Abstract
Much efforts are devoted to material structuring in a quest to enhance the photovoltaic effect. We show that structuring light in a way it transfers orbital angular momentum to semiconductorbased rings results in a steady charge accumulation at the outer boundaries that can be utilized for the generation of an open circuit voltage or a photogalvanic (bulk photovoltaic) type current. This effect which stems both from structuring light and matter confinement potentials, can be magnified even at fixed moderate intensities, by increasing the orbital angular momentum of light which strengthens the effective centrifugal potential that repels the charge outwards. Based on a full numerical time propagation of the carriers wave functions in the presence of light pulses we demonstrate how the charge buildup leads to a useable voltage or directed photocurrent whose amplitudes and directions are controllable by the light pulse parameters.
Introduction
The feasibility of light carrying orbital angular momentum (OAM)^{1,2,3,4,5,6,7} opened the way for exciting new applications ranging from electronics and life sciences to quantum information, astronomy, or optical telecommunications^{8,9,10,11,12,13,14,15,16,17,18,19,20}. For instance, OAM beams allow to trap, rotate and manipulate microscopic objects^{21,22,23}, atoms, molecules^{24,25,26} as well as BoseEinstein condensates^{27}. An OAM beam may also drive electric current loops in quantum rings with an associated local, lightcontrolled magnetic field pulses^{28,29}. The phase front associated with OAM beam forms a helical shape. Thus in cylindrical coordinates with z direction being along the light propagation, the field spatial distribution contains a term . Here φ is the azimuthal angle and is the topological charge of the optical vortex. Allan et al.^{1} showed that helical beams (realized for example as LaguerreGaussian (LG) modes) carry OAM with respect to z direction, the amount of which is per photon.
Numerous techniques are available for generating OAM beams: They can be created from usual light sources^{1,30,31}, by computergenerated holograms screened on a spatial light modulator (SLM)^{32,33,34}, astigmatic mode converters^{2}, spiral phase plates^{35} and conversion of spin angular momentum to OAM in inhomogeneous anisotropic plates^{36}. These methods have different strengths and limitations. In general beam generation in connection with SLM has a low efficiency and the overall beam quality is restricted by the pixel size of the nematic liquid crystal cells. The other techniques are static and therefore cannot be controlled dynamically. A newer approach for generating and manipulating OAM beams is realized with a ring resonator based geometry^{37}. Optical vortices with radii independent on the topological charge can be generated based on the widthpulse approximation of Bessel functions^{38,39}.
A key element of OAM light when interacting with matter is the change in time of the carriers’ OAM. This implies a torque exerted on the charge carriers^{23,40,41,42,43,44} rendering so qualitatively new ways to steer the orbital motion by light. For instance as demonstrated in ref. 45, an electronic wave packet in a semiconductor stripe irradiated with an OAM light spot acquires a transverse drift whose direction and amplitude are governed by the parameters of the OAM beam.
Here we explore a further effect of an OAM beam focused on a micro sized GaAsAlGaAsbased quantum ring^{46,47} causing intra conduction band transitions which is shown to result in a centrifugal drift of the carriers and thus to a timesustainable charge imbalance between the inner and out ring boundaries. This charge separation (mimics an intraband photovoltaic effect) is exploitable for the generation of an open circuit voltage which can be tuned in magnitude by increasing the light topological charge at a fixed frequency and intensity, i.e. without additional heating.
In a next step we demonstrate a structured lightinduced photogalvanictype (or bulk photovoltaictype) mechanism. Such an effect occurs conventionally for nonstructured light in media with noncentrosymmetric crystal structure^{48,49}. Typical examples are doped lithium niobate or bismuth ferrite. We suggest and show by fullfledge numerical simulations that the proposed optical vortexinduced photogalvanic effect is systematically controllable by changing the properties of light and by appropriate nano structuring of the system. A possible realization are quantum rings^{46,47} with spiral phase plates (SPP)^{35} deposited atop. A Gaussian light ray traversing SPP acquires OAM that is subsequently delivered to the ring carriers. Combined with the quantum confinement effect, the rings act as a lightdriven charge wheel enhancing the current in an attached wire. The potential of the current predictions are endorsed by recent experiments^{50} on ndoped bulk GaAs irradiated with OAM pulses. It was shown experimentally that the sample attains indeed an orbital angular momentum. The theory proposals presented here point to the advantage of nanostructuring the sample and exploiting the quantum confinement effects in addition to the structure of the light wave fronts.
Results
We employ experimentally feasible OAM laser beams impinging vertically onto a ballistic GaAsAlGaAsbased nanosize ring such as those reported in refs 46,47. We investigate the intraband quantum dynamics of the conduction band and calculate the transient and steadystate time evolution of the charge. To utilize the aforementioned centrifugal photovoltaic effect for direct current generation, we study in a further setup the OAMlaser driven charge current in wires attached to the ring. The thickness of the ring (and later of the wires) is small such that no dynamic occurs along the zdirection due to quantum size effects and thus one may safely restrict the considerations to the xyplane in which the ring structure is embedded. We assume a uniform effective mass m^{*} = 0.067 m_{e} and an average ring radius r_{0} = 150 nm with a width Δr = 50 nm and a Fermi energy ε_{F} = 8 meV. With these predetermined values the electronic structure of the electrons in the conduction band of the considered rings^{46,47} is well captured. We note that the frequency and the intensity of the light are chosen such that only intraband dynamics in the conduction band is triggered. In Fig. 1 the numerically calculated stationary, unperturbed subbands relevant for our study and the local density of states (LDOS) are shown. An initial energy level is classified according to the quantum numbers l_{0} and m_{0}, where characterize the radial motion in the ring. The angular motion is quantified by the angular momentum m_{0}. As expected, the initial states are degenerated with respect to the clockwise and anticlockwise angular motion, i.e. and hence the system is currentless. Furthermore, the radial density distribution is angularly homogeneous and is radially symmetric with respect to r_{0}, as also demonstrated below, meaning that there is no voltage drop between the inner and outer ring boundaries. In what follows we will be interested in noninvasive excitations near the Fermi energy in which case the independent effective single particle picture is still viable^{51,52,53,54,55}.
Applying a weak monochromatic laser pulse carrying orbital angular momentum we trigger the time propagation of the singleparticle wave functions that evolves from the stationary state labeled with the quantum numbers l_{0} and m_{0} at the time t = 0. Technically we obtain by solving fully numerically for the timedependent Schrödinger equation in the presence of the confining potential and the spatially inhomogeneous laser vector potential with frequency ω and amplitude A_{0}. The OAM beam is propagating along the zdirection and is focused vertically on the ring. The light has a right circular polarization, i.e. the polarization vector is . Below we choose and an amplitude corresponding to a peak intensity of I_{TL} = 10^{6} W/m^{2}. The photon energy is ħω = 5 meV (λ = 247 μm) and the pulse duration is characterized by two optical cycles, i.e. τ = 1.65 ps. The beam waist is chosen in a way that the radial intensity profile is not larger than 200 nm, i.e. w_{0} = 55 nm. As discussed below, a possible experimental realization is to deposit on the ring of interest an appropriate spiral phase plate and irradiating the whole structure by a focused Gaussian beam. The transmitted light is converted into OAM light resulting in the electromotive effects presented below. Aside from this idea, it is worthwhile to mention recent achievements in the development of metamaterialbased lenses allowing for strong focusing^{56,57}. The key point of such a lens arrangement is that the spatial profile of the electric field in the focused light spot may be modified but the corresponding topological charge is conserved.
As discussed in^{23,40,41,42,43,44} an OAM beam transfers its OAM when interacting with a dielectric particle. This change in OAM causes a torque. The total torque, within the paraxial approximation, can be given by the photon flux multiplied by the total angular momentum of the beam. In our case of LG modes the amount of transferrable angular momentum is given by , where σ_{z} is the helicity of the circularly polarized light (in our case σ_{z} = −1). This same amount also applies to the vortexinduced break in the clockwise anticlockwise symmetry, meaning that for large large charge currents are achievable. We note that the “torque” associated with this OAM change of the carriers has its origin not only in the vector potential of the OAMLG beam but equally important in the confinement potential that hinders the charge density to escape. A demonstration is depicted in Fig. 2(a) where the ring local density of states (LDOS) is shown at a time t = 2 ps, which means after the laser pulse. We clearly notice that the initial leftright radial symmetry with respect to r_{0} as well as the clockwise and anticlockwise angular symmetries are broken hinting so on the appearance of a radial charge accumulation at the outer ring boundaries (due to the enhancement in the effective centrifugal potential) and the emergence of a charge current loop. We notice 11 nodal angular structures which are explainable by selection rules (i.e., conservation of angular momentum) and considering that we are exciting initially completely symmetric (degenerate) states with a circular polarized OAM beam with , i.e. the total amount of angular momentum transferred to the ring structure is −11 ħ. The direction of the observed whirl is invertible by changing the sign of the topological charge (not shown for brevity). A scheme to collect this charge imbalance as a vortexdriven open circuite voltage is illustrated in Fig. 2(c). How the driven charge may tunnel the boundaries of the rings is shown below.
Discussion
Figure 2(a) evidences a charge density drift of the initial equilibrium state to outer radii over the course of the application time of the OAM beam. This is due to the enhanced radially repulsive centrifugal force upon an effective increase in the angular momentum by which means that this photovoltaic effect can be enlarged by tuning (as long as the centrifugal potential does not overcome the confinement leading so to electron emission). To quantify this observation we calculate the charge density in the inner and outer area of the ring structure corresponding to an initial state with the quantum numbers l_{0} and m_{0} as for the inner area and for the outer ring area. For the whole conduction subbands these quantities are found as
In Eq. (1) stands for the nonequilibrium distribution function. The relaxation processes (electronphonon scattering, simultaneous scattering by impurities and phonons or electronelectron scattering) are introduced phenomenologically by means of a single (averaged) quantity, the relaxation time τ_{rel}. The nonequilibrium distribution function is evaluated within the relaxation time approximation by solving the Boltzmann equation^{58} (we recall that we are considering relatively weak, lowenergy excitations around ε_{F})
The FermiDirac distribution is for a given temperature T and Fermi energy ε_{F} corresponding to the equilibrium. An averaged relaxation time of 25 ps is assumed^{55} at a constant Fermi energy. The evolution of the energy of the particle that develops from the initial stationary state l_{0}, m_{0} can be obtained by calculating the time dependent matrix elements from which we infer the levels that are involved in the process. In Fig. 2(b) the time dependence of Q^{In} and Q^{Out} are depicted. At a time t = 0 both quantities are equal which reflects the radial symmetry of the confinement potential around r_{0}. Over the course of time the charge is redistributed (cf. Fig. 2(a)) in a way that the density is pressed to outer radii by the vortex beam. At times well below the relaxation time, the evolution is unitary. The frequencies of oscillations exhibited in Fig. 2(a) are readily explained by the frequencies of the OAMselectionrulesallowed transitions between levels near ε_{F}. The generic long time behavior at finite temperatures might to a certain extent be inferred from our previous study ref. 59 on current relaxation in similar rings. There, it was shown that the current relaxation (related to the population dynamics) is due to longitudinal acoustic phonons. The recent experiment^{50} on OAMexcited transients in ndoped GaAs seems to indicate a longlived component pointing to a possible OAM dependence of the electronphonon coupling constant.
Now we wish to extract the charge accumulation in the ring for a useable directed current. To this end we wire the ring to a conductive straight channel at one side (cf. Fig. 3). This channel is not affected by the OAM beam, i.e. the light is focused only on the ring and the carriers are allowed to tunnel to the wire. Such a potential landscape has already been realized experimentally and can be modified at will by appropriate gating^{60,61}. Theoretically, we need to change the potential landscape such that the carries inhibit the wire and the ring equally (we assume the ring and the wire are made of the same material at the same chemical potential). Technically, the confining potential is such that the ring radius is 150 nm, i.e. it is characterized for x ≤ 200 nm for the potential given in Ref 51. At 200 nm a 100 nm wide wire, i.e. V(x > 200 nm, y) = 0, is attached (cf. Fig. 3). Taking into account a width Δr = 50 nm the effective barrier region between the quantum ring and the wire is around 25 nm wide. Since the modified potential V(x, y) has no radial symmetry we characterize the numerically calculated singleparticle states by the quantum numbers n with the energy E_{n}. The calculation of these states reveals that the shape of the LDOS in the ring region, i.e. is not dramatically different to the case without the conductive bar (in the equilibrium state anyway). Figure 3 illustrates nicely the action of the OAM (acting solely on the ring): the ring charge density acquires a twist and a radial drift and tunnels to the wire crashing at the wire outer boundary (at x = 300 nm). Due to the internal twist the reflected and the incoming waves in the wire form an interference pattern with an asymmetric density distribution with respect to the center of the ring. In fact this pattern is current carrying. The charge density flows mainly in negative ydirection after reaching the conductor, which reflects the sign of the topological charge . We carefully checked the symmetry, i.e. that a change of the polarization direction and the sign of the topological charge leads to a flow of density in positive ydirection. Quantitatively, the timedependent current associated with a single particle state with the quantum number n we obtain by calculating the probability current density in ydirection as
We position a detector in the conducting bar at y_{d} = −140 nm and calculate the timedependent current through this detector with the help of
The bounds of the integration are the borders of the conductor at x_{1} = 200 nm and x_{2} = 300 nm. The detector is marked by the black horizontal line in Fig. 3(a). The total current of the system calculated as the weighted sum over all contributions of the partial currents I_{n}(t) generated by the individual particles initially residing in the states specified by the quantum numbers n is given by
where f(n, t) is the aforementioned nonequilibrium distribution function given by eq. (2).
In Fig. 3(c) the time dependence of the total current is depicted for different topological charges. The results reveal that a higher topological charge leads to a higher current. It is interesting to compare Fig. 3(c) with Fig. 2(b). Due to inertia related to the finite effective mass of the carriers, the voltage drop does not build instantaneously as the field is applied. The current in the wire however (cf. Fig. 3(c)), builds up yet much later in a (transport) time determined by the effective velocities of the tunneling, rescattering and interfering current carrying states. Enhancing ω or the topological charge the current in the wire merges faster which is evidenced by the results for and . The small oscillations in the current in the wire are related to the oscillations of Q^{In/Out}(t). The currents have a maximum around t = 1.7 ps which is the time where the OAM light laser pulse is switched off. After that the currents decrease over the course of time which is associated with the weakening flux of the density out of the ring.
From the above it is evident that we can multiply the induced current by fabricating well separated rings and attaching them in series to the wire. Each of the ring should then be irradiated with an OAM beam (e.g., by depositing on each ring an appropriate spiral phase plate that generates locally OAM light). Similarly, one may clamp the ring serially between two wires and drive currents in both wires (in opposite directions) by OAM irradiations. The LDOS in such a case for the same pulse parameters and a topological charge is depicted in Fig. 4 endorsing this scenario which can be viewed as a photogalvanictype effect with the additional caveat that, via nanostructuring, we can steer the photogalvanic current both in direction and magnitude, as illustrated schematically in Fig. 4b.
Summarizing, On the basis of fullfledge quantum dynamical simulations we demonstrated that a focused laser pulse carrying orbital momentum irradiating a ring structure results in a radial centrifugal drift of the carrier which leads to a voltage drop between the inner and outer ring boundaries. Wiring the ring to a conductive straight channel and irradiating the ring with the OAM beam splashes a directed current in the wire whose direction, duration and strength is tunable by the pulse parameters such as the topological charge, the pulse width and the intensity. We also suggested possible ways to enhance the current and extract it in an effective way.
Methods
For the rings we use a radial confinement potential^{51} , where and . The key parameters of this potential are as follows: the average radius of the ring is given by r_{0} = (a_{1}/a_{2})^{1/4}, the width of the ring at the Fermi energy ε_{F} is , where and m^{*} is the electron effective mass. For r near r_{0}, the potential of the ring is parabolic: . Taking then V(r) describes a quantum dot. To uncover the centrifugal photovoltaic effect we perform full numerical propagation on spacetime grid of carriers wave function as governed by
where a gauge is used in which the scalar potential vanishes. In the plane z = 0, the OAM beam is taken as LaguerreGaussian (LG) mode with an onaxis phase singularity of the strength, i.e. vortex topological charge . In addition to the LG modes are described by the radial index p and the waist size w_{0}. Here we use the simplest form of the LG modes with p = 0 in which case the intensity profile is ringshaped around z = 0 (the case p ≠ 0 adds no further qualitative information). The corresponding pulse vector potential in polar coordinates with and is . Due to computational limitations of the present full numerical timepropagation scheme we did not inspect larger or more complex structures. However, it is conceivable that the predicted effects are of a general nature and are akin both to the light vortex and the confinement effects.
Additional Information
How to cite this article: Wätzel, J. and Berakdar, J. Centrifugal photovoltaic and photogalvanic effects driven by structured light. Sci. Rep. 6, 21475; doi: 10.1038/srep21475 (2016).
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Acknowledgements
We acknowledge financial support through the Deutsche Forschungsgemeinschaft under SPP 1840. Consultations on the experimental realization with J. Schilling, A. Sprafke and R.B. Wehrspohn are gratefully acknowledged.
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Both authors contributed equally to the development of the idea, analysis of the results and to writing the manuscript. J.W. conducted the numerical calculations.
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Wätzel, J., Berakdar, J. Centrifugal photovoltaic and photogalvanic effects driven by structured light. Sci Rep 6, 21475 (2016). https://doi.org/10.1038/srep21475
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