Towards a Dephasing Diode: Asymmetric and Geometric Dephasing

Robert S. Whitney,1 Alexander Shnirman,2,3 and Yuval Gefen4

1Institut Laue-Langevin, 6 rue Jules Horowitz, B.P. 156, 38042 Grenoble, France2Institut fur Theoretische Festkorperphysik and DFG-Center for Functional Nanostructures (CFN), Universitat Karlsruhe,

D-76128 Karlsruhe, Germany3Institut fur Theoretische Physik, Universitat Innsbruck, Technikerstr. 25, 6020 Innsbruck, Austria

4Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel(Received 10 September 2007; published 28 March 2008)

We study the effect of a noisy environment on spin and charge transport in ballistic quantum wires withspin-orbit coupling (Rashba coupling). We find that the wire then acts as a dephasing diode, inducing verydifferent dephasing of the spins of right and left movers. We also show how Berry phase (geometric phase)in a curved wire can induce such asymmetric dephasing, in addition to purely geometric dephasing. Wepropose ways to measure these effects through spin detectors, spin-echo techniques, and Aharanov-Bohminterferometry.

DOI: 10.1103/PhysRevLett.100.126806 PACS numbers: 73.63.Nm, 03.65.Yz, 73.23.Ad, 85.75.�d

A very promising idea for future (quantum or classical)information processing is ‘‘spintronics’’ [1– 4], where elec-trons’ spins (not their charges) are used to encode infor-mation. However, spins do not obey the same conservationlaws as charges; charges do not change sign, but spins canflip. Current conservation enforces symmetries on chargetransport (Onsager relations). For example, two-terminaldevices always have the same left-to-right and right-to-leftconductance in the linear-response regime of negligibleinteraction or charging effects (diodes do not exist withoutinteractions). By contrast, asymmetries between left-to-right and right-to-left spin-transport can occur in thelinear-response regime for two-terminal devices, if thereis spin-orbit coupling (Rashba or Dresselhaus). Coherenceis a crucial aspect of quantum transport, so here we inves-tigate analogous asymmetries in the dephasing (decay ofcoherence) of spins. We then study how Berry (geometric)phases—present in curved wires [5]—modify such asym-metric dephasing [6].

The coherence of a superposition of two spin states atthe Fermi-surface is quantified in terms of the purity, P �tr��2�, where � is a 2� 2 density matrix. A pure superpo-sition has maximal purity, P � 1, while an equal classicalmixture has minimal purity, P � 1=2. We use the term‘‘dephasing diode’’ for a two-terminal device in which spinsuperpositions of left movers experience very differentdephasing from right movers. For an ideal dephasing di-ode, an electron injected into the device from the left lead(a right mover) in an equal coherent superposition of spinstates (P � 1) would emerge completely dephased (P �1=2), while an electron injected into the device from theright (a left mover) in any superposition with P � 1 wouldemerge at the left without being dephased at all (stillhaving P � 1). This dephasing could be observed by mea-suring either certain spin components of the current, or by aconventional current measurement in an Aharonov-Bohm(AB) interferometer. In this Letter, we provide illustrationsof dephasing diodes with straight and curved ballistic

wires. For the latter, the Berry phase gives a geometry-induced contribution to dephasing, whose sign depends onthe curvature’s sign.

Noise causes dephasing, and real devices have manysources of noise (thermal or quantum), including electron-electron and electron-phonon interactions. However, aclear experimental observation of asymmetric dephasingrequires control of the noise power (seeing the asymmetrychange with the noise power). Thus, we propose taking awire with low intrinsic noise, modeled by ballistic non-interacting electrons, and applying man-made noise to themagnetic fields and gates. The response time of gates ormagnets is typically longer than the time-of-flight of elec-trons from source to detector along a ballistic wire [7].Hence, we study the effect of extremely slow (man-madeor intrinsic) noise on noninteracting electrons with spin-orbit coupling. For simplicity, here we consider onlyRashba coupling, BR � @�pxey � pyex�=�mlR�, where lRis the spin-precession length [8], and magnetic fields are inunits of energy. Biasing a back-gate gives control over lRby modifying the potential gradient along the z-axis [9].Noise can be applied to either the applied magnetic-field orthe Rashba spin-orbit coupling (via a noisy voltage on theback gate). By measuring any three orthogonal spin polar-izations h�ii, i 2 f1; 2; 3g, one gets the purity P� 1

2�1�h�1i

2�h�2i2�h�3i

2� [10]. Our qualitative predictions aresummarized in Table I. Dresselhaus coupling yields similarresults, but it is hard to control experimentally (i.e., notaffected by the back gate) so we do not consider it furtherhere [12].

Asymmetric dephasing in a straight wire.—[SeeFig. 1(a).] We neglect the contribution of the motion acrossthe wire (y-direction) [13], so the Hamiltonian for anelectron in the lowest mode of the ballistic wire is

H � �2m��1p2x � E0 � @�mlR��1px�y �

1

2B �; (1)

where E0 is the transverse mode’s energy. Hence, the

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eigenmodes have momenta, p�, given by

0 � p4� � �2p

20 � 4�@=lR�

2�p2� � �2@mBy=lR�p�

� p40 � �mB�

2; (2)

where B � jBj and p0 � �2m�EF � E0��1=2. Here, � �

1 � �"; #� is the spin state, orientated along the effectivefield �Bx; 2@�mlR��1p� � By; Bz�, see Fig. 2.

Let us first consider Bx � Bz � 0, then the solutions ofEq. (2) are pR;L

� � �@=lR�� �p20 � �@=lR�

2 �mBy��1=2

with the upper and lower sign for R and L movers,respectively. We assume p2

0 >mBy, then an R or L moverin a superposition of two spin states, " and # (e.g., spin in thex-direction), acquires a phase difference of �R;L �

�jpR;L" j � jpR;L

# j�L=@ between the two spin states whentransversing the wire. There is no energy term because bothstates have the same energy. Given pR;L

� , we have �R;L �

���1By���1

�By 2l�1

R �L where the length scale �B �

@�p20 � �@=lR�

2 �mB��1=2. Noise smears this phase differ-ence causing dephasing. We consider Gaussian-distributednoise in B and lR which is much slower than the time-of-flight between source and detector. We use h i to indicatetaking the expectation value, and averaging over �Bi, �lRwith weight exp�� 1

2

Pi��Bi=�i�

2 � 12 ��lR=�R�

2�.Assuming that the noise is weak, �i � Bi and �R � lR,we can expand �L;R to first order in �Bi and �lR. Defining�1 as along the eigenbasis (here the y-axis), we have

h�2;3i / jhexp�i�R;L�ij

� exp��

1

8�L��By ���By�m�B=@2�2

�

� exp��

1

2�L��By ���By 2lR��R=lR

3�2�

(3)

This yields a Gaussian decay of the purity with L, i.e., withthe time-of-flight along the wire [11]. Such Gaussian de-cays are typical of slow noise (inhomogeneous broadening)[14]. Noise in B dephases the spin of R and L movers in

the same way (symmetric). However, noise in lR dephasesR movers differently from L movers [upper vs lower signin Eq. (3)]. This asymmetry can only be large if j�By �

��By j 2lR.In contrast, for By � 0, Eq. (2) is a quadratic equation

for p2�. For every L mover with momentum p�, there is a

R mover with momentum �p�. In this case, �R � �L,so there is no asymmetry in dephasing.

Geometric dephasing in a curved wire.—[See Fig. 1(b).]The electron travelling along the wire is subject to aspatially varying effective field (B� BR, where B �Bzez). We go to cylindrical coordinates (r, �, z) [15]dropping terms that go like �F=r. Transforming the spinorusing U� � exp�i 1

2 ��z� gives the Hamiltonian

H cyl �p2�

2m� E0 �

@p�mlR

�r ��Bz �

@p�2mr

��z (4)

where the �-dependence of the frame causes the p��zterm. The eigenmodes’ momenta, p� � p����, are givenby Eq. (2) with By � Bx � 0 and Bz replaced by �Bz ��2mr��1p��. We next assume a large radius of curvature,r� @p0��@p0=lR�

2 � �mBz=2�2��1=2 (so the frame’s an-gular velocity� precession rate in the effective field) andexpand p� � jp1� j � @c�r

�1, where p1� is the momen-tum for r � 1. To order 1=r, we obtain c� � � cos�where we define � as the angle between the z-axis and aneffective field f0; 2@�p2

0 � �@=lR�2�1=2=�mlR�; Bzg. This is

readily generalized to other wire shapes [cf. Figure 1(c)],with r varying along the wire coordinate, ~x, [r�~x� to be keptlarge]. Noting that �c" � c#� � 2c", we obtain

�R;L � @�1Z�jpR;L" j � jpR;L

" j�r�~x�d��~x�

� �jp1" j � jp1# j�L=@ 2c"�: (5)

The first term in �R;L goes like the wire length, L (i.e.,proportional to the time-of-flight along the wire), and ishence a dynamic phase. The second term is proportional to

TABLE I. Contributions to dephasing for the various systems: ‘‘none’’ indicates the absence of dephasing [11]; ‘‘sym.’’ (‘‘asym.’’)indicates that the terms are symmetric (antisymmetric) under right movers $ left movers. Terms labeled ‘‘strong’’(‘‘weak’’) are ofsimilar size to (much less than) the main symmetric term, for typical experimental parameters. ‘‘Geom.-induced’’ and ‘‘purelygeometric’’ indicate terms which vanish for a straight wire; the former go like L� while the latter go like �2.

Wire lR-noise [cf. Equation (1)] Bz-noise By-noise Bx-noise

Straight, finite By sym:� weak asym: none sym. noneStraight, finite Bz sym. sym. sym. sym.Straight, finite Bx sym. sym. sym. sym.Curved, finite Bz sym:� weak geom:-induced asym:

� weak purely geometric sym.sym:� weak geom:-induced asym:

�weak purely geometric sym.Curved + spin-echo sym:� strong geom:-induced asym:

+ strong purely geometric sym.sym:� strong geom:-induced asym:

� strong purely geometric sym.AB setup, finite Bz sym:� strong geom:-induced asym:

� strong purely geometric sym.sym:� weak asym:

� weak geom:-induced asym:�weak purely geometric sym:

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the total change in angle, �, is independent of the time-of-flight, and is thus a Berry (geometric) phase; it is the samefor all wires in Fig. 1(c). As � is a directed angle, �!�� means a curve in the opposite sense (i.e., clockwise !counter-clockwise). Introducing noise (in Bz and lR) nowadds the factor ��lR

ddlR� �Bz

ddBz���jp1" j � jp

1# j�L=@

2c"�� to �R;L. Averaging over this noise as before yieldsdecay (dephasing) of the purity with the exponent

�1

2

���R

ddlR� �Bz

ddBz

���jp1" j � jp

1# j�L=@ 2c"��

�2:

(6)

The L2 and �2 terms in the exponential (dynamic andpurely geometric terms, respectively) are both unchangedunder R$ L. The cross term, L � (a mixed dynamic-geometric term), changes sign under R$ L, causing a(geometry-induced) left-right asymmetry.

Coherent oscillations carry the phase of Eq. (5), theiramplitude decays with the exponent in Eq. (6). Ambiguityin choosing the measurement axis (any axis perpendicularto the axis of the eigenbasis) causes ambiguity in the phase(known as gauge-dependence), but not the amplitude.Thus, the geometric contribution to dephasing is gauge-independent even when the BP is not [6].

Spin-echo.—To maximize the geometric or asymmetriceffects, one may use a spin-echo technique, sketched inFig. 1(d). If � � �L � 0, the wires left and right of thespin flipper are identical, and any spin component acquiresopposite phases before and after the spin flip. Thus, fornonzero � and �L, the phase is given by Eq. (5) with Lreplaced by �L. Dephasing is given by Eq. (6) with L!�L. Varying �L changes the relative size of the �L2,�L�, and �2 contributions to dephasing. For �L � 0,there is only purely geometric dephasing. Asymmetry ismaximized for a small �L, such that the �L2 and �L�terms are similar in size.

Aharonov-Bohm interferometer.—Measuring the flux-sensitive current through such an interferometer[Fig. 1(e)] allows us to study dephasing [16], while avoid-ing the need to measure spin-components of the current.We generate right or left movers via a voltage bias to eitherthe L or the R lead. Asymmetry in dephasing will mani-fest itself as a difference in the visibility (magnitude of theAB oscillations in the current) between right and leftmovers, We assume that the multiterminal (open) interfer-ometer is sufficiently open that no higher windings aroundit occur. This also avoids the symmetry constraints im-posed by the two-terminal Onsager-Buttiker relations. Weconsider injected electrons which are spin-polarized along

B Byx

y

R

R

E

(a) sketch of dispersion curve (b) field felt by spin at Fermi energy ( = =0)

Bp σ

ml

p

ml

p p hp

hp

Bz

yp

Fermi surface

FIG. 2 (color online). (a) A sketch of the dispersion curve forH in Eq. (1); dashed parabolas have Bx � Bz � 0 [21]. (b) Theeffective field felt by spin states at EF (for Bx � By � 0). SincepR" > pR

# , the two states have nonorthogonal spins (their overallorthogonality is due to pR

" � pR# ).

FIG. 1 (color online). The geometries we consider, with ap-plied field, B and Rashba field, BR. In (c), the BP changes signfor wires reflected in the x-axis (�! ��). In (d), the sourceinjects a superposition of " and # eigenmodes, and the spin fliptakes "$# . In (e), spin-polarized electrons are injected at L.Some are detected at R, but most escape into D1;2.

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B � Bzez, thus the phase difference between the two pathsis �R;L � �AB � jp1" jL=@ c"� where L � L2 � L1

and � � �2 ��1. The Aharonov-Bohm phase �AB ��eBzA=�g�B@� where A is the area enclosed by thepaths. The current at the detector is

jIR;Lj � jI1j � jI2j � 2jI1I2j1=2AR;L cos�R;L (7)

where Ii is the part of the current in arm i which enters thedetector lead. In the absence of noise, AR;L � 1. For I1 �I2, the visibility of the AB oscillations is maximal.Averaging over noise in Bz and lR, we find that

AR;L � exp��

1

2�2Bz

�eAg�B@

L@

djp1" j

dBz��

dc"dBz

�2

�1

2�2R

�L@

djp1" j

dlR��

dc"dlR

�2�: (8)

This is asymmetric, with the upper (lower) sign for R (L)movers. There are contributions to dephasing due to thecurvature, L� (asymmetric) and �2 (symmetric).

Discussion.—Quantum wires in GaxIn1�xAs=InP havelR ’ 5�F ’ 200 nm [17]. Then the ratio of asymmetric tototal dephasing in a straight-wire [cf. Equation (3)] will besmall, equal to 2�2

F=�lRlB� 8%, where lB � vF@=B lR.However, for spin-echo setups with �L lR, this ratio canbe of order one [cf. Equation (6) and thereafter]. Then thedephasing is not very strong, nonetheless we estimate thatone can tune �L such that R movers lose at least 50% oftheir coherence (purity P< 3=4), while L movers are notdephased. For the AB setup with lR-noise [cf. Equa-tion (8)], the situation is the same as for the spin-echo(with L playing the role of �L), while for Bz-noise, theratio of asymmetric to total dephasing (in the exponent) istiny �FL=A.

To summarize, we have analyzed the effect of noise oncurrent-carrying electrons, both spinful and spin-polarized,subject to a spin-orbit interaction. The dephasing may bemeasured through the spin components of the emergingelectrons, or the visibility of AB oscillations. We havedemonstrated both geometric dephasing and left-rightasymmetry in dephasing. The noise studied here fluctuatesat a pace slower than the time-of-flight, leading to dephas-ing with the exponential of L2, L�, �2. By contrast,for faster fluctuations (studied for a spin or qubit in a time-dependent field [6]), the dephasing would be exponentialwith L, � (see also [18–20]). Other L$R asymmetriesoccur in real devices; however, those discussed here have aunique signature; they are controlled by the noise power.

We thank Yu. Makhlin and G. Schon for very usefuldiscussions. This work was supported by Swiss NSF, FGSchwerpunktprogramm No. 1285 ‘‘Halbleiter Spintronik,’’EU Transnational Access program (No. RITA-CT-2003-5060965), and US-Israel BSF, Minerva Foundation and

Albert Einstein Minerva Center for Theoretical Physics.

[1] Semiconductor Spintronics and Quantum Computation,edited by D. D. Awschalom, D. Loss, and N. Samarth(Springer, Berlin, 2002).

[2] E. I. Rashba, in Future Trends in Microelectronics: Up toNano Creek, edited by S. Luryi, J. M. Xu, andA. Zaslavsky (Wiley, Hoboken, 2007).

[3] H.-A. Engel, L. P. Kouwenhoven, D. Loss, and C. M.Marcus, Quant. Info. Proc. 3, 115 (2004).

[4] A. Cottet et al., Semicond. Sci. Technol. 21, S78 (2006).[5] Y. Meir, Y. Gefen, and O. Entin-Wohlman, Phys. Rev.

Lett. 63, 798 (1989); D. Loss, P. Goldbart, and A. V.Balatsky, Phys. Rev. Lett. 65, 1655 (1990); A. G.Aronov and Y. B. Lyanda-Geller Phys. Rev. Lett. 70, 343(1993).

[6] R. S. Whitney, Y. Makhlin, A. Shnirman, and Y. Gefen,Phys. Rev. Lett. 94, 070407 (2005).

[7] The gate RC times * 10�6 s (although they may beengineered to be shorter), while the time-of-flight alongballistic wires (L< 0:1 mm, vF 105 m=s) is &10�9s.

[8] R. Lassnig, Phys. Rev. B 31, 8076 (1985); W. Zawadzkiand P. Pfeffer, Semicond. Sci. Technol. 19, R1 (2004);E. Bernardes, J. Schliemann, M. Lee, J. C. Egues, andD. Loss, Phys. Rev. Lett. 99, 076603 (2007).

[9] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.Rev. Lett. 78, 1335 (1997); G. Engels, J. Lange, Th.Schapers, and H. Luth, Phys. Rev. B 55, R1958 (1997).

[10] This equation for P is approximate when Fermi-surfacestates have slightly nonorthogonal spins, Fig. 2(b).

[11] Weak-noise in x or z, with Bx � Bz � 0 on average, givesonly (2nd order) dephasing, O�@=pL� smaller.

[12] Equal Rashba and Dresselhaus terms do not suppressdephasing in our model cf. J. Schliemann, J. C. Egues,and D. Loss, Phys. Rev. Lett. 90, 146801 (2003).

[13] This term reads H y � �2m��1p2

y � �mlR��1py�x �

V�y�. For �F � lR, this is a fast oscillator coupling tothe slow spin’s x-component. It slightly reduces the effec-tive fields in y, z-directions, we absorb this into B, BR.

[14] See, e.g., Spin Dynamics: Basics of Nuclear MagneticResonance, edited by M. H. Levitt (Wiley, New York,2001);A. Kamenev and Y. Gefen, Phys. Rev. B 54, 5428 (1996).

[15] F. E. Meijer, A. F. Morpurgo, and T. M. Klapwijk, Phys.Rev. B 66, 033107 (2002).

[16] F. Marquardt and C. Bruder, Phys. Rev. B 65, 125315(2002).

[17] Th. Schapers, V. A. Guzenko, M. G. Pala, U. Zulicke,M. Governale, J. Knobbe, and H. Hardtdegen, Phys.Rev. B 74, 081301(R) (2006).

[18] P. San-Jose, G. Zarand, A. Shnirman, and G. Schon, Phys.Rev. Lett. 97, 076803 (2006).

[19] R. S. Whitney and Y. Gefen, Phys. Rev. Lett. 90, 190402(2003); R. S. Whitney, Yu. Makhlin, A. Shnirman, andY. Gefen, arXiv:cond-mat/0401376.

[20] P. J. Leek et al., Science 318, 1889 (2007).[21] M. Governale and U. Zulicke, Phys. Rev. B 66, 073311

(2002).

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