Low-Dimensional Confining Structures on the Surface of Helium Films Suspended on Designed Cavities

Abstract

We investigate the formation of quantum confined structures on the surface of a liquid helium film suspended on a nanostructured substrate. We show theoretically that, by nanostructuring the substrate, it is possible to change the geometry of the liquid helium surface, opening the possibility of designing and controlling the formation of valleys with different shapes. By applying an external electric field perpendicular to the substrate plane, surface electrons can be trapped into these valleys, as in a quantum dot. We investigate how the external parameters, such as the electric field strength and the height of the liquid helium bath, can be tuned to control the energy spectrum of the trapped surface electrons.

Appendices

Appendix A: Smoothness of the Surface Profile

In order to obtain Eq. (2), one must consider that the surface profile ξ(x,y) varies smoothly. If such an approximation is not considered, one obtains the following equation for the surface profile:

$$\begin{aligned} & \kappa_x\frac{\partial^2\xi}{\partial x^2} + \kappa_y\frac{\partial^2\xi }{\partial y^2} - 2 \biggl(\frac{\partial\xi}{\partial x} \biggr) \biggl(\frac{\partial\xi}{\partial y} \biggr)\frac{\partial^2 \xi}{\partial x \partial y} \\ &\quad {}= \biggl[\frac{\sigma g}{\alpha}\bigl(\xi(\vec{r})-h+H\bigr)-\frac{\gamma}{\alpha\xi^{3}(\vec{r})} \biggr]\kappa_{xy}, \end{aligned}$$

(12)

where

$$\begin{aligned} \begin{aligned} \kappa_x &= \biggl[1+ \biggl(\frac{\partial\xi}{\partial y} \biggr)^2 \biggr],\qquad \kappa_y = \biggl[1+ \biggl(\frac{\partial\xi}{\partial x} \biggr)^2 \biggr],\\ \kappa_{xy} &= \biggl[1+ \biggl(\frac{\partial\xi}{\partial x} \biggr)^2 + \biggl(\frac{\partial\xi}{\partial y} \biggr)^2 \biggr]^{3/2}.\end{aligned} \end{aligned}$$

(13)

Indeed, if (∂ξ/∂x)²≪1, (∂ξ/∂y)²≪1 and (∂ξ/∂x)(∂ξ/∂y)≪1, Eq. (12) approaches Eq. (5), as expected. Among all the systems investigated in this work, the maximum value observed for the derivatives of ξ is ≈0.06, which would lead to squared derivatives of order of ≈0.0036 in the worst case scenario. This is still around three orders of magnitude smaller than 1, which guarantees that κ _x , κ _y and κ _xy in Eqs. (12) and (13) can be fairly well approximated to 1 and, therefore, validates Eq. (5) for all systems studied in this paper.

Appendix B: Two-Electrons in a Parabolic Quantum Dot

For two interacting electrons in a parabolic dot subjected to an external magnetic field, the Hamiltonian reads

$$\begin{aligned} H =&\frac{1}{2m}[\vec{p_{1}}-e \vec{A_{1}}]^{2}+\frac{1}{2m}[\vec {p_{2}}-e \vec{A_{2}}]^{2} \\ &{}+ \frac{1}{2}m\omega_{0}^{2}r_{1}^{2} +\frac{1}{2}m\omega _{0}^{2}r_{2}^{2}+ \frac{e^{2}}{4\pi\epsilon_{0}|r_{1}-r_{2}|}. \end{aligned}$$

(14)

To solve the Schrödinger equation for this Hamiltonian, we must change the coordinates system, so that it becomes separable. Using the center-of-mass $\vec{R}=(\vec{r_{1}}+\vec{r_{2}})/2$ and relative $\vec {r}=\vec{r_{1}}-\vec{r_{2}}$ coordinates system, the Hamiltonian for this system is re-written as H=H _r +H _R , where

$$ H_{r}=\frac{1}{2\mu}(\vec{p}+q\vec{A})^{2}+ \frac{1}{4}m\omega _{0}^{2}r^{2}+ \frac{e^{2}}{4\pi\epsilon_{0}r} $$

(15)

and

$$ H_{R}=\frac{1}{2M}(\vec{P}+Q\vec{A})^{2}+ m \omega_{0}^{2}R^{2} $$

(16)

with $q=\frac{e}{2}$, $\mu=\frac{1}{2}m$, Q=2e and M=2m. Besides, the introduction of this coordinates system gives rise to different momentum operators

$$\begin{aligned} \vec{p} =&\frac{\hbar}{i}\vec{\nabla}_{r}=\frac{1}{2}( \vec{p_{1}}-\vec{p_{2}}) \end{aligned}$$

(17)

$$\begin{aligned} \vec{P} =&\frac{\hbar}{i}\vec{\nabla}_{R}=\vec{p_{1}}+ \vec{p_{2}}. \end{aligned}$$

(18)

Assuming the potential vector $\vec{A}$ as the symmetric gauge, for a magnetic field applied in z-direction, one finds

$$\begin{aligned} H_{r} =&-\frac{\hbar}{2\mu}\nabla_{r}^{2}- \frac{i\hbar\omega_{c}}{2}\frac {\partial}{\partial\varphi}+\frac{1}{2}\mu\omega^{2}r^{2}+ \frac {e^{2}}{4\pi\epsilon_{0}r} \end{aligned}$$

(19)

$$\begin{aligned} H_{R} =&-\frac{\hbar^{2}}{2M}\nabla_{R}^{2}- \frac{i}{2}\hbar\omega _{c}\frac{\partial}{\partial\phi}+\frac{1}{2}M \omega^{2}R^{2}, \end{aligned}$$

(20)

where ω _c is the cyclotron frequency and $\omega^{2}=\omega ^{2}_{0}+\omega^{2}_{c}/4$.

The separability of the Schrödinger equation allow us to write two-particle wave function in the form $\varPsi(\vec{r},\vec{R})=\varPsi _{r}(\vec{r})\varPsi_{R}(\vec{R})$ and to separate the Schrödinger equation for the Hamiltonian equation (14) into two eigenvalue equations

$$\begin{aligned} H_{R}\varPsi_{R}(\vec{R}) =&E_{R} \varPsi_{R}(\vec{R}) \end{aligned}$$

(21)

$$\begin{aligned} H_{r}\varPsi_{r}(\vec{r}) =&E_{r} \varPsi_{r}(\vec{r}). \end{aligned}$$

(22)

Assuming $\varPsi_{R}(\vec{R})=\varGamma(R)\varPhi(\phi)$ and $\varPsi _{r}(\vec{r})=\varUpsilon(r)\varOmega(\varphi)$, one takes advantage of the circular symmetry of the problem to solve it for ϕ and φ as Φ(ϕ)=e ^iLϕ and Ω(φ)=e ^ilφ. As the wave function must be periodic in the angular direction, it is obtained that L,l=0,±1,±2,… .

The equation for radial part of the center of mass, Γ(R), can be reduced, after a few transformations, to a generalized Laguerre equation

$$ \zeta F''+ \bigl[ \bigl(|L|+1 \bigr)-\zeta \bigr]F'+ \biggl(\frac {\varepsilon_{R}}{2\omega\hbar}-\frac{\omega_{c}}{4\omega}L- \frac {|L|}{2}-\frac{1}{2} \biggr)F=0 $$

(23)

where ζ=R ²/2a ² with $a=\sqrt{\hbar/2M\omega}$ and $\varGamma(R) = e^{-\frac{\zeta}{2}}\zeta^{\frac{|L|}{2}}F(\zeta)$. For the eigenfunctions to converge, the last term which multiplies F(ζ) in Eq. (23) should be an integer, thus we obtain for the eigenenergies

$$ \varepsilon_{R}= \bigl(2N+|L|+1 \bigr)\hbar\omega+\frac{1}{2}\hbar \omega_{c}L $$

(24)

where N is an integer.

Unfortunately the equation for ϒ(r) cannot be solved by analytical methods, therefore we appeal to a numerical method [35] based on a finite differences scheme. The equation we intend to solve can be written in a dimensionless form as

$$ \varUpsilon''+\frac{1}{r}\varUpsilon'+ \biggl(\varepsilon_{r}-\frac {a_{0}\mu\omega^{2}}{2R_{y}}r^{2}- \frac{l^{2}}{r^{2}}-\frac{2}{r}-\frac {l\hbar\omega_{c}}{2R_{y}} \biggr)\varUpsilon=0, $$

(25)

where space and energy variables were divided by the Bohr radius a ₀ and the Rydberg energy R _y , respectively. By numerically solving this equation, one obtains the contribution of the relative coordinates ε _r to the eigenenergy of the system E=ε _R +ε _r .