e−iχ−= r−+. (9c)The reflection matrix refers to a basis of incident and reflected plane waves that carry unitflux perpendicular to the boundary, calculated using the proper spin-dependent velocity op-erator [16].By following the classical trajectories in the stadium billiard, and splitting them uponreflection with probabilities Rσσ′ , we calculate the probability f(x, y, p̂) that an electron at10-610-510-410-310-2λF/lso10-1FW/L=0.1          0.2          0.30.1 0.2 0.3W/L00.050.1γslope γ(a)(b)Fig. 2 – (a) Dependence of the Fano factor on the spin-orbit interaction strength for different widthsof the opening in the billiard. The data points are calculated from eq. (10). The linear fits in thelog-log plot (dashed lines) confirm the predicted scaling logF ∝ log(λF /lso). (b) Filled circles: slopeγ = d logF/d log(λF /lso) extracted from panel (a). The empty circles are the theoretical predictionγ = 1/ατdwell.A. Ossipov et al.: How spin-orbit interaction can cause etc. 119-1.5 -1 -0.5 0(W/L)log10(λFL/lsoW)-1.2-1-0.75log 10F λF/lso=10-2           10-3           10-4           10-5           10-6Fig. 3 – Dependence of the Fano factor on W/L for different fixed values of λF /lso. The data pointsfollow closely the predicted scaling logF ∝ (W/L) log(λF L/lsoW ).position x, y with direction p̂ of its momentum originated from the upper left opening [17].The Fano factor is then given by [13–15]F =∫dΩ f(1− f)2∫dΩ f, (10)where dΩ = dxdy dp̂.The results of the simulations are presented in figs. 2 and 3. We first varied λF /lso atfixed W/L to test the scaling F ∝ (λF /lso)1/ατdwell predicted by eq. (4). We kept λF /lso  1,to ensure that the classical Lyapunov exponent α = 0.86 vF /L [18] and mean dwell timeτdwell ∝ L2/vFW (calculated numerically) are not affected significantly by the spin-orbitinteraction. The log-log plot in fig. 2a confirms the scaling logF ∝ log(λF /lso). The slopeγ, plotted in fig. 2b as a function of W/L (filled circles), is close to the predicted theoreticalvalue γ = 1/ατdwell (empty circles) if the ratio W/L becomes sufficiently small. There is noadjustable parameter in this comparison of theory and simulation. We then tested the scalingF ∝ (L/W )1/ατdwell at fixed λF /lso. The data points in fig. 3 all fall approximately on astraight line, confirming the predicted scaling law logF ∝ (W/L) log(λFL/lsoW ).This completes our test of the scaling (4) in the regime lso  W . The scaling (5), in theopposite regime lso  W , was verified in ref. [19] using the quantum kicked rotator. We havetried to observe the crossover from the scaling (4) to (5) in that model, but were not successful—presumably because we could not reach sufficiently large system size.In conclusion, we have identified and analyzed a mechanism by which spin-orbit interactionin a ballistic system can produce electronic shot noise. The origin of the current fluctuations isa quantum mechanical effect, the splitting of trajectories, which persists in the limit of classicalorbital dynamics. Since the strength of the Rashba spin-orbit interaction can be varied by agate voltage in a two-dimensional electron gas [20], the most natural way to search for theeffect would be to measure the shot noise as a function of the spin-orbit precession length120 EUROPHYSICS LETTERSlso. One would then see an increase in the Fano factor with decreasing lso, starting when lsodrops below the point contact width W . Since the splitting of trajectories requires lso to belarger than the boundary penetration depth ξ, the noise would go down again when lso dropsbelow ξ (assuming ξ  W ). This non-monotonic dependence of the noise on the spin-orbitinteraction strength would be an unambiguous signature to search for in an experiment. Inorder to observe the effect, an experimental system should be sufficiently clean to guaranteethat the noise induced by quantum short-range disorder [15] is weak enough.∗ ∗ ∗This problem originated from discussions with P. W. Brouwer and V. I. Fal’ko. We havealso benefited from discussions with H. Schomerus. Our research was supported by the DutchScience Foundation NWO/FOM and by the European Community’s Marie Curie ResearchTraining Network (contract MRTN-CT-2003-504574, Fundamentals of Nanoelectronics).REFERENCES[1] Bergmann G., Phys. Rep., 107 (1984) 1.[2] Brouwer P. W., Cremers J. N. H. J. and Halperin B. I., Phys. Rev. B, 65 (2002)081302(R).[3] Zumbühl D. M., Miller J. B., Marcus C. M., Campman K. and Gossard A. C., Phys.Rev. Lett., 89 (2002) 276803.[4] Zaitsev O., Frustaglia D. and Richter K., Phys. Rev. Lett., 94 (2005) 026809.[5] Winkler R., Spin-orbit Coupling Effects in Two-Dimensional Electron and Hole Systems(Springer, Berlin) 2003.[6] Agam O., Aleiner I. and Larkin A., Phys. Rev. Lett., 85 (2000) 3153.[7] Silvestrov P. G., Goorden M. C. and Beenakker C. W. J., Phys. Rev. B, 67 (2003)241301(R).[8] Whitney R. S. and Jacquod Ph., Phys. Rev. Lett., 94 (2005) 116801; 96 (2006) 206804.[9] Oberholzer S., Sukhorukov E. V. and Schönenberger C., Nature, 415 (2002) 765.[10] Beenakker C. W. J. and van Houten H., Phys. Rev. B, 43 (1991) R12066.[11] Govorov A. O., Kalameitsev A. V. and Dulka J. P., Phys. Rev. B, 70 (2004) 245310.[12] Silvestrov P. G. and Mishchenko E. G., cond-mat/0506516.[13] Blanter Ya. M. and Sukhorukov E. V., Phys. Rev. Lett., 84 (2000) 1280.[14] Pilgram S., Jordan A. N., Sukhorukov E. V. and Büttiker M., Phys. Rev. Lett., 90(2003) 206801.[15] Sukhorukov E. V. and Bulashenko O. M., Phys. Rev. Lett., 94 (2005) 116803.[16] Molenkamp L. W., Schmidt G. and Bauer G. E. W., Phys. Rev. B, 64 (2001) 121202(R).[17] It is equivalent and computationally more efficient to use Birkhoff coordinates s, p‖, with s theposition along the boundary and p‖ the component of the momentum parallel to the boundary.Then eq. (10) holds with dΩ = ds dp‖.[18] Dellago Ch. and Posch H. A., Phys. Rev. E, 52 (1995) 2401.[19] Tworzydlo J., Tajic A., Schomerus H. and Beenakker C. W. J., Phys. Rev. B, 68 (2003)115313.[20] Nitta J., Akazaki T., Takayanagi H. and Enoki T., Phys. Rev. Lett., 78 (1997) 1335.

The shot noise in the electrical current through a ballistic chaotic
quantum dot with N-channel point contacts is suppressed for
$N\rightarrow\infty$, because of the transition from stochastic
scattering of quantum wave packets to deterministic dynamics of
classical trajectories. The dynamics of the electron spin remains
quantum mechanical in this transition, and can affect the electrical
current via spin-orbit interaction. We explain how the role of the
channel number N in determining the shot noise is taken over by
the ratio $l_\textrm{so}/\lambda_{F}$ of spin precession length
$l_\textrm{so}$ and Fermi wavelength $\lambda_{F}$, and present
computer simulations in a two-dimensional billiard geometry
(Lyapunov exponent α, mean dwell time $\tau_\textrm{dwell}$,
point contact width W) to demonstrate the scaling
$\ensuremath{\propto(\lambda_{F}/l_\textrm{so})^{1/\alpha\tau_\textrm{dwell}}}$
of the shot noise in the regime $\lambda_{F}\ll l_\textrm{so}\ll W$

J. Tworzydło

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How spin-orbit interaction can cause electronic shot noise

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