Model for petahertz optical memory based on a manipulation of the optical-field-induced current in dielectrics

A new physical insight into the ultrafast information communication can be gained from the reversible and robust petahertz (PHz, 1015 Hz) current induced by a strong few-cycle optical waveform in large band-gap dielectrics. We explore an asymmetric conduction of the petahertz current using a heterojunction of low-hole-mass and low-electron-mass dielectrics and devise various functionalities enabling the petahertz signal processing, like diode, switch, and diode transistor. We then propose a model of one-bit optical nonvolatile random-access memory (RAM) by assembling those functionalities and demonstrate its petahertz operation. Further, we suggest the scalability up to a four-bit data manipulation based on the 2 × 2 array of four one-bit RAM elements.

Delicate light-wave-controlled electron dynamics observed in diverse nanostructures reveal the wave nature of light in the context of extreme ultrafast quantum mechanics [13][14][15][16]. Schiffrin et al [20] have experimentally demonstrated that a strong few-cycle laser pulse induces the electric current and macroscopic charge separation in SiO 2 , a dielectric whose band gap E g is as large as ∼9 eV, without optical breakdown. This is a remarkable discovery in that such reversible and robust current could provide a great potential for the future light-wave electronics enabling the processing of petahertz (PHz, 10 15 Hz) electric signals. Recently, Mashiko et al [21] have presented the optical driving of carrier dynamics with a 1. 16 PHz bandwidth in a wide-band-gap (∼3.35 eV) semiconductor, GaN. Further, stimulated theoretical investigations on optical-field-induced currents in dielectrics have proceeded based on the independent particle models [22,23], adiabatic band response [24], and first-principles time-dependent density functional theory [25].
An open question at the present stage will be whether the optical-field-induced current can be manipulated and made directly accessible to the signal processing [26]. One of the first meaningful attempts was a proposal of the petahertz diode for rectifying the optical-field-induced current [27], which is given by a heterojunction of two dielectrics with different polarities, i.e., low-hole-mass (LHM; m m h e * *  ) dielectric with a large hole mobility and low-electron-mass (LEM; m m e h * *  ) dielectric with a large electron mobility (figure 1). m h * and m e * represent the effective hole and electron masses. Those dielectric heterojunction devices may be key ingredients for the signal processing ensuring the petahertz manipulation.
Ultrafast optical memory is a challenge critical for an advancement of the information processing. In this letter, we propose a model for the petahertz optical memory based on a manipulation of the optical-fieldinduced current in dielectrics. A basic petahertz device made of the LHM-LEM or LEM-LHM dielectric Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
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heterojunction is explored to have various functionalities like diode, switch, and diode transistor in an optoelectronic circuitry. Assembling those functionalities, we devise a prototype of one-bit optical nonvolatile random-access memory (RAM) and demonstrate its extreme ultrafast operation working at the petahertz control speed. We also discuss the feasibility of an integration of the one-bit RAM element in terms of the 2×2 elemental array up to the four-bit memory, which sets a viable milestone for an incorporation of the subfemtosecond electron dynamics into the ultrafast processing and communication.

Results
We consider a single heterojunction made of LHM and LEM dielectrics of figure 1(a) or (b), a basic petahertz device. Under a strong few-cycle optical pulse, the time-dependent macroscopic current can be studied by introducing a model Hamiltonian , ) ( )with the lattice constant a, respectively. The indices of c and v denote the conduction (electron) and valence (hole) bands. Depending on the desired functionality of the heterojunction device,  t ¢( ) could be written as optical transitions due to A P (τ) or A ⊥ (τ), or both, i.e., For the laser parameters, we take τ W =4 fs and ω=1.7 eV and the beam spot size to be comparable to the lateral dimension of the device. A P (τ) contributes to the intraband transition through t c p c where c l ñ | and v l ñ | are the l-centered electron and hole states, respectively. In contrast, bothA P (τ) and A ⊥ (τ) additively contribute to the interband transition with d  describes the tunneling across the LHM-LEM heterojunction in the form  Dynamics are initiated by the optical pumping and described by the time-dependent Schrödinger equation Explicit dynamical responses under several geometries of optical pumping are investigated up to the second order of T  . t Y ñ | ( ) is the quantum state that describes an entire heterojunction, given in the following: on the LEM side similarly. One can then attain the electron and hole occupations in the LHM side, n c c , , where λ is simply a control parameter replacing T T T T , In the end of calculation, putting λ=1, we guarantee that c are band occupations relevant to the lowest-order tunneling between the two dielectrics. Now the current across the heterojunction can be simply given by J P . A single in-plane pulse A P (τ) produces the rectified current from LHM to LEM dielectric (figures 2(a) and (b)), where the basic device plays a diode function [27]. The periodically oscillating population (gray line of  ( ) ]across the junction is obtained from the current J(τ; Δτ) from LHM to LEM dielectric. Furthermore, a comparison between the cases of δ=0 and δ=π reveals a sharp contrast, especially at Δτ=0, as shown in figures 2(d) and (e). Note that Q(0) under the case of δ=0, a constructive generation of charge transfer, can be understood through the optical interference induced by a complete overlap of the two pulses [28]. Consequently, we witness that the double-pulse optical pumping not only rectifies the optical-fieldinduced current but also amplifies (δ=0) or suppresses (δ=π) the current and therefore note that the basic device operates as a diode transistor in figure 2(c).
Meanwhile, a single out-of-plane pulse A ⊥ (τ) creates a role of switch to flow or stop the current under a constant bias electric field across the device ( figure 3). At E bias ≈0, carriers excited by A ⊥ (τ) in both sides of dielectrics can hardly make a directional charge flow, i.e., can make just negligible charge transfers (inset of figure 3(b)), while at E bias >0, the LHM-to-LEM current gets to be in the forward bias ( figure 3(b)). To be analytic under the bias field E bias , we note that the electron energy would change as E x E a ka 2 sin s D~W -( ) at the out-of-plane injection pumping [29]. We now put together two basic devices (i.e., LEM-LHM and LHM-LEM devices) and a parallel capacitor into a one-bit RAM ( figure 4(a)) and examine the ultrafast dynamics of memory operations at the petahertz control speed. Figure 4(b) shows that an application of the in-plane pulse A P (τ) (i.e., write pulse) to LEM-LHM device lets the memory turned on by changing the charge state in a capacitor from '0' to '1'. This clearly indicates the ultrafast operation of writing a one-bit information onto the memory. If necessary, the geometry of figure 2(c) may be adopted to amplify or suppress Q C (τ) to the capacitor. Incorporating the optical-field-induced current J t ( ) and the transient dc resistance R(τ) across the junction, we can explore the charge . R(τ) would then be assumed to be ∝1/J dc (τ) with a parametrized minimum value of R min , where J dc (τ) is proportional to the optically excited carrier density n(τ) (gray line of figure 2(b)) in the geometry of figure 4(b). R wire is the wire resistance. Next, an application of the out-of-plane pulse (i.e., read pulse) to LHM-LEM device makes it possible to check the charge storage by inducing a small charge flow from the capacitor, which facilitates the read process by ( )]across the junction should reflect the insulator-semimetal transition depending on E ⊥ ( figure 3(b)) and be, in addition, . Let us note that figure 4(c) manifests the operation of reading a one-bit information stored in the memory. The erase process for removing all the stored information and refreshing the memory is basically same as the read process. That is, a strong out-of-plane pulse (i.e., erase pulse) would make all the charges flow away from the capacitor and then exhausted ( figure 4(d)).
For both the write and read (also erase) processes, the condition of R wire =R min is required for a proper performance as shown in figure 5. Otherwise, the write and read processes would not be made complete within c v t t t á ñ = á ñ = á ñ is assumed. Let us note that E bias ≈0 can hardly make a directional charge flow.
the pulse duration. However, even a wire of a good metal may fail to satisfy R R wire min  2 . Hence, a superconducting wire ensuring R wire =0 could be considered. Furthermore, the value of the capacitance C is assumed to be 10 pF.  =´W -(erase) are adopted according to the optical-field strengths E P =2.08 V Å -1 (write), E ⊥ =0.93 V Å -1 (read), and 4.17 V Å -1 (erase). C is taken to be 10 pF.  In order for the proposed memory to be realistically meaningful, it should be scalable [30][31][32]. In figure 6(a), we indicate the feasibility of its integration by laying out four one-bit memory elements in a 2×2 array. Employing five optical channels, i.e., bit(1), bit(2), write(1), write (2), and read, we attempt a schematic demonstration of four-bit data based on the 2×2 array. A synchronous application of bit(1) and write(1) pulses given by A t  ( ) and A ⊥ (τ), respectively, write '1' in the bit A, whereas a read pulse by A ⊥ (τ) reads out the bits. A schematic example of the four-bit RAM operation under a controlled switch-on of optical channels is demonstrated over three steps in figure 4(b), where the four-bit data are processed as (step I) write '1' in the bit A →write '1' in the bit D →(step II) read out the bits of '1001' →erase the bits →(step III) write '1' in the bit B →read out the bits of '0100'. Definitely, the 2×2 integrated petahertz memory could be generalized to a n×n integration, which handles n 2 -bit data by employing n 2 1 + optical channels of bit(1), ..., bit(n), write(1), ..., write (n), and read.

Conclusions
Hiring a theoretical model of the basic petahertz device made of LHM and LEM dielectrics, we have proposed a superb manipulation of the optical-field-induced petahertz current, which contains a variety of functionalities, e.g., rectifying, switching, amplifying, and suppressing the current. Combining two basic petahertz devices and a parallel capacitor, we have further proposed a one-bit RAM element. Through an investigation of the ultrafast dynamics of fundamental memory tasks such as write, read, and erase processes, we have confirmed that the proposed memory element in fact works at the petahertz clock speed. Finally, we have suggested its scalability or integrability by offering a 2×2 elemental array consisting of four one-bit memory elements. We believe that the scalable petahertz optical memory, the proposition made here, would be a key ingredient to the new horizon of the future solid-state light-wave electronics.