ABSTRACT
In this chapter recent advances in plasma-wave devices for terahertz applications are described. A plasma wave is defined as a collective charge density wave by the electron dynamics in semiconductors. First, a 2D plasmon is introduced as the quantum of the 2D plasma waves. Nonlinear hydrodynamic properties of 2D plasmons in compound semiconductor heterostructure material systems are presented to demonstrate intense broadband emission and ultrahigh sensitive detection of terahertz radiation. The device structure is based on a high-electron-mobility transistor and incorporates interdigitated dual-grating gates. The second topic focuses on graphene, a monolayer carbon-atomic honeycomb lattice crystal exhibiting peculiar carrier transport and optical properties owing to a massless and gapless energy spectrum. A theoretical
and experimental study on active graphene plasmonics toward the creation of graphene terahertz injection lasers is described. 9.1 Introduction“Terahertz” is still an unexplored frequency band; there is no commercially available microelectronic device that can generate, detect, or manipulate electromagnetic waves over the entire terahertz (THz) frequency band [1]. The quest for the creation of compact, efficient, and tunable THz sources as well as ultrahigh-sensitive and fast-response THz detectors is currently one of the hottest topics in this research field. Fundamental limits reached by the available sources of electromagnetic radiation based on the classical electronic oscillations radiating at millimeter-wave frequencies, and on electron transitions between quantized energy levels corresponding to infrared (IR) frequencies, give rise to the so-called THz gap [1, 2]. 2D plasmons in submicron transistors have attracted considerable attention due to their nature of promoting emission and detection of electromagnetic radiation in the THz regime. The channel of a transistor can act as a resonator for plasma waves, the charge density waves of collectively excited 2D electrons. The plasma frequency depends on the resonator dimensions and the density of 2D electrons; it can reach the sub-THz or even THz range for gate lengths of a micrometer and submicrometer size. Therefore different micro-and nanodevices/structures supporting low-dimensional plasmons were intensively studied as possible candidates for solid-state THz sources [1-18] and detectors [19-30]. Mechanisms of plasma-wave excitation/emission can be divided into two types, (i) incoherent and (ii) coherent. The first is related to thermal excitation of broadband nonresonant plasmons by hot electrons [2, 8, 9, 13, 17, 18]. The second is related to the plasma-wave instability mechanisms like the Dyakonov-Shur (DS) Doppler-shift model [7] and/or the Ryzhii-Satou-Shur (RSS) transit-time model [14, 16], where coherent plasmons can be excited either by hot electrons or by optical phonon (OP) emission under near-ballistic electron motion [31]. On the other hand, hydrodynamic nonlinearities of 2D plasmons in high-electron-mobility transistors (HEMTs) are
promising fast and sensitive rectification/detection of THz radiation [19]. When an incoming THz radiation excites plasma waves, the local carrier density as well as the local carrier velocity is modulated by the radiation frequency. This results in generation of a quadratic current component in proportion to the product of the modulated components of the local carrier densities and velocities. The time average of this component is nonzero, leading to rectification. With an asymmetric boundary condition, a rectified component gives rise to a photovoltaic effect. Recently, plasma-wave properties have been successfully used for resonant and nonresonant (broadband) sub-THz and THz detection [20-30]. They can be applied to real-time THz imaging/spectroscopic analysis as well as future THz wireless communications [1]. First a 2D plasmon-resonant microchip emitter featured with an interdigitated dual-grating gate (DGG) structure was proposed [32-35]. The original structure uses a symmetrical dual-grating gate (S-DGG) in which interfinger spaces are all identical, providing room-temperature 0.5-6.5 THz emission with 1 mW order radiation power [34, 35] and rather low detection responsivity of the order of tens of volts/watt [32]. The major causes of broadband emission are considered to be multimode of coherent/incoherent plasmons [36], oblique modes [37], gated and ungated plasmon modes [38], hot plasmons [36], and chirped plasmon modes [34]. Recently an asymmetric dual-grating gate (A-DGG) structure has been proposed and has demonstrated the world’s first coherent monochromatic THz emission and ultrahigh-sensitive THz detection with a responsivity of 2.2 kV/W and extremely low noise performance with a noiseequivalent power (NEP) of 15 pW/√Hz at 1 THz [39-43]. These values are lower than those of any commercial room-temperature THz detectors such as Golay cells (200 pW/Hz0.5) [44] or Schottky barrier diodes (100 pW/Hz0.5) [36]. Very recently a physical model has been developed that explains this photoresponse dependence on gate bias [45]. On the other hand, graphene, a one-atom-thick planar sheet of sp2-hybridized honeycomb carbon crystals, has attracted considerable attention because of its unique carrier transport and optical properties [46-55]. The groundbreaking discovery of graphene activated the research and development of graphene-
based devices in wide aspects among electronic, optoelectronic, and THz photonic devices. Graphene channel transistors increase their cutoff frequencies approaching the THz range [56-59], whereas graphene photodetectors demonstrated high-speed operation[60] in the mid-IR range, which is also expected to operate in the THz range [61]. The conduction band and valence band of graphene have a symmetrical conical shape around the Brillouin zone edges, which are called K and K¢ points, and contact each other at “Dirac points” at the K and K¢ points. Electrons and holes in graphene have a linear dispersion relation with a zero band gap, resulting in peculiar features such as massless relativistic fermions with back-scattering-free ultrafast transport [46-49, 50-52, 54, 55, 62-65] as well as negative dynamic conductivity in the THz frequencies under optical or electrical pumping [66-72]. Graphene 2D plasmons hold unique optoelectronic properties and produce extraordinary light-matter interactions. So far graphene 2D plasmons have been intensively studied theoretically [73-95], and very recently electromagnetic responses of graphene surface plasmons have been experimentally observed [96-100]. When graphene is patterned into micrometer to submicrometer ordered structures the plasmon modes fall in the THz range so that it can provide intense THz emission if the cavity boundary conditions allow plasmon instability or the THz dynamic conductivity in the plasmon cavity takes negative values via optical or electrical pumping [76, 86-90]. In this chapter the fundamental basis and recent advances in plasma-wave devices for THz applications are described. 9.2 THz Emission Using 2D Plasmons
9.2.1 TheoryThe 2D plasma-wave kinetics can be formulated by the hydrodynamic Euler equation and the continuity equation [4]: m t
e V
m ∂ ∂
+ ∂ ∂
Ê ËÁ
ˆ ¯˜ = -
∂ ∂
- u
u u r r
u t
, (9.1) ∂
∂ +
∂ ∂
= n t
n r
u( ) 0 , (9.2)
where m is the electron effective mass, u(r, t) is the in-plane electron spatiotemporal local velocity, r is an arbitrary in-plane vector, V(r, t) is the local potential at r, τ is the electron momentum relaxation time, and n(r, t) is the spatiotemporal local density of electrons. The first term of the right-hand side in Eq. 9.1 is the Coulomb force, and the second term is the Drude friction. 2D electron channels in HEMTs consist of gated and ungated regions, as shown in Fig. 9.1. The ungated 2D plasmon receives the in-plane longitudinal Coulomb force so that it holds a square-root dispersion relation, which is identical to that for general surface plasmons. The gated 2D plasmon receives the transverse Coulomb force via the gate capacitor, which is far stronger than the in-plane force due to the geometrical situation so that it holds a linear dispersion. In a simple case of gradual channel approximation with infinite channel width (perpendicular to the source-drain direction), the 2D plasma-wave dynamics deduced to 1D systems [4]. The plasma-wave phase velocity s is given by s eV m= 0/ , where V0 is the gate swing voltage [19]. Assuming V0= O[1 V] and m = O[0.1 m0] (m0 is the electron rest mass in vacuum) for InP-based heterostructure HEMTs, s becomes O[1 × 106 m/s], which is at least 2 orders of magnitude higher than the electron drift velocity of any compound semiconductors with superior transport properties. Thus, when we consider a submicrometer gate-length HEMT, the fundamental mode of gated 2D plasmons stays at a frequency in the THz range. This is the main advantage for use in plasmon-resonant modes that can operate in frequencies far beyond the transit frequency limit of transistors [101]. When a single-gate HEMT is situated in source-terminated and drain-opened configuration with the DC potential at the drain terminal with respect to the source terminal, the drain end of the channel becomes depleted so that the drain-side impedance is mainly given by the depletion capacitance and takes a high value at high (THz) frequencies. In such a case, the Doppler-shift effect occurs on the plasma-wave propagation/reflection at the drain boundary, promoting DS instability [4]. Consider the case in which the plasma wave is excited in a HEMT with a constant DC drain bias, causing a background constant DC electron drift flow with velocity vd, and that the gate length L is shorter than the coherent length of electrons. The plasma-wave-originated local displacement current δjp is given by the product of the perturbation of the local
electron charge density edn and the plasma-wave velocity. The forward (backward) component d jp (d jp ) traveling to (from) the drain boundary is given by d d d dj e n s v j e n s vp d p d = ◊ + = ◊ -( ), ( ). Since the open-drain boundary conserves the current before and after reflection, d dj jp p = , d d dn n s v s v n = ◊ + - >( )/( )d d . This increment of the electron charge density d d dn n n= -( ) directly reflects the increment of the gate potential dVs via the gate capacitor C: dVs = edn/C. Since the source-terminated boundary gives a lossless reflection (reflection coefficient is –1) the gate potential becomes infinite after infinitesimal repetition of plasma-wave reflections, leading to DS instability. When the plasma wave is excited by the incoming THz radiation with angular frequency ω, e-iwt, the effect of the instability is derived into the imaginary part of ω, w¢¢, as ¢¢ = - +
- w
s v
Ls
s v
ln . (9.3)
Figure 9.1 DS-type plasma-wave instability in a 2D electron channel under source-terminated and drain-opened boundaries with DC drift velocity vd [4]. The reciprocal Doppler-shifting plasma waves reflecting at an asymmetric drain-opened boundary promote the increments of their intensity, leading to self-oscillation of instability. The plasma-wave increment is a dimensionless parameter in which the imaginary part of angular frequency is normalized to the fundamental resonant frequency. The positive values of the increment give rise to instability in an idealistic lossless case. In reality with a finite τ value, the Drude loss factor should be considered to obtain an overall gain, which is shown as a threshold level. Reproduced from Ref. [101] by permission of SPIE, © 2013.
