INTRODUCTION
In recent years, solidstate devices operating in the socalled Terahertz (THz)
gap region of electromagnetic (EM) waves are highly demanded for possible application
in the field of terahertz information and telecommunication system (Chamberlain
and Miles, 1997; Miles et al., 2001; Shur
et al., 1999). THz gap is defined by the frequencies in the range
of 0.5 ~ 10 THz. Below THz, devices classified as transit time devices and include
such familiar examples as Bipolar Junction Transistors (BJTs), Heterojunction
Bipolar Transistors (HBTs), Field Effect Transistors (FETs), High Electron Mobility
Transistors (HEMTs) and Transferred Electron (Gunn) devices operating up to
a few tens GHz are commercially available (Chamberlain and
Miles, 1997; Miles et al., 2001; Shur
et al., 1999). These devices are in general benefited from the advanced
development of conventional electronics where reduced dimensions or sophisticated
layer structures are utilized for operation beyond 100 GHz. In all of these,
the time taken for carriers to move a characteristic distance determines the
maximum frequency of operation. The performance of such conventional compound
semiconductor devices are also compared in terms of their power amplification
and operating frequency. As we know, the output powers of such conventional
devices decrease with the increase of frequency where the downsizing of the
characteristic distance of traditional devices is required in order to increase
the operating frequency. In fact, for example, the maximum cutoff frequency
obtained thus far still remains slightly above 500 GHz even with the use of
very short gate length of a few nanometer and gate channel distances of a few
nanometer (Watanabe et al., 2007). This downsizing
activity does not seem to promise any merit due to not only the skyrocketing
production cost but also the physical problems such as the shortchannel effect
and large gate leakage current. It is clear that those devices can not possess
both high output power and high frequency operation where presently they are
selectively used according to the required output power and operating frequency.
Those devices also seem to have difficult time approaching THz range.
Above the THz gap, there are welldeveloped solidstate sources such as near
infrared lasers and Light Emitting Diodes (LEDs). These devices may be classified
as transition devices, since the charge carriers undergo a transition from a
higher to a lower energy state with the direct emission of radiation at a frequency,
f given by E = hf where E is the energy state separation. It is evident that
the THz gap serves to mark the boundary between electronics and photonics sources
(Chamberlain and Miles, 1997; Miles
et al., 2001; Shur et al., 1999).
To fill the THz gap by using conventional electron approach or transit time
devices seems to be very difficult due to the limitation that comes from the
carrier transit time where extremely small feature sizes are required. One way
to overcome this limitation is to employ the traveling wave type approach in
semiconductors like classical traveling wave tubes (TWTs) where no transit time
limitation is imposed (Solymar and Ash, 1966). TWTs
are wellknown as an amplifier of microwave energy. It accomplishes this through
the interaction of an electron beam and an EM waves propagating through a slowwave
circuit. As the electron beam travels down this interaction region, an energy
exchange takes place between the electron beam and slow EM waves.
In this analogy, carriers in semiconductor correspond to the electron beam in TWTs. The interaction principle of carrier plasma waves and EM waves is thought to be in the same manner with TWTs which can lead to the amplification of EM waves. The possible mechanism of amplification can be briefly described as follows. The propagated EM waves will produce the electric fields which their directions become opposite at every halfwavelength, resulting in the acceleration and deceleration of electrons. In the case of the electron drift velocity, υ_{d} equals to the phase velocity of EM waves, υ_{phf} (υ_{d} = υ_{phf}), the amount of electrons in the acceleration region is equal to the amount of electrons in deceleration region, resulting in no energy exchange between electrons and EM waves. In the case of the electron drift velocity, υ_{d} slightly larger than the phase velocity of EM waves, υ_{phf} (υ_{d}>υ_{phf}), the bunching point of electrons will slightly shift into the deceleration region. Here, the amount of electrons in deceleration region become larger than the amount of electrons in acceleration region, resulting in the energy exchange between electrons and EM waves which leads to the amplification of EM waves. It is noted here that the behavior of these flowing electrons contribute to the socalled drifting carrier plasma wave.
Motivated by such semiconductor traveling wave amplifier concepts, there were tremendous theoretical works carried out by many researchers to evaluate the possibilities of carrier plasma wave interactions in semiconductor. The idea is to replace the electron beam in a traveling wave tube with drifting carriers in a semiconductor. These drifting charge carriers in the semiconductor would interact with the slow electromagnetic waves resulting in a convective instability. Hence, there would be the possibility of constructing a new type electromagnetic wave amplifier by injecting a signal at one end of the semiconductor and taking out an augmented signal at the other end.
Solymar and Ash (1966) published a onedimensional
analysis of an ntype semiconductor traveling wave amplifier predicting high
gain per centimeter. They assumed a single species of charge carrier with infinite
recombination lifetime obtaining a characteristic equation for the interaction
that is reducible to the well known traveling wave tube case. This onedimensional
analysis may be valid for the coupling that takes place directly in the semiconductor
bulk but in the case of using external circuit, the coupling is realized only
through a surface of semiconductor which sandwiching a thin insulating layer,
contacts with the slowwave circuit. Thus, the coupling through semiconductor
surface is essentially of two or three dimensions and hence, two or three
dimensional analysis would be required for the understanding of amplification
by this process.
Sumi (1967) and Sumi and Suzuki
(1968) published an analysis of semiconductor traveling wave amplification
by drifting carriers in a semiconductor in which he predicted 100 dB mm^{1}
gain for an InSb device operated at 4 GHz at liquid nitrogen temperature. The
analysis consisted of evaluating the transverse admittance at the surface of
a collisiondominant semiconductor and equating it to the transverse admittance
at the surface of a developed helix (slowwave structure). In this analysis,
all the electromagnetic fields in the semiconductor are included for the estimation
of propagation constants and the amplification is attained beyond the threshold
that the electronic gain exceeds all the semiconductor loss. Therefore, there
is no need to take into account an additional semiconductor loss. Robinson
and Swartz (1967) presented an experimental evidence to claim that Sumi’s
assumption of no surface charges or currents at the semiconductor surface is
incorrect, thus invalidating the dispersion equation so found, i.e., the equating
of transverse admittance is invalid in the presence of surface current.
Zotter (1968) corrected algebraic errors in Sumi’s
paper and numerically evaluated the available gain for different semiconductor
materials, predicting an even higher gain per millimeter. Vural
and Steele (1969) have extended Sumi’s analysis to consider the interaction
with a generalized admittance wall including the effects of surface charge and
currents. Ettenberg and Nadan (1970) published an analysis
by following essentially the same method of Solymar and
Ash (1966) which applicable for two carrier species, e.g., electrons and
holes and derived a maximum resistivity for a given material for which the single
dominant carrier approximation remains valid. In their analysis, carrierlattice
collisions, diffusion and carrier recombination were taken into account.
Although those theories were very different but they agreed on one point, the
gain may be very high (several hundred dB mm^{1}). Motivated by the
possibilities of amplification with such high gain, in particular demonstrated
theoretically by Solymar and Ash (1966) and Sumi
(1967) and Sumi and Suzuki (1968), some innovative
experimental study was performed. Sumi and Suzuki (1968)
demonstrated a coupling between drifting carriers in ntype InSb and slow electromagnetic
waves in meandertype and helixtype circuits using metalinsulatorsemiconductor
(MIS)structured device in the frequency range of 24 GHz at 77 K. Freeman
et al. (1973) reported the electronic gains in the meandermeander
line signal of up to 40 dB cm^{1} with Ge at 4.2 K. Unfortunately,
these experiments did not show any net gain and only an interaction much weaker
than predicted by theory was observed. Many effects may contribute to divergence
between theory and experiment. In these experiments, insulator with a few microns
in thickness was used. If the thickness of insulator is made thinner, the interactions
should give better results where the interactions become stronger with the reduction
of distance between slow waves and carrier waves. We have shown that the magnitude
of the negative conductance which is an indicator of plasma wave interaction
increases with the reduction of the distance between slowwave structure and
2DEG channel (Hashim et al., 2003). Thiennot
(1972a, b) also showed by using Sumi’s schematic
model that the thicknesses of semiconductor and insulator and electrostatic
surface charge have an important effect which may cause the divergence.
Swanenburg (1973) observed the phenomena of negative
conductance in the frequency range of 2575 MHz at temperature of 25 K using
ntype Si with interdigital structure. Baudrand et al.
(1984) reported the coupling with both even and odd harmonics where large
net gain of 13 dB mm^{1} was obtained at the coupling with third harmonics
in the frequency range of 12 GHz, using also ntype Si with interdigital structure.
But, these results seem to be unreasonable due to even space harmonics do not
exist in slow waves propagating through the interdigital structure. Also, in
this experiment, voltage of 400450 V was applied to accelerate carriers which
this method seems to be improper for practical application.
Thompson et al. (1991) also claimed a gain
of 13 dB mm^{1} at 8 GHz with an applied transverse dc field of 1.5
kV cm^{1} using ntype GaAs with interdigitated fingers and dc segmented
fan antenna. In their experiments, they observed the change of reflection coefficients
between the biased and unbiased states of the device which they assumed to be
caused by the traveling wave interaction without any theoretical explanation.
At best only marginal internal electronic gain was observed and it was not
clear that the gain mechanism corresponded to the predicted mode of operation.
In contrast, in 1974 the Rayleigh (acoustic) wave amplifier, similar in principle,
has achieved an external gain of 50 dB and a bandwidth of 30% (Coldren
and Kno, 1974). In this case, the acoustic wave velocity is of order 10^{5}
cm sec^{1}, in an easily accessible range for electron drift in semiconductor.
The device was operated in the frequency range of 550800 MHz.
Although some innovative results were demonstrated in the previous theoretical
and experimental work by various group in the 1960s to 1990s, those activities
faded out with inconclusive results. This is mainly due to the strongly collisiondominant
nature of semiconductor plasma. Further accurate theoretical approach and proper
device design supported by the remarkable progress in semiconductor materials,
fabrication techniques and measurement technologies should open new hope towards
the realization of solidstate THz device utilizing plasma wave interaction.
In addition, the operating frequency of the devices operating based on the plasma
wave interaction is related to the phase velocity of the fundamental space harmonic
component which should be comparable or smaller than the electron drift velocity.
Thus, pitch sizes of one hundred nanometer of slowwave circuit like the interdigital
slowwave circuit should lead to the operation in THz region. Besides this drifting
plasma concept, Dyakonov and Shur (1993) has proposed
the nondrifting plasma concept and successfully demonstrated the THz detection
using IIIV high electron mobility transistor (HEMT). We also successfully showed
the THz detection using AlGaAs/GaAs HEMT at room temperature (Hashim
et al., 2008). A brief analysis and discussion on the relationship
between drifting plasma and such nondrifting plasma is also described in details
(Mustafa and Hashim, 2010).
The objective of this study is to present a series of our fundamental works carried out since a past few years in order to reevaluate and revive again the possibility of realizing a solidstate amplifier operating in THz region based on this drifting plasma wave interactions. In this study, a description of a new method to analyze the properties of semiconductor drifting plasma in a semiconductorinsulator structure based on the transverse magnetic (TM) mode analysis is presented. Here, the components of waves, the electromagnetic fields and the ω and k dependent effective permittivity which is used to describe the dielectric response of the semiconductor plasma to the TM surface wave excitation are derived. In particular, those parameters are determined using the combination of wellknown Maxwell’s equations and carrier kinetic equation based on semiconductor fluid model. Following that, the properties of semiconductor drifting plasma in a twodimensional electron gas (2DEG) structure on semiinsulating substrate using the developed TM mode analysis are presented. Here, the electromagnetic fields and the ω and k dependent effective permittivity of the 2DEG drifting plasma are also determined. Then, the analyzed device structure fabricated on nAlGaAs/GaAs highelectronmobilitytransistor (HEMT) structure and the formulation procedures to explain the interactions between drifting plasma waves in semiconductor and electromagnetic waves propagating through the interdigital slowwave structure are presented. It is noted that the approach is also applicable to other IIIV compound HEMT material system. Next, the main theoretical results to show the interactions between plasma waves and electromagnetic waves are presented and their characteristics are analyzed. Here, the admittance of the interdigitalgated structure is calculated. The preliminary experimental studies of plasma wave interactions using a socalled interdigitalgated nAlGaAs/GaAs highelectronmobilitytransistor (HEMT) structure carried out by our group are presented. Finally, we conclude the findings or contributions of our work and some remarks for future research.
TRANSVERSE MAGNETIC MODE ANALYSIS OF PLASMA WAVES IN SEMIINFINITE SEMICONDUCTORINSULATOR STRUCTURE
Electromagnetic fields in semiconductor drifting plasma: The analysis
is a threedimensional analysis based on the generalization the transverse magnetic
(TM) mode analysis by Sumi (1967) in such a way that
the inertia effect of the electron is included. At high frequencies, a quantum
mechanical treatment may become necessary. However, this is beyond the scope
of the present paper. The physical situation under investigation is shown in
Fig. 1.
A semiconductor plasma slab with a finite thickness d_{s} is separated
by an insulator layer with a thickness d_{I}. The electromagnetic fields
near the surface of an ntype semiconductor with a uniform electron flow in
the zdirection with a drift velocity, υ_{d}, shown in Fig.
1 is analyzed.

Fig. 1: 
Semiconductorinsulator interface and its coordinate 
The electromagnetic fields are determined by combining Maxwell’s equations,
the chargecurrent equations and the kinetic equation for electrons based on
the phenomenological fluid model of the plasma given by the following equation
in the effective mass approximation (Sumi, 1967; Sumi
and Suzuki, 1968).
Here,
is the electric field, q is the electron charge,
= υ_{d} is the electron velocity,
is the magnetic field, υ_{th} is the thermal velocity, n is the
electron density, m* is the electron effective mass and v is the collision frequency.
It is shown that the plasma wave propagates in the drifting direction with a
factor of exp[j(ωtkz)] consists of two kinds of surface wave components
(Hashim, 2006). One is the quasisolenoidal wave (Swave)
without space charge perturbation (div
= 0) and the other is quasilamellar wave (Lwave) with space charge perturbation
(rot
= 0). The transverse decay constants, Γ_{s} and Γ_{l},
of these components satisfy the following relations (Iizuka
et al., 2004):
where, c is the light velocity, ω_{p }is the plasma frequency,
and λ_{D} is the dielectric relaxation frequency and λ_{D}
is the Debye length. Among ω_{p}, λ_{D} and related
characteristic parameters in Eq. 2a and b,
the following wellknown relations hold:

Fig. 2: 
Swave and Lwave in semiconductorinsulator structure 
where, ε is the permittivity, T_{e} is the electron temperature, μ is the mobility, k_{B} is the Boltzman constant and D is the diffusion constant.
As shown in Fig. 2 the Swave represents the solenoidal electromagnetic field penetrating into a semiconductor whereas the Lwave represents Debye screening of fields by carriers where space charge can exist only in a surface layer with a thickness given approximately by the Debye length. In the zero temperature limits, this layer tends to a surface charge layer with zero thickness. The electromagnetic fields are obtained as follow:
Here, A_{l} and A_{s} are the coefficients determined by the
boundary condition at semiconductorinsulator interface. The first term and
the second term of the righthand side of Eq. 46
represents the quasilamellar component and the quasisolenoidal component,
respectively.
Boundary condition at semiconductorinsulator interface: In reality, due to various causes, the surface states will exist at the semiconductorinsulator interface. It is generally believed that the response time of the surface states is very slow, lying in the kHz to MHz region. In this analysis, the surface recombination of carriers at the semiconductor insulator interface is ignored with the reason that the frequency range dealt in this work is high enough compared to the frequency range of surface recombination. Thus, the boundary conditions are determined as follows.
Here, the subscript 1 represents the insulator layer and subscript 2 represents the semiconductor layer. Using these boundary conditions, the ratio of A_{l}/A_{s} for the electric field in the z direction is obtained as follows:
The following Eq. 11 is obtained from Eq.
6 and 10:
The above Eq. 11 shows that the swave component and lwave component of electromagnetic fields have to be excited in order to be satisfied.
Effective permittivity of a semiinfinite semiconductor drifting plasma:
The amplitude ratio of the above two components can be determined from the boundary
condition that there can be no surface charge at finite temperatures and this
leads to an expression for the ω and k dependent effective permittivity
of the plasma, ε_{eff}(ω, k) (Iizuka et
al., 2004; Hashim, 2006). Here, ε_{eff}(ω,
k) is defined in such a way that the transverse admittance of the TM wave looking
into the semiconductor plasma is given by:
The TM mode analysis gives the following general expression of the effective
permittivity in the slow wave approximation (ω/k<<c) and under
assumption of an ideal semiconductor surface at the flat band condition without
surface states (Iizuka et al., 2004; Hashim,
2006):
TRANSVERSE MAGNETIC MODE ANALYSIS OF PLASMA WAVES IN SEMIINFINITE TWODIMENSIONAL ELECTRON GAS STRUCTURE
Electromagnetic fields in semiconductor drifting plasma: The developed
threedimensional analysis described previously is applied to analyze the characteristics
of twodimensional electron gas (2DEG) drifting plasma at the AlGaAs/GaAs heterointerface
(Hashim et al., 2003; Mustafa
and Hashim, 2010). In order to determine the electromagnetic fields in 2DEG
semiconductor drifting plasma, the extended Transverse Magnetic (TM) mode analysis
of the plasma wave interactions for the device geometry shown in Fig.
3 is performed basically following the procedures similar to those used
in the previous section.
Here, we also consider that a TM wave is propagating with a uniform electron flow in the z direction, with a drift velocity, υ_{d}, along the 2DEG layer shown at the bottom of Fig. 3, embedded in semiinfinite GaAs and AlGaAs layers. A basic dispersion equation for TM waves can be derived by combining Maxwell’s equations with the equation of electron motion in the effective mass approximation based on the fluid model of semiconductors.
The decay constants of solenoidal component, Γ_{s} and lamellar
component, Γ_{l}, are obtained as follows:
It was shown that only quasisolenoidal surface wave (swave) components, which
represent the solenoidal electromagnetic field distribution penetrating into
the semiconductor region as well as that penetrating into the upper dielectric
region, exist in the space and the charge modulation in the 2DEG layer can be
incorporated as a boundary condition connecting these two swave components,
i.e., one in the lower semiconductor halfspace and the other in the upper dielectric
halfspace, these two swave components are schematically shown in Fig.
4.

Fig. 3: 
2DEG AlGaAs/GaAs heterointerface and its coordinate 

Fig. 4: 
Swave in AlGaAs/GaAs structure 
The effect of lamellar component is very small due to the confinement of carriers
in 2DEG layer. Using the 2DEG layer, the loss due to the transverse carrier
diffusion of the swave can be suppressed due to strong quantum confinement.
The swave component of E_{y}, E_{z} and H_{x} are summarized
as follows (Hashim, 2006; Mustafa
and Hashim, 2010).
where,
and
are the coefficients determined from the boundary condition.
Effective permittivity of drifting plasma in 2DEG on semiinsulating substrate:
Similarly, from the TM wave analysis of the 2DEG layer, the admittance of plasma
for 2DEG is defined (Hashim, 2006; Mustafa
and Hashim, 2010) as:
where, the ω and k dependent effective permittivity of the 2DEG plasma was calculated as:
THEORETICAL ANALYSIS OF INTERACTION BETWEEN PLASMA WAVES AND ELECTROMAGNETIC SPACE HARMONIC WAVES
Device structure and theoretical formulation: A theoretical analysis procedure to describe the presence of interactions between surface plasma waves of carriers in a 2DEG at AlGaAs/GaAs heterostructure and electromagnetic space harmonics slow waves using interdigitalgated HEMT plasma wave devices are presented, the schematic physical device structure is shown in Fig. 5.
The carrier plasma waves are assumed to propagate along the 2DEG layer with
the phase factor of exp(j(ωtkz)) in the z direction as shown at the bottom
of Fig. 5. The effective permittivity derived in the previous
section was utilized in the calculation of the twoterminal admittance of the
interdigital structure shown in Fig. 5. With reference to
Fig. 6, a periodic Green’s function, Gr(z, z'), for the
multiconductor strip lines (Hasegawa et al., 1971;
Silvester, 1968) was defined as the potential at point
z on a metal finger in response to an array of the unit positive and negative
charge lines placed at positions of z' + mp with m = 0, ±1, ±3,
±5,….. Then, Gr(z, z') was calculated as follows (Hashim
et al., 2003; Mustafa and Hashim, 2010):

Fig. 5: 
Physical device structure under study 

Fig. 6: 
Schematic for admittance analysis of interdigital gates 
where, ε_{0} is the permitivity of vacuum and
is the effective permitivity of the 2DEG plasma.The interdigital admittance
was evaluated by solving the following Fredholm’s integral equation on
a computer using matrix algebra (Seki and Hasegawa, 1984)
in order to obtain the charge distribution, ρ(z), on the finger, where
Φ_{0} is the potential of the finger.
Finally, the interdigital twoterminal admittance was evaluated by the following equation:

Fig. 7: 
Schematic for space harmonic analysis of interdigital 
where, G and C are the conductance and capacitance of the interdigital structure
loaded with 2DEG plasma, respectively.
Space harmonics in interdigital slowwave structure: The basic characteristics
of the interdigital slowwave structure were theoretically considered in terms
of the existence of space harmonics in this structure. The crosssectional structure
for consideration is shown in Fig. 7 (Hashim,
2006; Hashim et al., 2007a, b).
This structure is divided into three regions as follows:
Region I (b ≤ y < +∞): 
dielectric layer with dielectric permittivity constant, ε_{0}. 
Region II (0 ≤ y < b): 
dielectric layer with dielectric permittivity constant, ε_{1}. 
Region III (∞ ≤ y < 0): 
semiconductor layer with dielectric permittivity constant, ε_{2}. 
The channel of carrier flow is assumed to be at the plane of y = 0 and the
interdigital slowwave structure is assumed to be located at the plane of y
= b where its thickness is ignored (infinitely thin) and has a unit length in
x direction. As shown in Fig. 7, these interdigital fingers
are arranged in z direction. The difference of phase angle between two adjacent
fingers is assumed to be equal to π. In this analysis, we found that only
odd space harmonics propagates in the interdigital slowwave structure.
Appearance of negative conductance: Using the procedure mentioned previously,
the twoterminal interdigital admittance of the present device was calculated
numerically on a computer. Calculation was carried out for wide range of parameters
and negative conductance was obtained in various cases (Hashim
et al., 2003).

Fig. 8: 
Calculated conductance as a function of drift velocity for
10 GHz, 100 GHz and 1 THz 
It was found that large negative conductance values can be obtained under a
condition when the drift velocity slightly exceeds the phase velocity, υ_{phf
}= f (frequency) x 2p, of the fundamental space harmonic component of electromagnetic
wave.
Examples of calculated conductance are plotted as a function of drift velocity
for 10, 100 GHz and 1 THz at 300 K in Fig. 8 for the case
of υ_{phf} = 1x10^{7} cm sec^{1}, n_{so }=
1x10^{11} cm^{2} and the AlGaAs thickness, b = 60 nm. Here,
the occurrence of negative conductance peak is seen when the electron drift
velocity slightly exceeds phase velocity, υ_{phf}. Since the value
of pitch, p, reduces with the increase of frequency for the same value of phase
velocity, υ_{phf} , the available value of negative conductance
per area is predicted to be very large, being of the order of 300 S cm^{1}
at 1 THz for υ_{phf} = 1x10^{7} cm sec^{1}, n_{so
}= 1x10^{11} cm^{2} and b = 60 nm.
EXPERIMENTAL STUDY OF PLASMA WAVE INTERACTIONS USING INTERDIGITALGATED HEMT STRUCTURES
Device structure and measurement method: A preliminary experimental
study on the presence of interactions between surface plasma waves of carriers
in a 2DEG at AlGaAs/GaAs heterostructure and electromagnetic space harmonics
slow waves using interdigitalgated HEMT plasma wave devices are presented by
Hashim et al. (2005, 2007a,
b). A mesa pattern was formed on the MBEgrown AlGaAs/GaAs
sample using electron beam lithography (EBL).

Fig. 9: 
Physical device structure of AlGaAs/GaAs interdigitalgated
HEMT device with dc connected fingers 
The etchant used (H_{2}SO_{4} : H_{2}O_{2}
: H_{2}O with a ratio of 8:1:1) produces anisotropic edge profiles.
Then, the ohmic contact patterns are formed by photolithography or EBL and Ge/Au/Ni/Au
metals were deposited using vacuum deposition method. After that, Schottky interdigital
gate patterns are formed by EBL and then Cr/Au metals are deposited using the
same deposition method.
A fabricated device with dc connected interdigital finger structure is schematically shown in Fig. 9. This device is similar to conventional HEMT in which a set of interdigital electrodes act as Schottky gate. However, the use of the device is very different. We are interested in the twoterminal admittance of the interdigital gate itself, which should be strongly modulated and which even becomes negative in its real part due to the wave interactions between plasma waves and electromagnetic space harmonic waves. As shown at the top of Fig. 9, the interdigital slowwave circuits consist of two comblike electrodes and have 25 pairs of fingers/channel with a finger pitch, p, of 5 or 10 μm. Here, the finger width and spacing are chosen to be the same and equal to a, so that p is equal to 2a. In the present device design, two channels were formed.
The device has the overall structure of a loaded coplanar waveguide (CPW), which facilitates onchip microwave probing. The channel width, W, was 50 μm and the thickness of the AlGaAs barrier layer was 50 nm. The carrier mobility and the carrier sheet density obtained by Hall measurements at room temperature are 7540 cm^{2}/Vs and 4.6x10^{11} cm^{2}, respectively. A plan view of the fabricated device is shown in Fig. 10.
The twoterminal admittance, Y, of the plasma wave device was determined from the Sparameter reflection measurement, as shown in Fig. 10 over the frequency range from 1 to 15 GHz at room temperature using a vector network analyzer HP8510C and a Cascade onwafer microwave microprober of GroundSignalGround type. It was shown theoretically in the previous section that travelingwave interaction becomes more and more favorable at higher frequencies, particularly in the THz region. However, direct measurement of twoterminal admittance at such ultrahigh frequencies is very difficult. Therefore, the measurements are carried out only at low microwave frequencies. During the measurement, the source and drain were biased with dc voltage, V_{S} and V_{D}, respectively to cause drift current to flow in the channel while the dc voltage to the set of interdigital fingers was kept at zero.
Large conductance modulation and comparison with theory: The measured
admittance of the twoterminal interdigital structure are summarized in Fig.
11a and b as a function of the drainsource voltage,
V_{DS}, for a device with a pitch of 5 μm under zero gate voltage
for all the interdigital fingers (Hashim et al.,
2005, 2007a, b). The measured
values of conductance and capacitance experience remarkably large changes with
changes of the drainsource voltage for all the measured frequencies. The conductance
decreases rapidly at 5 and 10 GHz when the drainsource voltage slightly exceeds
4V. Obviously, the observed behavior of the admittance of the interdigital gates
cannot be explained at all by the conventional transport theory with the transit
time picture. These results indicate the presence of the effect of the interactions
between the surface plasma waves of 2DEG carriers and electromagnetic space
harmonic waves in the fabricated interdigitalgated HEMT device.

Fig. 10: 
Planview of fabricated AlGaAs/GaAs interdigitalgated HEMT
device with dc connected fingers 
However, a suitable theoretical analysis is required to confirm the presence
of such an effect quantitatively. The calculation results of interdigital admittance
based on our theoretical approach explained previously are shown in Fig.
12a and b. The calculated values of conductance and capacitance
also experience remarkably large changes with changes of the drainsource voltage
for all the calculated frequencies. Here, small values of negative conductance
are obtained.
Effect of field nonuniformity along the channel: Theoretical curves
in Fig. 12a and b show that drastic variations
in conductance and capacitance take place with the drainsource voltage over
all frequencies. The calculated values of conductance and capacitance have nearly
the same magnitudes as the experimental ones (Hashim et
al., 2005, 2007a, b).
Thus, the general behavior of the admittance is reproduced surprisingly well
by calculation in spite of various assumptions used in the theory. The major
assumptions in the theory are: (1) the interdigital pattern is infinitely repeated;
(2) the thickness of the electrode pattern is infinitely thin, but its conductance
is infinitely large; (3) the phase of the electromagnetic field on the metal
is maintained the same in the finger direction (xdirection) for all frequency
components and (4) the carriers have the same drift velocity along the whole
channel.
In spite of the above general remarks, however, agreements on the details are obviously not adequate between theory and experiment. Particularly, the appearance of a small negative conductance is predicted

Fig. 11: 
Measured (a) conductance and (b) capacitance characteristics
as a function of drainsource voltage 

Fig. 12: 
Calculated (a) conductance and (b) capacitance characteristics
as a function of drainsource voltage 
We believe that the major cause for the lack of agreement in the detailed behavior
comes from assumption (4) in which the carriers have the same drift velocity
along the whole channel. This is not the case in any semiconductor fieldeffect
transistor under a strong drain bias. The nonoccurrence of negative conductance
in the experiment can be explained in terms of cancellation of the small negative
conductance obtained under the high field portion of the channel by the large
positive conductance coming from the low field portion of the channel.
To take account of this effect, we have estimated the electric field distribution, E(z) under the interdigital gates by fitting the dc drain IV characteristics using the gradual channel approximation and the fielddependent mobility. Here, we replace the entire interdigital pattern with one gate electrode, since the surface potential of the airgap region is similar to that underneath the gate with zerobias due to strong Fermi level pinning. From the estimated field distribution, we obtained the drift velocity under each interdigital finger, v_{d}^{finger}, using the following equation:
where μ_{0} is the lowfield mobility and υ_{s} is the saturation velocity. Then, the interdigital admittance is calculated by
where, G^{finger} and B^{finger} are the conductance and susceptance of the individual interdigital fingers, respectively. For the values of G^{finger} and B^{finger}, we used the values assuming uniform velocity distribution, taking advantage of gradual changes in velocity along the channel and the quasiperiodic nature of the interdigital pattern.

Fig. 13: 
Calculated (a) conductance and (b) capacitance characteristics
as a function of drainsource voltage taking into account the uniformity
of field distribution along the HEMT channel 
The recalculated conductance and capacitance, taking into account the non
uniformity of the field in this fashion, are shown in Fig. 13a
and b, respectively. The recalculated results show much better
agreements with the experimental ones shown in Fig. 11a and
b in spite of a very simple theoretical treatment of a very
complicated problem. Thus, it can be concluded that the observed changes in
both conductance and capacitance are due to the coupling between the drift plasma
waves in the 2DEG carriers and the electromagnetic space harmonic slow waves
through the interdigital pattern.
Behavior of capacitively coupled device: We believe that uniform carrier
drift velocity should be realized if the potential array is arranged in the
same slope or in the steplike distribution (Hashim, 2006;
Hashim et al., 2007a, b).
Ideally, such a potential distribution can be realized by introducing a different
individual bias to each finger. However, this method is totally not suitable
for real application. Here, we come with the idea that the most possible way
is to make all fingers being capacitively coupled. This structure also called
as dc segmented rf coupled structure. This structure will keep uniform electric
field in the channel when the dc bias is applied to the interdigital gates which
modulates the potential in the channel.
The schematic structure of fabricated device with capacitively coupled fingers
is shown in Fig. 14. The structure of interdigital fingers
on the channel area is not changed which is similar to a dc connected structure
shown in Fig. 9. The plan view of the fabricated device is
shown in Fig. 15. The fingers are capacitively coupled through
SiO_{2} layer (thickness = 300 nm) only at certain part as indicated
in Fig. 15. SiO_{2} layer on the channel was etched
out. The other device parameters such as pitch size, number of fingers, carrier
mobility and carrier sheet density are same with the previous dc connected structure.

Fig. 14: 
Schematic device structure with capacitively coupled interdigital
fingers 

Fig. 15: 
Planview of fabricated device with capacitively coupled interdigital
fingers 
Only the channel width was designed to be 40 μm. In addition, five meandertype
inductors with the width of 3 μm were applied to block the rf signal from
going through to the dc probe. It was confirmed that the rf leakage is less
than 1%.
The measurements were also carried out at low microwave frequencies in the range of 10 to 40 GHz and their setups were also shown schematically in Fig. 15. During the measurement, as shown in Fig. 15, the source and drain were biased with dc voltage, V_{S} and V_{D} respectively to cause drift current to flow in the channel while the dc voltages to the set of interdigital gates, V_{GS} and V_{GD}, were adjusted to a desired gate voltage, V_{G} according to the following group of Eq. 25:
By isolating fingers by capacitors and by using meander inductors, such a special dc biasing described above is realized so as to achieve a uniform field distribution along a channel assumed in theory, since conventional HEMTs produce a highly nonuniform field distribution in the channel. In addition, this structure allows the adjustment of the gate voltage which was not realized using previous dc connected structure.
The measured dc IV characteristics of a device measured by this special biasing method, as shown in Fig. 16 shows the shift of pinch off voltage and extension of linear region as compared with conventional FET gate operation shown in Fig. 17. Thus, it can simply confirm that uniformity of field distribution was improved. Here, the calculation results for both cases were carried using the following equation:

Fig. 16: 
dc IV characteristics measured by special biasing method 

Fig. 17: 
dc IV characteristics measured by conventional FET method 
where, W_{G} is the channel width, L_{G} is the gate length, I_{DS} is the drainsource current and V_{T} is the threshold voltage. It is indicated in Fig. 16 that the calculated results for special biasing method show no pinch off behavior as what seen in the experimental ones. It is noted here, for the special bias method the second term in the bracket of Eq. 26 is zero. This discrepancy is believed due to the fingers at the both edges as shown in Fig. 14, were made directly connecting of to the main gate electrodes where these fingers have pinched off the channel. By making all the fingers not only being capacitively coupled among each other but also to the main gate electrodes, the pinch off of the channel should be prevented. This design will be carried out in the future work.
The measured conductance is shown as a function of drainsource voltage, V_{DS}
for various gate voltages in Fig. 18a and for various frequencies
in Fig. 18b, respectively (Hashim et
al., 2007a, b). The change of conductance using
this structure takes place at low V_{DS} as compared with dc connected
interdigital finger structure. Thus, it can be assumed that uniformity of field
distribution was improved.

Fig. 18: 
Measured conductance as a function of drainsource voltage
(a) gate voltage dependence and (b) frequency dependence 
The dc IV measurements indicated that ample carriers are in the channel at
zero gate bias which totally pinches off at 2V. Conductance modulation by V_{DS}
for channels with ample carriers could be clearly seen and this absolutely only
can be explained in terms of interactions between the input rf signals and 2DEG
surface plasma waves. Absolute conductance values were smaller than the theoretical
prediction, due to the small capacitance between interdigital fingers attenuating
the propagation of rf signal at these frequencies.
CONCLUSIONS
A new method based on the Transverse Magnetic (TM) mode analysis for analyzing plasma wave interactions in semiconductorinsulator structure and semiconductor with 2DEG structure was presented. In this analysis, two kinds of excited wave components were shown to exist in the semiconductorinsulator structure which are known as quasilamellar wave (lwave) and quasisolenoidal wave (swave). On the other hand, the excited surface wave component in 2DEG semiconductor was shown to be only dominated by quasisolenoidal wave which should improve the interactions between carrier plasma waves and electromagnetic waves. The effective permittivity which is used to describe the dielectric response of the semiconductor plasma to the TM surface wave excitation was derived for both structures. It was shown that only the odd mode of space harmonics propagate through the interdigital slowwave structure. An occurrence of negative conductance was predicted when the drift velocity of carriers slightly exceed the phase velocity of electromagnetic space harmonic slow waves. Although the magnitude of negative conductance peak is small at low microwave frequencies, it increases drastically with frequency toward THz region. This means that broadband and high power amplification may be realized by an innovative device design in the millimeter wave and submillimeter wave regions where power level of conventional semiconductor devices fall very rapidly with the increase of frequency. A large modulation of conductance due to interactions between surface plasma waves of 2DEG carriers in AlGaAs/GaAs heterostructure and electromagnetic space harmonic slow waves was observed experimentally although no net negative conductance was observed due to nonuniformity of field distribution under interdigital gates. Net negative conductance can be obtained if a uniform field distribution can be realized. The result seems to prove the existence of surface plasma wave interactions even under the strongly collision dominant situation in the microwave region and provides great hope for increased interactions at THz frequencies with nearly collision free conditions. The uniformity of field distribution was improved by applying a dc segmented rf coupled slowwave structure instead of dc connected structure. In our present design, the fingers at the edge of channel were connected directly to the main gate electrodes. The pinch off of current can be delayed at higher voltage if the fingers at the edge of channel are totally isolated from the main gate electrodes. We believe that the pinchoff characteristics in our present designed device are resulted from this matter.
ACKNOWLEDGMENTS
The author (A.M.Hashim) would like to thank Professor S. Kasai, Professor T. Hashizume and Professor H. Hasegawa for fruitful discussion and guidance on theoretical and experimental work. This work was partly supported by Japan’s Ministry of Education, Culture and Technology (MEXT), Malaysia’s Ministry of Science, Technology and Innovation (MOSTI) and Malaysia’s Ministry of Higher Education (MOHE) through 21st century COE Program Mememedia Technology Approach to the RandD of Next Generation ITs, Science Fund Vote 79174 (030106SF0283) and Fundamental Research Grant Scheme Vote 78205 and 78417, respectively. The author also would like to thank Research Center for Integrated Quantum Electronics, Hokkaido University and Institute of Ibnu Sina, Universiti Teknologi Malaysia for fabrication facilities.