Fluid photonic crystal from colloidal quantum dots

TL;DR

This paper investigates how resonant optical forces on semiconductor quantum dots in colloids can create fluid photonic crystals with spatially periodic fluxes, enabling manipulation and separation based on exciton frequencies.

Contribution

It demonstrates the formation of a fluid photonic crystal driven by optical forces on quantum dots and explores their resonant dielectric response for selective separation.

Findings

Resonant increase in optical forces near exciton energies.

Formation of fluid photonic crystal with circulating fluxes.

Quantum dot separation based on exciton frequency.

Abstract

We study optical forces acting upon semiconductor quantum dots and the force driven motion of the dots in a colloid. In the spectral range of exciton transitions in uantum dots, when the photon energy is close to the exciton energy, the polarizability of the dots is drastically increased. It leads to a resonant increase of both the gradient and the scattering contributions to the optical force, which enables the efficient manipulation with the dots. We reveal that the optical grating of the colloid leads to the formation of a fluid photonic crystal with spatially periodic circulating fluxes and density of the dots. Pronounced resonant dielectric response of semiconductor quantum dots enables a separation of the quantum dots with different exciton frequencies.

Equations

$$\text{E}(\text{r})=(E_1{\mathrm{e}}^{i{\text{q}}_1\cdot \text{r}}+E_2{\mathrm{e}}^{i{\text{q}}_2\cdot \text{r}})\hat{\text{y}},$$

$$I=\frac{c{n}_{\omega }}{8\pi }|\text{E}{|}^2=\frac{c{n}_{\omega }}{8\pi }[E_1^2+E_2^2+2E_1{E}_2\cos (2xq\sin \theta )].$$

$$F_j(\omega )=\frac{1}{2}\sum _{j^{\prime }}\mathrm{Re}\left[\alpha E_{j^{\prime }}(\text{r})\frac{\partial E_{j^{\prime }}^{\ast }(\text{r})}{\partial r_j}\right],$$

$$\alpha (\omega ,q)=\frac{\pi a_B^3{\omega }_{LT}}{{\omega }_0-\omega -i(\Gamma +q^3{a}_B^3{\omega }_{LT}/6)},$$

$$F_x^{(\mathrm{gr})}=-q\sin \theta (\mathrm{Re}\alpha )E_1{E}_2\sin (2xq\sin \theta ).$$

$$F_x^{(\mathrm{gr})}=\frac{2{\pi }^2}{c{n}_{\omega }}\frac{a_B^3{\omega }_{LT}({\omega }_0-\omega )}{({\omega }_0-\omega {)}^2+(\Gamma +q^3{a}_B^3{\omega }_{LT}/6{)}^2}\frac{dI}{dx}.$$

$$F_x^{(\mathrm{sc})}$$

$$F_z^{(\mathrm{sc})}$$

$$\text{F}=F_{0x}\sin (kx)\hat{\text{x}}+F_{0z}[1+\cos (kx)]\hat{\text{z}},$$

$$F_{0x}$$

$$F_{0z}$$

$$\frac{\partial n}{\partial t}+\nabla \cdot \text{j}=0.$$

$$\text{j}={\text{j}}^{(\mathrm{drift})}+{\text{j}}^{(\mathrm{diff})}.$$

$${\text{j}}^{(\mathrm{drift})}=\mu n\text{F},$$

$${\text{j}}^{(\mathrm{diff})}=-D\nabla n,$$

$$\frac{\partial \delta n}{\partial t}-D\Delta \delta n=-\mu n_0\nabla \cdot \text{F}$$

$$\text{j}=\mu n_0\text{F}-D\nabla \delta n.$$

$$j_z{|}_{z=\pm h/2}=0,$$

$$\delta n(x,z,t)=a(z,t)+b(z,t)\cos (kx).$$

$$\frac{\partial a}{\partial t}-D\frac{d^{2}a}{d{z}^2}=0,\frac{\partial b}{\partial t}-D\left(\frac{d^{2}b}{d{z}^2}-k^{2}b\right)=-\mu n_0{F}_{0x}k,$$

$${\frac{da}{dz}|}_{z=\pm h/2}=\frac{n_0{F}_{0z}}{k_{B}T},{\frac{db}{dz}|}_{z=\pm h/2}=\frac{n_0{F}_{0z}}{k_{B}T}.$$

$$a(z,t)$$

$$b(z,t)$$

$$\delta n=\frac{n_0{F}_{0z}}{k_{B}T}z+\frac{n_0}{k_{B}T}\left[F_{0z}\frac{\sinh (kz)}{\cosh (kh/2)}-F_{0x}\right]\frac{\cos kx}{k}$$

$$j_x$$

$$j_z$$

Full text

Optofluidics, photonic crystal, colloidal quantum dots, non-conservative optical force

I Introduction

Optical fields interacting with micro- and nanoparticles induce sizeable mechanical forces acting upon the particles, which enables their trapping and precise manipulation Ashkin1970 ; Ashkin1975 . Such non-invasive optomechanical approaches have been demonstrated for dielectricBurns1990 ; Cizmar2006 ; Yang2009 , metallicSvoboda1994 ; Hansen2005 ; Huang2015 ; Petrov2016 , semiconductorPan2007 ; Jauffred2008 ; Jauffred2010 particles, as well as biological cells Ashkin1987 ; Wang2011 ; Zhong2013 . Using photonic interference schemes, one can create the arrays of optical traps and form the lattices of dielectric particles Grigorenko2008 or biomolecules Soltani2014 . Among other achievements of optomechanics are measurements of interaction forces between molecules Meiners2000 and optical forces with femtonewton resolution Zensen2016 .

Highly interesting objects for optical trapping and manipulation are colloidal nanocrystals (or quantum dots, QDs), Pan2007 ; Jauffred2008 ; Jauffred2010 since they are important for sensor and biological applications, particularly as superior fluorescent labels Dubertret2002 ; Dahan2003 ; Ebenstein2009 ; Yum2009 . The optical force acting upon a QD is determined by the dot polarizability Chaumet2000 which has a resonant behavior at the frequency of exciton transitions in the QD, when the radiation excites electron-hole pairs (excitons) in the dot Wang2006 ; Ivchenko_book . Far from the exciton resonance, the polarizability is given by the background dielectric contrast between the QD material and the environment and commonly described by the Clausius-Mossotti relation. Optical trapping of colloidal CdTe- and CdSe-core QDs in such conditions has been demonstrated in Refs. Pan2007, ; Jauffred2008, ; Jauffred2010, . Close to the exciton resonance, the QD polarizability and, correspondingly, the optical force are drastically enhanced. Currently, there is a lack of theoretical studies of this effect as well as the optical force driven dynamics of quantum dots in the colloid.

Here, we study the optical force acting upon a QD and the force driven fluxes of QDs in a colloid induced by optical grating. We consider that the dielectric response of a QD is caused by exciton transitions and has a resonant behavior. The force can be split in two contributions: the gradient force and the scattering/absorption force determined by the real and imaginary parts of the susceptibility, respectively Chaumet2000 ; Rohrbach2001 . Close to the excitonic resonance, both contributions increase and are comparable to each other. In the presence of both contributions, the total force is non-conservative and cannot be described by a potential. Therefore, the optical grating of a colloid leads to the formation of spatially periodic density of the dots as well as the emergence of spatially periodic circulating fluxes of the dots (Brownian vertexes Sun2009 ). The emerging modulation of the QD density and fluxes results, in turn, in the periodic modulation of the optical properties of the colloid, such as the refractive index. As a result, the system demonstrates the properties of a photonic crystal. By considering the interplay of optical forces and the processes of viscous friction and diffusion of dots, we study the occurrence of such a fluid photonic crystal from colloidal dots. We calculate the steady-state distributions of the dot density and fluxes as well as the time scales of the photon crystal formation.

II Theory

II.1 Optical force acting upon a quantum dot

We consider a cell with a colloidal solution of semiconductor QDs excited by two coherent laser beams of the $s$-polarization and the same frequency $\omega$, as shown in Fig. 1. The laser fields interfere and produce an optical grating. The total electric field of the radiation in the solution is given by $\bm{{\cal E}}(\textbf{r},t)={\rm Re}[\textbf{E}(\textbf{r}){\rm e}^{-i\omega t}]$, where

[TABLE]

$E_{1}$ and $E_{2}$ are the (real) amplitudes of the laser electric fields in the solution, which are determined by the amplitudes of the incident fields, $E_{10}$ and $E_{20}$, respectively, and the Fresnel transmission coefficient, $\textbf{q}_{1}=q\,(\sin\theta,0,\cos\theta)$ and $\textbf{q}_{2}=q\,(-\sin\theta,0,\cos\theta)$ are the wave vectors, $q=\omega n_{\omega}/c$, $\theta$ is the angel of refraction, which is related to the angle of incidence $\theta_{0}$ by Snell’s law $\sin\theta_{0}=n_{\omega}\cdot\sin\vartheta$, $n_{\omega}$ is the refractive index of the solution, $c$ is the speed of light, and $\hat{\textbf{y}}$ is the unit vector along the $y$ axis. The intensity of the radiation $I$ for the geometry we consider depends on the $x$ axis only and is given by

[TABLE]

The radiation interacts with a QD and produces a force acting upon the dot. For the case of small QDs compared to the light wavelength (typical QD sizes are a few nm while the light wavelength in a colloid is few hundred of nm), the projections of the optical force acting upon the dot located at the r point are given by Ashkin1975 ; Chaumet2000

[TABLE]

where $j$ and $j^{\prime}$ run over the Cartesian coordinates $x$, $y$, and $z$, $E_{j}(\textbf{r})$ is the $j$-projection of the electric field in the colloid $\textbf{E}(\textbf{r})$ given by Eq. (1), and $\alpha$ is the QD polarizability. The optical force is proportional to the QD polarizability and scales quadratically with the electric field. This is because the force originates from the interaction of the oscillating electric dipole moment of the QD, induced in turn by the ac electric field and determined by the dot polarizability, with the ac electric field. In the regime of weak light-matter coupling, which we consider, for the electric field in Eq. (3) it is sufficient to use the unperturbed field of the laser beams (1). We note, however, that the optical force requires more sophisticated self-consistent calculations if the particle itself considerably affects the electric field distribution as it can happen, e.g., in the case of strong light confinement Juan2009 ; Rubin2011 ; Rubin2011_1 ; Sadreev2016 .

The dielectric response of the QD caused by exciton transitions within the dot has a resonant behavior and the polarizability has the form Ivchenko_book

[TABLE]

where $\omega_{0}$ is the resonance frequency corresponding to the exciton transitions, $a_{B}$ is the Bohr radius of the exciton, $\omega_{LT}$ is the frequency corresponding to the longitudinal-transverse splitting of exciton states in the host semiconductor of the QD, which determines the strength of light-matter coupling in semiconductors at inter-band optical transitions. The radiative decay rate of excitons is described by the term $q^{3}a_{B}^{3}\omega_{LT}/6$, the parameter $\Gamma$ stands for the non-radiative decay rate of excitons. The background dielectric contrast of the QD material and the liquid substance is neglected.

Straightforward calculations show that the optical force has two contributions: $\textbf{F}=\textbf{F}^{({\rm gr})}+\textbf{F}^{({\rm sc})}$. The first (gradient) contribution $\textbf{F}^{({\rm gr})}$ is determined by the real part of the QD polarizability. Substitution of Eq. (1) for the electric field in Eq. (3) yields

[TABLE]

Taking into account Eq. (4) for the QD polarizability and Eq. (2) for the intensity of radiation, we obtain

[TABLE]

This contribution to the optical force is directed along or opposite to the light intensity gradient depending on the detuning between the laser field frequency and the exciton frequency.

The second contribution $\textbf{F}^{({\rm sc})}$, commonly referred to as the scattering and absorbing force, is determined by the imaginary part of the QD polarizability and has non-zero both projections. They can be obtained by the substitution of Eqs. (1) and (4) in Eq. (3), which yields

[TABLE]

[TABLE]

The scattering contribution to the total optical force reaches a maximum at the frequency of the exciton resonance.

We consider in what follows that the amplitudes of the incident laser beams are equal to each other, i.e., $E_{1}=E_{2}$. In this case, $F_{x}^{({\rm sc})}$ vanishes and the scattering force points along the line of the equal intensity of radiation. The total force can be presented in the form

[TABLE]

where

[TABLE]

$k=2q\sin\theta$ and $I_{0}=(cn_{\omega}/4\pi)E_{1}^{2}$. For the parameters $\hbar\omega_{0}=2.35$ eV, $a_{B}^{3}\omega_{LT}=0.25\cdot 10^{-6}$ cm3/s (Ref.Ivchenko_book, ), $\hbar\Gamma=30$ $\mu$eV (Ref. Fernee2014, ) and $q^{3}a_{B}^{3}\omega_{LT}/6\ll\Gamma$ corresponding to CdSe quantum dots, the optical force $F_{0}$ can be estimated as $0.1$ fN for the radiation intensity $I_{0}=1$ kW/cm2.

II.2 Circulating currents and separation of dots

In the presence of the scattering contribution, the total optical force is non-conservative (does not conserve mechanical energy) and cannot be described as a gradient of the light-induced potential. Mathematically, it follows from the fact that $\nabla\times\,\textbf{F}\neq 0$ for the force F given by Eq. (9) and the net work done by the optical force in moving a dot around a closed loop is non-zero. Therefore, the optical grating of a colloidal solutions leads not only to the formation of spatially periodic density of the dots but also to the emergence of spatially periodic circulating fluxes of the dots. The emergence of a Brownian vortex of a similar nature in a single-beam optical tweezer has been demonstrated in Ref. Sun2009, . The one-dimensional motion of dielectric particles in an viscous medium in an optical waveguide was considered in Ref. Sadreev2016, .

The kinetics of quantum dots in the solution is governed by the interplay of the optical forces and the forces of viscous friction as well as the processes of dot diffusion. The local concentration of the dots $n(x,z,t)$ and the dot flux $\textbf{j}(x,z,t)$ are related by the continuity equation

[TABLE]

The dots flux consists of the drift and diffusion terms,

[TABLE]

The drift term is proportional to the optical force driving the dots and given by

[TABLE]

where $\mu$ is the mobility of dots in the solution. The diffusion term is determined by the spatial inhomogeneity of the dot concentration and has the form

[TABLE]

where $D$ is the diffusion coefficient. In thermal equilibrium, the mobility and the diffusion coefficient are connected with each other by the Einstein relation $D=\mu k_{B}T$, where $k_{B}$ is the Boltzmann constant and $T$ is the temperature. The diffusion coefficient of quantum dots with radii $15$ - $20$ nm in the glycerol/water solutions with the viscosity $45$ - $55$ cP at room temperature was experimentally determined to be $0.5$ - $0.7$ $\mu$m2/s, Ref. Lessard2007, . These experimental values were found to be in a good agreement with the Stokes-Einstein relation for the diffusion coefficient of spherical particles in a liquid with a low Reynolds number.

To solve the equation set (11)-(14) we present the local concentration of the dots as the sum $n=n_{0}+\delta n$, with $n_{0}$ being the average concentration and $\delta n$ being the correction. Then, in the regime linear in the light intensity, Eqs. (11)-(14) yield

[TABLE]

and

[TABLE]

Equations (15) and (16) are to be solved with boundary conditions. We consider the boundary conditions of zero fluxes through the cell bottom and top, i.e.,

[TABLE]

where $h$ is the cell depth.

For the particular form of the optical force given by Eq. (9), the solution of the diffusion Eq. (15) can be presented in the form

[TABLE]

This decomposition leads to the equations

[TABLE]

and the boundary conditions

[TABLE]

where we took into account the relation $D=\mu k_{B}T$.

Equations (19) with the boundary conditions (20) and an arbitrary initial distribution of quantum dots in the solution can be solved by the Laplace transform method. We assume that the solution was initially in thermal equilibrium and then, at $t=0$, the laser radiation is switched on. In this case, the functions $a(z,t)$ and $b(z,t)$ at $t\geq 0$ are given

[TABLE]

where $k_{n}=(\pi/h)(1+2n)$. The maps of the dot concentration $\delta n(x,z,t)$ and the dot fluxes $\textbf{j}(x,z,t)$ can be readily found from Eq. (18) and (16), respectively.

The steady-state distributions of the dot concentration and the dot fluxes induced by cw radiation can be obtained by considering the limit of the functions $a(z,t)$ and $b(z,t)$ at $t\rightarrow\infty$. This yields

[TABLE]

and

[TABLE]

The steady-state spatial modulation of dot concentration is produced by both $F_{x}$ and $F_{z}$ components of the optical force while the steady-state fluxes are induced by the non-conservative $F_{z}$ component only.

III Results and discussion

Figure 2 demonstrates the steady-state maps of the quantum dot density $n(x,z)$ and the dot circulating fluxes $\textbf{j}(x,z)$ induced by the optical grating of a colloidal solution. The density distribution $n(x,z)$ is encoded by color, the flux distribution $\textbf{j}(x,z)$ is shown by black arrows. Panels a, b, and c correspond to a negative, zero, and a positive detuning $\omega_{0}-\omega$ between the dot resonant frequency $\omega_{0}$ and the radiation frequency $\omega$. Typically, the resonant frequencies of the dots in a solution have a dispersion due to, e.g., the dispersion of the dot sizes. The optical grating leads to the formation of a spatially periodic density of the dots in the longitudinal direction $x$ and inhomogeneous distribution of the dots in the vertical direction $z$. The modulation $\delta n/n_{0}$ is about $1\%$ for the radiation intensity $1$ kW/cm2. At fixed temperature and radiation intensity, the modulation of the QD density is determined by the frequency detuning $\omega_{0}-\omega$, the radiative and non-radiative decay rates of excitons, $\Gamma$ and $q^{3}a_{B}^{3}\omega_{LT}/6$, respectively. The non-radiative decay rate is typically much larger than the radiative decay rate at room temperature. Therefore, the higher degree of the QD density modulation and the higher spectral sensitivity of the QDs are expected for QDs with suppressed non-radiative channels of exciton recombination.

Quantum dots with the exciton frequencies detuned from the radiation frequency (panels a and c) are acted upon by both the scattering component $F_{z}$ and the gradient component $F_{x}$ of the optical force. The scattering component is directed along the $z$ axis but its magnitude depends on the longitudinal coordinate $x$, see Eq. (10). It leads to the formation of a vertical profile of the dot density and a spatially periodic dot density in the longitudinal direction at the cell top and bottom. More importantly, the scattering force gives rise to persistent circulating fluxes of the dots shown by black arrows. The circulating fluxes are more pronounced in cells with the depth $h$ comparable to and exceeding the wavelength $2\pi/k$ since in shallower cells the vertical motion of the dots in suppressed. The gradient component $F_{x}$ leads to the formation of a spatially periodic density of the dots in the longitudinal direction. This effect is most clearly visible at $z=0$. The direction of the gradient force depends on the sign of the detuning, therefore the density modulation at $z=0$ is of opposite signs for the dots with positive (panel a) and negative (panel c) detuning.

For quantum dots with the resonant frequency corresponding to the radiation frequency (panel b), the gradient component $F_{x}$ vanishes and the kinetics is solely determined by the scattering force $F_{z}$. The scattering force gives rise to the inhomogeneous distribution of the dots as well as the circulating fluxes of the dots.

Figure 3 shows some stages of the steady-state distribution formation after the laser radiation is switched on. As it follows from Eqs. (21), the time scale of the formation of longitudinal periodic density of the dots is determined by the time $\tau=1/(Dk^{2})$. For the wave vector $k=0.5\cdot 10^{4}$ cm*-2* and the diffusion coefficient $D=0.5\cdot 10^{-8}$ cm2/s, this time $\tau\approx 0.1$ s. Figures 3a, 3b, and 3c show the snapshots of the density and flux distributions at $t=0.002$ s, $0.02$ s, and $0.1$ s, respectively. Just after the radiation is switched on, the dots are homogeneously distributed in the solution. At this stage, the dot fluxes are completely determined by optical forces since there are no contributions related to the dot density gradients. At large times, the density gradients are formed, the fluxes become closed, and the distributions approach the steady-state ones.

IV Summary

We have presented a theoretical study of optical forces acting on semiconductor quantum dots in a colloid and formation of a fluid photonic crystal from the dots by optical grating. In such a system, the density of the dots and the fluxes of the dots are periodically modulated in space suggesting the periodic modulation of the optical properties of the colloid. By considering the interplay of the optical forces and the processes of viscous friction and diffusion of dots, we have calculated the steady-state distributions of the dot density and fluxes as well as the time scales of the fluid photonic crystal formation. The results can be employed for creating dynamic photonic structures with tunable parameters and further study of their optical properties. Other possible applications include optically-induced mixing of colloids and the separation of quantum dots with different resonant frequencies.

This work was supported by the RF President Grant for young scientists MD-4550.2016.2 and the RFBR grant 16-02-00204.