Analytical modeling odime

Mohammad A. Khorram

Dep. of Electrical Enginering, Univ

Abstract — Plasma wave propagation alongDimensional Electron Gas (2DEG) layer of a studied. It is shown that the wave can be usefuof THz signals. An analytical solution oHydrodynamic equations is presented. This meinsight into electromagnetic modes allowed tothe 2DEG as electrons are in motion with constvelocity. Besides, wave impedances of the modAfterwards, a simple matching network desigoutput ports of the 2DEGs is developed.

Index Terms — Amplifiers, analytical mMaxwell equations, plasma waves.

I. INTRODUCTION

Detection and generation of THz signals plasma waves inside two Dimensional Electrlayers of High Electron Mobility Transistorsbeen observed in several experiments [1]-[2theoretical models have been proposed toobservations [3]-[5]. In [3], plasma wave anequations are compared and a THz wavpredicted in a gated 2DEG layer of a HEMTshort circuited drain and source, respectivelused to explain the THz wave generation annon-resonant) detection in gated 2DEG of HEroom temperature THz source implemented inbased HEMT, tunable with gate voltage betwTHz, has been reported [2]. On the other hanresonances in un-gated 2DEG layers have alsin [4] with the same boundary conditions as in

In [3] and [4], modeling of the plasma wavexecuted by solving Poisson and Hydrody(Euler and continuity). This solution is corwavelength of the plasma wave and the devicmuch smaller than transverse electromagnetithe same frequency [5]. However, it is not ablmechanism of the wave amplification exactldepth field analysis is performed. Therefoboundary conditions are introduced to estabtransfer from the bias source to the plasma wsimple and direct design procedure is not vimethod.

In this paper, the plasma wave propagatigated 2DEG in the presence of drift curremethod based on a solution of Maxwell eqwith the Hydrodynamic one is used to d

of THz wave propagation inside ensional electron gas layers

mi, Samir El-Ghazaly, Shui-Qing Yu, Hameed N

versity of Arkansas, Fayetteville, 3217 Bell Eng72701, US

g an ungated two hetrostructure is

ul in amplification of Maxwell and ethod provides an o propagate along tant average drift es are illustrated. gn for input and

models, HEMTs,

with the aid of ron Gas (2DEG)

s (HEMTs), have 2]. Also, several o describe these nd shallow water ve generation is T with open and ly. The model is nd (resonant and EMT. Recently, a n an AlGaN/GaN ween 0.75 to 2.1 nd, plasma wave so been proposed n [3]. ve propagation is

ynamic equations rrect because the ce dimensions are ic wavelength at le to describe the ly because no in ore, the specific blish the energy wave. Besides, a iable through the

on along an un-ent is studied. A quations coupled define the wave

characteristics. To this end, the 2Dsheet positioned at the interface of tgap semiconductors. Next, the coupto plasma waves is simply taken insurface currents on the sheet assatisfied. To consider drift motionbias source, linearization of Hydroto reflect the movement into the helps us perform an exact propagdefine required conditions for the wpropagation impedance investigatifurther design of a matching netwoTHz amplifier.

In section II, the formulation of tMaxwell and Hydrodynamic equanew dispersion relation is derivpropagation mode characterization iproperties of each mode are discusinterpretation for the wave amplificspecific example is described in related propagation modes and wav

II. DISPERSION RELATION CALCULA

DRIFT CURR

Consider a 2DEG layer placed atinside a semi-infinite hetrostructurea constant motion of electrons alocharacterized by the average electro

Fig. 1. Schematic view of a 2DEhetrostructure (not shown) with a covelocity

While developing the solutionpropagation along x axis, it has b

ungated two

Naseem

gineering Center, AR

DEG is treated as a charge two wide and narrow band-pling of the 2DEG carriers nto account by introducing s Maxwell equations are

ns of electrons induced by odynamic equations is used

surface conductivity. This gation mode analysis and wave amplification. Also, a on is done that facilitates

ork required for an efficient

the problem is presented as ations are being solved. A ved and also a complete is performed. In section III, ssed and a simple physical cation is presented. Next, a section IV to define the

e impedances.

ATION IN THE PRESENCE OF

RENT

0z = plane and embedded e as in Fig. 1. There is also ng the 2DEG toward x+ ,

on drift velocity 0v .

EG layer implemented in a onstant average drift electron

n for the plasma wave een shown that xTE mode

978-1-61284-757-3/11/$26.00 C2011 IEEE

does not exist if the 2DEG surface and the surrounding media are isotropic [6]. Therefore, electromagnetic field equations and the related dispersion relations are presented just for nonradiative xTM case. Here, relative permittivities of the wide and narrow bandgap semiconductors are assumed to be equal rε .

Accordingly, wave function and field formulations for 0z ≥ part are:

( )exp j t x zψ φ ω γ δ= − − (1)

where jγ α β= + and

2

, ,

0.

x z y

y z x

E E Hj j

E H H

δ γδψ ψ δψωε ωε

= − = = −

= = =

(2)

From Helmholtz equation, it is obtained that: 2

2 22 0r

cε ωδ γ+ + = (3)

where 0 rε ε ε= ( 12 80 8.85 10 , 3 10 /F m c m sε = × = × ).

Continuity of tangential component of the electric field along

the interface 0

x

zE

=is the first boundary condition. Second

boundary condition is simply acquired by relating surface

current ( )0 0ˆx y y

z zJ z H H+ −= =

= × − to the tangential electric

field by0

x x

zJ Eσ

== , where σ is the xx component of

surface conductivity tensor. Considering the boundary conditions, it is derived that:

2 .jσ εω δ

− = (4)

In [6], the surface conductivity approximation from Drude

model 2

0n qj m

σω ∗= is replaced into (3) and the dispersion

relation of a normal mode of propagation is attained from (3) and (4) as:

2

2ja

ωγ = ± (5)

where 2

0* .

4n q

amε

=

Here, in order to take the electron motion into consideration a new surface conductivity but not the value from Drude model approximation is employed. To this end, the surface conductivity model developed in [7] is used. In [7], linearization of Hydrodynamic equations (Euler and continuity) is performed to include the carrier movement. Hydrodynamic equations are well known to be valid as the mean-free path for electron-electron collision is smaller than the device length and the mean-free path for scatterings from phonons and impurities [3]. Electrons in the 2DEG layers

simply satisfy these two conditions. Conductivity of the 2DEG layer in the presence of the electrons motion has been obtained with the aid of the linearization in [7] as:

( )

20

*

0 01

n q jm j v j v

ωσω γ ω γ

τ

=⎛ ⎞− − +⎜ ⎟⎝ ⎠

(6)

where *m , 0n , q and τ are electron effective mass, 2DEG

electron density at steady state condition, unit charge (191.6 10q C−= × ) and momentum relaxation time, respectively.

In a collisionless case ( 01vγτ

), by replacing (4) and (6)

into (3):

( )2

4202 2

1 04

rj va c

ω εγ ω γ+ − + = (7)

is derived. In the THz frequencies, (7) can be reduced to:

( )42 204a j vγ ω γ− = − (8)

and therefore, four different modes with dispersion relations of:

( )

( )

20 0

1,2 20

20 0

3,4 20

2

2.

a v a avj

v

a v a avj

v

ω ωγ

ω ωγ

+ ± +=

− + ± −=

(10)

are obtained. It means that the normal xTM mode of propagation along the surface of the 2DEG, with the dispersion relation (5), is divided into four different modes as electrons are moving with the constant average drift velocity. It can be shown that the collisionless condition is not a critical one, and the four modes still can exist with slight changes as collisions are also included. From now to the end, only the collision less case is explained and the collision effects will be covered in a separate paper.

To finish the analysis, the wave impedance along the 2DEG z

xy

EZH

= − is also calculated for each mode as:

1,...,41,...,4 .xZ

jγ

ωε= (11)

III. CHARACTERISTICS OF EACH MODE

It is obvious (from (10)) that for frequencies lower than

02bfav

ω = (named breaking frequency afterwards) all four

modes are purely propagating since their propagation constants are imaginary. With typical value of a and 0v , first

two modes are propagating toward x+ while the other two are moving in the opposite direction. Also, it can be shown that the second mode has a similar behavior to the normal mode as the electron drift is not included. For frequencies above bfω ,

978-1-61284-757-3/11/$26.00 C2011 IEEE

the propagation constants of the third and fourth modes are complex numbers and have attenuation constant (positive or negative). In other words, as a plasma wave is launched properly along the 2DEG at frequency ranges above bfω ,

energy is being transferred between the bias source and the electromagnetic wave as being amplified or attenuated. It is obvious that bfω can be controlled by changing the 2DEG

charge density and the electron drift velocity. Obviously, the mechanism of energy transfer still needs more investigations. Perhaps, a complete analysis based on a time domain full wave model in conjunction with Boltzmann equations should be performed for better understanding.

The separation of the normal mode into four new modes is very similar to what has been proposed in travelling wave tubes [8]. A traveling wave tube is basically composed of a slow wave structure as a helix and an electron beam. As the electron beam is passed through the helix, the normal propagation mode is divided into three different modes of growing, attenuating and propagating ones [8].

IV. DISCUSSION OF A SPECIFIC EXAMPLE

In this section, the characteristics of the four propagation modes is investigated for a 2DEG created at the interface of

InGaAs/InP with 12 2 70 00.3 10 , 2 10n cm v cm s−= × = × and

electron effective mass 00.042m ( )310 9.1 10m kg−= × . Besides,

relative permittivity of both InGaAs and InP are assumed to be equal 12.6rε = .

With these values, calculated attenuation and phase constants (α and β ) of the four modes are shown in Fig. 3

and Fig. 4 for frequency range of 300GHz up to 3 THz. As depicted in Fig. 3, the two first modes are propagating along the electron drift velocity x+ while the two last modes are in the opposite direction x− . As shown in Fig. 4, the first and second modes are merely propagating ones but the third and fourth modes can have attenuation term as soon as operating frequency is higher than bfω . From the propagation direction

of each mode and the sign of the attenuation constants, it is clear that the third mode is an attenuating mode while the fourth one is an amplifying mode.

In Fig. 5, phase velocity of each mode normalized to the drift velocity is shown. As depicted, the first mode is the slowest and the second one is the fastest. Also, notice that bfω

is the point which the phase velocities of the third and the fourth modes become equal to the drift velocity.

Next, magnitude of real part of wave impedances normalized to vacuum wave impedance 0 377Z = Ω , are

illustrated in Fig. 6. As shown in Fig. 6, the real parts of the third and fourth modes are equal at the frequencies higher than

bfω .

In Fig. 7, imaginary part of the wave impedances normalized to 0Z are presented. As seen, the first two modes

don’t have imaginary part. Additionally, last two modes have

complex wave impedance for frequencies above bpω . After

this point, third mode has inductive impedance while the fourth one is highly capacitive considering that they both propagate toward x− .

Fig. 3. Phase constants of the four modes versus frequency

Fig. 4. Attenuation constant of each mode versus frequency

Fig. 5 Phase velocity of each mode normalized to the constant electron drift velocity versus frequency in a logarithmic plot

978-1-61284-757-3/11/$26.00 C2011 IEEE

Fig. 6 Real part of each mode’s wave impedawave impedance in vacuum in a logarithmic plot

Fig. 7 Imaginary part of wave impedance noimpedance in vacuum

After considering the appropriate sign

impedances, a simple matching network shoproposed. Notice that to couple the wave source placed at x L= to the plasma wave wproperties, a matching network with highly impedance is required. The same matching required for the load placed at 0z = to handbetween the load and the wave impedance. Ifurther investigations and designs are needeimprove the THz wave coupling to 2DEG.

Fig. 8 A simple matching network placed atcircuit is not included)

ance normalized to t

ormalized to wave

for the wave own in Fig. 7 is from the signal

with fourth mode inductive output network is also

dle the mismatch It is obvious that ed to be done to

t both ports (Bias

VII. Conclus

An analytic method is proposed to along 2DEG layers of hetrostructuremotion. The electrons movement separation into four new modes wmode at THz frequency ranges. Theplasma waves in this range is predsophisticated matching network dpointed out that for better understdesigns, application of a full wasatisfied by Boltzmann equation is i

ACKNOWLEDGE

Research was sponsored by the Aand was accomplished under CoopeW911NF-10-2-0072. The views anthis document are those of the ainterpreted as representing the expressed or implied, of the Army RU.S. Government. The U.S. Govreproduce and distribute reprints notwithstanding any copyright notat

REFERENC

[1] T. A. Elkhatib, V. Y. Kachorov

Veksler, K. N. Salama, X. Zhanplasma wave detection of terahertzelectron-mobility transistors conneMicrowave Theory & Tech., voFebruary 2010.

[2] A. El Fatimy, N. Dyakonova, Y. S. Vandenbrouk, K. Madjour, D. Poisson, S. Delage, P. Prystaw“AlGaN/GaN high electron mobitunable room temperature terahervol. 107, pp. 024504-024507, Janu

[3] M. Dyakonov and M. Shur, “Sballistic field effect transistor: Newgeneration by dc current,” Physic15, pp. 2465-2468, October 1993.

[4] M. Dyakonov and M. Shur, “Cuwaves generation in ungated two-Applied Phys. Letters., vol. 87, pp.

[5] V. V. Popov, O. V. Polischuk excitation of plasma oscillationsdimensional electron layer,” J. o033510-033517, 2005.

[6] M. Nakayama, “Theory of surfaccarriers,” J. of the Physical Society393-398, February 1974.

[7] S. A. Mikhailov, “Plasma instaelectromagnetic waves in low-dimPhysical Rev. B, vol. 58, no. 3, pp.

[8] L. J. Chu and J. D. Jackson, “Fitubes,” Proceeding of the I.

sion

study plasma wave modes es in the presence of carrier

causes the normal mode with one being a growing erefore, an amplification of dicted and also, a need for designs is addressed. It is tanding and more accurate ave time domain method inevitable.

EMENT

Army Research Laboratory erative Agreement Number

nd conclusions contained in authors and should not be

official policies, either Research Laboratory or the vernment is authorized to for Government purposes tion herein.

ES

vskii, W. J. Stillman, D. B. ng and M. Shur, “Enhanced z radiation using multiple high ected in series,” IEEE Trans. ol. 58, no. 2, pp. 331-339,

Meziani, T. Otsuji, W. Knap, Theron, C. Gaquiere, M. A.

wko and C. Skierbiszewski, ility transistors as a voltage-rtz source,” J. of Appl. Phys., uary 2010. Shallow water analogy for a w mechanism of plasma wave al Review Letter, vol. 71, no.

urrent instability and plasma -dimensional electron layers,” 111501, 2005. and M. S. Shur “Resonant

s in a partially gated two-of Appl. Phys., vol. 98, pp.

ce waves coupled to surface y of Japan., vol. 36, no. 2, pp.

ability and amplification of mensional electron systems,” 1517-1532, July 1998. ield theory of traveling-wave .R.E., pp. 853-863, 1948.

978-1-61284-757-3/11/$26.00 C2011 IEEE