spin relaxation due to deflection coupling in nanotube quantum dots

Spin relaxation due to deflection coupling in nanotube quantum dots

Mark S. Rudner1 and Emmanuel I. Rashba1,2,3

1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA2Center for Nanoscale Systems, Harvard University, Cambridge, Massachusetts 02138, USA

3Department of Physics, Loughborough University, Leicestershire LE11 3TU, United KingdomReceived 5 January 2010; revised manuscript received 17 February 2010; published 23 March 2010

We consider relaxation of a single-electron spin in a nanotube quantum dot due to its coupling to flexuralphonon modes, and identify a new spin-orbit-mediated coupling between the nanotube deflection and theelectron spin. This mechanism dominates other spin-relaxation mechanisms in the limit of small energy trans-fers. Due to the quadratic dispersion law of long-wavelength flexons, q2, the density of states dq /d

−1/2 diverges as →0. Furthermore, because here the spin couples directly to the nanotube deflection, thereis an additional enhancement by a factor of 1 /q compared to the deformation-potential coupling mechanism.We show that the deflection coupling robustly gives rise to a minimum in the magnetic field dependence of thespin lifetime T1 near an avoided crossing between spin-orbit split levels in both the high- and low-temperaturelimits. This provides a mechanism that supports the identification of the observed T1 minimum with an avoidedcrossing in the single-particle spectrum by Churchill et al. Phys. Rev. Lett. 102, 166802 2009.

DOI: 10.1103/PhysRevB.81.125426 PACS numbers: 72.25.Rb, 71.70.Ej, 85.35.Kt

I. INTRODUCTION

Due to their outstanding mechanical properties and versa-tile electrical characteristics, carbon nanotubes offer an ex-citing platform both for studies of fundamental physical phe-nomena and for a variety of potential applications. Therelatively small nuclear charge of carbon and the low naturalabundance of carbon isotopes with nonzero nuclear spin sug-gest that the spin-orbit and hyperfine interactions, which arethe main sources of electron-spin relaxation in GaAs,1,2

should be weak in carbon nanotubes. Thus in recent years,the electronic spin properties of nanotubes have gained wideattention for potential applications in spintronics and quan-tum computing. Furthermore, the availability of isotopicallypurified starting materials opens the possibility of growing12C nuclear spin I=0 and 13C nuclear spin I=1 /2 nano-tubes to study the behavior of electron spins in the presenceor absence of a nuclear-spin bath.3

As techniques for preparing ultraclean samples and study-ing them in cryogenic environments have become available,few-electron quantum dots have emerged as a powerful toolfor studying the electron-spin properties of nanotubes.3–5 Inthe experiment by Kuemmeth et al.,4 single-electron andsingle-hole quantum-dot spectra were shown to display thecharacteristics of coherent coupling between the electron’sspin and its orbital magnetic moment,6–12 see Fig. 1a. Ex-periments in the spin-blockade regime of few-electrondouble quantum dots5,13,14 have also yielded important infor-mation about spin relaxation in nanotubes. In particular, theexperiment by Churchill et al.5 demonstrated a minimum ofthe spin lifetime T1 near the narrow avoided crossing be-tween levels circled in Fig. 1. Below we will identify amechanism of spin relaxation in nanotube quantum dots thatis based on the coupling of an electron’s spin to the nano-tube’s deflection. This mechanism provides a deeper under-standing of the T1 minimum. Our theory is developed for asingle-electron quantum dot, with the corresponding energy-level diagram shown in Fig. 1a. While we envision that the

basic results are more general, a full investigation of thedetails for many-electron systems is beyond the scope of thispaper.

In order for spin relaxation to occur, energy must be trans-ferred from the spin to the environment, which for confinedelectrons typically involves the phonon bath of the host crys-

Ω

0 0.5 1.0 1.5 2.0

0

0.4

-0.4

-0.8

Spin Follows TubeSpin Stationary

∆SO = 0 ∆SO Ω

Bz

t(z)

Magnetic Field (T)

Ene

rgy

(meV

)

Ωb)

c)

a)

Upper AvoidedCrossing

µorb ‖ t(z)E+

E−

FIG. 1. Spin relaxation due to deflection coupling to bending-mode phonons. a Single-electron quantum-dot energy-level dia-gram with the parameter values of Ref. 5. The four-dimensionalsubspace is spanned by the states s labeled by the valley index= 1 and the spin projection s= 1 along the laboratory z axis,see Eqs. 2 and 4. For the magnetic field B=0, the upper lowerKramers doublet includes states 1,1 and −1,−1 1,−1 and−1,1. Due to the flexon density of states singularity as →0, wefocus on spin-relaxation rates between levels with small energysplittings in the lower Kramers doublet denoted by E, and in thenarrow upper avoided crossing dashed circle. b Gedanken ex-periment to illustrate the coupling mechanism. In the absence ofspin-orbit coupling, the electron spin remains fixed in the laboratoryframe irrespective of the nanotube’s motion. When the strength ofspin-orbit coupling SO is much greater than the rate of motion ,coupling between the spin and orbital moments keeps the electronspin aligned with the direction of the tube axis. c For a nanotubewith constrained ends, bending-mode phonons cause spatial varia-tions in the direction of the nanotube axis tz, which locally coupleto the electron spin as described above.

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http://dx.doi.org/10.1103/PhysRevB.81.125426

tal. Due to strong coupling, phonons play an especially im-portant role in nanotubes.15–19 Crucially for us, clean nano-tubes possess quadratically dispersing bendingmodes12,15,16,20 with qq2. The corresponding density ofstates for each such mode has a Van Hove singularitydq /d1 / at low energies →0. Thus any relaxationmechanism that relies on coupling to such phonons is ex-pected to become especially efficient for small energy trans-fers.

Indeed, Bulaev et al.12 considered spin-orbit coupling tothe bending mode deformation potential, and used the singu-larity in the density of states to predict a spectacularenhancement of 1 /T11 / near the upper anticrossingpoint of Fig. 1a in the high-temperature regime. In the low-temperature regime, however, this mechanism predicts a sup-pression of spin relaxation 1 /T1. Because the experi-ments of Ref. 5 were performed in an intermediate regime /kBT100 mK, it is difficult to confidently link the T1minimum to the flexon deformation-potential coupling. Thedeflection-coupling mechanism described below resolves thisambiguity, as it ensures a T1 minimum for all temperatures.

Borysenko et al.21 also invoked the density of states sin-gularity but rather than the deformation potential, they con-sidered spin-flexon coupling through the Zeeman interactiondue to the g-factor anisotropy g. This coupling is weakbecause g10−2, see Refs. 22 and 23. Moreover, becausethe orbital moment7 orb was not taken into account, theupper avoided crossing did not appear in their spectrum andthe theory was only applied to the low magnetic field Kram-ers doublets.

Here we consider a different coupling mechanism be-tween flexural modes and electron spin. Similar to themechanism of Borysenko et al.,21 the coupling is propor-tional to the nanotube deflection, arising from local changesin the direction of the nanotube axis in the global labora-tory reference frame. Unlike their mechanism, however, thespin-orbit interaction of Refs. 4 and 5 is intrinsically suffi-cient to provide the coupling, which thus operates even inzero external magnetic field. Therefore, our mechanism isefficient near both the upper avoided crossing and the narrowKramers doublets, is not limited by small g, and isparametrically stronger than the deformation-potentialmechanism at low energies small q because it is propor-tional to the deflection z rather than the deformation= /zq.

The physical origin of the coupling mechanism can mosteasily be understood through a thought experiment, illus-trated in Fig. 1b. Imagine rigidly twirling a nanotube infree space about an axis perpendicular to the tube axis. In theabsence of spin-orbit coupling, an electron spin initially ori-ented in some direction in the global laboratory referenceframe will remain polarized in the same direction as thenanotube rotates around it. Now consider the opposite caseof strong coupling between the electron spin and orbitalmagnetic moments. If the electron is initialized to a statewith the orbital and spin moments aligned along the tubeaxis, then as the tube slowly rotates, the electron spin willadiabatically follow the direction of the tube axis. Forweaker coupling and/or in the presence of an external mag-netic field which provides a preferred direction for the elec-

tron spin in the laboratory frame, nontrivial spin dynamicswill be generated.

In the more relevant situation of a freely vibrating, elec-trically contacted, suspended, or substrate-supported nano-tube with immobilized ends, bending-mode phonons can ac-complish the same effect locally, section-by-sectionthroughout the tube, as indicated in Fig. 1c. Below we treatthis effect perturbatively for small deflections. To make con-nection to the Gedanken experiment depicted in Fig. 1cwhere the spin follows the direction of the tube axis, we notethat the electron-flexon coupling Hamiltonian Hs-ph see Eq.5 and accompanying discussion below used throughoutthis paper depends only on the instantaneous deflection ofthe nanotube. Because the local coupling strength dependson the deflection angle, it is proportional to the first deriva-tive of the nanotube displacement along the tube axis z, andhence to the first power of the phonon momentum q.

The plan for the rest of the paper is as follows. In Sec. II,we present the model in detail and define the transition ratesto be calculated. Then in Sec. III, we calculate the spin-relaxation rate between states of the field-split lower Kram-ers doublet for weak external field B0. In Sec. IV, wecalculate the spin-relaxation rate between the two nearly de-generate states at the upper avoided crossing of Fig. 1a anddemonstrate the scaling of the T1 minimum in the high-temperature and low-temperature limits. The main results ofSecs. III and IV are summarized in Eqs. 17 and 30, re-spectively. Last, in Sec. V, we discuss how this picture ismodified in “dirty” tubes where a substrate or coating maylead to flexon localization.

II. SPIN-PHONON COUPLING

We consider a single electron confined in a semiconduct-ing narrow-gap or large-gap nanotube quantum dot, whichin the leading approximation is described by the Hamil-tonian,

Hd = vF3kc1 − i2d

dz + Vz , 1

where vF106 m /s is the Fermi velocity of graphene, 3 isthe isospin Pauli matrix with eigenvalue in valley K,where K1=2 /3a1,3 and K−1=−K1 are the K and Kpoints of the graphene Brillouin zone and a is the latticeconstant, 1 and 2 are pseudospin sublattice space Paulimatrices, and Vz is a confining potential in the longitudinaldirection z. The transverse momentum kc, which sets theband gap Eg=2vFkc for free electrons when Vz=0, is de-termined by the boundary condition and warping effects thatarise from rolling the graphene sheet into a cylinder. Forlarge-gap nanotubes, kc is proportional to 1 /R, where R is theradius of the tube. For narrow-gap tubes, kccos 3 /R2 isnonzero due to curvature of the graphene sheet.24 Here isthe winding angle of the tube.6,12,25 To keep the discussiongeneral, for now we avoid imposing any particular form ofthe longitudinal confining potential.

Due to the twofold real spin and twofold isospin symme-tries, all eigenstates of Hd are fourfold degenerate. In particu-lar, this applies to the ground state,

MARK S. RUDNER AND EMMANUEL I. RASHBA PHYSICAL REVIEW B 81, 125426 2010

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Hds = E0s , 2

where s= s is factorized into orbital and spin partssatisfying 3=, szs=ss. Here sz is the real spin Paulimatrix associated with the z direction in the laboratoryframe. Note that describes both the quantized longitudinaland transverse orbital motion of an electron in valley K. Thespecific expression for E0 depends on the structure of thenanotube and on the potential Vz. Below we set the zero ofenergy to be E0=0.

We now add another piece HS to the full system Hamil-tonian H=Hd+HS, that is central to this paper. It includes thecoupling of an external magnetic field B to the electronicorbital and spin magnetic moments orb and B, spin-orbitcoupling, and intervalley scattering,

HS =SO

23t · s + KK1 − orb3t · B + Bs · B ,

3

where t= tx , ty , tz is the local tangent unit vector at eachpoint along the tube, t= tz, with components defined in thelaboratory frame, and SO and KK are the spin-orbit andintervalley coupling matrix elements. The constant SO inour model Hamiltonian HS absorbs both currently knownspin-orbit coupling terms.6,9–11 Orbital angular momentswere discovered by Minot et al. in Ref. 7, and spin-orbitcoupling was found and measured by Kuemmeth et al. andChurchill et al. in Refs. 4 and 5. We assume that at rest thetube is straight and oriented along the laboratory z axis.

Coupling to flexural phonons appears when we take intoaccount the fact that the tangent vector is actually an operatordepending on the phonon displacement coordinates uz ateach point along the tube. For small amplitudes, long-wavelength deflections are described by tz= z+duz /dz,where u is the nanotube displacement perpendicular to z.Substituting this expression for t into Eq. 3, we split theHamiltonian HS into a zero-order unperturbed part,5

H0 =SO

23sz + KK1 − orb3Bz + Bs · B 4

and a spin-phonon coupling part,

Hs-ph =SO

2du

dz· s 3 − orbdu

dz· B 3. 5

As usual in the theory of condensed-matter systems and inthe spirit of the Born-Oppenheimer approximation, the spin-phonon coupling Hamiltonian Hs-ph depends only on the in-stantaneous deflection uz. In terms of the flexon creationand annihilation operators aq

† and aq,

du

dzz =

q

iq

2Lzqxaq + aq

† eiqz, 6

where distinguishes two bending-mode polarizations x, Lzis the nanotube length, q and q are flexon momentum andfrequency, and is mass per unit length. As shown in Refs.15, 16, and 20, q=q2 for small q. Because of the nano-tube symmetry, the two polarization modes are degenerate.

The phonon normal-mode profiles and frequencies dependin a nonuniversal way on the details of the nanotube’s envi-ronment e.g., on boundary conditions and on the presence ofdisorder or coating particles on the tube’s surface. In Eq.6, we capture the generic behavior of the mechanism byassuming plane-wave normal modes. However, below wewrite the main results 17 and 30 in terms of a form factorMq, Eq. 8, which can be generalized to other normal-mode profiles. Indeed, a more general discussion of theflexon normal modes is provided in Sec. V.

Similar comments can be made regarding the influence ofelectron-electron interactions in many-electron quantumdots. A homogeneous background of closed electron shellsshould merely renormalize the coefficients without influenc-ing the basic qualitative results. However, a strongly inho-mogeneous electron background including charge puddlesmay influence electron-flexon coupling more profoundly.Such a regime is outside the scope of our investigation.

The spin Hamiltonian HS is defined on the four-dimensional subspace spanned by the eigenstates s ofHd, Eq. 2. The orbital term orb3t ·B results from ashift of kc due to the vector potential associated with B. Thisrenormalizes the gap Eg, which, in principle, affects the lon-gitudinal confinement see Appendix. Because this shift issmall in the regime of interest for experiments,4,5

orbt ·B /Eg1 /30, we ignore its effect on the longitudinalmotion. By ignoring coupling to higher orbital levels, weomit terms which are small in the inverse level spacing but inreturn obtain general analytical formulas which allow us toclearly extract the essential physics underlying thedeflection-coupling mechanism.

Armed with the perturbation Eq. 5, we calculate tran-sition rates between eigenstates of the zero-order Hamil-tonian Hd+H0 using Fermi’s golden rule,

Wfi =2

2 fHs-phi2Lz

2 dq

d

=Ef−Ei

. 7

Here i and f are the initial and final states satisfyingHd+H0n=Enn, and the overbar indicates averagingover the thermal phonon distribution.

As discussed further in the Appendix, due to the fact thatthe orbital states with = 1 are time-reversal conjugate,the density nz0z−z0, which is a real scalar, isindependent of the isospin index . Using this fact and theproperty that Eq. 5 is diagonal in the isospin index , anymatrix element fHs-phi includes the longitudinal formfactor,

Mq −

dznzeiqz, 8

which depends on the specific form of the longitudinal con-finement but otherwise does not depend on the compositionof i and f in terms of the basis states s.

In the Appendix, we calculate Mq for square-wellconfinement. For small momentum q→0, the form factorMq→1, simply reflecting the wave-function normaliza-tion. For large phonon momentum qLd2, where Ld is theelectronic length of the quantum dot, the factor eiqz oscillates

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rapidly relative to the electron wave function and leads to astrong suppression of the matrix element, as will be dis-cussed in Sec. V. Note that we distinguish between Lz, theunconstrained length of the nanotube relevant for phononnormal modes, and Ld, the quantization length of the elec-tronic wave function controlled by electrostatic gates, whichin general are independent quantities see, e.g., Ref. 19.

III. RELAXATION RATE IN THE LOWER KRAMERSDOUBLET

Due to the flexon density of states singularity as →0,we focus our attention on spin relaxation between states withsmall energy splittings. We begin by considering relaxationbetween the lower Kramers pair of states see Fig. 1a in asmall applied magnetic field. Although time-reversal symme-try prevents relaxation from occurring at BB=0 VanVleck cancellation, see Refs. 26–29, the large density ofstates contributes to a steep rise of the relaxation rate as B isincreased away from zero.

To apply Eq. 7 for the relaxation rate, the first step is tofind the two lowest-energy eigenstates of Hamiltonian H0.We take SO0, as observed in Ref. 4. Diagonalization isaccomplished in two steps: we first apply a unitary transfor-mation,

U =1

2 + SO

21 −

KK

KK + sz31 +

KK

KK

+1

2 − SO

21 +

KK

KK − isz21 −

KK

KK

9

to switch to a frame where UH0B=0U†=− 2 sz3 is diago-

nal. Here SO2 +4KK

2 . For SO0, a different unitarytransformation must be chosen.

After the transformation, the zero-order HamiltonianH0

U=UH0U† for B0 reads

H0U = −

23sz − orbBz

SO

3 + BBzsz − 2BB

KK

2sy

− BB

KKSO

KKsx − 2orbBz

KK

1sz, 10

where without loss of generality, we choose the perpendicu-lar component of the magnetic field B to be oriented alongthe x direction, Bx=B ,By =0.

To linear order in B /, the lowest two energy eigenvaluesare E=−

2 , with

=1

orbSO + B2Bz

2 + 4B2B

2 KK2 . 11

Explicit expressions for the corresponding four-spinor eigen-vectors are too cumbersome to reproduce here but willbe used below to calculate the necessary matrix elements. Inthe case KK→0, + and − correspond to states adia-batically connected to +↓ and −↑ as B→0, where +−

indicates =1=−1. More generally, the label in simply indexes the two states which are degenerate asB→0.

Substituting the expressions for into Eq. 7 yieldsthe rate W

EWA for emission absorption of a phonon with

momentum q and polarization ,

WE =

q2

LzqNq + 1+H

U−2Lz

2

dq

d,

WA =

q2

LzqNq−H

U+2Lz

2

dq

d, 12

where Nq is the thermal occupation number for the phononwith q=E−−E+, and

HU = UeiqzSO

2x · s3 − orbx · B3U† 13

is the U-transformed spin-phonon coupling operator of Eqs.5 and 6 with the phonon operators and related dimen-sional constants removed. Using the explicit expressions for, we find

+HxU− = −

2KK

3orb2 − BSO

2 BB2

+ SOorbSO + BorbBz2Mq ,

+HyU− =

2iKK

2 MqSOorbBz, 14

where Mq is the longitudinal form factor calculated in theAppendix.

First, because B, it is apparent that both matrix ele-ments in Eq. 14 vanish as B→0 as expected for a transitionbetween time-conjugate partners of a Kramers doublet. Sec-ond, both matrix elements vanish when KK→0, which isalso expected because in this limit, the two members of theKramers doublet belong to opposite valleys K and K, whichare not coupled by long-wavelength phonons. Finally, al-though it is not immediately obvious from Eq. 14, in thelimit B→0, the matrix elements for transitions involvingthe two polarization modes are equal in magnitude as re-quired by axial symmetry, +Hx

U−= +HyU−

=2MqSOKKorbBz /2.The relaxation rate =1 /T1 is the total rate of phonon

emission and absorption in both channels x,

=

WE + W

A . 15

To simplify the complicated expressions of Eq. 14, we takeadvantage of the fact that

B/orb 1, 16

typically3–5 B0.1orb, and that B always appearsmultiplied by B, and so retain only the leading term inorbBz. Using the energy conservation law ZE−−E+=q2orbBz and d /dq=2q, expression 12 evaluatesto


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1

T1

Mq2

2SOKK

2 2Z

3/2

coth Z

2kBT .

17

Not surprisingly, the Z dependence of expression 17 hasthe same form as that found by Borysenko et al.21 However,their relaxation rate g is suppressed by a very weakg-factor anisotropy g /g0.01.22 Comparing prefactors, wefind that the rates of relaxation due to these two mechanismsare related by the ratio

g

=

1

2g

g·

2

SOKK 2

. 18

For parameters in the regime which appears to be relevantfor experiments,4,5 i.e., SOKK, we find g /1, indi-cating that the spin-orbit-mediated mechanism studied hereinshould strongly dominate the behavior at low fields.

We now provide an estimate of based on Eq. 17, withrealistic experimental parameters. For , the coefficient of q2

in the flexon dispersion, we use the result of Mariani and vonOppen,16

= R2 + 2 + s

, 19

where R1 nm is the nanotube radius,5 s=10−7 g /cm2 isthe graphene sheet density,21 related to the linear density through =2Rs, and 4=9 eV Å−2 are Lamécoefficients.16 This gives 104 eV nm2. Note that thetheory of flexon vibrations in nanotubes was first studiedindependently by Suzuura and Ando15 and Mahan,20 whoused different models of the curved graphene sheet. Mahan’stheory and that of Suzuura and Ando, which was later ap-plied by Bulaev et al.12 and by Mariani and von Oppen,16

predict different functional dependence of on and .However, the numerical values obtained from each of thesetheories differ only by an insignificant factor of order 1.

In the low-temperature limit, is suppressed as Z3/2 as

B→0. The ratio of SO to KK is strongly sample depen-dent; for illustration we take SO /KK=2. For a typical fieldB=103 G nominally oriented along the tube axis, we find300 s−1 for T=0. For high temperatures, the relaxationrate is enhanced by kBT /Z due to the presence of thermalphonons, and the Z→0 limit of Eq. 17 goes as Z

1/2.In this section, we focused on relaxation within the lowest

Kramers doublet. When condition 16 is satisfied, however,the results above can also be applied directly to relaxationwithin the upper Kramers doublet.

IV. UPPER AVOIDED CROSSING

We now turn our attention to spin relaxation in the vicin-ity of the upper avoided crossing circled in Fig. 1a, wherea minimum in T1B was discovered by Churchill et al.5

Such an acceleration of spin relaxation in the high-temperature regime was first predicted in Ref. 12 for the caseof electron-phonon coupling through the deformation poten-tial. Here we build on the idea outlined in that work, and

consider how relaxation is affected in the presence of thedeflection-coupling mechanism described above. As we willshow below, for small level splittings , the rate is para-metrically enhanced by a factor of 1 /, thus producing arobust enhancement of spin relaxation both in the high- andlow-temperature regimes.

Because the secular equation corresponding to Hamil-tonian 4 is fourth order, the spectrum of the system formoderate values of B cannot in general be described in anysimple form. Furthermore, depending on whether or not thesystem possesses certain symmetries such as valley conser-vation KK=0 or axial symmetry B=0, level crossingsmay be exact or avoided. The various possible regimes arereviewed in Sec. IV A below see also Ref. 12. In Sec. IV B,we calculate the relaxation rate close to the upper avoidedcrossing for the general case KK0,B0, and then pro-vide results for the special limiting cases KK=0 andB=0 in Sec. IV C.

A. Spectrum for BÅ0

For B0, the spectrum of Hamiltonian 4 displays twoavoided level crossings, shown in Fig. 1a. Couplingbetween nominally degenerate states is provided by interval-ley scattering KK or by the magnetic field component per-pendicular to the tube axis, B. To provide a better under-standing of what controls the splitting at each crossingindividually, we first examine the two limiting casesKK=0,B0 and KK0,B=0, which are describedby simple analytical expressions.

In the first case, KK=0,B0, the four branches of thespectrum are given by

E1 = orbBz +1

2SO − 2BBz2 + 4B

2B2 ,

E2 = orbBz −1

2SO − 2BBz2 + 4B

2B2 ,

E3 = − orbBz +1

2SO + 2BBz2 + 4B

2B2 ,

E4 = − orbBz −1

2SO + 2BBz2 + 4B

2B2 , 20

shown in Fig. 2a. An exact crossing at the upperintersection can only occur if E1=E2, which implies2BBZ=SO,B=0; therefore the spectrum displays anavoided crossing for any B0. Similarly, an exact crossingat the lower central intersection occurs when E2=E3, whichrequires

4orb2 B2cos2 −

B2

orb2 − B

2 sin2 = SO2 , 21

where is the angle between the magnetic field B and thetube axis z. This condition is always satisfied for some valueof B when the magnetic field direction does not deviate toofar from the tube axis, tan2 orb /B2−1100.


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In the opposite case, KK0,B=0, we find

E1 = BBz +1

2SO − 2orbBz2 + 4KK

2 ,

E2 = − BBz +1

2SO + 2orbBz2 + 4KK

2 ,

E3 = BBz −1

2SO − 2orbBz2 + 4KK

2 ,

E4 = − BBz −1

2SO + 2orbBz2 + 4KK

2 . 22

An exact crossing E1=E2 exists at Bz satisfying

4B2Bz

2 +KK

2

orb2 − B

2 = SO2 , 23

whenever 4B2KK

2 orb

2 −B2SO

2 . Because Borb, thiscondition is unlikely to be violated in clean samples. The gapbetween E1 and E3 persists for any KK0.

More generally, both KK ,B0, as in Fig. 1a. Herewe gain further insight by focusing on the regime,

2BBz = SO − , 24

where is a small deviation from the upper crossing foundin the KK ,B=0 limit. For small , KK, and B, wediagonalize the Hamiltonian within the 22 subspace de-fined by the crossing levels, taking into account coupling tothe lower two levels to first order in perturbation theory. Theperturbative treatment is justified by the large energy de-nominators compared to the coupling matrix elements,orbBBz ,BB ,KK. Because KK shifts thelevel crossing point, which is otherwise determined by con-dition 24, we also require KK orb /BSO to ensurethat the perturbation criteria are not violated. This latter re-

striction is rather mild, however, due to orbB.This treatment yields eigenvectors 1 and 2 not

shown and eigenvalues,

E1,2 = orbBz +orbKK

2 R

2orb2 − B

2Bz

, 25

where +− is used for E12, see Fig. 2a. Here

R2 = orb2 − B

2Bz − BKK2 2 + 4orb

2 − B22Bz

2B2B

2 .

26

In writing R, we use Eq. 24 and keep only the lowest-orderterm in . In the limits KK=0 and/or B=0, Eqs. 25 and26 reduce to the earlier results 20 and 22.

B. Transition rates for KK ,BÅ0

Proceeding as in Sec. III, the amplitudes for transitionsfrom 1 to 2 due to coupling to phonons with polariza-tion are

2HxU1 =

SO

2RMqBKK

2 − orb2 − B

2Bz ,

2HyU1 = − i

SO

2Mq 27

with R as defined in Eq. 26. In the expressions above,small corrections proportional to B /Bz2 andKK

2 / 4orb2 −B

2Bz2 have been omitted.

Interestingly, the contribution from y-polarized phonons isconstant near the avoided crossing, with an amplitude equalto MqSO /2. When B=0, the two amplitudes are equal asrequired by axial symmetry; although the energy splittingE1−E2 goes through zero in this limit, the amplitude remainsfinite. Unlike the case of the Kramers doublets, here there isno time-reversal symmetry to prevent such nonzero ampli-tude.

Additionally, for B0, the amplitude for emitting anx-polarized phonon vanishes for the detuning =0 satis-fying

0 =BKK

2

orb2 − B

2Bz

. 28

Thus the coupling and hence the relaxation rate is asym-metric in detuning relative to the position of the anticrossing.For KK /SO1, the asymmetry is rather small. If it can bedetected, however, it will allow the value of KK to be ex-tracted.

It is convenient to express R in terms of 0,

R = orb2 − B

2BzBz,B ,

Bz,B = Bz − 02 + 4B2B

2 . 29

Here the energy splitting E1−E2=R / orb2 −B

2Bz=Bz ,B is controlled by a combination of Bz detuningand B avoided crossing gap. Summing the squares of the

FIG. 2. Color online Behavior of the spectrum in limiting re-gimes KK=0 or B=0. a Spectrum for KK=0,B0. Thelevels E1 and E2 exhibit an avoided crossing with splitting con-trolled by the field misalignment B. The crossing between E2 andE3 is exact, except when the field direction deviates very far fromthe tube axis see text. b Spectrum for KK0,B=0. Interval-ley scattering opens an avoided crossing between levels E1 and E3.The exact crossing between E1 and E2 persists as long as SO

2B /orbKK.


125426-6

matrix elements Eq. 27, using the energy conservationlaw q=Bz ,B and using the golden rule, Eq. 7,

=SO

2 Mq2

163/21 + − 0

2coth

2kBT ,

30

where =WE+W

A=1 /T1 is the total rate of transitionsbetween 1 and 2.

As expected, the relaxation rate Eq. 30 between statesnear the upper avoided crossing shows a resonant enhance-ment proportional to −1/2 −3/2 in the low- high- tempera-ture regime due to the combination of the divergent flexondensity of states at small energies and the deflection-couplingmechanism. The singularity in the rate is a factor of 1 /stronger than that found by Bulaev et al.12 for thedeformation-potential coupling mechanism. For larger valuesof , the competition between the two mechanisms is sensi-tive to various system parameters. Without an analytical ex-pression for for the deformation-coupling mechanism,however, it is difficult to make a more detailed comparison inthis regime.

Using the parameters of Ref. 5, SO=170 eV,10 eV, and T0.1 K, we find 104 s−1. Despitethe strong singularity for small energy splittings, this value isstill two orders of magnitude smaller than the observed re-laxation rate. However, the rate is highly sensitive to param-eter values, including the flexon temperature and especiallythe avoided crossing splitting . The latter depends on theangle of misalignment between the magnetic field and thetube axis, which was estimated to be 5° by electron micro-graph but if smaller could easily lead to enhancement of therate. Thus although the magnetic field dependence of T1 fromthis model supports the association of the T1 minimum ob-served by Churchill et al.5 with coupling to flexons, addi-tional experiments are needed to understand the quantitativedetails of this relationship.

C. Limiting cases KK=0,BÅ0 and KKÅ0,B=0

To gain a fuller picture of the behavior near the upperavoided crossing, we now examine the simple limitsKK=0 and B=0 of Sec. IV A. In the limit KK=0,B0, E1 and E2 exhibit an avoided crossing of splittingE1−E2=2BB. At this point, =0 and therefore2Hx

U1 vanishes; only flexons with y polarization con-tribute to relaxation. The finite-energy splitting at theavoided crossing, controlled by B, provides a low-energycutoff for the denominator in Eq. 30. Further away from theavoided crossing, the rate is suppressed by the decaying pho-non density of states but gains an extra factor of 2 from thephonons with x polarization that become active there.

In the other limit, when KK0 and B=0, there is anexact crossing when condition 23 is satisfied. Here bothflexon polarizations contribute equal transition matrix ele-ments SO /2. With constant matrix elements in the numera-tor and no low-energy cutoff in the denominator, expression30 diverges as B approaches the critical value defined byEq. 23. Of course, in any real experiment, the divergence

will always be cut off by something, such as a small mis-alignment of the field axis. However, as found for the generalcase in the previous section, the behavior in all regimes dis-plays an acceleration of spin relaxation near the upperavoided crossing. This is a general property of the deflection-coupling mechanism of spin relaxation.

V. DISCUSSION AND CONCLUSIONS

Above, we found that the direct coupling between an elec-tron’s spin and the deflection of its host nanotube provides anefficient mechanism for electron-spin relaxation near narrowavoided level crossings where the energy transfer is small.Efficient relaxation is made possible by the diverging densityof states for bending-mode phonons with quadraticdispersion,12,15,16,20 qq2. Such a dispersion law is pre-dicted for clean systems such as perfect suspended nano-tubes. What happens for dirty systems where the nanotube isplaced on a substrate, and/or covered by an irregular coatingleft over from sample processing?

The primary effects of disorder are twofold. Constraintson the tube’s motion alter the displacement profiles of thenormal modes. As a result, the spectrum of normal-modefrequencies is altered, with spectral density at very low fre-quencies generally getting redistributed to higher frequen-cies. We begin this section by discussing normal-mode pro-files and providing some additional discussion of thelongitudinal form factor Mq that appears throughout thetext. Then we discuss how changes in Mq and dq /d dueto flexon localization may affect the relaxation rates calcu-lated above.

By writing Eq. 6 for the nanotube deflection, we assumethat the flexon normal modes are described by plane waveseiqz. Even for a perfectly clean system, however, the bound-ary conditions at the mechanically constrained points of thetube can mix modes with momenta q to yield normal-modeprofiles which are sums of sines and cosines. While theseboundary conditions determine the nanotube length Lz, theelectronic length Ld is determined by a different set of con-straints arising from electrostatic potentials created by exter-nal gates and/or impurities; thus it is not possible to find acompletely general result for the form factor Mq, whichdepends on the shapes and relative displacements of both theelectron wave function and flexon normal-mode profile. In-deed the authors of Ref. 19 concluded that asymmetric sup-pression of vibrational sidebands in transport through ananotube quantum dot could be explained by the presence ofa vibrational mode localized near one end of the quantumdot. Thus we see that the general situation can be quite com-plicated; we consider the above results for plane-wave nor-mal modes to be applicable for the “typical” or “average”case.

In Sec. II, we pointed out that for small momentum en-ergy transfer q→0, the factor Mq→1. For large momen-tum transfer qLd2, however, Mq is strongly suppresseddue to the fast oscillations of eiqz relative to the electronwave function. Using the results of the Appendix for sym-metric square-well confinement, we plot Mq2 for a large-gap and a small-gap tube in Fig. 3. For the small-gap tube,


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where kLd0.75, confinement is softer and the electronwave function extends over a distance significantly largerthan the barrier separation Ld. Thus suppression sets in atqLd

2, where kLd=.

The asymptotic behavior of Mq for large q containstwo contributions. The classically allowed region provides acontribution Mq1 /q3 for qLd2, see Eq. A16.In the classically forbidden region, the exponential tails ofthe wave function contribute a Lorentzian decay lawMq1 / kc

2+q2 with a prefactor that tends to zero askcLd→. Power-law decay arises from the steps in the po-tential at z= Ld /2 but the suppression behavior is generic.For a smooth confinement potential, one expects exponentialdecay with a characteristic scale qLd

2.The resonant enhancement of relaxation due to the flexon

density of states singularity is thus not affected by Mq aslong as the minimum of the energy gap Bz ,B=q2 atthe upper avoided crossing corresponds to a phonon momen-tum q satisfying qLd

2. Using 104 eV nm2 seeSec. III and an energy gap of 10 eV as observed in Ref. 5,we find qLd

=1.6–3.2 for Ld=50–100 nm. Thus for the pa-

rameters of Ref. 5, the longitudinal overlap does not suppressrelaxation at the upper avoided crossing but will lead to sup-pression for larger energy splittings. Note that the same formfactor appears in the matrix elements for the deformation-potential coupling12 and for the g-factor anisotropy deflec-tion coupling,21 and thus does not affect the relative compe-tition between these mechanisms.

In the presence of disorder, the quadratically dispersingflexons are subject to the same conditions that lead to thefamiliar localization of all one-dimensional electron eigen-states in a random potential.30,31 For weak scattering charac-terized by q1, where is the localization length, the den-sity of states remains smooth and the normal-mode profilesoscillate within envelopes with exponentially decaying tails,e−x/. The exponential envelope cuts off the rapid oscilla-tions responsible for the suppression of Mq, allowing tosubstitute for Ld. Thus, localization can in some cases helppromote relaxation at energy splittings larger than thosewhere strong suppression by Mq sets in for clean samples.

Because the localization length is on the order of the mean-free path,31 which for short-range scatterers is proportional tothe phonon energy, q2 and the clean limit very weakscattering should generally be recovered for large enoughenergy transfers.

When scattering is strong, q1, the normal-mode spec-trum collapses to a collection of discrete resonances and thenormal-mode profiles are highly distorted. In this case, thelow-energy modes responsible for the Van Hove singularitythat promotes spin relaxation near the upper avoided cross-ing can be destroyed. Rather than observing a smooth in-crease in the relaxation rate as the intersection is approached,a complicated nonmonotonic system of resonances may thenbe found.

It is also interesting to note that rigid rotational motion ofthe entire sample, as described in Fig. 1b, can be regardedas a global q=0 deflection of the nanotube. Consequently,simple laboratory vibrational noise can be a source ofelectron-spin decoherence through the deflection-couplingmechanism.32

In summary, we have identified and analyzed an efficientmechanism of electron-spin relaxation in nanotube quantumdots that results from spin-orbit-mediated coupling betweenthe electron spin and nanotube deflection. Due to the flexondensity of states singularity at small energies, relaxation dueto this deflection-coupling mechanism is particularly effi-cient near level crossings or small avoided crossings. Thismechanism is expected to dominate over other mechanismssuch as deformation potential12 and anisotropicg-factor-induced deflection coupling,21 which are suppressedby small phonon momentum q- and g-factor anisotropy, re-spectively. Finally, we predict a robust minimum of the spinrelaxation time T1 near the upper avoided crossing in boththe high- and low-temperature regimes, thus offering a firmbasis for understanding the observed T1 minimum of Ref. 5.

Note added in proof. Recently, electrical spin manipula-tion in curved carbon nanotubes based on a related spin orbitcoupling mechanism was proposed.37

ACKNOWLEDGMENTS

We gratefully acknowledge H. O. H. Churchill, F. Kuem-meth, and C. M. Marcus for stimulating discussions. Thework of M.R. was supported by NSF under Grants No. PHY-0646094 and No. DMR-09-06475. E.I.R. received supportfrom the NSF Materials World Network Program.

APPENDIX: LONGITUDINAL FORM FACTOR M(q)

As mentioned in Sec. II, every matrix element fHs-phi includes a longitudinal form factor,

Mq = −

dznzeiqz,

which depends on the specific form of longitudinal confine-ment for the quantum dot but is independent of the compo-sition of if in terms of the basis states s. In thisappendix, we discuss the properties of Mq in more detail

FIG. 3. Color online Longitudinal form factor Mq2 for sym-metric square-well confinement as described in Appendix, with bar-rier separation Ld=100 nm, for Eg=0.03 eV dashed blue andEg=1 eV solid red. For the large-gap tube, kLd and Mq2closely follows the hard-wall limit, Eq. A16, with strongsuppression for qLd2. For the small-gap tube, softer confine-ment results in a larger effective length Ld

, kLd=. When plotted

against qLd, the profile nearly collapses onto the large-gap result

dotted line.


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and provide an explicit result for Mq in the case of square-well confinement.

The property that a unique longitudinal overlap integralMq can be factored out of all matrix elements of Hs-phindependent of the initial and final states relies on two keyfacts. First, the perturbation Hs-ph, Eq. 5, is diagonal in theisospin index. Thus for any initial and final states i and f, which can be written as n=,scn;ss, where nstands for i or f ,

fHs-phi = ,

s,s

cf;s ci;ssHs-phs

= ,s,s

cf;s ci;ssHs-phs . A1

Furthermore, because each state s= s is a product oforbital and spin parts, Eq. 5 gives

sHs-phs = eiqz · sHs-ph s , A2

where Hs-ph is the spin-phonon coupling Hamiltonian of Eq.

5 projected onto the orbital state with

eiqz = −

dz0eiqz0z − z0

factored out. Second, the density nz0= z−z0, whichincludes a sum over densities on the two sublattices, is a realscalar. Because with =1 and =−1 are time-reversalconjugate, nz0 is independent of . Hence

fHs-phi = Mq ,s,s

cf;s ci;ssHs-ph

s , A3

thus completing the proof that a single, unique contributionfrom the longitudinal degrees of freedom appears in all tran-sition rates.

In the presence of a magnetic field that breaks time-reversal symmetry, the density n,sz0= sz−z0s isgenerally not independent of and s. In particular, the term3orbBz in Eq. 4 leads to a renormalization of Eg of oppo-site sign in valleys K and K. For the device of Ref. 5, witha gap Eg30 meV and for fields up to 1.5 T, however,orbB /Eg1 /30. Thus, our calculation of the longitudinalwave functions to lowest order and the resulting factorizationof Mq is justified.

We now calculate Mq for a quantum dot in a carbonnanotube formed by a symmetric step potential as shown inFig. 4. Although the details of Mq depend on the form ofthe longitudinal confinement, this example illustrates its ba-sic properties. Electron eigenfunctions in a square-well po-tential were analyzed in detail in Ref. 12. Therefore, here wefocus on the main features and omit a step-by-step deriva-tion.

A pure “hard-wall” boundary condition cannot be definedfor this system due to the relativistic dispersion and internalspinor structure recall Klein phenomenon for relativisticparticles in one dimension. The strongest confinement isrealized when the potential step at the barrier is roughly

equal to half the band gap, V0Eg /2. Here the tails of thewave function decay exponentially over a distance propor-tional to Eg

−1. For large-gap tubes, Eg1 /R while for small-gap tubes, the curvature-induced gap scales according toEgcos 3 /R2, where R is the tube radius and is the wind-ing angle.

The orbital states with = 1 of Eq. 2 correspond tothe lowest quantized mode of longitudinal motion in the Kand K valleys, respectively. Because we assume that theexternal potential Vz only depends on the longitudinal co-ordinate z and not on the circumferential coordinate x, theseeigenfunctions of Hd can be factored as

r eiK·reikcxz , A4

where r= x ,z, the circumferential wave vector kc isproportional to Eg, Eg=2vFkc, and is a two-componentwave function describing amplitudes

Az and Bz,

on the graphene A and B sublattices. Recall thatK=2 /3a1,3 for = 1 corresponds to the K=1or K=−1 point of the graphene Brillouin zone, where a isthe lattice constant. In terms of the pseudospin Pauli matrices1 and 2, satisfies

vF1kc − i2d

dz + Vz = E, A5

see Eq. 1. In a homogeneous tube with constant Vz, theeigenstates are plane waves proportional to eikz and the spec-trum has a relativistic dispersion E= vF

kc2+k2.

In the classically forbidden region I, z−Ld /2, thepotential takes the constant value Vz=V0. To emphasize theessential behavior and to simplify the equations, we pick thespecial value V0=E, where E is determined self-consistentlythrough the allowed wave vector k which satisfies thecontinuity relations at the boundaries, see Eq. A10 below.With this choice, the wave function in the classically forbid-den region is described by a single exponential living ononly one sublattice: the spinor components ,I

A and ,IB

satisfy

FIG. 4. Color online We consider a quantum dot formed in asmall or large gap semiconducting nanotube. The gate-inducedpotential Vz is taken to be Vz=V0 for z−Ld /2 or zLd /2, andVz=0 for −Ld /2zLd /2. The band gap Eg arises either due toquantization of transverse motion for pure semiconducting tubes, inwhich case Eg1 /R, or due to curvature of the graphene sheet, inwhich case Eg1 /R2. Electron density nz is indicated by the reddotted curve.


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,IA z =

A

21 − ekcz+Ld/2,

,IB z =

A

21 + ekcz+Ld/2. A6

Similarly, in region III, xLd /2,

,IIIA z =

D

21 + e−kcz−Ld/2,

,IIIB z =

D

21 − e−kcz−Ld/2. A7

In these regions, the solutions in opposite valleys exist onopposite sublattices. However, the total density nz is in-deed the same for each case.

In the classically allowed region II, Vz=0 and the spinorcomponents are plane waves ,II

A eikzeik and ,IIB eikz

which satisfy

vF 0 kc − ik

kc + ik 0 eik

1 = Eeik

1 . A8

Here eik = kc− ik /kc2+k2 and E=vF

kc2+k2. Note that

−k=−k.Due to the degeneracy of states k and −k in region II, a

general state with energy E has the form

,IIA

,IIB = Be

ikzeik

1 + Ce

−ikze−ik

1 . A9

Using the continuity condition of the wave function at theboundaries between different regions, we fix the relative co-efficients A, B, C, and D, and find the allowed wave vec-tor k that gives a bound state,33

tan kLd = −k

kc, A10

which comes from the condition kLd=k mod .

We calculate Mq=MIq+MIIq+MIIIq piecewise. Inregion I, MIq=−

−zdzA2e2kcze2kc+iqz evaluates to

MIq =4N sin2kLde−iqz

2kc + iq, A11

where z=Ld /2 and the normalization constant N satisfies

1

N= 4Ld +

4

kcsin2 kLd −

2

ksin 2kLd. A12

In region III,

MIIIq =4N sin2kLdeiqz

2kc − iq= MI

q . A13

Finally, in region II, we must evaluate

MIIq = 4N−z

z

dzsin2 kz + z + sin2 kz − zeiqz,

A14

which after some steps yields

MIIq = 4N2 sin qLd/2q

− cos kLd sink + q/2Ld

2k + q

+sink − q/2Ld

2k − q . A15

Because the density nz is symmetric in z, Mq=MIq+MIIq+MIIIq is real and even in q.

For a high aspect ratio dot with kcLd1, we havekLd. In this case, the contributions MI and MIII from thetails vanish, and Eq. A15 simplifies to

MIIq =82 sinqLd/2

qLd42 − qLd2, A16

which equals 1 for q=0 and decays as qLd−3 for qLd2.This limit coincides with Eq. 56 of Ref. 12. Interestingly,for a small-gap nanotube with Eg30 meV andLd=100 nm as in Refs. 4 and 5, kLd0.75, indicating sig-nificant penetration into the classically forbidden region.

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spin relaxation due to deflection coupling in nanotube quantum dots

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