Feedback enhanced Dyakonov–Shur instability in graphene field-effect transistors

Graphene devices are known to have the potential to operate THz signals. In particular, graphene field-effect transistors (gFETs) have been proposed as devices to host plasmonic instabilities in the THz realm; for instance, Dyakonov–Shur (DS) instability which relies upon dc excitation. In this work, starting from a hydrodynamical description of the charge carriers, we extend the transmission line description of gFETs to a scheme with a positive feedback loop, also considering the effects of delay, which leads to the transcendental Laplace-transform transfer function, with complex frequency s, with terms of the form e−assechk(s)/s , for a given a∈R0+ arising from the delay time and with k∈N . Applying the conditions for the excitation of DS instability, we report an enhanced voltage gain in the linear regime that is corroborated by our simulations of the nonlinear hydrodynamic model for the charge carriers. This translates to both greater saturation amplitude—often up to 50% increase—and faster growth rate of the self-oscillations. Thus, we bring forth a prospective concept for the realization of a THz oscillator suitable for future plasmonic circuitry.


Introduction
Present day technology-and thus society itself-relies heavily on high frequency communication and nanoscale devices; yet, despite all the advances in this subject the THz range of the spectrum, with the potential to bridge the realms of electronics and photonics, still proves to be a challenge.Thus, new and efficient ways to generate, detect, or manipulate electromagnetic radiation in this range are ever sought after.
In this context, several designs of graphene nano-devices have been proposed [1][2][3][4][5][6][7][8][9] where an instability triggers the current to self-oscillate.Among the various mechanisms, the Dyakonov-Shur (DS) instability [10,11], driven by the large asymmetry of impedance at the contacts, has been frequently brought forth as a way to induce self-oscillations from dc current.However, such instability has a modest amplitude gain, to circumvent that we study the use of a feedback loop as a way to improve the gain [12] as well as the growth rate.
In this work, we consider the effect of positive feedback on a graphene field-effect transistor (gFET) under the conditions of the DS instability.For the analytical characterization, we resort to a transmission line description [13,14] that can be obtained from the hydrodynamic description of charge carriers.From such a looped transmission line, we show that the growth rate of the unstable modes is enhanced.Alongside this, fully nonlinear simulations performed by tethys simulation tool [15] evince these findings from a numerical approach while also proving the enhancement of the saturation amplitude.
A growing number of both theoretical [16][17][18][19][20] and experimental works [21][22][23][24][25][26] have shown that the hydrodynamic regime of carriers in graphene and similar 2D materials is achievable even around room temperature.Therefore, the mean-field description of the carriers number density n and momentum density p = nm ⋆ v, with m ⋆ the effective mass and v the mean velocity of the carriers, can be written as a set of conservation equations in the form [1,2,16,19,27,28]: where ϕ is the electrostatic potential, and P = hv F √ π n 3 /3 is the Fermi pressure of the electronic fluid with v F = 10 6 m s −1 is the Fermi velocity.Moreover, due to the linear nature of the bands in graphene the mass is defined via the Drude weight m ⋆ = h√ π n/v F .In a gFET and under the gradual-channel approximation [29] one can close the set of equations as ϕ = en/C ox with C ox the gate capacitance per unit area.Due to the effective screening of the gate, equation ( 1) result in linear plasmons with the dispersion relation which defines the plasmon velocity v p .Here and henceforth the quantities with zero subscript indicate equilibrium values.Assuming uniformity along the direction transverse to the charge flow, we can convert to the local description to the macroscopic quantities of interest of current and voltage at source (x = 0) and drain (x = L) by I(x) = −eWv(x)n(x) and V(x) = en(x)/C ox , with W the gFET width as rendered in figure 1.

DS instability
Additionally, it is well-known that such kind of system undergoes DS instability when the boundary conditions corresponding to an ac short ac open [10,11] (i.e.n 0 v 0 is the imposed current at the drain which is kept constant), are applied, resulting in oscillations of the electronic fluid with eigenfrequency [2,10].This instability was first studied in the context of regular two-dimensional electron gas (2DEG) in high mobility field-effect transistors but can be easily implemented in gFET.The DS instability rises from the multiple reflections of the plasma waves at the boundaries while being amplified by the driven current at the drain with a reflection coefficient Such reflection of the incoming density waves is induced by the condition at source, while the imposed drain current guarantees the necessary Doppler shift for the upstream current to interfere positively with the downstream current.
To obtain the frequency of the DS instability let the density be defined by the sum of two travelling waves, with momenta q ± = ω/(v 0 ± v p ), and arbitrary amplitudes A ± , plus the steady state, as n(x) = n 0 + n 1 (x) = n 0 + A + e iq+x + A − e iq−x then, recurring to the continuity equation in Fourier space, one has Applying the boundary condition at source n(t, 0) = n 0 reads A + = −A − , while the imposition of constant current at drain, n(t, L)v(t, L) = n 0 v 0 , leads to ω q+ A + e iq+L + ω q− A − e iq−L = 0. Eliminating the amplitudes in the previous relations yields the condition which can be solved in terms of the frequency and sorted in the real and imaginary parts of the complex frequency ω = ω r + iγ, returning where l is an odd integer number [10,30,31].One can observe that the instability occurs for the subsonic regime where v p > v 0 as the imaginary part of the frequency, γ, becomes positive.In fact, in the supersonic regime, the Doppler shift would prevent the opposite traveling waves from interacting and there would be no place for amplification along the channel.Despite the wide condition for instability, given the appropriate boundary conditions, the saturation amplitude, i.e. the gain, of the process is quite moderate.Thus, studying mechanisms that can improve the amplitude is a compelling issue.

Open-loop transfer function
From the hydrodynamical description given in equation ( 1) one can derive a transfer matrix description (see details in [13,14] and appendix) in the form where akin to a transmission line, with θ = v 0 /v p the ratio between the imposed drift velocity and the plasmons velocity, the characteristic impedance Z −1 0 = C ox Wv p , and the normalized complex frequency.Thus, we can retrieve the voltage gain making use of the drain impedance Z D = V D /I D .Note that the non-algebraic nature of the transfer is a regular feature of distributed parameters systems [32] and is the root of the unusual response to even dc input.
For the case of an ideal ac open circuit, as required for DS instability, Z D = ∞, which greatly simplifies the analysis of g(s).If, however, faced with non-ideal impedance matching, one can consider a typical a common-gate configuration [33] where the drain impedance then includes the effects of substrate resistance and drain-gate capacitance (see figure 1) as Such additional high-frequency pole can then be appended, since for the sake of simplicity, in the remainder of this work we assume Z D → ∞.
As we can see, all the poles of g(s)-in infinite numberare located along the imaginary axis, rendering the system marginally stable and in fact any perturbation leads to its instability, as it is expected given the DS conditions imposed.Moreover, note that, the unit step response of equation ( 10) is a square wave [34] as sechs s a trait that is clear in our simulations during the linear regime as will be discussed hereafter.

Closed-loop transfer function
Let us now assume that a feedback loop is imposed on our system, as represented in figure 1.In general, we have where the matrix Ĥ encodes the proportion of voltage and/or current that is looped, in this work we will consider only voltage feedback and so with β < 1 the voltage fraction that is reintroduced at the source and τ a delay factor.Solving for V D leads to the closed loop gain: Since for any experimental implementation of feedback, the source and drain of the gFET will have to be connected by conductive material, its effect can not be instantaneous, but rather to take a given time delay τ to occur, such delay is estimated to be of the order of the travel time of a a plasmon.The feedback drastically modifies the structure of the transfer function, and the location of the poles start to wander over the complex plane, as the feedback ratio increases, as can be seen in figure 2. At β = 0, corresponding to no feedback, the poles are located on the imaginary axis at odd multiples of π/2.For increasing β the poles digress away from their initial position.For moderate values of delay the poles that started from π/2 + 2πℓ, with ℓ ∈ Z, drift towards the unstable righthalf plane while the other ones recede, becoming decaying modes-a feature that might be exploited for upconversion.Furthermore, as expected the feedback pushes the overall gain up, as is made clear by the Bode plots on figure 3; the delay, however, appears to curb the self-amplification.
Regarding the unit step response of the looped system, we expand the transfer function in powers of the feedback amplification, which leads to So, while the first term yields the usual square wave, the subsequent terms in the series return unbounded functions (see table 1 and [34,35]), as L −1 [sech k (s)/s] ≍ t k−1 , this leads to a faster growth rate for the instability but evidently cannot represent physical modes of the solutions for t → ∞.In table 1, we provide closed-form expressions for the first few terms of the inverse Laplace transform, that attest to the polynomial growth for the envelope of the base square wave.Nonetheless, one must bear in mind that insofar we have discussed but linear analysis of a nonlinear system, and the transient growing terms of equation ( 15) saturates at a certain point.

Nonlinear simulations
Thus far, the transfer function analysis provided crucial insight into the stability and early stages of the gFET operation under DS instability.However, aiming for continuous THz generation, the time evolution for later times is critical.To surpass the limitations of the linear analysis, the hydrodynamic system of equation ( 1) needs to be numerically simulated so that the full nonlinear effects could manifest.We resorted to Tethys computer program [15], a finite-volume solver designed for the simulation of graphene electrohydrodynamics.
Under the DS conditions, we obtain the typical results (see [1,[36][37][38][39]) for the voltage, which are presented on figure 4 and where is clear the positive impact of the feedback albeit the modest β ratio.
Even in the absence of feedback, one can distinguish three regimes, viz.linear evolution, transient time, and saturation.For the first few periods, the system roughly follows a square wave, as predicted by equation ( 11), before reaching saturation, which for the β = 0 is dictated by Rankine-Hugoniot jump considerations that limit the maximum amplitude-and thus the power diverted to the ac response.
On the contrary, closing the feedback loop leads to a significant increase in the saturation amplitude and to a faster saturation itself.However, the waveform is distorted into a new shape due to the interplay of the higher order modes in the expansion of equation (15).More importantly, it is clear from the simulations presented in figure 4 that the growth rate is much higher, not only the saturation is attained much faster, but the amplitude envelope of the waveforms is steeper near

Conclusion
To conclude, since the discovery of graphene and its first applications to integrated circuit devices, the number of uses and designs is ever-growing, boosted also by the discovery and development of new 2D materials.
In that context, our study-of a graphene transistor under a feedback loop-paves the way for future self-driven oscillators in the THz range.We report that the effect of the closed loop on DS instability mitigates the usual technical problems, since it enhances both the growth rate and the amplitude of oscillations.The former is essential to overcome the eventual damping mechanisms ever present in real devices; while the latter improves the power and resilience of the signal.
Furthermore, resorting to the transfer matrix description, we characterized the delayed closed loop system.Firstly, as the feedback ratio increases the trajectory of the poles on the complex plane leads them away from the imaginary axis, showing that the feedback leads to further destabilization.The modes that depart from π/2 + 2π ℓ, ℓ ∈ Z, move rightwards; that is, they increase their growth rate.Whereas the ones that arise from 3π/2 + 2π ℓ move to the left half-plane.Such an effect explains the reported third harmonic intensification [12] and can be further tuned to place this third harmonic on the goal frequency.Secondly, expanding the transfer function in powers of the feedback ratio evinces the structure of the unit step response as a sum of square waves whose amplitude grows with a power law, since the inverse Laplace transform of G(s)/s scales as t 2 up to second order in the feedback ratio β.Nonetheless, as expected, the delay must be tailored to avoid destructive interference between the waves imposed by the loop and the source terminal and the counter-propagating ones inside the channel.
The simulation of the nonlinear stage complemented our linear analysis and proved the increase of the saturation amplitude, e.g. from a voltage ratio V D /V S ≈ 2 to V D /V S ≈ 3, by applying a 20% feedback.However, this saturation stage and its limit cycle are still open for further inquiries, namely: which Liénard equation would accurately describe it, and how the limit cycle varies with the system parameters.
Lastly, throughout this work, by using the classic transmission line model, we implicitly assumed Galilean invariance of the plasmons.This constitutes a good approximation, since for most systems the drift velocities are well below the Fermi velocity.Yet, in future works, the deviation from this assumption could be an interesting line of research to pursue.

Figure 1 .
Figure 1.Left panel: simplified model of the gFET considered, highlighting the gate-channel capacitance Cox as well as the main parasitic impedances represented.Right panel: schematic rendition of the positive feedback loop.

Figure 2 .
Figure2.Motion of poles of the closed-loop transfer function G(s) on the upper half plane of s-domain with ℑ(s) < 6π, for different feedback ratios and delays.At β = 0 (no feedback) the poles are located on the imaginary axis at odd multiples of π/2.For increasing β the poles digress away from their initial position, along the curves, with the ticks marking the feedback ratio (in 0.2 increments).The values of delay are τ = 0 (solid black lines); τ = 0.4 (red dashed lines); and τ = 1 (blue dot-dashed lines).The behavior of poles on the lower-half complex plane is similar due to G(s * ) = G * (s).

Figure 4 .
Figure 4. Nonlinear oscillations due to DS instability with delayed feedback.Direct simulation of the fluid model of equation (1) with θ ≡ v 0 /vp = 1/20.Note the growth of the saturation amplitude and the quicker onset of the saturated regime.

Table 1 .
Inverse Laplace transforms of functions of the form L −1 (sech k (s)/s).

Table 2 .
Saturation amplitudes and growth rates of non-linear simulations, retrieved by logistic fitting of the envelope of the oscillations.All simulations with v 0 /vp = 1/20.