A conclusion is drawn that a slot diode with two-dimensional electron channel provides a resonant circuit at terahertz frequencies that couples effectively to external electromagnetic radiation with loaded Q-factor exceeding unity even at room temperature. The diode resistance may be measured from contactless measurements of the characteristic electromagnetic lengths of the diode.

1. Introduction 2. Theoretical model High-frequency response of field-effect transistors and Consider a plane electromagnetic wave incident normally diodes with two-dimensional electron channels is strongly from vacuum onto a perfectly conductive plane z = 0 with affected by plasma oscillations in the channel. This a slot of width w, which is located on the surface of a phenomenon in its various manifestations can be used for dielectric substrate. We assume that the electric field of the the detection, frequency multiplication and generation of tewave, E0 exp(-it - ik0z ), where k0 = /c with c being rahertz (THz) radiation [1–13]. One of the main parameters the speed of light in vacuum, is polarised across the slot of a device, which determines its high frequency properties, (along the x-axis). The edges of the slot are connected by a is the device impedance. The high-frequency impedance two-dimensional electron channel with the areal conductivity (admittance) of a slot diode was calculated for capacidescribed by the Drude model as tively [14] and conductively [15] contacted two-dimensional electron channel in the frame of the electrostatic theory and Ne () =i, an equivalent circuit approach. In these approaches, the m( + i) radiative contribution to the impedance (radiation resistance, Rrad, of the diode) is not accounted for and inter-contact where is the electron momentum scattering rate, N is the geometrical capacitance, Cg, is either ignored altogether [14] sheet electron density, e and m are the charge and effective or treated as a free parameter [15]. However, at ultra-high mass of electron, respectively.

(terahertz) frequencies (i) the radiation resistance of the Our theoretical procedure involves the following steps.

diode may play a role as additional damping mechanism and We rewrite the Maxwell equations in the ambient medium (ii) the inter-contact geometrical capacitance may effetively and in the substrate in the Fourier transform representation shunt the channel. Furthermore, a typical length of the over the in-plane wave vector kx. The Fourier transforms side contacts in high-frequency high-electron-mobility tranof in-plane components of the electric and magnetic fields sistors (HEMT) [12] is comparable with the THz radiation satisfy the following boundary conditions at z = 0:

wavelength, what makes the electrostatic calculations of the inter-contact capacitance inaccurate. We calculate here (ind) (tot) (kx )E0 + Ex,a (kx ) =Ex,s (kx), the impedance of a slot diode with conductively contacted two-dimensional electron channel using the full system of (ind) (tot) (kx )H0 + Hy,a - Hy,s (kx ) = j(kx ), the Maxwell equations. In this way, we account for the radiation resistance and inter-contact capacitance from the where j(kx ) is the Fourier transform of the surface first principles. In addition to its own practical importance, electron current density, (kx ) is the Dirac -function, the the slot diode analyzed here is an idealization of long subscripts a and s label the fields in the ambient medium ungated parts of the HEMT with ultra-short nanometer gate.

and the substrate, respectively, and superscripts (ind) and Such HEMT was recently shown to exhibit resonant THz (tot) refer to induced and total fields, E0 and H0 are the emission tunable by a bias voltage [12].

amplitudes of electric and magnetic fields in the incident ¶ E-mail: popov@ire.san.ru wave. Then we relate the Fourier transform of the surface 158 V.V. Popov, G.M. Tsymbalov, M.S. Shur, W. Knap electron current density in the diode plane to that of the The wave vectors ka(s) have kx and kz = ± k2a(s) - k2 as x in-plane electric field in the same plane as their components. The integrals in the right sides of Eqs. (2) and (3) describe the scattered fields in terms of the planej(kx ) =G(kx )Ex(kx ) + (kx )E0, wave continuum, while the first summand in Eq. (2) is the Zwave reflected normally from perfectly conductive plane.

(tot) The sing before the radical in expression for kz is chosen where Ex (kx ) =Ex,s (kx ) is the Fourier transform of into correspond the outgoing waves for kx < k0a(s) and plane component of the total electric field in the diode evanescent waves for kx > k0a(s) is respective medium.

plane, Z0 is the free-space impedance. The kx-space surface Since only the outgoing plane waves (with admittance G(kx ) depends exclusively on the frequency and kx < k0a(s)) contribute to radiative losses, we can dielectric constants of the ambient medium, a (which we calculate the fluxes of energy scattered per unit length of assume to be 1), and the substrate, x :

the slot in each medium as a + s G(kx ) =, k0 a(s) Zka(s) Pa(s) = na(s) E(sc) (kx ) dkx, where Z0 k0 a(s) a(s)k-k0 a(s) a(s) =.

k2a(s) - kx where na(s) is the internal normal to the diode plane into respective medium. Then we can define the scattering Coming back to the real-space representation we have length as L(sc) = Pa(s)/P0 in each medium, where P0 is a(s) the energy flux density in incident wave. We can also 2Ej(x) = + dx Ex(x ) dkxG(kx ) exp[ikx(x - x )].

introduce the total scattering length as L(sc) = L(sc) + L(sc) Z0 a s - and the absorption length L(ab) = Q/P0, where Using Ohm law j(x) = ()Ex (x) for the two-dimensional w/electron channel and the condition Ex = 0 for the perfectly Q = |Ex (x, 0)|2 Re [ ()]dx conductive contact semi-planes, we arrive at the following integral equation for in-plane component of the total electric -w/field within the slot:

is the energy absorption rate (per unit length of the slot).

w/The scattering and absorption lengths obey the energy 2E ()Ex (x) = G(x, x )Ex (x )dx + (1) conservation law in the form L(sc) + L(ab) = L(ex), where Z-w/Re [EE(sc)(kx = 0)] a L(ex) = 4 (4) with the kernel |E0| is the extinction length, which is the ratio between the G(x, x ) = dkxG(kx) exp[ikx (x - x )].

amount of energy picked out of the incident wave per unit time (per unit length of the slot) and the energy flux density in incident wave. The formula analogous to Eq. (4) is known Integral equation (1) is solved numerically by the Galerkin as the optical theorem in the scattering theory [16].

method through its projection on an orthogonal set of the Legendre polynomials within the interval [-w/2, w/2]. As a result we find the induced electric field in the ambient medium as 3. Results and discussion The calculated spectra of the scattering, absorption and E(ind)(r) =E0 exp(ik0z ) + E(sc)(kx) exp(ikar)dkx, (2) a a extinction lengths are shown in Fig. 1 for parameters typical of two-dimensional electron channels in sub-100 nm gate length HEMT at room temperature [12]. All characteristic and the total electric field in the substrate as lengths exhibit maxima at frequencies of the plasmon resonances in the channel. Note that the extinction length E(tot)(r) = E(sc)(kx) exp(iks r)dkx. (3) a s exceeds by the order of magnitude a geometrical width of the slot even for a short electron relaxation time chosen for the calculations. Arrows in Fig. 1 mark the frequencies of The electric fields E(ind)(r) and E(tot)(r) have zero y-compo- ungated plasmons in isolated two-dimensional electron chana s nents and r is the two-dimensional radius vector r = {x, y}. nel with wave vectors qn =(2n - 1)/w(n = 1, 2, 3,...), Физика и техника полупроводников, 2005, том 39, вып. Resonant terahertz response of a slot diode with a two-dimensional electron channel current resonance is exhibited at higher plasmon resonant frequencies.

The normalized resistance, R/Z0, is, in essence, the matched width of the diode (measured along the slot), since the resistance of diode with the matched width is equal to the free-space impedance. The following theorem is valid:

4R = Z0L(ex). The extinction length calculated using this formula (dashed curve in Fig. 1) and by Eq. (4) (solid line) coincide within the accuracy of our numerical procedure.

Accordingly, we can introduce the electron resistance, Re, and radiation resistance, Rrad, where Re = Z0L(ab)/4 and Rrad = Z0L(sc)/4, respectively, so that the total resistance of the diode is given by R = Re + Rrad. Note that R does not vanish at high frequencies (but approaches Rrad instead).

One can see from Fig. 2 that higher resonances are not nearly so pronounced as predicted by the electrostatic description [15] because RradCg circuit effectively shunts plasma oscillations in the channel at high frequencies. However, a sharp fundamental resonance, which corresponds to Figure 1. Characteristic lengths vs. frequency for the slot the current resonance in the equivalent circuit description, diode with parameters: N = 3 · 1012 cm-2, = 4.35 · 1012 s-1, shows up even for room temperature parameters of the w = 1.3 µm, s = 13.88, m = 0.042m0.

diode.

The frequency of the fundamental plasma resonance in the slot diode increases with decreasing the slot width which are estimated by a simple approximate formula [17]:

(according to formula (5) it varies roughly as a square root from the inverse of the slot width). Figs. 3 and 1 e2Nqn exhibit absorption length and scattering length spectra for f =. (5) n 2 m0(1 + s ) 100-nm width of the slot for different electron scattering rates. Notice that the resonant absorption (the excess It is evident from Fig. 1 that the plasma oscillations in the of the absorption at the resonance over the non-resonant slot diode are softened because of the induction of image Drude background) grows considerably with decreasing charges in the perfectly conductive contact semi-planes by the width of the slot (cf. Figs. 1 and 3). Clearly, plasma oscillations.

the fundamental plasma resonance becomes narrower with In the equivalent circuit description, we can characterize decreasing the electron scattering rate. However, the width the slot diode with two-dimensional electron channel by its of the resonance remains finite due to the radiative damping impedance. Within the channel the total current is the sum of the electron current and displacement current caused by oscillating charges in both the channel and contacts of the diode. Far away from the slot, the current in the contact planes is purely conductive and is determined by the amplitute of incident wave. Since the total current is conserved along the circuit, we can define the diode impedance Z = R + iX (per unit length of the slot) as w/Z = Ex (x)dx, I -w/where I = 2E0/Z0 is the surface current density induced by the incident wave in the perfectly conductive contact planes.

The frequency dependence of the diode normalized impedance shown in Fig. 2 displays resonances at the plasmon excitation frequencies. The reactance, X, exhibits transition from inductive (X < 0) behavior caused by the kinetic inductance of the electron channel to a capacitive behavior (X > 0) at the frequency of the fundamental plasmon resonance, which corresponds to the current Figure 2. Impedance of the slot diode with two-dimensional resonance in the equivalent circuit description. However, no electron channel vs. frequency.

Физика и техника полупроводников, 2005, том 39, вып. 160 V.V. Popov, G.M. Tsymbalov, M.S. Shur, W. Knap error arisen from the non-resonant background contribution, we estimate the radiative damping of plasmons, rad, as the half width at half magnitude (HWHM) in the low-frequency slope of the scattering resonance curve for = 0 (curve in Fig. 4), which yields rad = 4.27 · 1012 s-1. Then the electron scattering contribution to the resonance linewidth can be obtained as a difference between the HWHM measured in the low-frequency slope of the scattering resonance at any given = 0 (or, which yields the same result, in the high-frequency slope of the corresponding absorption resonance) and the radiative damping. The scattering length at the plasma resonance monotonically grows with decreasing the electron scattering rate, while the resonant absorption exhibits maximum when the dissipative broadening (caused by the electron scattering in the channel) of the resonance linewidth becomes equal to the radiative broadening (curve 2 in Fig. 3 corresponds to this case). Both the weaker as well as stronger electron scattering result in smaller resonant absorption. Note that under the condition of maximal absorption the scattering length of the diode is approximately equal to its absorption Figure 3. Absorption length of the slot diode with parameters:

N = 3 · 1012 cm-2, w = 0.1 µm, s = 13.88, m = 0.042m0 as a length, which puts the limit of the absorption-to-scattering function of the frequency for different electron scattering rates:

length ratio of the diode close to unity.

= 2 · 1013 s-1 (curve 1); 7 · 1012 s-1 (curve 2); 2.3 · 1012 s-We conclude that a slot diode with two-dimensional (curve 3).

electron channel provides a resonant circuit at terahertz frequencies that effectively couples to external electromagnetic radiation with the loaded Q-factor exceeding unity even at room temperature. We also claim that the diode high-frequency resistance may be measured from contactless measurements of the chatacteristic electromagnetic lengths of the diode.

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