generation in synchronized semiconductor superlattices.

We study high-frequency generation in a system of electromagnetically coupled semiconductor superlattices fabricated on the same doped substrate. Applying a bias voltage to a single superlattice generates high-frequency current oscillations. We demonstrate that within a certain range of the applied voltage, the current oscillations within the superlattices can be self-synchronized, which leads to a dramatic rise in the generated microwave power. These results, which are in good agreement with our numerical model, open a promising practical route towards the design of high-power miniature microwave generators.


I. INTRODUCTION
Emergent phenomena resulting from the complex cooperative behavior of coupled elements are among the most topical and important subjects of modern physical science. These phenomena cannot be observed in individual elements and come into existence only due to an interaction between entities. Recent examples in condensed matter physics embrace such interesting findings as monopoles in spin ice [1], Skyrmion lattices in chiral magnets [2], and light-induced superconductivity [3,4]. Other instances of emergent behavior relate to dynamical self-organization such as dynamical phases of a driven Bose-Einstein condensate [5], synchronization of micromechanical oscillators coupled through an optical radiation field [6], and coherent terahertz emission from layered high-temperature superconductors [7,8].
Here we investigate cooperative effects in the generation of microwaves from an array of miniband semiconductor superlattices (SLs) [9,10] arranged on the same doped substrate. Once the bias voltage exceeds a certain threshold, each individual SL exhibits current oscillations produced by moving charge domains [11,12]. Our measurements for two nonidentical superlattices show that despite the frequencies of the single SLs being considerably different from each other, electromagnetic (EM) coupling mediated by the substrate can force all of the SLs to generate current oscillations at the same frequency, thus, producing frequency synchronization [13,14]. Moreover, within a certain range of the applied voltage, the synchronization phenomena can significantly (up to 300%) enhance the collective microwave power output. Our measurements are in a good agreement with the results of corresponding numerical simulations. Further numerical investigations show that synchronization is possible for three and four SLs on the same substrate, leading to an even more dramatic increase in the generated ac power. Earlier experimental work has demonstrated that charge domains traveling along miniband SLs are able to produce current oscillations with frequencies up to 300 GHz [15]. Thus, our results suggest a way to solve the long-standing problem of power amplification in solid-state generators operating in the subterahertz or terahertz regime.

II. EXPERIMENTAL RESULTS
In our experiments, we use three SLs grown on the same Si-doped 500-μm-thick GaAs substrate (Fig. 1). Each SL consists of 15 periods, which are separated from two heavily n-doped GaAs contacts by Si-doped GaAs layers of width 50 nm and doping concentration of 1 × 10 17 cm −3 . Each SL period, which is Si doped at 3 × 10 16 cm −3 , includes a 1-nm AlAs barrier, a 7-nm GaAs quantum well, and a 0.8 InAs monolayer at the center of each quantum well. The latter enhances electron injection from a doped contact layer into the first miniband and inhibits interminiband tunneling [16]. The SLs are processed into circular mesa structures with Ohmic contacts to the substrate and top cap layer. A sketch of the sample is shown in Fig. 1. In our study, we measure the microwave generation from the SLs labeled SL 1 and SL 2 in Fig. 1, which are separated by 100 μm. Each device has the same mesa diameter of 20 μm and an independent power supply, as shown in Fig. 1. A LeCroy SDA 18000 serial data analyzer is used to measure the spectra of oscillations generated by the SLs. All measurements are performed at room temperature. First, we consider how the EM output of each SL depends on the bias voltage when the second SL is not powered. Our measurements show that each SL is able to generate microwave current oscillations for an applied voltage V in the range 0.26-0.35 V. Figures 2(a) and 2(b) illustrate the fundamental frequency f [ Fig. 2(a)] and the power P [ Fig. 2(b)] of the microwave signal generated by SL 1 (purple curve) and SL 2 (green curve) as a function of V 1;2 . Although the SLs demonstrate very similar PðV 1;2 Þ dependences [ Fig. 2(b)] and I-V curves [inset in Fig. 2(b)], the frequencies f of the signals measured from different devices are significantly different at all values of the voltage applied. As Fig. 2(a) reveals, SL 1 generates signals in the frequency range 562-621 MHz, while SL 2 has a generation range 690-722 MHz. This difference in the generated frequencies can originate from variation of the contacts and leads attached to a particular SL and corresponding differences in the parasitic capacitance and inductances of the SL circuits [17,18].
Next, we investigate how the microwave output of each SL is affected by simultaneous generation from another SL. To do this, we fix the voltage V 2 applied to SL 2 at 295 mV and measure the spectra of the voltage oscillations on the contacts of both SLs for different voltages V 1 applied to SL 1 . Figure 3(a) illustrates the effect of changing V 1 on the fundamental frequencies f corresponding to the dominant spectral peaks of the two SLs. For small V 1 ¼ 260 mV (dashed line 1 in Fig. 3), the power of the harmonic at the fundamental frequency of SL 2 significantly exceeds that of SL 1 . Therefore, the frequency of the dominant peak in the spectrum of SL 1 coincides with the fundamental frequency of SL 2 . This is also confirmed by Fig. 4(a), which shows that the spectral peak induced by the contribution from SL 2 (around 700 MHz) is much higher than that corresponding to the generation from SL 1 (around 600 MHz). With increasing V 1 (dashed line 2 in Fig. 3 . For higher V 1 , the peak frequencies are locked, manifesting the onset of synchronization [14]. In order to study the collective high-frequency output from SL 1 and SL 2 , we use an unpowered element SL D as a detector ( Fig. 1), which is placed 400 μm away from SL 1 . The detector has a mesa diameter of 5 μm and an almost linear I-V curve for bias voltages in the range −300 to þ300 mV. The relative power of the voltage oscillations measured from SL D for different V 1 is displayed in Fig. 3(b). This characteristic is the ratio between the total ac power P 1þ2 measured from SL D , when both SL 1 and SL 2 are generating microwaves (V 2 ¼ 295 mV) and the power P 1 measured when only SL 1 is active 3. Experimental dependence of (a) the SL fundamental frequencies on V 1 and (b) the relative power of the signal measured from SL D versus V 1 for V 2 ¼ 295 mV. In (a) the purple squares correspond to the data measured from SL 1 and green circles refer to SL 2 . The synchronization region is highlighted in gray. The vertical dashed lines with numbers correspond to the values of V 1 discussed in the text.   Figs. 3(a) and 3(b) reveals that in the absence of synchronization, the microwave power collected from SL D is less than that in the case of a single generating element. However, under certain conditions, the frequency locking can yield a significant boost (up to 3 times) of the detected microwave power. We find that the synchronization phenomena associated with frequency locking can be achieved for a range of the voltages V 1 and V 2 , as shown by the shaded area in Fig. 5. Thus, the experiment shows that a stable synchronization regime is possible even for SLs with a large initial frequency mismatch.

III. THEORETICAL MODELING
To gain deeper insight into the results of the experiments, we theoretically model the system under study. For this aim, we describe the charge transport in each SL using a self-consistent system of Poisson and continuity equations, as discussed in Refs. [12,18]. Within this approach, the SL is split into layers [see Fig. 6(a)], and the discretized forms of the transport equations are solved numerically. Thus, the time evolution of charge density n m ðtÞ in the mth layer is determined by the equation where e is the electron charge, and J m−1 , J m are the volume current densities on the left and right boundaries of the mth layer. The latter can be calculated as whereF m is the mean field in the mth layer [11,12]. It is assumed that charge transport is realized within the lowest miniband when interminiband tunneling can be neglected. In this case, the miniband drift velocity v d for the finite temperature T and givenF m can be calculated using the Esaki-Tsu-Romanov formalism [19]: v d ðFÞ ¼ Δd 2ℏ where d ¼ 8.3 nm is the period of the SL, Δ ¼ 19.1 meV is the miniband width, and τ is an effective scattering time, which takes into account both elastic and inelastic scattering events [11,20]. Parameter k B represents the Boltzmann constant, and I n ðxÞ, where n ¼ 0, 1, is a modified Bessel function of the first kind. In our calculations, we fix T ¼ 4.2 K and τ ¼ 176 fs, whose value corresponds to recent experiments [17,18,20]. The dependence of electron drift velocity v d on electric field strength F is shown in Fig. 6(b). The function v d ðFÞ has a characteristic maximum, which is associated with onset of Bloch oscillations. For large F, the effect of Bloch oscillations on electron dynamics becomes stronger. This leads to increased localization of the electron orbits and decrease of v d . Although Eq. (3) is obtained for a static electric field F, it can also be used for a slowly oscillating electric field, when the miniband electrons can follow the ac field adiabatically [21][22][23], i.e., for 2πfτ ≪ 1, where f is the frequency of an applied electric field. We note that for the value of τ used in our model, this adiabatic limit spans up to several hundred gigahertz. The electric field F m at the left-hand edge of the mth layer [ Fig. 6(a)] is determined by the discretized Poisson equation Here, ε 0 and ε r ¼ 12.5 are the absolute and relative permittivities, respectively, and n D ¼ 3 × 10 22 m −3 is the n-type doping density in the SL layers [20]. To ensure convergence of the numerical solutions, we set N ¼ 480 and Δx ¼ 0.24 nm [12,18]. Ohmic boundary conditions determine the current J 0 ¼ σF 0 in the heavily doped emitter of electrical conductivity σ ¼ 3788 Sm −1 [20]. The voltage V n SL applied to the nth SL is a global constraint given by  (c) Equivalent circuit of SL n (n ¼ 1, 2, D), where C n , L n , and R n are the equivalent capacitance, inductance, and resistance, IðV n SL Þ is the current through the SL, with voltage V n SL dropped across it, and V n is the dc supply voltage. The load resistance is R n where the voltage U dropped across the contacts includes the effect of charge accumulation and depletion in the emitter and collector regions and the contact resistance R ¼ 17 Ω [24,25]. We calculate the current through the SL as where A ¼ 5 × 10 −10 m 2 is the cross-sectional area of the SL [11,12,20]. To take into account the impedance imposed by the contacts and the leads, we model each SL connected to an equivalent LRC (resonant) circuit [17,18,26]; see Fig. 6(c). The current IðtÞ generated by each SL depends on the voltage V n SL applied to this device, where n ¼ 1, 2, and D is the index of the given SL. This voltage includes a dc bias V n and the ac voltage V n c induced by the LRC circuit, which, according to Fig. 6(c), can be described by the following equations: Here, C n , L n , and R n are the equivalent capacitance, inductance, and resistance of the circuit, respectively; R n l is the load resistance. The circuit parameters corresponding to our present experiment are summarized in Table I. The dynamics of the EM field in the common substrate, which provides the coupling between SLs, is considered within the theory of microwave resonator excitation [27,28]. It is assumed that the substrate together with all leads, waveguides, and the measurement system form a single-mode resonator system, where the sth eigenmode E s is excited. In this case, the longitudinal component of the electric field Eðr; tÞ can be represented in the form E ¼ Re½C s ðtÞE s ðrÞe jω s t , where E s and ω s are the spatial field distribution and frequency of the sth eigenmode of the resonator, and C s ðtÞ ¼ AðtÞe jψðtÞ is slowly varying compared to ω s . The distance between the interacting SLs is less than several hundred micrometers, and the wavelength of the generated microwaves is around a few centimeters; therefore, we assume that the field distribution E s ðrÞ in the region of the interacting SLs is homogeneous, and the electric field E in this region depends only on time t. According to the resonator excitation theory, the nonstationary equations for the slowly varying amplitude A and phase ψ are given by where L is the length of each SL, including the contact regions, K is the impedance of the resonator, Q is the quality factor for the sth resonator eigenmode, M is the number of SLs (M ¼ 3 in our experiments), and I n ðtÞ is the current through the nth SL [see Eq. (6)]. The parameters of the resonator system are estimated from the analysis of the experimental data: is the bias voltage applied to the SL, and V n c is the voltage across the corresponding parasitic circuit [defined by Eqs. (7) and (8)]. Thus, the coupling between SLs is realized through the field E, while the electromagnetic properties of the substrate are taken into account via the parameters of the resonator.
We use the above model for numerical calculations of the device characteristics that are measured in experiment. Figure 7 shows the f values (a) and the relative power (b) of the voltage oscillations between the contacts of SL D calculated as functions of V 1 for fixed V 2 ¼ 295 mV. One can see that the plots in Fig. 7 are in good agreement with the measured data presented in Fig. 3. In both cases, the fundamental frequencies exhibit a similar dependence on V 1 . Also, synchronization associated with the frequency  locking leads to a dramatic increase in the power of the signal measured from SL D . The difference between theory and experiment [cf. Figs. 3(b) and 7(b)], especially near the boundary of synchronization, can be explained by the simplifications made in our theoretical model. In particular, the assumption that the contact parameters are independent of the frequency and amplitude of V SL n can lead to inaccurate treatment of the reactance in the coupling of the SLs, which affects their collective dynamics. However, despite these simplifications, the numerically calculated region of synchronization shown in Figs. 7 and 8(a) resembles closely the experimental results in Figs. 3 and 5, respectively.
In Fig. 8, the color map represents the ac power P 1þ2 of the voltage oscillations in SL D normalized to the maximal power P 1 ¼ 1.5 μV 2 of these oscillations in the case when only SL 1 generates microwaves (V 2 ¼ 0). Figure 8(a) reveals that synchronization generally provides a significant increase in power [bright area in Fig. 8(a)]. However, in certain areas (marked orange or red) of the synchronization region (bounded by white dashed line), the power collected from SL D is unaffected, or even weakened, by the frequency-locking phenomenon. We find that this depends on the phase difference Δϕ between the locked signals. A color map representing the ΔϕðV 2 ; V 1 Þ variation is shown in Fig. 8(b). If the voltage oscillations on the contacts of SL 1 and SL 2 are in phase, Δφ ≈ 0, they may constructively interfere, thereby increasing the power measured at SL D [compare Figs. 8(a) and 8(b)]. By contrast, antiphase synchronization Δφ ≈ AEπ will produce destructive interference of the generated signals and, thus, suppress the output from the detector. In the absence of synchronization, the phase difference changes with time, preventing the coherent summation of the output from the generating SLs. We note that the relative power in Figs. 3(b) and 7(b) does not achieve the theoretical maximum value M 2 ¼ 4 (here, M is a number of the generating SLs), which is expected when two identical signals being in phase are summed up in the same detector. This discrepancy occurs because the SLs involved in the interaction are not identical and, as we mention above, initially have a significant frequency mismatch. The latter prevents the generation of the identical current oscillations, even in the case of synchronization.
Finally, we check whether synchronization can be achieved for a larger number of interacting SLs located on the same substrate. In our numerical calculations, we assume that all additional SLs have parameters identical to SL 2 (see Table I). If the in-phase synchronization is achieved, we expect that the superposition of M-synchronized signals will produce an M 2 up scaling of the relative power. Our simulations predict that (i) synchronization of SLs coupled through the substrate is possible even for a larger number of elements, and (ii) synchronization can dramatically increase the ac power collected from the detector. The result of our calculations is presented in Fig. 9, where the dependences of the relative power P detected by SL D are shown versus V 1 for two (dotted curve), three (dashed curve), and four (solid curve) SLs (V 2;3;4 ¼ 295 mV). As expected, the figure reveals that the increase of output power is associated with the onset of synchronization, and the maximal power is realized when the phase difference between signals generated from different SLs is close to zero. For M ¼ 3, the relative power P 1þ2þ3 =P 1 achieves a value of approximately 8, which is close to, but still less than, M 2 . As before, such disagreement is caused by differences between SL 1 and the other SLs that are coupled to it. It is surprising, therefore, that when M ¼ 4, we observe a 20-fold increase of the output power, which considerably exceeds M 2 ¼ 16. We find that this is because the voltage induced at the contacts of SL D exceeds the threshold value, and so SL D starts to generate high-frequency power on its own. In addition, the total power pumped into the resonator becomes large enough for it to demonstrate a visible resonant response.

IV. CONCLUSION
In conclusion, we show both theoretically and in experiment that EM generation from SLs fabricated on the same substrate can be synchronized due to EM interaction through the substrate. Remarkably, the synchronization is possible even if the individual SLs have a large  fundamental frequency mismatch and can lead to dramatic increase of output power, e.g., up to 3 times for two SLs, and up to 20 times for four SLs. Thus, our results provide an efficient way to create powerful solid-state generators able to work at room temperature and at very high frequency (potentially up to the terahertz range). Our calculations also show that the electrodynamics of the substrate play an important role in coupling the SLs and in realizing synchronization with a particular phase difference. These findings opens a path to developing a type of active device comprising an array of generating nanostructures, which are fabricated on an appropriately shaped coupling substrate [29,30]. Moreover, synchronization mediated by the substrate has the potential to boost the power generated from other superlattice-based devices, e.g., quantum cascade lasers [31], arrays of Josephson junctions [32][33][34], or van der Waals heterostructures [35,36].