Article Cite This: J. Phys. Chem. C 2019, 123, 17043−17048
pubs.acs.org/JPCC
Origin of Spin-Dependent Tunneling Through Chiral Molecules Karen Michaeli*,† and Ron Naaman‡ †
Department of Condensed Matter Physics and ‡Department of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot 76100, Israel
Downloaded via UNIV AUTONOMA DE COAHUILA on August 5, 2019 at 14:00:38 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.
S Supporting Information *
ABSTRACT: The functionality of many biological systems depends on reliable electron transfer with minimal heating. Interestingly, nature realizes electron transport via insulating molecules, in contrast to man-made electronic devices which are based on metals and semiconductors. The high efficiency of electron transfer through these organic molecules is unexpected for tunneling-based transport, and it is one of the most compelling questions in the field. Furthermore, it has been shown that the electron tunneling probability is strongly spin-dependent. Here, we demonstrate that the chiral structure of these molecules gives rise to robust coherent electron transfer. We introduce spin into the analysis of tunneling through organic helical molecules and show that they support strong spin filtering accompanied by enhanced transmission. Thus, our study resolves two key questions posed by transport measurements through organic molecules.
1. INTRODUCTION Quantum coherent processes in biology have attracted a lot of attention in recent years.1,2 It is well known that electron transfer in biological systems occurs via tunnelingdirect or in several stepsthrough organic molecules,3,4 most of which exhibit a helical structure. However, the magnitude of the observed transmission over distances of nanometers5,6 and beyond7 is much higher than anticipated based on firstprinciples calculations of the electronic states.8 Recent experiments have revealed that transport through such helixshaped molecules strongly depends on the electron’s helicity, that is, the projection of its spin onto its propagation direction.9−11 Electrons of certain spin can traverse the molecule more easily in one direction than in the other (depending on the handedness of the molecule). These directions are reversed for electrons of opposite spin. This phenomenon, known as chiral-induced spin selectivity (CISS), is observed over a wide energy range of hundreds of meV. Previous theoretical attempts to explain this effect12−17 relied on large spin−orbit coupling, which is uncommon in organic materials. Two important questions arising from experimental observations are as follows: why is the transmission through helical molecules in the tunneling regime so large, and what causes the robust CISS? In this article, we demonstrate that these two properties are strongly interlinked and propose a resolution to both questions. Our work focuses on short helixshaped molecules, such as double-stranded DNA and oligopeptide, where the dependence of the conductivity on the length of the molecule is consistent with direct tunneling. We show that the helical geometry induces correlations between the spins of the transferred electrons and their flow direction. This kind of spin−orbit coupling alone is too weak © 2019 American Chemical Society
to account for the observations; however, it can induce strong CISS in combination with the large dipole electric field characteristic of these molecules. Moreover, we demonstrate that in the tunneling regime, spin selectivity goes hand in hand with a dramatic enhancement of overall transmission through the molecule.
2. MODEL To illustrate these properties, we construct a continuum effective model for electrons in the helical molecules that contains the minimal set of necessary ingredients. The first is a potential VH(r⃗) that confines electrons to propagate within a spiral tube18 centered around
i bs y i 2πs yz i 2πs yz zz + yR zz + zjjĵ zzz P (⃗ s) = xR ̂ cosjjj ̂ sinjjj ̃ ̃ (1) k R { k R { k R̃ { as illustrated in Figure 1a. Here, s is the coordinate along the helix, b is its pitch, R its radius, and the helix parameter R̃ = ± (2πR )2 + b2 is positive (negative) for a right (left)handed spiral. VH models the periodic component of the potential generated by the atoms that comprise the molecule. The second ingredient is a dipole potential VD(r) that grows linearly along the central axis of the molecule. This field is a consequence of the dipolar nature of the hydrogen bonds, and in many cases also by the amino acids terminating the molecule or by potential difference between the donor and the acceptor. Together, they give rise to a substantial voltage difference of 0.1 − 1 V across the molecule. The dipole potential favors localization of electrons and is the basis of the prediction that the tunneling conductivity should be Received: May 27, 2019 Published: June 5, 2019 17043
DOI: 10.1021/acs.jpcc.9b05020 J. Phys. Chem. C 2019, 123, 17043−17048
Article
The Journal of Physical Chemistry C trivial shift of the energy levels ΔE =
Crucially, sign and magnitude of the field are determined by the angular momentum S , and hence, this term preserves timereversal symmetry. A similar coupling between (iso-) spin and spatial degrees of freedom was shown to emerge from curvature effects in strained graphene.19 Because the constant component of the spin−orbit term along the z-direction has a negligible effect on the spin-dependent transport, we will suppress it from here on. Below, we show that the rotating component of the magnetic field gives rise to the CISS effect. The Hamiltonian can be further simplified by the spiñ dependent gauge transformation Ψσ → eiπσs/RΨ̅ σ, where σ = ± denotes spin pointing along the ±z-direction. Under this transformation, the Hamiltonian becomes:
significantly lower than measured. In the helical molecules, VH and VD both arise from the atomic potential, and they correspond to its periodic and nonperiodic components, respectively. The behavior of electrons in our model is governed by the single-particle Hamiltonian iℏ2 ℏ2∇2 + VH(r ) + VD(r ) + σ ⃗· (∇⃗ VH(r ) × ∇⃗) 2m 4m 2 c 2 (2)
The last term in the above Hamiltonian is the spin−orbit coupling which arises as a leading relativistic correction to the Schrödinger equation. The precise form of VH is not crucial for our analysis; for specificity, we here take it to be an isotropic harmonic potential in the plane perpendicular to the helix axis. Thus, VH(r) ≈
/N , S(s) = EN + VD(s) −
ℏ2ρ2 , 2ma0 4
where (ρ,θ) are the spherical coordinates in this plane and a0 ≪ R̃ is the radius of the tube. To lowest order in a0/R̃ , each eigenstate can be written as a product of an s-dependent function and the wavefunction of a two-dimensional harmonic oscillator in the (ρ,θ) plane. The latter are labeled by the level index N ∈ and S = −N , −N + 2 ... N − 2, N . The quantum number S denotes the eigenvalues of −id/dθ, that is, the angular momentum operator pointing perpendicular to the plane. For the helix-shaped cylinder, this vector L⃗ helix is tangent to the helix vector P⃗ defined in eq 1, and hence changes as a function of position s. The coupling between the angular momentum and s due to the helical geometry becomes evident when the Hamiltonian is expanded to order (a0/R̃ )2 /N , S(s) = EN + VD(s) −
+i
ℏ2 (N ma0 2
ℏ2γ S ji ∂ π y jj − i σz zzz + κ Sσy + ΔES m k ∂s R̃ {
(5)
Without the dipole field, VD = 0, the electronic spectrum consists of bands, labeled by N and S , that disperse with momentum k along s. In other words, the Hamiltonian in eq 5 with V D = 0 is diagonal in momentum space, and ΨS(k) = ∫ ds e iksΨS(s) is determined by the Schrö dinger equation ÄÅ 2 ÉÑ 2 ÅÅ ℏ i Ñ ÅÅ jjk − π σ yzz + κ Sσ ÑÑÑΨ(k) = E (k)Ψ(k) ÅÅ j zz yÑ N ,S S ÑÑ S ÅÅÇ 2m k R̃ { ÑÖ (6) Here, k = −i∂/∂s − γ S is the momentum along the helix axis. The solutions to eq 6 are
4
ℏR
Here, we introduced the parameters γ = ̃ 2 and κ = 4m3a 4c 2R̃ R 0 that are smaller by a factor of (a0/R̃ )2 than the leading contribution EN =
2
3. RESULTS
(3) b
ℏ2 ij ∂ π y jj − i σz zzz 2m k ∂s R̃ {
The rotating field has been transformed into a Rashba-like spin−orbit coupling and a constant Zeeman field.20
ℏ2γ S ∂ ℏ2 ∂ 2 + + κ σ⃗ ·L⃗ helix i 2m ∂s 2 m ∂s
+ ΔES
The physical
interpretation of the other two terms is also clear: iγS∂s describes the centripetal potential felt by a particle moving along the helix. It corresponds to the classical effect that a particle propagating through a spiral pipe gets pushed further up along the sides of the tube at higher velocity. Finally, κσ⃗ · L⃗ helix is the familiar atomic spin−orbit coupling due to the confining potential. For a helical system, as shown in Figure 1c, it resembles a Zeeman magnetic field with a component that rotates in the x−y plane as a function of position along the helix É ÅÄÅ 2πs 2πs b ÑÑÑÑ κ σ⃗ ·L⃗ helix = κ SÅÅÅÅσx sin − σy cos − σz Ñ ÅÅÇ (4) R̃ R̃ 2πR ÑÑÑÖ
Figure 1. Helix-shaped tube and the corresponding coordinate system. (a) Electrons are confined to within a helical tube of radius R and pitch b. (b) Helical coordinate system is defined by the position along the spiral axis s as well as n and t that span the perpendicular plane. (c) Spin orbit coupling in eq 4 acts as an effective Zeeman field rotating as a function of position along the helix.
/=−
ℏ2b2 S2 . 2mR̃ 4
ij k jj i 1 + jj 2 jj k + 4R̃ 2κ 2 S2/π 2 1 jjj ΨS +(k) = j 2 jjjj k jj 1 − jj 2 j k + 4R̃ 2κ 2 S2/π 2 k
+ 1) (see Supporting Informa-
tion). The first three terms describe electrons in a (straight) cylinder with a potential VD(s) = eEDs that increases linearly along the central axis. The remaining terms are unique to this problem and reflect the helical geometry. The last term is a 17044
yz zz zz zz zz zz zz zz zz zz z {
(7a)
DOI: 10.1021/acs.jpcc.9b05020 J. Phys. Chem. C 2019, 123, 17043−17048
The Journal of Physical Chemistry C
k jij jj− i 1 − jj 2 k + 4R̃ 2κ 2 S2/π 2 1 jjjj − ΨS (k) = j 2 jjjj k jj 1 + jj 2 j k + 4R̃ 2κ 2 S2/π 2 k with energies EN , S ±(k) = EN + ΔES −
zyz zz zz zz zz zz zz zz zz zz {
Article
(7b)
ℏ2γ 2 S2 ℏ2π 2 ℏ2k 2 + + 2m 2m 2mR̃ 2
ij π ℏ2k yz 2 2 jj z j 2mR̃ zz + κ S k { 2
±
In Figure 2a, we plot the spectrum, illustrating that the Rashbalike term splits the energy of spin-up and -down electrons, Figure 3. Transmission through a helix-shaped molecule without a dipole field. (a) Setup considered in the derivation of the transmission probability assuming a helical molecule attached to two straight cylindrical leads with the same parameters as the spiral system. The spin polarization as a function of energy of incoming electrons with N = 1 is shown in panel (b). The blue curve corresponds to transmission of electrons with S = 1 or S = − 1 between the left and right leads through a right-handed molecule. The green curve illustrates that the opposite polarization is obtained when the current is sent from right to left. The shaded regions mark the energies of the partial gap. (c) Transmission per spin clearly shows the reduced probability for one helicity compared to the other.
spin σ and energy ε (as well as quantum numbers N and S ) to be transmitted through a molecule of length L. The scattering matrix is found by matching the boundary condition at the points where the molecule is connected to the leads. In Figure 3b, we show that the spin polarization,
Figure 2. Electronic band structure for VD = 0. (a) N = 1 bands, schematically shown as a function of linear momentum k along the helix axis, that are split by a large energy difference from the N = 0 band. (b) Exact spectrum for the N = 1 band featuring a partial gap opening for κ ≠ 0 (upper panel) but not for κ = 0 (lower panel). For the derivation, we assumed b = R = 0.3 nm as well as a delocalized bandwidth of ℏ2/2m ≈ 1 eV Å2. The arrows indicate the spin direction of the electronic state.
7N , S =
π 2ℏ2
Δsoc ± |κ S| = ± |κ S|. Typical parameters for organic 2mR̃ 2 molecules, R, b ≈ 0.3 nm and κ = 5 meV, correspond to Δsoc /|κ S| ≈ 10 for S = 1, where a delocalized band with the free electron mass me is assumed. For |κ S| < Δsoc , the states within the partial gap are quasi-helical; the spin is almost perfectly locked to the momentum direction as shown in Figure 2b. As we show next, the coupling of the spin to the electron velocity in this energy window gives rise to CISS. To find the spin-dependent transmission coefficients,24 we consider the setup illustrated in Figure 3a; the molecule (spiral tube) is connected to straight cylindrical leads of the same radius a0. Thus, the only difference between the molecule and the leads is in the curvature effects, which are absent in the latter. Consequently, the electronic states in the leads are characterized by the same quantum numbers N, S , and k. However, the energy spectra of the two differ, most crucially in that the ones of the leads are spin-degenerate
ing matrix
24
ℏ2k 2 . 2m
TN , S, ↑ + TN , S, ↓
, becomes of order unity at energies within
the partial gap. Because of the helical nature of the electronic states in this energy window, the polarization is reversed for transmission in the opposite direction. Moreover, the helicity of electronic states inside the partial gap, and consequently the polarization direction, is determined by the handedness of the helix (the sign of the Rashba-like term). We emphasize that the sign and magnitude of the spin polarization, 7N , S , are independent of the sign of S , that is, 7N , S = 7N , −S . Moreover, we demonstrate in Figure 3c that within the partial gap, one helicity has a higher transmission probability than the other. Finally, we show in Figure 4 that for incident current with spin polarized in the x-direction, the outgoing states are nevertheless spin-polarized along the z-direction. One-dimensional electronic systems described by Hamiltonians similar to eq 6 with VD = 0 have attracted a lot of attention in recent years as a platform for engineering topological superconductors.21−23 In these systems, it is essential to apply an external magnetic field to create the Zeeman term. This is because in strictly one-dimensional systems, helical states can only emerge once time-reversal symmetry is broken. Although for a given value of S and N our model is effectively one-dimensional, the Zeeman term, κ Sσy , is inherited from the curved geometry, which can only exist in three-dimensional space. Moreover, once both the positive and negative values of S are taken into account, it is clear that our model does not break time-reversal symmetry. Importantly,
while the Zeeman field opens a partial gap at energies
EN↑, ↓, S(k) = EN +
TN⃗ , S, ↑ − TN , S, ↓
We studied the spin-dependent scatter-
and the probability TN , S, σ(ε) for an electron with 17045
DOI: 10.1021/acs.jpcc.9b05020 J. Phys. Chem. C 2019, 123, 17043−17048
ÄÅ ÉÑ Å ÑÑ ÑÑ 1 ÅÅÅ ijjj κ 2 S2R̃ yzzz Å ÑÑA i(ξ) exp 1 |
