Hydrodynamical study of Terahertz emission in magnetized graphene field-effect transistors
HHydrodynamical study of Terahertz emission in magnetized graphenefield-effect transistors
Pedro Cosme a) and Hugo Terças b) Instituto Superior Técnico, 1049-001 Lisboa, Portugal andInstituto de Plasmas e Fusão Nuclear, 1049-001 Lisboa, Portugal
Several hydrodynamic descriptions of charge transport in graphene have been presented in the late years.We discuss a general hydrodynamic model governing the dynamics of a two-dimensional electron gas in amagnetized field-effect transistor in the slow drift regime. The Dyakonov–Shur instability is investigated,including the effect of weak magnetic fields (i.e. away from Landau levels). We show that the gap on thedispersion relation prevents the instability from reaching the lower frequencies, thus imposing a limit on theMach number of the electronic flow. Furthermore, we discuss that the presence of the external magnetic fielddecreases the growth rate of the instability, as well as the saturation amplitude. The numerical results fromour simulations and the presented higher order dynamic mode decomposition support such reasoning.Keywords: Graphene hydrodynamics; Dyakonov–Shur instability; Magnetic field; Graphene field-effect tran-sistor
I. INTRODUCTION
In recent years, the scientific community has witnessedthe emergence of integrated-circuit technology with bi-dimensional (2D) materials. In this scope, grapheneis undoubtedly one of the most prominent materials.Among the many applications of graphene, the possi-bility of resorting to plasmonics instabilities to triggerthe emission, or conversely, the detection, of THz radia-tion has been an active field of study . The exploredmechanisms for the creation and control of plasmons ingraphene commonly rely on graphene field-effect transis-tors (GFET), which allow to control the Fermi level whilebeing easily combined in integrated circuitry.One of the defining characteristics of graphene is itshigh electron mobility, as a consequence of the weak scat-tering between electrons and phonons, defects, or impu-rities, which leads to large electron–impurity mean freepath (cid:96) imp . Indeed, ultra-clean samples of graphene en-capsulated by hexagonal boron nitride (hBN) or hBN–graphene–WSe structures exhibit a mobility µ > . × cm V − s − . Yet, the electron–electron scattering issignificant, resulting in a short mean free path (cid:96) ee atroom temperature. Thereby, it is possible to design asystem of size L under the condition (cid:96) ee (cid:28) L (cid:28) (cid:96) imp . Insuch a regime, the collective behavior of carriers can beaccurately described hydrodynamically , with somerecent experimental results validating this approach .Given the massless nature of graphene electrons, a rel-ativistic description is required for velocities near theFermi velocity v F . However, for the usual operation con-ditions of GFETs, the velocity of the carriers is expectedto saturate far below v F . As such, we here modelgraphene plasmons making use of a hydrodynamic setof equations valid in the regime v (cid:28) v F . Moreover, a) Electronic mail: [email protected] b) Electronic mail: [email protected] x = 0 x = L GS Dgraphene + − U gate I DS B Figure 1. Schematic representation of a graphene channelfield-effect transistor with a top gate (G). The presentedsetup also shows the Dyakonov–Shur impedance realizationat source (S) and drain (D). The magnetic field is perpendic-ular to the channel. we operate at room temperature, such that the Fermilevel is large enough to prevent interband transitions, E F (cid:29) k B T .The Dyakonov–Shur (DS) instability has been ex-tensively studied for high-mobility semi-conductorsas a mechanism for emission/detection of THzradiation and has recently been considered in gra-phene devices . However, few works have approachedthe issue under the influence of magnetic fields . Inthis work, we investigate the DS instability taking placein GFETs in the regime of weak magnetic fields, i.e. awayfrom the Landau levels. Due to the appearance of a gap,the difference of frequency between the forward and back-ward plasmon modes is decreased, leading to an atten-uation of the DS frequency and growth rate. We alsoshow that the emergence of a transverse (Hall) currentin the channels in the nonlinear regime is responsible forthe decreasing of the electron saturation amplitude. II. HYDRODYNAMIC MODEL FOR GRAPHENEELECTRONS
The fact that the electrons in graphene behave as mass-less Dirac fermions poses the major difficulty for the de-velopment of hydrodynamic models: not only do carriers a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n have zero mass, but also the effective inertial mass tensordiverges . A naive approach would dictate to define aneffective mass as m (cid:63) = (cid:126) k F v F = (cid:126) √ πnv F , (1)where (cid:126) k F is the Fermi momentum and n is the electron2D number density. This definition is extensively usedin the literature , and recent developments based onquantum kinetic theory propose corrections to it . Sincethe electronic fluid is compressible, the effective mass isnot a conserved quantity, contrary to customary fluids.For typical conditions in GFETs, the effective mass isexpected in the range . / c (cid:28) m (cid:63) (cid:28)
270 keV / c , (2)lying fairly below the free electron mass.Starting from the Boltzmann equation for the distri-bution function f = f ( r , p , t ) ∂∂t f + v F p | p | · ∇ r f + F · ∇ p f = (cid:98) C [ f ] , (3)one can derive the hydrodynamic model for elec-tronic transport in graphene. Here, the collision op-erator can be taken in the Bhatnagar–Gross–Krookapproximation , (cid:98) C [ f ] = ( f Equilibrium − f ) /τ . How-ever, since we are interested in mesoscopic effects withsmall Knudsen number, vτ /L (cid:28) , and time scales muchlonger than the collision time, we can safely set (cid:98) C [ f ] ≈ .This does not imply the absence of electron-electron col-lisions in the electronic fluid, but rather that they occurfast enough to maintain the local equilibrium.By integrating the zero-order momentum of Eq. (3),yields the continuity equation ∂n∂t + ∇ · ( n v ) = 0 . (4)Furthermore, the first momentum of Eq. (3) leads to ∂ v ∂t + ( v · ∇ ) v nm (cid:63) ∇ · P − F m (cid:63) = 0 , (5)where P is the pressure stress tensor and F the resultantexternal force. As we can see, the variation of the effec-tive mass introduces a / factor to the convective term.Such correction breaks the Galilean invariance of the sys-tem, leading to an unusual expression for the dispersionrelation in the presence of a Doppler shift .The hydrostatic diagonal terms of the pressure, P = P δ ij , is given by the 2D Fermi-Dirac pressure P = 2( k B T ) π (cid:126) v F F (cid:18) E F k B T (cid:19) , (6)where F is the complete Fermi-Dirac pressure, which atroom temperature, E F (cid:29) k B T , gives P = E F π ( (cid:126) v F ) + O (cid:18) k B TE F (cid:19) (cid:39) (cid:126) v F π (cid:0) πn (cid:1) . (7) As such, the pressure term in (5) reduces to nm (cid:63) ∇ P = v F n ∇ n. (8)The off-diagonal elements of the pressure in Eq. (5)describe the viscous terms of the fluid. The kinematicviscosity near the Dirac point is ν (cid:39) v F (cid:96) ee / ∼ . × − m s − ; however, at room temperature T (cid:28) T F thisvalue increases to ν ∼ . s − , and the cor-responding Reynolds number of the electron fluid is Re ∼ Lv . s − . (9)A suitable choice of the system parameters can be madesuch that Re (cid:29) , rendering the viscous effects negligible.As a matter of fact, our simulations performed for mod-erate values of the Reynolds number have not shown anysignificant difference from the inviscid case, apart fromthe expected suppression of higher frequency content andsubsequent smoothing of the waveforms.For a magnetized graphene electron gas in the field-effect transistor configuration, as depicted in Fig. 1, theforce term results from the combined effect of the gateand the cyclotron (Lorentz) force, F = − ∇ U gate − em (cid:63) v × B , (10)where U gate is the gate voltage, U gate = en (cid:18) C g + 1 C q (cid:19) , (11)with C g and C q denoting the geometric and the quan-tum capacitances . For typical carrier densities n (cid:38) cm − , the quantum capacity dominates, C q (cid:29) C g ,and U gate (cid:39) en/C g = end /(cid:15) . A. Enhanced diamagnetic drift
In the presence of a magnetic field, the system is sub-ject to Lorentz force and, taking the steady state of eq.(5) leads to v F n ∇ n + s √ n n ∇ n + e v × B m (cid:63) = 0 , (12)where s = (cid:0) e dv F √ n /ε (cid:126) √ π (cid:1) / is the screened plasmonsound velocity. The drift velocity perpendicular to B canbe retrieved as v ⊥ = S m (cid:63) n e ∇ n × BB , (13)with S = s + v F / which is analogous to a diamagneticdrift in plasmas. Here, however, the drift is not only . . . . . | ω | / ω c kv /ω c | / − s | k | / s | kω − ( k ) ω + ( k ) ω ( k ) Figure 2. Magneto-plasmon dispersion in graphene FETs. So-lutions of the dispersion relation in EQ. (15) with
S/v = 10 (solid lines) alongside the solutions in the absence of magneticfield (dashed lines). due to the pressure gradient but has the added con-tribution of the force drift since F ∼ ∇ n as well. Thus,the fluid has a larger diamagnetic drift compared to whatwould be expected from the pressure itself. In the caseof wave or shock propagation along the GFET channel,as the density gradient will be mostly in the x directionand, therefore, the diamagnetic drift will give rise to atransverse Hall current. B. Magneto-plasmons in graphene FETs
Considering an uniform field B = B ˆ z perpendicularto the graphene layer and writing v = v x ˆ x + v y ˆ y whilelooking for propagation along x , k = k ˆ x , linearizationof Eqs. (4) and (5), with v = ( v + v x ) ˆ x + v y ˆ y and n = n + n , reads in Fourier space ( ω − kv ) ˜ n = kn ˜ v x , (14a) (cid:18) ω − kv (cid:19) ˜ v x = k S n ˜ n − iω c ˜ v y , (14b) (cid:18) ω − kv (cid:19) ˜ v y = iω c ˜ v x , (14c)where ω c = eB/m (cid:63) is the cyclotron frequency. Note thatas the effective mass is much smaller than the electronmass, m (cid:63) (cid:28) m e , it is possible to access high cyclotronfrequencies with modest fields; for a typical excess densityof cm − ω c /B = 9 THzT − . Furthermore, combin-ing (14) yields the relation ( ω − kv ) (cid:34)(cid:18) ω − kv (cid:19) − ω c (cid:35) = S k (cid:18) ω − kv (cid:19) . (15)With this dispersion relation, the propagating solutions ω ± ( k ) coalesce to ω c as k → , opening a gap at the originas patent in Fig. 2, whereas for large k we recover theunperturbed solutions ω (cid:39) (3 / v ± S ) k . Moreover, athird solution ω ( k ) (cid:39) kv / is also present. . . . ω r ω c = 1 v /Lω c = 2 v /Lω c = 3 v /Lω c = 5 v /L γ S/v Figure 3. Numerical solutions for frequency and growth rate(in units of v /L ) of Dyakonov–Shur instability for several cy-clotron frequencies ω c (coloured dots) and analytical solution(18) corresponding to B = 0 (dashed black line). Althoughthere is no significant change in the real part of the frequency,the growth rate diminishes slightly. III. DYAKONOV–SHUR INSTABILITY
The hydrodynamic model in Eqs. (4) and (5) containsan instability under the boundary conditions of fixed den-sity at the source n ( x = 0) = n and fixed current densityat the drain n ( x = L ) v ( x = L ) = n v , dubbed in the lit-erature as the Dyakonov–Shur (DS) instability . Thelatter arises from the multiple reflections of the plasmawaves at the boundaries, which provide positive feedbackfor the incoming waves driven by the current at the drain.From an electronic point of view, the peculiar bound-ary conditions correspond to an AC short circuit at thesource, forcing the voltage (and so the carriers density)to remain constant, and an AC open circuit at the drainsetting the current constant . Thus, these conditionscan be implemented with a low-reactance capacitor onthe source and a high-reactance inductor on the drain ,as outlined in Figure 1.The asymmetric boundary conditions described aboveimply that the counterpropagating wave vectors need tocomply with the relation k + k − = e i ( k + − k − ) L , (16)where k ± = ω ∓ Sgn ( ω ) (cid:113) s ( ω − ω c ) + (cid:0) ω c (cid:1) (cid:0) (cid:1) − s . (17)This condition leads to complex solutions, ω = ω r + iγ ,where ω r is the electron oscillation frequency and γ isthe instability growth rate . Numerical inspectionof Eq. (16) provides the results depicted in Fig. 3. Inthe unmagnetized case, the instability condition can beanalitically solved ω r = | S − (cid:0) v (cid:1) | LS π,γ = S − (cid:0) v (cid:1) LS log (cid:12)(cid:12)(cid:12)(cid:12) S + v S − v (cid:12)(cid:12)(cid:12)(cid:12) . (18)Plasmonic dynamical instability takes place for S/v > / , i.e. in the subsonic regime. The fact that the insta-bility develops in such a regime is advantageous from thetechnological point of view, as it allows the operation ofthe GFET far from the velocity saturation . More-over, when S (cid:29) v the frequency is dominated by the S/L ration as ω r ∼ πS/ L while γ ∼ v / L . Then,given the dependence of S with gate voltage, and as v n ∼ I DS /W e , with I DS representing the source-to-drain current and W the transverse width of the sheet,the frequency can be tuned by the gate voltage and in-jected drain current, not being solely restricted to thegeometric factors of the GFET.In the presence of the magnetic field, the solutions of(16) reveal that the growth rate of the instability de-creases slightly, which is more evident around the tran-sonic regime, while at the subsonic case the influence ofthe magnetic field on the growth rate is less noticeable(Fig. 3). This observation contradicts what has beenpreviously reported in Ref. . Regarding the frequency,the magnetic field introduces a small shift from the un-magnetized scenario.The reason for our results to differ from those presentedin lies in the treatment of the wave vector solutions. Inthe cited work the cyclotron frequency ω c is a priori nor-malized to S/L . Such approach simplifies the problemas it artificially linearises (17). However, this obscuresthe analysis as in a ω vs. S plot, the cyclotron frequencywould also be varying. Moreover, the gap of the dis-persion relation opened by the magnetic field suppressesfrequencies below ω c ; hence, as one approaches the sonicregime S ∼ v , the real part of the frequency drops andreaches the cut-off. Thus, leaving the solutions on Fig.3with an endpoint. IV. NUMERICAL SIMULATION
In order to perform the simulations revealing the late-stage (nonlinear) evolution of the plasmon wave in theFET channel, the hydrodynamical equations have beenrecast into a conservation form plus a magnetic sourceterm. Resorting to the mass flux density p = m (cid:63) n v , thecontinuity and momentum equation can be written in theequivalent form ∂n∂t + ∇ · p √ n = 0 , (19a) − − ω c = 0 v /L − − ω c = 5 v /L − − ω c = 15 v /LI DS I Hall I / ( e n v L ) tL/v Figure 4. Evolution of drain-to-source and Hall currentsacross the graphene channel for distinct values of cyclotronfrequency. The presence of magnetic filed diverts part ofthe current to the transverse direction and diminishes thegrowth rate of instability. All three simulations performedwith S = 20 v and v F = 10 v . ∂ p ∂t + ∇ · (cid:18) p ⊗ p n / + v F v n / + S v n (cid:19) ++ ω c ω p × ˆz √ n = 0 . (19b)This hyperbolic system of differential equations has beensolved with a finite volume Lax-Wendroff method ,the two-step Richtmyer scheme for nonlinear systems .The simulation of system (19), as well as the computa-tion of the observable electronic quantities of the GFET,has been carried with a software specifically developed forthe task . Our simulations confirm that the magneticfield reduces the instability growth rate, as expected forthe subsonic regime (Fig.3). The average value and oscil-lation amplitude of the quantities along the channel arealso reduced (Tab.I), as the diamagnetic current removesa fraction of the electrons participating in the longitudi-nal oscillation. A typical situation for the current densityat source can be seen in Fig.4. The latter reveals that I H a ll / ( e n v L ) ω c L/v min I Hall max I Hall h I Hall i Figure 5. Hall current response with the applied magneticfield. All simulations performed with S = 20 v and v F =10 v . Table I. Average values and extrema of the drain-to-sourceand Hall currents (in units of en v L ) at the nonlinear regimewith the imposition of a cyclotron frequency ω c (in units of v /L ). All simulations were performed with S = 20 v and v F = 10 v . ω c (cid:104) I Hall (cid:105) min I Hall max I Hall (cid:104) I DS (cid:105) min I DS max I DS — — — . − .
884 21 . .
017 0 .
701 1 .
322 2 . − .
851 21 . .
039 3 .
486 6 .
539 2 . − .
183 21 . .
796 6 .
979 12 .
507 1 . − .
053 18 . .
979 10 .
703 17 .
134 1 . − .
356 13 . the magnetic drift is responsible for a transverse current,which could be exploited for a directional coupler oper-ating in the THz regime . In the present case, we aredealing with plasmons, but it may also be applicable tothe case of surface-plasmon polaritons . Indeed the ap-plied magnetic field can control not only the average I Hall value but also amplify the amplitude of its oscillation aspatent on Fig.5.To further analyze and quantify the impact of ω c on theelectronic fluid, the numerical results were evaluated withhigher order dynamic mode decomposition (HODMD) resorting to PyDMD software . The direct outputs ofthe fluid equations have been firstly integrated to ob-tain the average drain-to-source current; this enables theanalysis to be performed on a lower dimensionality quan-tity that retains the dynamic of the system. Then, theHODMD algorithm was applied to the linear growth por-tion of the signal, i.e. before the nonlinear saturationeffects, which corresponds to t (cid:46) . L/v . AlthoughHODMD can perfectly deal with the transition to thesaturation regime, the eigenmodes and complex frequen-cies thus retrieved do not necessarily reflect the valuespredicted by linear theory. Figure 6 shows an exam-ple of such results where the overall decrease of growthrate is evident, with the growth rates from the ω c = 0 case exceeding the subsequent results with magnetic field.Moreover, the predicted slight drift of the main frequencytowards higher values can also be observed. V. CONCLUSIONS
The theoretical study of electronic transport in gra-phene is a challenging task, covering several regimesand interactions, and resorting to complex techniques.Nonetheless, the hydrodynamic models provide a semi-classical description capable of recovering the behaviorand properties of such quantum fluids while also allow-ing numerical simulation with well-established methods.However, it is vital to stress that conventional fluid equa-tions — for instance, the Cauchy momentum equation —can not be bluntly applied and that the variation of theeffective mass with the numerical density introduces acorrection in the nonlinear convective term, breaking the − − − = ( ω m ) < ( ω m ) ω c = 0 v /Lω c = 5 v /Lω c = 10 v /Lω c = 15 v /L Higher amplitude mode
Figure 6. Higher order dynamic mode decomposition fre-quencies, (cid:60) ( ω m ) , and growth rates, (cid:61) ( ω m ) (in units of v /L ),the modes with higher amplitude are displayed with strongercolor. Dashed line marking the theoretical growth rate from(18). The decomposition was obtained from the linear regime( t (cid:46) L/v ) of the average drain-to-source current for dif-ferent values of cyclotron frequency ω c with S = 20 v and v F = 10 v . symmetry of the dispersion relation in the presence of abase drift of the fluid.The presented model evince that the presence of a weaktransverse magnetic field dramatically changes the na-ture of the plasmons for small k , opening a gap in thedispersion relation, imposing a cut-off on the feasiblefrequencies of such systems. Furthermore, our numeri-cal results point out that the magnetic field impairs thegrowth of the DS instability, a result that, to our knowl-edge, has not yet been reported in this context. Such re-duction of the growth rate is practically unnoticeable forthe deep subsonic flows on which technological applica-tions are bound to operate. Yet, the frequency itself canbe increased for moderate values of Mach number beforereaching the gap cut-off. Moreover, our results suggestthat the DS configuration in a magnetized FET has thepotential to function as a directional coupler operatingin the THz regime . In future studies, other magneticeffects could be addressed, either with DS mechanismor exploring other instability processes. Namely, driftinstabilities considering the enhanced diamagnetic driftarising from the gated scenario. Lastly, the presence ofmagnetic field would also lead to the emergence of anodd viscosity contribution with potentially interestingeffects, such as topologically protected edge states andnew exotic dynamics. ACKNOWLEDGMENTS
The authors acknowledge the funding provided by Fun-dação para a Ciência e a Tecnologia (FCT-Portugal)through the Grant No. PD/BD/150415/2019 and theContract No. CEECIND/00401/2018.
AIP PUBLISHING DATA SHARING POLICY
The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest.
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