Optically controlled stochastic jumps of individual gold nanorod rotary motors

Lei Shao, Daniel Andrén, Steven Jones, Peter Johansson, and Mikael Käll

Phys. Rev. B 98, 085404 – Published 2 August 2018

Abstract

Brownian microparticles diffusing in optical potential-energy landscapes constitute a generic test bed for nonequilibrium statistical thermodynamics and have been used to emulate a wide variety of physical systems, ranging from Josephson junctions to Carnot engines. Here we demonstrate that it is possible to scale down this approach to nanometric length scales by constructing a tilted washboard potential for the rotation of plasmonic gold nanorods. The potential depth and tilt can be precisely adjusted by modulating the light polarization. This allows for a gradual transition from continuous rotation to discrete stochastic jumps, which are found to follow Kramers dynamics in excellent agreement with stochastic simulations. The results widen the possibilities for fundamental experiments in statistical physics and provide insights into how to construct light-driven nanomachines and multifunctional sensing elements.

Received 1 May 2018
Revised 16 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.085404

Physics Subject Headings (PhySH)

Laser applications Optomechanics Plasmonics Polarization of light

Brownian dynamics Optical tweezers

Nonlinear DynamicsAtomic, Molecular & OpticalPhysics of Living SystemsGeneral Physics

Authors & Affiliations

Lei Shao^1,*, Daniel Andrén¹, Steven Jones¹, Peter Johansson^2,†, and Mikael Käll^1,‡

¹Department of Physics, Chalmers University of Technology, S-412 96 Göteborg, Sweden
²School of Science and Technology, Örebro University, S-701 82 Örebro, Sweden

^*Also at Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China; shaolei@cuhk.edu.hk
^†Also at Department of Physics, Chalmers University of Technology, S-412 96 Göteborg, Sweden.
^‡mikael.kall@chalmers.se

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Issue

Vol. 98, Iss. 8 — 15 August 2018

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Images

Figure 1
Tilted washboard potential from elliptically polarized light. (a) An elliptically polarized plane wave can be decomposed into one linear and one circular polarization component. (b) Schematic of a gold nanorod in an optical trap. The nanorod is oriented with its long axis perpendicular to the direction of incidence. (c) Nanorod potential-energy distribution induced by the linear polarization component of an elliptically polarized light field. The circular polarization component tends to continuously rotate the rod with a torque of $M_{C}$ . (d) The nanorod thus experiences a rotational tilted washboard potential-energy landscape as a function of orientation angle $φ$ . The potential barrier height in the preferred rotation direction can be calculated to $Δ U = 0.8 k_{B} T$ for parameters mimicking experimental data (nanorod with dimensions of $65 \times 147 {nm}^{2}, λ_{Laser} = 830$ nm, $P_{Laser} = 6$ mW, $T = 326$ K) for the case when the elliptically polarized wave has $Δ ϕ = 40^{\circ}$ . The barrier height in the opposite rotation direction is $9.4 k_{B} T$ .
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Figure 2
Transition from continuous rotation to discrete jumps of gold nanorods. (a) Scanning electron microscopy image of the nanorods (scale bar is 200 nm). (b) Scattering spectrum of a trapped nanorod. The strong peak at $\sim 740$ nm is caused by the long-axis surface plasmon resonance of the nanorod. (c)–(h) Measured (black) and simulated (red) rotational dynamics of (c)–(e) the rod undergoing continuous rotation in the presence of an almost perfectly circularly polarized laser field ( $Δ ϕ = 88^{\circ}$ ) and (f)–(h) discrete rotational jumps due to an elliptically polarized field ( $Δ ϕ = 37^{\circ}$ ). (e) and (h) show the nanorod orientation angle $φ$ versus time, while (c) and (d) and (f) and (g) show the corresponding scattering intensity time traces and intensity autocorrelation functions (ACFs), respectively. The data are based on measurements and simulations of cross-polarized backscattering from the nanorod. The blue stars in (f) and (h) mark individual jumps. (i) Polarization-dependent rotation of a nanorod. We showed both measured (black) and calculated (red) rotational frequency of a nanorod as the laser polarization continuously varies from almost circular to linear. The blue curve indicates the barrier height at varying $Δ ϕ$ . In the highlighted area ( $30^{\circ} < Δ ϕ < 45^{\circ}$ ), where the potential barrier $Δ U \in (0, 4 k_{B} T)$ , we observe the nanorod undergoing a transition from continuous rotation to discrete jumps. The nanorod stops continuous rotation when $Δ ϕ$ decreases to around $40^{\circ}$ , where an effective rotational potential barrier $Δ U \approx k_{B} T$ .
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Figure 3
Stochastic rotational jump dynamics of a gold nanorod. The gold nanorod was trapped using an elliptically polarized laser beam with $Δ ϕ = 37^{\circ}$ and a power of 6 mW. (a) and (b) Experimentally measured and simulation-calculated probability distributions of nanorod flips in a fixed time interval of 5 ms and (c) and (d) transit times of such a jump. The nanorod rotational effective Brownian temperature $T_{r}$ was set at 360 K in simulation to achieve good agreement with experiments. The laser trap forms a tilted washboard rotation potential with a barrier $Δ U = 1.4 k_{B} T_{r}$ , and near a local maximum the potential landscape can be modeled with an inverted harmonic potential [inset in (a)]. The rod was agitated thermally to overcome this barrier to jump by an angle of $π$ , and each jump process was recorded by one scattering intensity peak shown in Fig. 2. The numbers of jumps follow Poisson distribution [blue curves in (a) and (b)]. The full distributions of transit times are well fitted by the formula derived from Kramers theory [blue curves in (c) and (d)], with the coefficients of determination $R^{2} = 0.979$ and 0.994, respectively.
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Figure 4
Temperature- and medium-viscosity-dependent nanorod stochastic jump dynamics. (a) Rotational effective Brownian temperature $T_{r}$ -dependent average number of flips in a time interval of 5 ms and (b) transit time calculated from simulation (red dots). The blue curve in (a) and the red curve in (b) are fitting results, with the coefficients of determination $R^{2} =$ 0.975 and 0.998. (c) Medium viscosity $η$ -dependent average number of flips and (d) transit time calculated from simulation (red dots). The blue curve in (c) and the red curve in (d) are exponential and linear fitting results, with the coefficients of determination $R^{2} =$ 0.999 and 0.968.
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