Electromagnetic Properties of Nanohelices

Abstract

Subjecting a nanohelix to an electric field normal to its axis (a transverse electric field) gives rise to superlattice behavior with tunable electronic properties. We theoretically investigate such a system also with an applied longitudinal electric field along the nanotube axis and find Bloch oscillations and negative differential conductance. Furthermore, we study dipole transitions across the transverse-electric-field-induced energy gap, which can be tuned to the eulogized terahertz frequency range by experimentally attainable external fields, and predict a photogalvanic effect by shining circularly polarized light onto our helical quantum wire. Finally, an exact treatment of the bound state spectra of electrons confined to a helix in a transverse electric field is presented.

Appendices

Appendix 1

Here we consider a particle constrained to a nanohelix of $\mathcal{N}$ turns in a transverse field, defined by the potential

$$\displaystyle{ V = 2u\cos (z/{\mathchar'26\mkern-12mu d} ),\quad 0 \leq z/{\mathchar'26\mkern-12mu d}\leq 2\pi \mathcal{N} }$$

(2.27)

and $V = \infty$ otherwise. With normalization constant c _n , the eigenfunctions are

$$\displaystyle{ \psi _{n} = \frac{c_{n}} {\sqrt{{\mathchar'26\mkern-12mu d}} } S(4\lambda,4u,z/2{\mathchar'26\mkern-12mu d} ),\quad 0 \leq z/{\mathchar'26\mkern-12mu d}\leq 2\pi \mathcal{N} }$$

(2.28)

and ψ _n  = 0 otherwise. The eigenenergies are given by solutions to the transcendental equation

$$\displaystyle{ S(4\lambda,4u,\pi \mathcal{N}) = 0, }$$

(2.29)

which recovers the infinite square well result $\lambda = (n/2\mathcal{N})^{2}$, where n is an integer, in the limit u → 0. We plot in Fig. 2.6 the energy spectra of the low-lying states: it is noticeable how in progressing from the infinite square well limit towards the harmonic oscillator limit the neighboring pairs of states coalesce.

Fig. 2.6

Plot of bound state energies for a helix of (a) $\mathcal{N} = 2$ (b) $\mathcal{N} = 3$ turns as a function of transverse electric field for the four lowest states: the ground state (solid red line) along with the first excited (dotted blue line), second excited (dashed green) and third excited state (dot-dashed orange line)

Subjecting this system to an external double well potential [77, 78], instead of using the hard-wall boundary condition, is an interesting open question.

Appendix 2

We consider an electron-hole pair bound together by the Coulomb interaction V, with the two particles situated on the nanohelix at positions (x _e, h , y _e, h , z _e, h ). In relative coordinates $z = z_{e} - z_{h}$ and with the center-of-mass variable $Z = (M_{e}^{{\ast}}z_{e} + M_{h}^{{\ast}}z_{h})/(M_{e}^{{\ast}} + M_{h}^{{\ast}})$, where the geometrically renormalized masses are $M_{e,h}^{{\ast}} = M_{e,h}(1 + R^{2}/{\mathchar'26\mkern-12mu d} ^{2})$, the two-body problem is separable and one finds the total energy of the exciton is

$$\displaystyle{ \varepsilon = \frac{\hslash ^{2}K_{z}^{2}} {2M_{T}} +\varepsilon _{B}, }$$

(2.30)

where the total renormalized mass is $M_{T} = M_{e}^{{\ast}} + M_{h}^{{\ast}}$. The first term in Eq. (2.30) is the kinetic energy of the translational motion (with wavevector K _z ) in the center-of-mass frame and the second term is the binding energy, which is the eigenvalue of the following Schrödinger equation for relative motion

$$\displaystyle{ -\frac{\hslash ^{2}} {2\mu } \frac{d^{2}} {dz^{2}}\phi -\frac{e^{2}} {4\pi \epsilon _{0}} \frac{1} {\sqrt{z^{2 } + 4R^{2 } \sin ^{2 } (z/2{\mathchar'26\mkern-12mu d} )}}\phi =\varepsilon _{B}\phi, }$$

(2.31)

where the reduced renormalized mass is $1/\mu = 1/M_{e}^{{\ast}} + 1/M_{h}^{{\ast}}$. The interaction in Eq. (2.31) can be represented as a shifted Coulomb potential

$$\displaystyle{ V = \frac{-e^{2}} {4\pi \epsilon _{0}} \frac{1} {\gamma {\mathchar'26\mkern-12mu d}+ \vert z\vert }, }$$

(2.32)

where $\gamma =\gamma (R/{\mathchar'26\mkern-12mu d} )$ is a fitting parameter, which has the advantage that it both avoids the unphysical divergence at zero separation and yields an analytic solution. With this approximation, the exciton on a helix problem maps onto the Loudon model [79], with eigenvalues determined via

$$\displaystyle{ W_{\frac{\mu /M_{e}} {\kappa a_{B}},1/2}\left (2\gamma \kappa {\mathchar'26\mkern-12mu d} \right ) = 0,\quad \text{(odd states)} }$$

(2.33a)

$$\displaystyle{ \left (\gamma \kappa {\mathchar'26\mkern-12mu d}-\frac{\mu /M_{e}} {\kappa a_{B}} \right )W_{\frac{\mu /M_{e}} {\kappa a_{B}},1/2}\left (2\gamma \kappa {\mathchar'26\mkern-12mu d} \right ) = W_{\frac{\mu /M_{e}} {\kappa a_{B}} +1,1/2}\left (2\gamma \kappa {\mathchar'26\mkern-12mu d} \right ),\quad \text{(even states)} }$$

(2.33b)

where $\kappa = \left (-2\mu \varepsilon _{B}/\hslash ^{2}\right )^{1/2}$, the Bohr radius $a_{B} = \frac{4\pi \epsilon _{0}} {M_{e}e^{2}} = 0.529\,\r{A}$ and $W_{\alpha,\beta }(\xi )$ are Whittaker functions of the second kind.

The initial length scales of the problem $R,{\mathchar'26\mkern-12mu d}$ are much larger than the Bohr radius a _B , leading to a characteristic energy scale much smaller than the Rydberg energy, $\text{Ry} = 13.6\,\text{eV}$. The energies in the main part of the paper are measured in $\varepsilon _{0} = \frac{\text{Ry}} {(1+R^{2 } /{\mathchar'26\mkern-12mu d} ^{2 } )}\left (\frac{a_{B}} {{\mathchar'26\mkern-12mu d} } \right )^{2}$, so for excitonic effects to be safely neglected we require the binding energies to be significantly higher.

As an illustrative example we put $R = {\mathchar'26\mkern-12mu d}$ and M _e  = M _h (such that μ = M _e and γ = 0. 203) and choose ${\mathchar'26\mkern-12mu d}= 100a_{B}$. The resulting energy levels are shown in Fig. 2.7, alongside the true and approximated Coulomb interactions. One notices the binding energies are at the scale $\varepsilon _{B} \sim \text{Ry}/10$, much larger than $\varepsilon _{0} \sim \text{Ry}/10^{4}$, such that the excitons are not of primary importance due to their high stability. The dominance of the binding energy means that even when screening is included, for example though the relative dielectric constant of the host material, the equality $\varepsilon _{B} >\varepsilon _{0}$ will hold.

Fig. 2.7

Energy levels of bound electron-hole states (solid green lines) in a Coulomb interaction (solid blue line) approximated by a shifted potential (red dotted line). Here $R = {\mathchar'26\mkern-12mu d}= 100a_{B}$ and γ = 0. 203

It should also be noted a two-body analysis of two electrons on a helix [36] shows pairing can occur when $d \ll R$. A proper description of interactions in the framework of Luttinger liquid theory [80] is certainly required to gain more insight into the many body problem.

Appendix 3

Despite remarkable advances in the synthesis of nanohelices, there is still some degree of inhomogeneity in the radius of helices. We consider the effect of a changing helix radius [81, 82] along the helix axis, in the variable-radius coordinates

$$\displaystyle{ \mathbf{r} = \left (R(z)\cos (z/{\mathchar'26\mkern-12mu d} ),R(z)\sin (z/{\mathchar'26\mkern-12mu d} ),z\right ). }$$

(2.34)

Remarkably, the equation of motion equation becomes a free Schrödinger equation in the new dependent variable

$$\displaystyle{ \xi (z) =\int ^{z}h(z')\mathrm{d}z',\quad h(z) = \left (1 + R(z)^{2}/{\mathchar'26\mkern-12mu d} ^{2} + R'(z)^{2}\right )^{1/2} }$$

(2.35)

where R′(z) represents a derivative with respect to z, such that the solutions are simply

$$\displaystyle{ \psi _{n} = \frac{c_{n}} {\sqrt{{\mathchar'26\mkern-12mu d}} } \sin \left (k\xi \right ),\quad \varepsilon _{n} = \frac{\hslash ^{2}} {2M}\left ( \frac{n\pi } {\xi _{\mathcal{N}}}\right )^{2} }$$

(2.36)

where $k = \left (2M\varepsilon /\hslash ^{2}\right )^{1/2}$ and $\xi _{\mathcal{N}} =\xi (2\pi \mathcal{N}{\mathchar'26\mkern-12mu d} )$. The homogeneous helix R(z) = R easily recovers the solution

$$\displaystyle{ \varepsilon _{n}^{\text{homo}} = \frac{\hslash ^{2}} {8M^{{\ast}}{\mathchar'26\mkern-12mu d} ^{2}}\left ( \frac{n} {\mathcal{N}}\right )^{2} }$$

(2.37)

as found in the literature [58]. Considering a bump inhomogeneity described by $R(z) = R\left (1 +\gamma \exp (-z^{2}/{\mathchar'26\mkern-12mu d} ^{2})\right )$, one finds the relative energy eigenstates do not deviate dramatically from the inhomogeneous case. For example, $\varepsilon ^{\text{bump}}/\varepsilon ^{\text{homo}} \approx 0.95$ for a helix of parameters $R/{\mathchar'26\mkern-12mu d}= 1$ and $\mathcal{N} = 1$ turns, with bump parameter γ = 0. 3.