Self-Diffusion in a Spatially Modulated System of Electrons on Helium

Abstract

We present results of molecular dynamics simulations of the electron system on the surface of liquid helium. The simulations are done for 1600 electrons with periodic boundary conditions. Electron scattering by capillary waves and phonons in helium is explicitly taken into account. We find that the self-diffusion coefficient superlinearly decreases with decreasing temperature. In the free-electron system, it turns to zero essentially discontinuously, which we associate with the liquid-to-solid transition. In contrast, when the system is placed in the fully commensurate one-dimensional potential, the freezing of the diffusion occurs smoothly. We relate this change to the fact that, as we show, a Wigner crystal in such a potential is stable, in contrast to systems with a short-range inter-particle coupling. We find that the freezing temperature nonmonotonically depends on the commensurability parameter. We also find incommensurability solitons in the solid phase. The results reveal peculiar features of the dynamics of a strongly correlated system with long-range coupling placed into a periodic potential.

Appendices

Appendix

Numerical Integration of the Equations of Motion

To study properties of the electrons on helium, we integrate classical equations of motion numerically. We use HOOMD-Blue [45, 46] as the base code for our simulations, with the integrator, interaction potentials, and external forces developed specifically for our study.¹ The code is designed to be used on a graphics processing unit (GPU). This gives up to 16-fold acceleration compared with a single CPU core. For integration, we use the standard velocity Verlet algorithm, which is symplectic [47]. The algorithm is implemented numerically as follows:

$$\begin{aligned} \varvec{v}_i\left( t + \Delta t/2\right)&= \varvec{v}_i(t) + \frac{1}{2}\varvec{a}_i(t)\Delta t,\nonumber \\ \varvec{r}_i\left( t + \Delta t\right)&= \varvec{r}_i(t) + \varvec{v}_i\left( t + \Delta t/2\right) \Delta t,\nonumber \\ \varvec{v}_i(t + \Delta t)&= \varvec{v}_i\left( t + \Delta t/2\right) + \frac{1}{2}\varvec{a}_i(t + \Delta t)\Delta t, \end{aligned}$$

(A.1)

where $\varvec{v}_i$ is the ith particle velocity, and the acceleration $\varvec{a}_i(t)$ is evaluated based on the forces derived from the positions $\varvec{r}_j(t)$ of all particles at time t. In the calculation, time is discretized and $\Delta t$ is the discretization step.

We modified the algorithm to incorporate discrete scattering events, which are described by an abrupt random change in the electron velocity $\varvec{v}\rightarrow \varvec{v}'$, or the corresponding change in the electron momentum $\hbar \varvec{k}\rightarrow \hbar (\varvec{k}+\Delta \varvec{k})$, that obeys a certain probability distribution. As mentioned in Sect. 2.2.1, electron positions do not change in scattering events, in the classical description. Therefore, the electron potential energy is not changed either. This is automatically satisfied in the calculation if the velocity is changed at integer time points. Importantly, in elastic scattering, which is the dominating scattering mechanism, $|\varvec{v}(t) + \Delta \varvec{v}| = |\varvec{v}(t)|$, where $\Delta \varvec{v}$ is the velocity change at time t, and then the total energy is conserved to the first order in $\Delta t$. Indeed,

$$\begin{aligned} \varvec{v}(t+\Delta t)&= \varvec{v}(t) + \Delta \varvec{v}+ \frac{1}{2}(\varvec{a}(t) + \varvec{a}(t+\Delta t))\Delta t,\nonumber \\ \varvec{r}(t + \Delta t)&= \varvec{r}(t) + \Delta t\left[ \varvec{v}(t) + \Delta \varvec{v}+ \frac{1}{2}\varvec{a}(t)\Delta t \right] . \end{aligned}$$

(A.2)

Then, it is easy to see that the energy difference over one step vanishes in the first order in $\Delta t$,

$$\begin{aligned} E(t + \Delta t) - E(t) = \frac{m}{2}\left( \varvec{v}^2(t + \Delta t) - \varvec{v}^2(t)\right) + U\left( \varvec{r}(t + \Delta t)\right) - U\left( \varvec{r}(t)\right) = O(\Delta t^2). \end{aligned}$$

(A.3)

Since the scattering events in our simulations are rare, about one scattering event per $10^5$ time steps per particle, this accuracy is sufficient for our purpose.

We have chosen $\Delta t \approx (2\pi /\omega _\mathrm{p})/50$ for most of the simulations, where $\omega _\mathrm{p}$ is the short-wavelength plasma frequency given in Eq. (5). Reducing $\Delta t$ proved to have no visible effect on the studied phenomena.

As described in Sect. 2.2.1, scattering processes are characterized by their rates $w(\varvec{k}\rightarrow \varvec{k}')$, which are calculated separately for the ripplon and phonon scattering. Different scattering events are uncorrelated. The scattering by ripplons is effectively elastic. Therefore, its rate $w^\mathrm{(rp)}(\varvec{k}\rightarrow \varvec{k}')$ depends only on the absolute value of $\varvec{k}$ and the angle $\theta$ between $\varvec{k}$ and $\varvec{k}'$. We tabulate the integrated over $\theta$ scattering rate for a given magnitude $k =|\varvec{k}|$ of $\varvec{k}$ on a grid of 200 points. This rate is used to determine the probability to scatter at every time step. We then tabulate the inverse cumulative distribution of the scattering angles $\theta$ on 400 points and use the inverse transform sampling to generate a random value of $\theta$ if a scattering event occurs at a give time step.

In inelastic scattering, both $k'$ and $\theta$ must be determined for a scattering event. As Fig. 1 shows, the scattering rate varies quickly with $k'\equiv |\varvec{k}'|$ and rather smoothly with $\theta$ for each value of k. Therefore, it is convenient to implement the following radial scheme of generating $\varvec{k}'$. For each value of k, we compute the probability distribution of $k'$ by integrating the scattering rate over the scattering angle $\theta$. If a scattering event occurs at a given time step, we first generate the absolute value $k'$ based on this distribution. Then, for every pair $k, k'$ we compute the probability distribution of $\theta$. We draw a random $\theta$ using again the inverse transform sampling after $k'$ is generated in the previous step. The distributions are computed and tabulated in two arrays with sizes $200\times 200$ and $200\times 200\times 10$. In addition to these distributions, we also tabulate the total rate of scattering from a state with wave vector $\varvec{k}$ into any other state to find the overall probability to scatter at each time step, as in the case of the ripplon scattering. For both scattering mechanisms, we use interpolation schemes to obtain continuous scattering distributions from the tabulated values.

An important test of the developed scheme is the stationary electron distribution function. The distribution over the electron momentum should be of the Maxwell–Boltzmann form with the temperature of the helium excitations. We checked that this is indeed the case both for noninteracting electrons and in the presence of the electron–electron interaction.