Continuum model for chiral induced spin selectivity in helical molecules

0.21 MB·English

by  Ernesto Medina

Continuum model for chiral induced spin selectivity in helical molecules
Ernesto Medina,1,2,3 Luis A. Gonz´alez-Arraga,3 Daniel
Finkelstein-Shapiro,3 Bertrand Berche,2,1 and Vladimiro Mujica3
1Centro de F´ısica, Instituto Venezolano de Investigaciones Cient´ıficas, 21827, Caracas, 1020 A, Venezuela.
2Groupe de Physique Statistique, Institut Jean Lamour,
Universit´e de Lorraine, 54506 Vandoeuvre-les-Nancy Cedex, France.
3Department of Chemistry and Biochemistry, Arizona State University, Tempe, AZ 85287, USA
5
A minimal model is exactly solved for electron spin transport on a helix. Electron transport is
1
0 assumed tobe supported bywell oriented pz typeorbitals on base molecules forming a staircase of
2 definitechirality. Inatight bindinginterpretation,theSOCopensupan effectiveπz−πz coupling
viainterbasepx,y−pz hopping,introducingspincoupledtransport. Theresultingcontinuummodel
n spectrumshowstwoKramersdoublettransportchannelswithagapproportionaltotheSOC.Each
a doublydegeneratechannelsatisfiestimereversalsymmetry,nevertheless,abiaschoosesatransport
J
directionandthusselectsforspinorientation. Themodelpredictswhichspinorientationisselected

5 depending on chirality and bias, changes in spin preference as a function of input Fermi level and
2 scatteringsuppressionprotectedbytheSOgap. Wecomputethespincurrentwithadefinitehelicity
andfindittobeproportionaltothetorsionofthechiralstructureandthenon-adiabaticAharonov-
]
Anandanphase. To describe room temperature transport we assume that the total transmission is
l
l theresult of a product of coherent steps limited bythe coherence length.
a
h
-
s I. INTRODUCTION sation is proposed to be due to incoherent superposition
e
of short coherent length polarization[10].
m
Spin selectivity is an exciting prospect for many tech- On the other hand in the experiments performed by
.
t nologicalapplicationsthatrequirespinimbalanceasare- biasing a DNA molecule attached to metallic contacts[7]
a
m source for informationstorageand processing[1, 2]. Spin have been addressed theoretically by approaches based
polarized currents, are traditionally derived from opti- onquantum transport[8, 13–15] in anchiralelectric field
-
d calorientation[3]whenmaterial-opticinterphasesarere- generated by the charge distribution of the molecule, or
n quired or ferromagnets[4] involving gross power sources helical potentials. The electrons are considered to follow
o to control. Spin polarized currents derived by more sub- paths on the helix sampling the chiral electric field. An
c tle means are then desirable in the spin-torque driven importantfeatureofthemodelistherelationbetweenthe
[
writingonmagneticsurfaces,valvedevicesandmoregen- fieldintensitiesproducingSOCandtheelectronmobility.

1 erally in spin based information processing gates. Themodelspredictastrongpolarisationeffectinspiteof
v There is now well established experimental evidence thesmallSOCbecausethelowmobilityenhancestheres-
1 that chiral molecules have the tantalizing property of ident times of electrons in the potential. Considerations
0
polarizing electrons scattered either through 2D films or ofthe limitedcoherencelengthduetoroomtemperature
2
driven through individual chiral molecules[5, 6]. These operations have been discussed in ref.14.
6
0 represent two limiting cases for transport: For 2D films, We aim here to theoretically describe the single
. electrons are emitted from a metallic surface and are molecule experiments in more detail: For electrons with
1
asymptoticallyfree,havingandenergyabovethefilmpo- energies below the molecular barrier, a current is driven
0
5 tentialbarrier. Ontheotherhand,fortransportthrough by a potential difference, through single DNA molecules
1 a single molecule, electrons tunnel betweenmetallic con- in an STM set up[7]. The experimental results consist
: tacts driven by a potential bias or a two terminal set of a spin dependent barrier to tunnelling of electrons
v
up[7, 8]. through the chiral molecule. A preferred spin direction
i
X The former, scattering experimental setup, is un- results in a larger current than the opposite spin direc-
r derstood theoretically in terms of multiple scattering tion. Thereis agapbetweenbarrierheightsthatisinde-
a
through the chiral target in the presence spin-orbit cou- pendent of the molecule length and is thus proportional
pling (SOC)[8–11]. The theoreticalmodel requiresinter- to the SO coupling. A striking characteristic of the I-V
ference between scalarpotential events and spin-coupled curve is that it is anti-symmetric (V V), and the
→ −
scattering. Chiral structure and SOC define an en- preferred spin filtering direction in one bias direction is
ergy window in which the electron spin is effectively oppositetothepreferredspinoftheother[7]. Finally,the
polarized[10, 12] in the range of 1-10 eV. A higher en- bias needed to produce the same spin polarised current
ergylimitisrequiredformultiplecoherentscatteringand increases with molecule length.
electron probing of the chiral structure, while a minimal Herewepresentamodelforthetunnellingofelectrons
energy is required for the SO coupling to be significant. through a chiral molecule which can be solved exactly
The spin-orbit coupling strengths required are compati- explaining many experimental features. The model in-
blewiththoseexpectedformthe atomiccarboncores,in cludes the intrinsic SOC whose source are the atomic
the energy scale of the meV. Room temperature polari- cores of the carbon atoms sitting on the bases of the
2
DNAstructure(withadefiniteorientationinspace). The II. DERIVATION OF THE INTRINSIC
strength of such a coupling for tunnelling electrons can ELECTRIC FIELD HAMILTONIAN ON A HELIX
bereadilyderivedfromatightbindingapproach,andin-
volves both the intrinsic atomic value and contributions ThemodelHamiltonianweproposeconsistsofanelec-
from the coupling between nearest neighbour bases sites tronconfinedona helix ofN turns of radiusa andpitch
of the DNA molecule. We can anticipate the results for b as seen in Fig.1. An internal electric field, of atomic
thismodelbyperformingasymmetryanalysis: Boththe origin, is assumed to exist as the source of the electron
SOcouplingandthehelixbreakinversionsymmetry,but spin-orbitcoupling. Inanatomicmodelforthehelix,the
time reversal symmetry is preserved. This implies that SO interaction offers a p to p hopping route, first or-
z x,y
thespectrumofthespin-orbitactivehelixshouldbecom- derintheatomicintrinsicSOcouplingandthus,atleast
posed of Kramers degenerate doublets, separated by the inthe meV energyrange. Assuming a motionstrictlyon
effective spin-orbit coupling gap. The quantum numbers thehelixthroughasequenceofnearestneighbourstates,
of the helix of definite chirality comprise the kinetic en- we have the following Hamiltonian (see also ref.[17])
ergyindex,therotationsenseoftheelectronanditsspin.
EachKramersdoubletpreservestimereversalsymmetry, H=Hkinetic+HSO
sothatthey comprisebothrotationandbothspinquan- 1
= (p2 +p2 +p2)+(α/~)(p σ p σ ) (1)
tum numbers. On choosing a bias direction, only the 2m∗ x y z x y− y x
channel (one of the two states in the doublet) for that
wherethe firsttermis purekinetic energy,while the sec-
sense of propagation is selected and has an associated
ond termis the SO coupling for an intrinsic electric field
spin. Thus, the bias breaks time reversal symmetry by
inthez direction. Thestrengthoftheelectricfieldisem-
only populating one channel and spin selectivity results.
bedded in the parameter α = e~2E/(4m2c2) which has
dimensions of energy times a length. Nevertheless, this
electricfieldisanontrivialquantitytoassess,sinceitnot
onlycontainsameasureofthefieldfeltbytheelectronsin
This paper is organised as follows: We first derive the
their excursion to their nuclei, but also a quasi-resonant
Hamiltonian for a continuous one dimensional helix of a
coupling contribution to the neighbouring states[18, 19].
fixed number of turns and chirality, in the presence of
We will discuss these contributions later.
spin-orbit coupling, whose source is a local atomic core
Due tothesymmetryofthe problemitispreferableto
electricfieldinthez direction. Electronsareconstrained
work in cylindrical coordinates, and after putting care-
tofollowthehelicalpathonthecorrespondingeigenchan-
ful attention to hermiticity issues, well discussed in the
nels. Two energies are defined; that of the free electron
literature[20, 21], we arrive at
problem and the SO energy whose ratio determines the
adiabaticity of spin transport[16]. Following, we obtain p2 αa iαa
the channel energies and the corresponding exact eigen- H= 2m∗(a2ϕ+b2) − ~(a2+b2)σρpϕ+ 2(a2+b2)σϕ
functionsasafunctionoftwoquantumnumbers(current
(2)
direction and spin) and a chirality index. We then show where p = m∗(a2 +b2)ϕ˙ = i~∂ , a is the helix ra-
thatthespectrumimpliesthattherearealwaystwodou- dius, b iϕs the helix pitch, and−σ =ϕσ cosϕ+σ sinϕ
bly degenerate levels and each degenerate pair combines and σ = σ sinϕ+σ cosϕ.ρThe hxelix curvatyure is
time reversed states. Whenever one biased direction is ϕ − x y
given by the ratio κ = a/(a2 + b2) and the torsion is
chosen, time reversal is broken and a preferred spin is
τ = b/(a2+b2). The momentum in the z direction will
filtered for thatdirection. The opposite spin state in the
then be p = τp . m∗ is the electron effective mass,
same direction is at a higher energy so that it requires z ϕ
which assumes the carbon states form a narrow band of
a higher bias to be occupied, a feature that suppresses
states. Itis physically convenient[16] to identify two dis-
backscattering[7]. The resulting spin selected transport
tinct frequencies ω = ~/m∗(a2+b2) related to the free
isevidencedthroughthecomputationofthespincurrent electronkineticene0rgyandω =2αa/~(a2+b2)propor-
alongthehelix,leadingtospinaccumulationobservedin SO
tional to the helix curvature and the SO coupling. One
the experiments.
can then simplify the the Hamiltonian in Eq.2 into the
simple quadratic form
~ω ω 2
The final section addresses the problem of the SOC H = 0 i∂ϕ+ SOσρ . (3)
2 2ω
strength, where we report a tight-binding result which (cid:18) 0 (cid:19)
might address the strength of the spin-orbit energy gap The ansatz for the wave function for spinfull electrons is
observed. We also argue that the tunnelling throughthe onahelixofN turnswithhardwallboundaryconditions
DNAmoleculeoccursthroughanincoherentsequencedi- at the helix ends
viding the molecule into coherence-length long segments
to explain the scaling of the barrier potentials. We end Aλ,ζe−iϕ/2
Ψλ,ζ =eiλ(n/2N)ϕ s , (4)
with the conclusions. n,s Bλ,ζeiϕ/2
(cid:18) s (cid:19)
3
0.30
When the spin-orbit interaction is absent (α=0), space
and spin inversion symmetries are recovered (four fold
degeneracy) once one combines n˜ = n/2N and n˜ + 1
labelled eigenvalues.
Figure 1 shows a sequence of levels starting from the
0.25
groundstate(Kramersdoublets)attwosuccessivevalues
E+,+ Eñ-,’++1,- ionfdn˜exal+on1g,awsitahfutnhcetiriodneogfenthereaStOe, asttreαn=gth0.pTahrtenoerrdsewriinthg
Eλ,ζ Eñ-,,++ ñ,- Eñ+’,++1,+ of the levels are indicated according to the spin orienta-
ñ,s 0.20 Δ tion and sense of the current of the chirality ζ =+1.
hω To obtain a physical intuition on the nature of the
0 z E-,+ wave functions[16, 21] , we derive explicitly some of the
ñ+1,-
coefficients in the ansatz put forward. We explicitly do
θ E+,+
the λ = +,ζ = + case. Using equations (3) and (4) we
0.15 ñ+1,+
find from the secular equation
Β
SO ω s
B+ = 0 1 A+ , (6)
+,s ω cosθ − +,s
SO
ζ=+1 (cid:16) (cid:17)
0.10
0.0 0.2 0.4 0.6 0.8 where cosθ = 1/ 1+(ωSO/ω0)2. In order to conform
to a normalised spinor we choose A+ = cos(θ/2), and
ω ω p +,+
SO 0 thus B+ =sin(θ/2), so we have
+,+
ω
SO
tanθ = . (7)
FIG. 1. Energies of electrons on a counter-clockwise helix ω
0
(ζ = +1) as a function of the SO interaction strength ωSO.
Theenergyisinunitsof~ω0. Eachlevelisdoublydegenerate This angle results from the existence of a SO “magnetic
with the index labels indicated. When ωSO = 0 each energy field”[16,22] B~SO = α(~k zˆ)(2c/e)=(2cα/a~e)λLρˆ,
is fourfold degenerate as expected since the time inversion where L is the angu−lar m×omentum of the electron| o|n
symmetry turns into independent space and spin inversion the hel|ix,|and λ gives the direction of the effective field
symmetries. ∆ indicates the energy gap between Kramers
depending on the rotation sense of the electron.
degeneratestates. TheinsetshowstheSOmagneticfieldand The choice for the second eigenfunction is A+ =
the orientation of the spin for an eigenstate of the spin-orbit +,−
sinθ/2, leading to the two eigenspinors
coupled system. −
cosθe−iϕ/2
Ψ+,+ =ein˜ϕ 2 ,
n,+ sinθeiϕ/2,
where λ = +1( 1) labels for the counter clockwise, (cid:18) 2 (cid:19)
(clockwise) electr−ons, s= 1 labels the spin and ζ = 1 sinθe−iϕ/2
± ± Ψ+,+ =ein˜ϕ − 2 , (8)
labels the chirality of the helix. Although the chiral in- n,− cosθeiϕ/2
(cid:18) 2 (cid:19)
dex doesnotappearinthe Hamiltonian,itchoosesthe z
direction of propagation for a particular λ index. Using The corresponding eigenfunctions for λ= are
−
thewavefunctionansatzonecanderivetheexactenergy
cosθe−iϕ/2
of the model as Ψ−,+ =e−in˜ϕ 2 ,
n,+ sinθeiϕ/2
(cid:18) 2 (cid:19)
2
Enλ,,sζ = ~ω20 2nN − ζλ2ss1+(cid:18)ωωS0O(cid:19)2  , (5) Ψ−n,,−+ =e−in˜ϕ(cid:18)−csoinsθ2θ2eei−ϕi/ϕ2/2(cid:19). (9)
The angle θ here tells aboutthe inclination ofthe spinor
 
where n is a positive integer valued index. The basis with respect to the vertical z axis[16] (see Fig.1) and
functions chosenin Eq.4areconvenientwhen addressing n˜ = n/2N with n an integer. When the SO coupling is
biased conditions for the system. very strong, θ π/2, and the spin is in the plane, while
→
Notethatleftandrightpropagatingelectronswiththe for very small SO coupling, the spin is aligned with the
same s-index are not degenerate, but time reversalsym- z axis.
metryissatisfiedi.e. Eλ,ζ =E−λ,ζ (simultaneouschange
n,s n,−s
of λ ands). This symmetry reflects the fact that the SO
III. SPIN FILTERING AND SUPPRESSED
interaction is not symmetric under space inversion but
BACKSCATTERING
preserves time reversal symmetry (simultaneous change
ofλands)soweretainKramersdegeneracy. Thechiral-
ity label ζ also reflects inversion asymmetry as changes The rotationandspin eigenvalue correspondingto the
in the chiral sign, at fixed λ and s, change the energy. wavefunctions is depictedin Fig.2for bothchirallabels.
4
selection of a propagation direction by applying a bias
(as in I V experiments) or by local population im-
ψn−˜,,++ ψn˜+,,+ ψn˜++,+1,+ ψn−˜+,+1, balance,e−ffectivelybreakstime reversalsymmetrybyse-
− − lectingoneofthetimereversedpartnersandgeneratinga
1 s
+ netspincollectedontheotherendofthechiralmolecule.
= Notealsothatscatteringinthehelixissuppressedbythis
ζ gapifthe spinis notconcurrentlyflipped by the scatter-
E> E< ing event[22]. This is exactly the same mechanism for
z
protected transport in chiral edge state in graphene[23].
To further connect with the experimental results, we
ψn˜+,,+− ψn−˜,,− ψn−˜+,−1,+ ψn˜++,−1, note that if we change the bias direction for the same
− − chirality label, the preferred spin is the opposite spin di-
1
− rection by choosing the other partner of the Kramers
= doublet of lowest energy. This change of selected spin is
ζ proven in the experiments by the fact that spin injected
E E back into the Ni magnet has to go where the DOS is
> <
higher, which is the opposite spin state[22].
The symmetry of the I V curve in experiments is
−
FIG. 2. Electron propagation direction coupled to spin ori- directly related to the fact that the same degenerate en-
entation and the corresponding energies for the two possi- ergy corresponds to the forward and backward bias, the
ble chiralities of the helix. Each energy is doubly degenerate change only being which of the two partners in the dou-
(Kramers doublet). If the electron propagates on a ζ = +1 blet is selected. The barrier for electrons to be injected
helix in the positive z direction (as indicated), the lower en-
into the molecule depend on the workk function of the
ergy state corresponds to the spin up state, while the spin
metal, and this is also very similar between Ni and Au
down state has to pay theenergy gap ∆.
( 5 eV), so this is also a source of symmetry.
∼
IV. CHARGE AND SPIN CURRENTS
Wehavealsoevaluatedtheenergiescorrespondingtothe
splitdoubletsseparatedbythegap∆. Forfixedchirality,
sayζ =+1,thereisadoublydegenerate(changingsand Another way to pose spin selectivity is by evaluating
λ simultaneously) low energy configuration and a high thespincurrentsthroughthehelix. Inacoherentregime
energy configuration for each n˜. one can assess spin transport in the helical molecule by
computingtheexpectationvalueofthe velocityoperator
2
~ω 1 ω 2 J~ = Ψ†e~v Ψ, where e is the electron charge and ~v
E = 0 n˜+ 1+ SO charge
> 2  2s (cid:18) ω0 (cid:19)  iisnttehreacvteiloonc,ittyheopveerloatcoitry. oInpetrhaetoprreissennoctesoifmtphleysp~pi1n/-omrb(iat
  2 diagonalmatrix)astherearisesanadditionalanomalous
~ω 1 ω 2
E = 0 n˜+1 1+ SO . (10) velocity term. We start from the quantum mechanical
< 2  − 2s (cid:18) ω0 (cid:19)  definition of the velocity ~v = ~i[H,~r]. The azimuthal
velocity component ϕ˙, is then
 
AsshowninFig.2,electronspropagatinginthepositivez
direction will havea lowerenergy if their spin is s=+1. ~v = −i~∂ϕ12×2−maασρ. (13)
ϕ
The same energy corresponds to their Kramers partner m∗√a2+b2
which propagates in the opposite direction. In order to
The Hamiltonian takes a simple form when expressed in
propagate a s = 1 state in the positive direction, we
− termsofthevelocity: H= 1m∗v2. Thisresultisaman-
need to pay an energy price ∆ 2 ϕ
ifestation of the possibility to write the SOC as a gauge
1 E2 field[24]andthus agaugeinvariantvelocityasdefinedin
∆= n˜+ SO , (11)
2 2~ω Eq.13. Note that all the manifestations of spin filtering
(cid:18) (cid:19) 0
havebeen observedin biasedsetups. In the free electron
for ω /ω < 1 and E = ~ω . On the other hand,
SO 0 SO SO case, photoelectrons are emitted from the metallic sur-
if the energy ~ω < ~ω due to poor mobility of the
0 SO face in contactwith one end of the molecules, and travel
electrononthehelix(smallω duetolargeeffectivemass)
0 in a preferred direction. Also, in the localized regime,
then the energy gap is directly related to the SO energy
molecules are biased in a particular direction selecting
the preferred sense of transport of the electrons. This
1
∆= n˜+ ESO, (12) bias, as argued before, breaks time reversal symmetry
2
(cid:18) (cid:19) and chooses between the states depicted in Fig.2. We
Thus,establishingasenseofelectronpropagationorbias computethechargecurrentsforthe biasedsysteminthe
willselectapreferredspindirection(lowestenergy). The positive z direction
5
ofmagnitudefromplanargraphenetocarbonnanotubes,
justbybendingthegraphenesheets. Ingraphene,thein-
~b n˜+1 1 1+ ωSO 2 trinsic SOC is second order in the atomic coupling (µeV
J+,+ = " − 2r (cid:16) ω0 (cid:17) #, (14) sceocuopnlidnngeifgohrbsoiunrglceouwpallilnegd)wnahniloetiutbiseli(nmeaeVrinfrtohmeadtoirmecict
n˜+1,+ − m∗√a2+b2
p (i) p (j)coupling[18,25]wherei,j arenearestneigh-
x z
−
versus the opposite spin state in the same propagation bours). ThelattersituationisalsopresentinDNAifone
direction regards the available orbitals to be on it bases, which
are endowed with available p orbitals at such angle as
z
2 to generate both V and V couplings and linear in
~b n˜+ 1 1+ ωSO ppπ ppσ
J+,+ = " 2r (cid:16) ω0 (cid:17) #. (15) aptroemseinctSpOapCe.rT, hbeutdetrhievarteisounltiss sbheoywonndotnhterivsciaolpfeeaotfutrhees
n˜,− − m∗√a2+b2
related to structural effects.
We computed the effective coupling between DNA
Thus, different spin states propagate at different ve-
inter-basenearestneighbourp (i)orbitals,assumingone
locities generating a net spin filtering effect. The net z
orbital per base, and a single helix. The same features
spin current vanishes for either SO zero or zero chirality
shouldapplytothedoublehelix. Wefoundthefollowing
(b=0).
form for the SO coupling
The difference is a net spin up current in the positive
z direction 2λ(Vπ Vσ)
V pp− pp , (18)
=J+,+ J+,+ SO ∝ (επ εσ )~
J n˜+1,+− n˜,− 2p− 2p
~b ωSO 2 The parameter λ is the bare atomic SOC, Vpπp and Vpσp
= 1+ 1 (16) are the Slater-Koster p orbital couplings perpendicular
m∗√a2+b2 s (cid:18) ω0 (cid:19) −  and parallel (respectively) to the line joining the carbon
  atoms of the model. επ εσ is the difference in en-
The longitudinalspintransportcurrentis welldefined 2p − 2p
ergy between the bonded p in the plane of the base
x,y
and can be calculated as
andthe p orbitalperpendicular to the plane. Note that
z
+,+ =(Ψ+,+ )†1 v ,sz Ψ+,+ the energy contrast in the denominator may be small,
Jn˜,z n˜+1,+ 2{ ϕ } n˜+1,+ potentially increasing the SOC by a large amount. This
1 is especially true when interbase coupling is weak. An-
+(Ψ+,+)† v ,sz Ψ+,+
n˜,− 2{ ϕ } n˜,− other wayto expressthe denominatorusing lowestorder
~2b perturbationtheoryisbyrenormalizingtheinplanecou-
= (cosθ 1). (17) pling bythe s p overlap;επ εσ (Vσ)2/(ε ε ),
2m∗√a2+b2 − where ε are−the bare atom2pic−va2lpu∼es ansdp Vσ 2ips−an2isn-
2i sp
The longitudinal spin current depends on the spin-orbit terbase coupling that needs to be computed from a first
coupling through θ. The spin filtered in the z direction principles study.
disappears when the SO coupling is zero (θ =0). Again, The spin selectivity demonstrated by experiments is
the pitch dictates the strength of the vertical spin cur- a room temperature phenomenon. Thus any coherent
rent and both the pitch and the SOC must be present. mechanism must be limited by decoherence lengths[10,
Note that the factor (1 cosθ) = ϕ /π is the non- 14]. The gap for degradation of spin selectivity found
AA
−
adiabatic Aharonov-Anandan phase found by Frustaglia in experiments is very high, 0.5 eV (ref.7), compared
andRichterinadetailedanalysesofconductancethrough to thermal effects at room temperature 25 meV. This
SOcoupledrings[16]. Thisphaseoffersanewinsightinto gappreventselasticbackscatteringandexponentiallyre-
spin filtering of chiral molecules since in the strong SO ducesinelasticscatteringwiththesamespin. Aspincou-
limit it is related to the Berry phase of the spin and for pled scattering mechanism, nonetheless, could degrade
the general case it is controlled by non adiabatic spin the spin selectivity rapidly because of the existence of
precession. the Kramers doublets as a backscattering channel (see
Fig.2).
If one proposes a decoherence length operating at
V. STRENGTH OF THE SOC AND ROOM
room temperature, one can also suggest a mechanism
TEMPERATURE SPIN SELECTIVITY
which preserves spin selectivity analogous to that pro-
posed in ref.[14] and [10], where relaxation of phase co-
Recentexperimentalfindingsdemonstratingspinselec- herence occurs within a few nanometers while spin se-
tivityhaveshownthatthegap∆inEq.12,ismuchlarger lectivity is preserved. This mechanism entails an expo-
that expected from even the purely atomic contribution. nentialdecayofthetransmissionwithadecayrate[26]of
We haveattempted to explainthis anomaloussize ofthe β = (1/d )ln[t /(E (E eV/2))]whereE isthe
coh eff α F α
− − −
SO gap through a tight-binding approach[18, 19, 25]. It energyofthespinprefferedchannel,t isthehoppingin-
eff
iswellknownthattheSOCcanbeenhancedthreeorders tegralbetweensitesseparatedbyacoherencelengthd ,
oh
6
E isthe FermienergyandV theappliedbias. Thenin- model,shouldincreaseaslongasthe moleculeisnotdis-
F
creasing the number of d by having longer molecules rupted. 3) Current decay at fixed bias when changing
coh
will make the input current decrease exponentially, so the molecule length. 4) Depending on the strength of
and increased bias is needed to achieve the same out- the SOC and the length of the molecule, the number of
put current. This mechanism might be the source of the same-spin channels can increase making the selectivity
increased barrier for spin selectivity as the molecule is more robust. 5) The suppressed backscattering mecha-
elongated. nismcouldbetestedbychangingDNAbasesintroducing
Again, there is no mechanism for the coupling to random defects that do no couple to spin. The spin se-
backscattering channels with the same spin (See Fig.2) lectivity should be robust against such inhomogeneities.
so that there will be degradationof spin current but not The model here can be made more quantitative in
of spin polarization. many directions. The derivation of the Hamiltonian di-
rectly from the tight-binding descriptions is desirable so
thatonecanproperlyaccountforthegeometryoftheor-
VI. SUMMARY AND CONCLUSIONS
bitals participating in transport. We have solved a vari-
ationoftheproblemmoreakintothegeometryofthep
z
We have examined a model for chiral spin selectivity orbitals in carbon nanotubes, where they rotate on the
on a spin orbit active helix with N turns. The origin outside of a cylinder. Although the resulting Hamilto-
of the SO electric field comes from the atomic cores of nianisdifferentindetailthesamephysicsdescribedhere
thecarbonatomswhichprovide,throughthep orbitals, follows. This is expected from the symmetry arguments
z
a narrow band for transport. The resulting channels of thatresultinKramersdoubletsseparatedbyagap. Con-
the model helix are Kramersdoublets involvingopposite templatingthedoublehelixstructureofDNA,couldalso
propagating and opposite spin projections at the same bring about new interference effects that might enhance
energy. An appliedbias or otherwise preferredtransport or reduce spin selectivity[14] and bring the model quan-
directionwillthenselectaspinandeffectivelybreaktime titatively closer to the experiments.
reversal symmetry (choosing one in a doublet pair), and Finally, incorporatingthe metallic contacts to the chi-
transferring a particular spin. The resulting spin cur- ral molecule through the thiol groups, should introduce
rent is the spin selectivity mechanism. The spectrum of levelbroadeningtothespectrumofthemoleculeandde-
the model also insures suppressed backscattering by an terminetheescaperateofelectronsandpossiblecharging
energy gap controlled by the SO energy and the inter- phenomena.
base coupling of the p orbitals, as suggested by a tight
z
bindingcalculation. Aformforthecouplingisproposed,
thatcouldenhancetheSOCinlowestorderperturbation
ACKNOWLEDGMENTS
theory.
Thereareafewpredictionsofthetheorythatcouldbe
tested experimentally: 1) The spinselected, after chang- The authors thank Ron Naaman, Mark Ratner and
ing the chirality of the transport structure, seems to be Abe Nitzan for enlightening discussions. EM would like
the same althoughthe stateis different. 2)There should to gratefully acknowledge support from the Fulbright
be changes in the strength of the SOC (as measured Foundation and IVIC Venezuela.
through the experimentally measured gap), by DNA de-
formations that alter the inter-base hopping integrals.
These could be induced by changing torsion (tugging on
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