Spin-Selective Transport of Electron in DNA Double Helix

0.36 MB·English

by  Ai-Min Guo

Spin-Selective Transport of Electron in DNA Double Helix
Ai-Min Guo1 and Qing-feng Sun1,∗
1Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
(Dated: January 25, 2012)
The experiment that the high spin selectivity and the length-dependent spin polarization are
observed in double-stranded DNA [Science 331, 894 (2011)], is elucidated by considering the com-
bination of the spin-orbit coupling, the environment-induced dephasing, and the helical symmetry.
2 Weshow that the spin polarization in double-stranded DNA is significant even in the case of weak
1 spin-orbit coupling, while no spin polarization appears in single-stranded DNA. Furthermore, the
0 underlyingphysicalmechanismandtheparameters-dependenceofthespinpolarizationarestudied.
2
PACSnumbers: 87.14.gk,85.75.-d,87.15.Pc,72.25.-b
n
a
J
Molecular spintronics, by combining molecular elec- length, and no spin polarization appears for the ssDNA.
3
tronics with spintronics to manipulate the transport of The physical mechanism arises from the combination of
2
electron spins in organic molecular systems, is regarded the dephasing, the SOC, and the helical symmetry. No

] as one of the most promising research fields and is now spinpolarizationcouldbeobservedifanyaforementioned
l
l attractingextensiveinterest[1–4],owingtothelongspin factor is absent. In addition, the spin polarization could
a
relaxation time and the flexibility of organic materials. beconsiderablyenhancedbyappropriatelyincreasingthe
h
- Unconventionalmagneticpropertiesofmolecularsystems dephasing strength or by decreasing the helix angle.
s
reportedinorganicspinvalvesandmagnetictunneljunc- The charge transport through the dsDNA, illustrated
e
m tions, are attributed to the hybrid states in the organic- in Fig. 1, can be simulated by the Hamiltonian:
magnetic interfaces [5–9] and to single-molecule magnet
.
t [4]. Organic molecules would not be suitable candidates H=HDNA+Hlead+Hc+Hso+Hd. (1)
a
m forspin-selectivetransportbecauseoftheir nonmagnetic Here H = N [ 2 (ε c† c + t c† c ) +
properties and weak spin-orbit coupling (SOC) [10]. DNA Pn=1 Pj=1 jn jn jn jn jn jn+1
d- However,veryrecently,G¨ohleret al. reportedthespin λnc†1nc2n+H.c.] is the Hamiltonian of usualtwo-leg lad-
n der model including spin degree of freedom [17], with N
selectivity of photoelectron transmission through self-
co assembledmonolayersofdouble-strandedDNA(dsDNA) the DNA length, c†jn =(c†jn↑,c†jn↓) the creation operator
[ deposited on gold substrate [11]. They found that well- of the spinor at the nth site of the jth chain of the ds-
organized monolayers of the dsDNA act as very efficient DNA, εjn the on-site energy, tjn the intrachain hopping
1
v spin filters with high spin polarization at room temper- integral,andλn the interchainhybridizationinteraction.
8 ature for long dsDNA, irrespective of the polarizationof Hlead + Hc = Pk,β(β=L,R)[εβka†βkaβk + tβa†βk(c1nβ +
8 the incident light. The spin filtration efficiency increases c2nβ)+H.c.]describetheleftandrightnonmagneticleads
8 with increasing length of the dsDNA and contrarily no andthecouplingbetweentheleadsandthedsDNA,with
4 spin polarization could be observed for single-stranded nL = 1 and nR = N. Hso and Hd are, respectively, the
.
1 DNA (ssDNA). These results were further substantiated Hamiltonians of the SOC term and the dephasing term,
0 by direct charge transport measurements of single ds- which will be discussed in the following.
2
DNAconnectedbetweentwoleads[12]. Althoughseveral Whenachargeismovingunderanelectrostaticpoten-
:1 theoretical models were put forward to investigate the tial V, an SOC arises Hso = 4m~2c2∇V ·(σˆ×ˆp~), with the
v spin-selective properties of DNA molecule based on sin- electron mass m, the speed of light c, the Pauli matrices
Xi gle helical chain-induced Rashba SOC [13, 14], the mod- σˆ = (σx,σy,σz), and the momentum operator ˆ~p. In the
r els neglect the double helix feature of the dsDNA and dsDNA, the differences of the potential are usually big-
a are somewhat inconsistent with the experimental results geralong the radialdirectionrˆthan that alongthe helix
that the ssDNA could notbe a spin filter. Until now the axis (z-axis in Fig. 1) [18]. On the other hand, since the
underlying physical mechanism remains unclear for high differences of V are especially large between the interior
spin selectivity observed in the dsDNA [15, 16]. and the exterior of the dsDNA, dV/dr is very large at
In this Letter, a model Hamiltonian, including the the boundary r = R with R the radius [19]. Hence, it
small environment-induced dephasing, the weak SOC, is reasonableto consider the r-component of V only and
and the helical symmetry, is proposed to investigate the the SOC can be simplified in the cylindrical coordinate
quantumspintransportthroughthe ssDNA anddsDNA systemHso =−α~σˆ·(rˆ×ˆ~p)withα≡ 4m~22c2ddrV(r). Con-
connected to nonmagnetic leads. We interpret the ex- sidering a charge propagating in one helical chain of the
perimentalresultsthattheelectronstransmittedthrough dsDNA (e.g., the dotted line in Fig. 1), the momentum
thedsDNAexhibithighspinpolarization,thespinfiltra- ˆ~p=pˆ ˆl withˆl theunitvectoralongthehelicalchaindi-
k k k
tion efficiency will be enhanced by increasing the DNA rection. ThusH isreducedtoH =−α[σ pˆ +pˆ σ ],
so so 2~ ⊥ k k ⊥
2
a† = (a† ,a† ) is the creation operator of the vir-
jnk jnk↑ jnk↓
tual lead and t is the coupling between the nucleobase
d
z and the virtual lead.
Letus demonstrate analyticallythatthe ssDNAcould
notbehaveasaspinfilter. Incontinuousreal-spacespec-
trum,theHamiltonianofthessDNAcontainingtheSOC
term is written as H = pˆ2k − α[σ pˆ +pˆ σ ]+V(l)
h ss 2m 2~ ⊥ k k ⊥
with V(l) the potential energy of the helical chain.
By taking a unitary transformation with the operator
U(l) = e(imα/~2)Rlσ⊥dl [24], Hss is transformed into
la H′ = U†H U = pˆ2k − mα2 +V(l), which is indepen-
ss ss 2m 2~2
l dent of spin. Therefore, no spin polarization could be
R
observed in the ssDNA, regardless of the SOC term, the
l y existence of the dephasing, and other model parameters.
x r This result can be obtained also by using the discrete
Hamiltonians of Eqs. (1) and (2). Similarly, we can also
FIG. 1: (color online). Schematic view of the dsDNA with verify that any kind of SOC could not give rise to spin
radiusR,pitchh,helixangleθ,andarclengthla. Thecircles polarization in the ssDNA.
represent the nucleobases, where the full ones assemble one
AccordingtotheLandauer-Bu¨ttiker(LB)formula,the
helical chain and the open ones form the other helical chain.
current in the qth lead (real or virtual) with spin s can
The arc length satisfies lacosθ = R∆ϕ and lasinθ = ∆h,
be written as [25]: I = (e2/h) T (V −V ),
with ∆ϕ and ∆h being the twist angle and the stacking dis- qs Pm,s′ qs,ms′ m q
tance between neighboring base pairs, respectively. We set where Vq is the voltage in the qth lead and Tqs,ms′ =
R = 1 nm, ∆h = 0.34 nm, and ∆ϕ = π5, which are typical Tr[ΓqsGrΓms′Ga]isthetransmissioncoefficientfromthe
values of B-form DNA. Other parameters can be obtained: mth lead with spin s′ to the qth lead with spin s. The
h=3.4 nm, θ≈0.5 rad, and la ≈0.71 nm. Green function Gr = [Ga]† = [EI − H − H −
DNA so
Σr ]−1 and Γ =i[Σr −Σa ], with E the incident
Pqs qs qs qs qs
electronenergy(ortheFermienergy). Σr istheretarded
whereσ (ϕ)=σ sinϕsinθ−σ cosϕsinθ+σ cosθwith qs
⊥ x y z self-energy due to the coupling to the qth lead. For the
θ the helix angle and ϕ the cylindrical coordinate. Since
real left/right lead, Σ = −iΓ /2 = −iπρ t2 ;
the dsDNA consists of two helical chains, the total SOC L/Rs L/R L/R L/R
isH =−α 2 [σ(j)pˆ(j)+pˆ(j)σ(j)]withσ(1) =σ (ϕ) while for the virtual leads, Σrqs = −iΓ/2 = −iπρdt2d,
so 2~Pj=1 ⊥ k k ⊥ ⊥ ⊥ with the dephasing parameter Γ and ρ being the
and σ⊥(2) = σ⊥(ϕ +π). The interchain SOC has been density of state of the leads. Since tLh/eR/ndet currents
neglected because it is very small due to the potential through the virtual leads are zero, their voltages can
symmetry. By using the second quantization [20], Hso be calculated from the LB formula by applying an ex-
can be written as: ternal bias V between the left and right leads with
b
V = V and V = 0. Finally, the conductance for spin-
Hso =Xitsoc†jn[σn(j)+σn(j+)1]cjn+1+H.c., (2) upL (G b) and spRin-down (G ) electrons can be obtained
↑ ↓
j,n G =(e2/h) T V /V , and the spin polariza-
s Pm,s′ Rs,ms′ m b
where t = − α , σ(1) = σ (n∆ϕ), and σ(2) = tion is Ps =(G↑−G↓)/(G↑+G↓).
so 4la n+1 ⊥ n+1 ForthedsDNA,ε issettoε =0andε =0.3,t
σ (n∆ϕ+π). l and∆ϕare,respectively,thearclength jn 1n 2n jn
⊥ a
is taken as t = 0.12 and t = −0.1, and λ = −0.3.
and the twist angle between successive base pairs. 1n 2n n
All these parameters are extracted from first-principles
On the other hand, a charge transmitting through
calculations [26, 27] and the unit is eV. The helix angle
the dsDNA will experience inelastic scattering from the
and the twist angle are set to θ = 0.5 rad and ∆ϕ = π,
phononsduetothefluctuationofeachnucleobasearound 5
which are typical values of B-form DNA. The SOC is
itsequilibriumpositionandotherinelasticcollisionswith
estimated to t = 0.01, which is an order of magnitude
the absorbed counterions in the dsDNA due to the neg- so
smaller than the intrachain hopping integral. In fact,
atively charged sugar-phosphate backbones [21]. Such
allthe results arequalitativelysameevenforsmallert .
inelastic scattering will give rise to the lose of the phase so
Fortherealleads,theparametersΓ =Γ =1arefixed.
and spin memory of the charge. To simulate the phase- L R
Forthevirtualleads,thedephasingstrengthisverysmall
breakingprocess,Bu¨ttiker’svirtualleadisintroducedby
with Γ = 0.005, because the monolayers of the dsDNA
connecting to each nucleobase [22, 23], with the Hamil-
arerigidandthelongestDNAinexperimentisshort[11].
tonian of the dephasing term being:
For this value of Γ, the phase coherence length is longer
Hd = X(εjnka†jnkajnk+tda†jnkcjn+H.c.). (3) than the dsDNA and the electron transport through the
dsDNA will keep its phase coherence [22]. However, the
j,n,k
3
finitedephasingisindispensable. Thevaluesofallabove-
mentionedparameterswillbeusedthroughouttheLetter
except for specific indication in the figure.
Figure 2(a) shows the conductances G and the cor- 0.6
↑/↓ (a)
responding spin polarization P . One notices two trans- =0.005
s
=0.5
mission bands—the highest occupied molecular orbital 0.3
= /5
(HOMO) and the lowest unoccupied molecular orbital PS
(LUMO)—in the energy spectrum, where several trans- h), 0.0
mission peaks are found for both spin-up and spin-down 2 (e/ 0.8

electrons(holes)due tothe coherenceofthe system. For G (b)
the HOMO band, G↑ and G↓ are almost identical, while h), 0.4 =0
bfoerll-tshheapLeUdMcOonfibgaunrda,tiGon↑ iasnodbGse↓rvaerdeinvetrhyedciffuervreenot.f PA 2 (e/ 0.0
s G
vs energy E, where the spin polarization of the dsDNA (c)
0.4
at N = 30 can achieve 0.45, which is comparable with = /2
the experimental results [11]. 0.2 =0
To explore the physicalscenario to high spin polariza-
0.0
tionobservedinthe dsDNA, Figs.2(b) and2(c)plotthe -0.3 -0.2 -0.1 0.4 0.5 0.6
conductance G and P in the absence of the dephasing Energy
↑ s
(Γ = 0) and of the helical symmetry (θ = π), respec-
2
tively. It clearly appears that the spin polarization van-
ishes when Γ=0 or θ = π, although the conductance is FIG. 2: (color online). (a) Energy-dependence of conduc-
still very robust in both c2ases. When Γ=0, the dsDNA tance G↑ (solid line), G↓ (dotted line), and spin polarization
Ps (dashed line) for realistic situation. (b) and (c) show G↑
decoupleswiththevirtualleadsandthechargetransport
and Ps in the absence of the dephasing and of the helical
throughthe dsDNA is completely coherent. In this case, symmetry,respectively. Here N=30.
the SOC can not generate any spin polarization due to
the time-reversal symmetry, the helical symmetry, and
the phase-lockingeffect [28]. Asmalldephasingis neces- has two effects: (i) it promotes the openness of the two-
sary for the existence of finite spin polarization. Indeed, terminal device and produces the spin polarization [28];
it is reasonableto assume a small Γ because the dephas- (ii) it makes the chargelose its phase andspin memories
ing occurs inevitably in the experiment. On the other andthenP isdecreasedbyfurtherincreasingΓ. Accord-
s
hand,thespinpolarizationstronglydependsontheDNA ingly,forlargeΓwiththephasecoherencelengthL [22]
φ
helix. If there is no helix (θ = π), no spin polarization shorter than the dsDNA length, P will be quite small.
2 s
could be observed (P = 0). If the right-handed helical P <0.05 for Γ=0.5 and P →0 if Γ→∞. Due to the
s s s
dsDNA is transformed into the left-handed one (Z-form interplaybetween the abovetwo effects, a small Γ, rang-
DNA) with θ →π−θ, P (π−θ)=−P (θ) exactly. ing from 0.0002 to 0.01, where L is much larger than
s s φ
WethenfocusonthespinpolarizationP andtheaver- 100, is optimal for large P . In addition, Fig. 3(c) shows
s s
agedonehP i,wherehP i≡(hG i−hG i)/(hG i+hG i) the averaged conductance hG i vs N. hG i is declined
s s ↑ ↓ ↑ ↓ ↑ ↑
with hG i averaged over the LUMO band. Figures 3(a) by increasing N or Γ, because large N or Γ will enhance
s
and 3(b) show P at a fixed energy E and hP i vs length the scattering. However, hG i remains quite large for
s s ↑
N, respectively, for several values of the dephasing pa- N =100 and Γ=0.012. Therefore, the dsDNA is a well
rameter. One notices that P and hP i are enhanced by spin filter due to the large P and hG i.
s s s ↑
increasing N at first and then saturate or slightly de- Let us further study the spin polarization by varying
cline after a critical length N . With increasing Γ, N other model parameters. Figures 4(a) and 4(b) show P
c c s
shrinks monotonically [Fig. 3(d)]. The behavior of N at E = 0.488 with N = 80 and hP i with N = 30, re-
c s
vs Γ can be fitted well by a simple function N ∝ Γ−1. spectively,asfunctionsoftheSOCt andthedephasing
c so
For relatively large Γ (diamond and triangle symbols), strength Γ. P and hP i are zero exactly when Γ = 0 or
s s
P and hP i increase faster in the beginning and satu- t = 0. Of course, t is a key factor for the spin polar-
s s so so
rate at shorter length with smaller values because the ization or equivalently t is “the driving force” of P . If
so s
device is more open. While for smaller Γ, P and hP i there is no SOC, no spin polarization would appear for
s s
increase slower with increasing N in a wider range of N whatever other parameters are. In general, strong SOC
and have larger saturation values. Let’s see Γ = 0.0004 usuallyleadto largehP i[Fig.4(b)]. However,forafixed
s
for instance (circle symbols). P and hP i will keep ris- energy, P will not increase monotonically with t , as
s s s so
ing even for N > 100, and P = 0.34 at N = 40 and seeninFig.4(a). AlargeP canbe obtainedforlongds-
s s
P =0.5atN =80. Theseresultsarequantitativelycon- DNA evenfor quite small t , because the spin polarized
s so
sistent with the experiment [11]. In fact, the dephasing electrons will accumulate gradually when electrons are
4
molecule[29]. ItisobviousthathP iiszerointheabsence
s
of the helical symmetry (θ = π) and hP i is increased
0.12 2 s
0.50 (a) monotonically by decreasing θ. This indicates that the
S S
P P helix of the dsDNA plays a vital role to the existence of
0.25 0.06 thespinpolarization. Finally,wepresenttheinfluenceof
(b) the hopping integrals t1 and t2 on the spin polarization,
as illustrated in Fig. 4(d). We can see that hP i is small
0.00 0.00 s
0 30 N60 90 0 30 N60 90 whent1andt2haveidenticalsignandbecomelargewhen
t andt haveoppositesign. Sincethesignofthehopping
75 =0.0001 1 2
h) 0.6 =0.0004 integralissensitivetothetypeofneighboringnucleobases
2 (e/ 0.4 NC50 =0.004 [27]andto the twistangle∆ϕ [30], the spin polarization
G =0.012 couldbeimprovedbysynthesizingspecificDNAmolecule
0.2 25 and putting force along the helix axis of the dsDNA.
(c) (d)
In summary, we propose a model Hamiltonian to sim-
0 30 60 90 0.006 0.012 0.018
N ulate the quantum spin transport through the dsDNA.
This two-terminal dsDNA-based device would exhibit
FIG. 3: (color online). Length-dependence (a) of Ps at E = high spin polarization by considering the SOC, the de-
0.488, (b) of hPsi, and (c) of hG↑i for different values of the phasing, and the helical symmetry, although no spin po-
dephasing parameter. (d) The critical length Nc vs Γ. Here larization exists in the ssDNA. The spin polarization in-
Nc isextractedfrom thecurveofhPsi-N andthesolid lineis creases with increasing the DNA length. Additionally,
the fitting curve with Nc ∝ Γ−1. The legends in (d) are for thespinpolarizationcouldbeimprovedbyproperlymod-
panels (a), (b),and (c).
ifyingthehoppingintegralanddecreasingthehelixangle.
This work was supported by China-973 program and
NSF-China under Grants No. 10974236and 10821403.

0.02 0.02
(a) (b)
0 0
0.01 0.01 ∗ Electronic address: [email protected]
[1] A. R. Rocha, V. M. Garc´ıa-su´arez, S. W. Bailey, C. J.
0.6 0.13 Lambert,J.Ferrer,andS.Sanvito,NatureMater.4,335
(2005).
0.000.00 0.01 0.02 0.000.00 0.01 0.02 [2] A. Fert, Rev.Mod. Phys. 80, 1517 (2008).
ts o tso [3] V. A. Dediu, L. E. Hueso, I. Bergenti, and C. Taliani,
0.50 0.2 Nature Mater. 8, 707 (2009).
(c) (d)
0 -0.15 [4] M. Urdampilleta, S. Klyatskaya, J-P. Cleuziou, M.
/ t2 Ruben, and W. Wernsdorfer, Nature Mater. 10, 502
0.25 0.0 (2011).
0.13 0.15 [5] Z. H. Xiong, D. Wu, Z. V. Vardeny, and J. Shi, Nature
(London) 427, 821 (2004).
[6] C. Barraud, P. Seneor, R. Mattana, S. Fusil, K. Bouze-
0.00 -0.2
0.00 0.01 0.02 -0.2 0.0 0.2 houane, C. Deranlot, P. Graziosi, L. Hueso, I. Bergenti,
t1 V. Dediu, F. Petroff, and A. Fert, Nature Phys. 6, 615
(2010).
[7] J.Brede,N.Atodiresei, S.Kuck,P.Lazi´c,V.Caciuc, Y.
FIG.4: (coloronline). (a)Ps vstheSOCtso andthedephas-
Morikawa, G. Hoffmann, S. Blu¨gel, and R. Wiesendan-
winigthΓNat=E3=0 a0s.4f8u8ncwtiitohnsNof=ts8o0a.n(db)Γ,,(oc)f,Γanadnd(dπθ),sahnodwohfPts1i [8] Nge.r,APtohdyisr.eRseeiv,.JL.eBttr.e1d0e,5,P0.4L7a2z0i4´c,(2V0.10C)a.ciuc, G. Hoff-
and t2, respectively.
mann,R.Wiesendanger, andS.Blu¨gel, Phys.Rev.Lett.
105, 066601 (2010).
[9] S. Schmaus, A. Bagrets, Y. Nahas, T. K. Yamada, A.
transmittingalongthedsDNA.Inaddition,weobservea Bork, M. Bowen, E. Beaurepaire, F. Evers, and W.
largeareawithredcolorinFig.4(b), wherehP iexceeds Wulfhekel, NatureNanotech. 6, 185 (2011).
s
0.1forshortdsDNA.ThisimpliesthatthedsDNAwould [10] F. Kuemmeth,S.Ilani, D.C. Ralph,and P.L. McEuen,
be an efficient spin filter in a wide parameters range. Nature (London) 452, 448 (2008).
[11] B. G¨ohler, V. Hamelbeck, T. Z. Markus, M. Kettner,
Figure 4(c) plots the averaged spin polarization hP i
s G. F. Hanne, Z. Vager, R. Naaman, and H. Zacharias,
vs Γ and πθ by fixing the radius R and the arc length Science 331, 894 (2011).
la to account for the rigid sugar-phosphate backbones. [12] Z. Xie, T. Z. Markus, S. R. Cohen, Z. Vager, R. Gutier-
Thehelix angleθ canbe changedbystretchingtheDNA rez, and R.Naaman, NanoLett. 11, 4652 (2011).
5
[13] S.Yeganeh,M.A.Ratner,E.Medina,andV.Mujica, J. 115346 (2008).
Chem. Phys.131, 014707 (2009). [23] H.Jiang,S.Cheng,Q.-F.Sun,andX.C.Xie,Phys.Rev.
[14] R. Gutierrez, E. D´ıaz, R. Naaman, and G. Cuniberti, Lett. 103, 036803 (2009).
arXiv: 1110.0354 (2011). [24] Q.-F. Sun, J. Wang, and H. Guo, Phys. Rev. B 71,
[15] G. L. J. A. Rikken,Science 331, 864 (2011). 165310 (2005).
[16] M. Di Ventra and Y. V. Pershin, Nature Nanotech. 6, [25] Electronic Transport in Mesoscopic Systems, edited by
198 (2011). S.Datta(CambridgeUniversityPress,Cambridge,U.K.,
[17] G.Cuniberti,E.Maci´a,A.Rodriguez,andR.A.Ro¨mer, 1995).
inCharge Migration inDNA: Perspectives from Physics, [26] Y. J. Yan and H. Y. Zhang, J. Theor. Comp. Chem. 1,
Chemistry and Biology, edited by T. Chakraborty 225 (2002).
(Springer-Verlag, Berlin, 2007). [27] K. Senthilkumar, F. C. Grozema, C. F. Guerra, F. M.
[18] D. Hochberg, G. Edwards, and T. W. Kephart, Phys. Bickelhaupt, F. D. Lewis, Y. A. Berlin, M. A. Ratner,
Rev.E 55, 3765 (1997). and L. D. A. Siebbeles, J. Am. Chem. Soc. 127, 14894
[19] Inotherwords,hereweconsidertheSOCinducedbythe (2005).
boundary-confiningpotential.Alltheresultsarequalita- [28] Q.-F.SunandX.C.Xie,Phys.Rev.B71,155321(2005).
tively same if otherkinds of SOCare considered. [29] J. Gore, Z. Bryant, M. N¨ollmann, M. U. Le, N. R. Coz-
[20] Q.-F. Sun, X. C. Xie, and J. Wang, Phys. Rev. B 77, zarelli, and C. Bustamante, Nature (London) 442, 836
035327 (2008). (2006).
[21] X.-Q. Li and Y.-J. Yan, Appl. Phys. Lett. 79, 2190 [30] R.G.Endres,D.L.Cox,andR.R.P.Singh,Rev.Mod.
(2001). Phys. 76, 195 (2004).
[22] Y. Xing, Q.-F. Sun, and J. Wang, Phys. Rev. B 77,