Hydrodynamic instability of conﬁned two-dimensional electron … · 2015-05-15 · their study is...

Hydrodynamic instability of confined two-dimensional electron flowin semiconductors

Williams R. Calderón-Muñoz,1,a! Debdeep Jena,2 and Mihir Sen11Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame,Indiana 46556, USA2Department of Electrical Engineering, University of Notre Dame, Notre Dame, Indiana 46556, USA

!Received 26 March 2009; accepted 30 May 2009; published online 10 July 2009"

Hydrodynamic instabilities in two-dimensional electron flow in ungated semiconductors are studiedhere. The driving force for the electrons is an imposed voltage difference that generates aunidimensional electric field inside the semiconductor and its surroundings. The governingequations are linearized for small perturbations around the steady-flow solution. The eigenvaluespectrum determining the rates of growth and wave numbers of the oscillations is calculated. Theelectron flow undergoes oscillatory instability and becomes more unstable as the voltage differenceis increased. The results show that it is possible to obtain oscillation frequencies of the order ofterahertz, indicating the possibility of radiative power at this frequency. © 2009 American Instituteof Physics. #DOI: 10.1063/1.3158551$

I. INTRODUCTION

Terahertz radiation sources are of current interest due totheir advantages in leading edge applications such as bioim-aging and sensing.1,2 Features such as compactness and tun-ability are required in a terahertz source for applications inchemical and biological systems, imaging, and radio as-tronomy, among others. Semiconductors are among the pos-sible candidates as potential sources. A necessary step forthis application is a fundamental understanding and controlof charged-particle interactions and dynamics in solid statedevices. This motivated both theoretical and experimentalstudies of electron flow in semiconductors and it has beenfound that instabilities can produce oscillations even whenthe imposed electric field is steady.

Electrons in semiconductors scatter as a result of colli-sions between themselves and with the lattice and impurities.The most common theoretical approach for the analysis ofelectron flow in semiconductors is to neglect energy transferbetween electrons and the lattice, for which hydrodynamicmodels provide a useful description. This has been an impor-tant perspective to explain physical phenomena involved inelectron transport in semiconductors and several inroadshave been made in this direction. Dyakonov and Shur3 foundanalogies with shallow water equations that predict plasmaoscillations at terahertz frequencies and radiation emissionsin ballistic transport in an AlGaAs/InGaAs field effect tran-sistor !FET". Subsequently, this description was generalizedand applied to high electron mobility transistors !HEMT".4–6

The mechanism of current saturation in a FET due to chok-ing of electron flow and plasma waves was seen to showsimilarities to shallow water phenomena in fluid dynamics.Under this idea, nonlinear oscillations due to ballistic trans-port in FETs and effects similar to hydraulic jumps were alsodescribed.7 Nonlinear dynamic response and how the bound-

ary conditions influence the nonlinear effects were studied intwo-dimensional electron plasmas in FETs by using the hy-drodynamic model.8 It was found that current and plasmawaves in an ungated two-dimensional electron layer maypresent instability similar to that for a gated electron layer.9

Transit-time effects in plasma instabilities were related to theelectron drift across the high field region in HEMTs.10 Mi-croscopically bounded plasma due to current-driven plasmainstability has been reported in lower-dimensional solid-statesystems.11 Plasma oscillations were analyzed in both gatedtwo-dimensional layers and HEMTs.12,13 Instabilities in mul-tilayered semiconductor structures have been studied nu-merically and theoretically.14 Drift wave instabilities havebeen found in semiconductor electron-hole plasmas.15

Experiments have also been performed to detect and un-derstand the mechanisms of terahertz radiation. Subterahertzand terahertz radiation have been found in silicon FETs andnanometer-scale gate-length HEMTs due to plasmawaves.16,17 A better understanding of the strengths and limi-tations of experimental techniques, for instance in the two-color diode laser,18 also helped in the study of terahertz ra-diation. In addition, new techniques and algorithms todetermine the radiation spectrum of terahertz sources havebeen analyzed recently.19 Experimental and theoretical stud-ies have shown nonresonant and resonant detection of tera-hertz radiation in both Si metal oxide semiconductor fieldeffect transistors !MOSFETs" and gated two-dimensionalstructures such as GaAs HEMTs.20,21 Under particular con-ditions, a nanoscale FET made of InGaAs/InAlAs can pro-duce terahertz emission.22,23

Calderón-Muñoz et al.24 determined analytically the spa-tial and time dependent instabilities in one-dimensional elec-tron flow in ungated semiconductors. In this earlier work, onassuming plane-wave propagation of electrons between thecontacts, the instabilities that can arise perpendicular to thedirection of particle flow were neglected. From analogy tofluid dynamics, such instabilities are likely to be present, and

a"Currently on leave from the Departamento de Ingeniería Mecánica, Uni-versidad de Chile, Casilla 2777, Santiago, Chile.

JOURNAL OF APPLIED PHYSICS 106, 014506 !2009"

0021-8979/2009/106"1!/014506/10/$25.00 © 2009 American Institute of Physics106, 014506-1

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:132.236.27.111 On: Sat, 09 May 2015 19:51:59

http://dx.doi.org/10.1063/1.3158551

http://dx.doi.org/10.1063/1.3158551

http://dx.doi.org/10.1063/1.3158551

their study is the goal of this work. Analysis of instabilitiesin semiconductors based on a two-dimensional model, whichmay be very useful for the study of potential terahertzsources, has not been done. The two-dimensional geometrycan be implemented experimentally due to the simplicity ofthe configuration and the boundary conditions can be pre-scribed by specifying the kind of contacts at the source andat the drain. In this paper we analyze the electrostatic andhydrodynamic equations in a high FET and characterize theinstabilities present in them.

II. MATHEMATICAL MODEL

The problem is defined by a doped two-dimensionalsemiconductor as shown in Fig. 1. The semiconductor has alength L and width H and represents the !x! ,y!" plane at z!

=0. The two contacts at x!=0 and x!=L are larger than thesemiconductor width.

The driving force through the device is given by an elec-tric field due to a voltage difference between the two contactsat x!=0 and x!=L. The electric field surrounds the devicedue to the size of the contacts. x!, y!, and z! are the Cartesiancoordinates and t! is the time. The electric field can havecomponents in the three Cartesian coordinates. The imposedelectric field in the x!-direction implies that the !y! ,z!" planedescribes an equipotential area. We are interested in captur-ing instabilities in the !x! ,y!" plane, neglecting variations inthe direction normal to that plane. Electron-lattice interac-tions are neglected, which implies that the heat generationsources due to electron transport in the semiconductor arenot taken into account. Under these assumptions, the two-dimensional hydrodynamic equations for electron flow in thesemiconductor are25

#2V!

#x!2 +#2V!

#y!2 +#2V!

#z!2 −e

!s!n! − ND" = 0, !1a"

#n!

#t! +#!u!n!"

#x! +#!v!n!"

#y! = 0, !1b"

#u!

#t! + u!#u!

#x! + v!#u!

#y! −e

me

#V!

#x! +u!

"= 0, !1c"

#v!

#t! + u!#v!

#x! + v!#v!

#y! −e

me

#V!

#y! +v!

"= 0, !1d"

where V!!x! ,y! ,z! , t!" is the voltage, n!!x! ,y! , t!" is thecharge density, and u!!x! ,y! , t!" and v!!x! ,y! , t!" are the driftelectron velocities in the x! and y! directions, respectively.The system of Eq. !1" includes Gauss’ law Eq. !1a", thecontinuity equation Eq. !1b", and the momentum conserva-tion equations in the x! and y! directions, Eqs. !1c" and !1d",respectively. The system parameters are the doping concen-tration ND, the permittivity !s, the charge of an electron e, itseffective mass me, and the momentum relaxation time ". Theboundary conditions establish a voltage gradient along thex!-direction, a fixed charge density at x!=0, a constantcharge in the semiconductor as a whole, and charge reflexionalong the edges y!=0 and y!=H. This is made possible byimposing an Ohmic contact between the semiconductor andthe metal at x!=0 and an inductive boundary condition atx!=L. The Ohmic contact does not allow fluctuations in themajority carrier density, i.e., electrons, due to the infinitelyhigh surface recombination velocity, whereas an inductiveboundary condition does. These forms of contacts allow pre-scription of the voltage at both ends, the charge density at thesource, and charge neutrality of the semiconductor layer.This last condition is enforced at all times and implies theabsence of space charge injection effects.3,24 Then, theboundary and charge neutrality conditions are

V!!0,y!,z!,t!" = − V0, V!!L,y!,z!,t!" = 0, n!!0,y!,t!" = n0,

!2a"

v!!x!,0,t!" = 0, v!!x!,H,t!" = 0, !2b"

#V!

#y! !x!,0,0,t!" = 0,#V!

#y! !x!,H,0,t!" = 0,

%0

L %0

H

n!!x!,y!,t!"dx!dy! = NDLH . !2c"

For convenience, the governing equations can be nondi-mensionalized. Defining the aspect ratio as R=H /L and writ-ing V=V! /V0, n=n! /ND, x=x! /L, y=y! /H, z=z! /H, u=u!&me /eV0, v=v!&me /eV0, and t= t!&eV0 /meL2, the nondi-mensional version of Eq. !1" is

#2V

#x2 +1R2

#2V

#y2 +1R2

#2V

#z2 − #!n − 1" = 0, !3a"

V ∗ = −V0

E∗

electron flow

y∗

x∗

z∗

V = 0H

L

∗ FIG. 1. Schematic of semiconductor material.

014506-2 Calderón-Muñoz, Jena, and Sen J. Appl. Phys. 106, 014506 "2009!


#n

#t+

#!un"#x

+1R

#!vn"#y

= 0, !3b"

#u

#t+ u

#u

#x+

1R

v#u

#y−

#V

#x+

&#

$u = 0, !3c"

#v#t

+ u#v#x

+1R

v#v#y

−1R

#V

#y+

&#

$v = 0. !3d"

The dimensionless parametric groups are #=eNDL2 /V0!s, $=&"2e2ND /!sme=%p", and %p=&e2ND /!sme, which is thefundamental plasma frequency of free electrons in the semi-conductor. The only tunable parameter is # and it is inverselyproportional to the applied bias V0; the phenomena related tocollective excitations of the electron gas are captured by $.The nondimensional boundary and charge neutrality condi-tions are

V!0,y,z,t" = − 1, V!1,y,z,t" = 0, n!0,y,t" = 1, !4a"

v!x,0,t" = 0, v!x,1,t" = 0, !4b"

#V

#y!x,0,0,t" = 0,

#V

#y!x,1,0,t" = 0, !4c"

%0

1 %0

1

n!x,y,t"dxdy = 1, !4d"

where we use n0=ND.The charge neutrality condition Eq. !4d" can be rewritten

by integrating Eq. !3a" in the !x! ,y!" plane to give

%0

1 ' #V

#x!1,y,0,t" −

#V

#x!0,y,0,t"(dy + %

0

1 ' #V

#y!x,1,0,t"

−#V

#y!x,0,0,t"(dx = 0. !5"

Substituting Eqs. !4c" into Eq. !5", we get

%0

1 ' #V

#x!1,y,0,t" −

#V

#x!0,y,0,t"(dy = 0. !6"

III. TWO-DIMENSIONAL ELECTRON FLOW

The steady-state solution of Eq. !3" in the semiconductorsatisfying the boundary conditions !4" is

V!x,y,0" = x − 1, n!x,y" = 1, u!x,y" =$&#

, v!x,y" = 0,

!7"

where ! " indicates time independence. The steady-state so-lution captures the drift flow and the electron velocity isindependent of position. Also, the electron density and elec-tric field are independent of position, since the potential var-ies linearly with x.26,27 In dimensional form the electron con-centration is n!!x ,y"=ND, which is the doping density ofthe semiconductor and the electron velocity u!!x ,y"= !e" /me"!V0 /L" is proportional to the electric field. Any in-

stability that may exist is due to the growth of fluctuationsfrom this steady state.

Applying small perturbations to the time-independentsolution, we have V= V!x ,y ,0"+V!!x ,y ,z , t", n= n!x ,y"+n!!x ,y ,z , t", u= u!x ,y"+u!!x ,y , t", and v= v!x ,y"+v!!x ,y , t". Substituting in Eq. !3" and linearizing, we get

#2V!

#x2 +1R2

#2V!

#y2 − #n! = 0, !8a"

#n!

#t+

$&#

#n!

#x+

#u!

#x+

1R

#v!

#y= 0, !8b"

#u!

#t+

$&#

#u!

#x−

#V!

#x+

&#

$u! = 0, !8c"

#v!

#t+

$&#

#v!

#x−

1R

#V!

#y+

&#

$v! = 0. !8d"

In a two-dimensional semiconductor slab, the perturbation inthe electron density in the z!-direction can be consideredvery small. This implies that any variation in the electricfield in the semiconductor in the z!-direction is small in mag-nitude and can be neglected if compared to the steady-stateelectric field and its perturbations in the !x! ,y!" plane. Theelectric field outside the semiconductor generated by thecharge in the semiconductor can be neglected since thestrong electric field in the x!-direction is predominant. Due tothis, Eq. !8" only includes the perturbations for the voltage,electron density, and longitudinal and transversal velocitiesin the !x! ,y!" plane.

The boundary and charge neutrality conditions for theperturbations in the semiconductor are

V!!0,y,t" = 0, V!!1,y,t" = 0, n!!0,y,t" = 0, !9a"

v!!x,0,t" = 0, v!!x,1,t" = 0, !9b"

#V!

#y!x,0,t" = 0,

#V!

#y!x,1,t" = 0, !9c"

%0

1 ' #V!

#x!1,y,t" −

#V!

#x!0,y,t"(dy = 0. !9d"

The system in Eq. !8" and the boundary conditions in Eq. !9"describe the evolution of the perturbations in voltage, elec-tron density, and longitudinal and transversal velocities fromthe steady state of the two-dimensional electron flow in thesemiconductor.

In order to find the temporal and spatial modes that char-acterize the perturbations, we use normal modes of the form

)V!n!u!v!*T = )V!y"n!y"u!y"v!y"*Texp!kxx + %t" , !10"

where the wave number vector k=kxi+kyj, the frequency is%, and the amplitudes denoted by ˜ are all complex. We willwrite kx=kr

x+ ikix, ky =kr

y + ikiy, and %=%r+ i%i. Therefore, Eq.

!8" becomes



d2V

dy2 + !kx"2R2V − #R2n = 0, !11a"

dvdy

+ R'% + kx $&#

(n + kxRu = 0, !11b"

kxV − '% + kx $&#

+&#

$(u = 0, !11c"

dV

dy− R'% + kx $

&#+

&#

$(v = 0. !11d"

If !%+kx$ /&#+&# /$"=0, then Eqs. !11a" and !11c" give V= n=0, which is not of interest, so we will assume that !%+kx$ /&#+&# /$"$0. By differentiating Eq. !11d", the sys-tem of equations can be reduced to

dV2

dy2 − !ky"2V = 0, !12"

with

!ky"2 = − !kx"2R2. !13"

By using Eq. !11d" with the boundary conditions in Eqs. !9b"and !9c", we get

dV

dy!0" = 0,

dV

dy!1" = 0. !14"

The general solution for Eq. !12" is

V = C1 exp!kyy" + C2 exp!− kyy" . !15"

By applying the boundary conditions !14", this takes theform

V = D cosh!ky,my" , !16"

with ky,m= & i2'm. Therefore, the general solution can bewritten as

V = +m=0

(

Dm cos!iky,my" . !17"

It is easy to show that the proposed solution in Eq. !10" cansatisfy the boundary conditions in Eqs. !9a" and !9d" inde-pendently of the value of ky. Rewriting

)Vnuv*T = )Vnuv*Texp!kyy" ,

where the amplitudes denoted by ˆ are also all complex, Eq.!8" becomes

,− !kx"2 − !ky"2/R2 # 0 0

0 % + kx$/&# kx ky/R− kx 0 % + kx$/&# + &#/$ 0

− ky/R 0 0 % + kx$/&# + &#/$-.V

n

u

v/ = .

0

0

0

0/ .

For a nontrivial solution, the determinant should vanish, sothat the characteristic equation is

− #!kx"2R2 + !ky"2$!%&#$ + $2kx + #"

)0'% +$&#

kx('% +$&#

kx +&#

$( + #1 = 0. !18"

The roots of Eq. !18" are

k1x = i

ky

R, k2

x = − iky

R, k3

x = a + ib, k4x = a − ib , !19"

where a=−!# /2$2"!2$% /&#+1" and b= !# /2$2"&4$2−1.Since ky,m= & i2'm, we have two cases.

A. mÅ0

Now there are oscillations in both x- and y-directions.The modes in the x-direction are described in Eq. !19" andthe modes in the y-direction are ky,m= & i2'm. We can writethe amplitudes of the perturbations as

V! = #Aeiky,mx/R + Be−iky,mx/R + Ce!a+ib"x + De!a−ib"x$ekyy+%t,

#V!

#x= 0Ai

iky,m

Reiky,mx/R − Bi

ky,m

Re−iky,mx/R

+ C!a + ib"e!a+ib"x + D!a − ib"e!a−ib"x1ekyy+%t,

n! =1#0Ce!a+ib"x'!a + ib"2 +

!ky,m"2

R2 (+ De!a−ib"x'!a − ib"2 +

!ky,m"2

R2 (1ekyy+%t.

The boundary conditions can be written as



,1 1 1 1

eiky,m/R e−iky,m/R ea+ib ea−ib

0 0 !a + ib"2 +!ky,m"2

R2 !a − ib"2 +!ky,m"2

R2

iky,m

R!1 − eiky,m/R" − i

ky,m

R!1 − e−iky,m/R" !a + ib"!1 − ea+ib" !a − ib"!1 − ea−ib"

-.A

B

C

D/ = .

0

0

0

0/ .

For a nontrivial solution the determinant must vanish, whichgives

'!a + ib"2 +!ky,m"2

R2 (0!a − ib"!1 − ea−ib"!e−iky,m/R

− eiky,m/R" + iky,m

R!ea−ib − e−iky,m/R"!1 − eiky,m/R"

+ iky,m

R!ea−ib − eiky,m/R"!1 − e−iky,m/R"1 − '!a − ib"2

+!ky,m"2

R2 (0!a + ib"!1 − ea+ib"!e−iky,m/R − eiky,m/R"

+ iky,m

R!ea+ib − e−iky,m/R"!1 − eiky,m/R" + i

ky,m

R!ea+ib

− eiky,m/R"!1 − e−iky,m/R"1 = 0. !20"

Equation !20" is satisfied by

a + ib = & iky,m

R, a − ib = & i

ky,m

R.

If b= &2'p, with p being a natural number, Eq. !20" is alsosatisfied by a=0. Due to the complexity of Eq. !20" othersolutions may be possible, in which case the spectra mayinclude additional temporal modes. b may be real, zero, orimaginary if 4$2 is greater than, equal to, or less than unity.

1. For 4!2>1

The solution for % is

%m =&#

2$0&

4$2'm

#R− 1 & i&4$2 − 11 , !21"

which indicates temporal modes that are either growing ordecaying with an oscillatory component. Also, if b= &2'p,the temporal mode is %=−&# /2$.

2. For 4!2*1

Now,

%m =&#

2$0&

4$2'm

#R− 1 & &1 − 4$21 . !22"

In addition to this, if b= & i2'p, the temporal mode is %=−&# /2$. The temporal modes % are real and therefore theevolution in time is either growing or decaying without os-cillations. Furthermore, a and b are real, which implies nospatial oscillations. From the nonlinear nature of the prob-lem, the growth must have an upper limit, though the linearanalysis does not provide this value.

B. m=0

This represents oscillations only along the x-direction. Acomplete treatment of this has been described in Ref. 24. Thetemporal modes are described by Lambert W functions #TheLambert W function is defined as the solution of the equationW!z"eW!z"=z$. There are three operating conditions that canbe determined by the value of 4$2. This also defines thenature of the spatial modes in the x-direction: no oscillations!purely real", constant amplitude oscillations !purely imagi-nary", and oscillations with spatial growth or decay !com-plex". By requiring a nontrivial solution over a wavelike so-lution for the perturbed system, the temporal modes are

TABLE I. Temporal modes.

Condition %

4$2+1 %=−!$ /&#")!# /2$2"+b cot b+W#−!be−b cot b /sin b"$*4$2=1 %=−!$ /&#"#!# /2$2"+1+W!−e−1"$

4$2,1%=−!$ /&#")!# /2$2"+b! coth b!+W#

−!b!e−b! coth b! /sinh b!"$*

−50 0 50−50

−40

−30

−20

−10

0

10

20

30

40

50

ωr/ωp

ωi/

ωp

FIG. 2. Spectrum of eigenvalues for GaAs semiconductor where effectivemass of electron is 6.6% of its actual mass, -s=113.28)10−12 C2 /m2 N,L=100 nm, n0=5)1017 cm−3, ND=5)1017 cm−3, "=0.4)10−12 s, andV0=1 V, which gives #=7.072, $=17.365, and %p=43.41)1012 s−1. Theshaded area represents the stable region and the unshaded the unstable.



obtained. The solutions of Eq. !20" are in Table I, where b!= !#&1−4$2" /2$2.

IV. DISCUSSION

A. Spectrum of eigenmodes

The spectrum presents both stable and unstable regions.It is tunable mainly through the applied voltage, but the as-

pect ratio R can also determine instability when we havespatial oscillations along the y-direction, i.e., if ky,m$0.Since ky,m is purely imaginary, we get oscillations along they-direction with a constant amplitude in space. The numberof oscillatory temporal modes over a range of temporal modeamplitudes is independent of the value of the oscillatorycomponent for ky,m=0. Otherwise, for ky,m$0 it presents two

t = 0

0

0.5

1

0

0.5

1

0n′

xy

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

y

t = (π/2)/ωi

0

0.5

1

0

0.5

1

0n′

xy

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

y

t = (π)/ωi

0

0.5

1

0

0.5

1

0n′

xy

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

y

t = (3π/2)/ωi

0

0.5

1

0

0.5

1

0n′

xy

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

y

t = (2π)/ωi

0

0.5

1

0

0.5

1

0n′

xy

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

y

FIG. 3. Electron density eigenfunction !left" and elec-tric field !right" for #=10, $=1, R=0.5, and %i20.87%p !arbitrary scale" at the first unstable eigen-mode m=1.



delta functions at &!&# /%p"&%p−1 /4"2. Taking 4$2+1, thespectrum for GaAs is shown in Fig. 2.

B. Aspect ratio dependency

We are interested in imaginary components of the tem-poral modes %, which describe the oscillatory behavior in

time. Given an aspect ratio R, there is a critical positivemode above which the system is unstable. From Eq. !21", thecritical positive mode is

mc =#R

4$2'. !23"

t = 0

0

0.5

1

0

0.5

1

0n′

xy 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

t = (π/2)/ωi

0

0.5

1

0

0.5

1

0n′

xy 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

t = (π)/ωi

0

0.5

1

0

0.5

1

0n′

xy 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

t = (3π/2)/ωi

0

0.5

1

0

0.5

1

0n′

xy 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

t = (2π)/ωi

0

0.5

1

0

0.5

1

0n′

xy 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

FIG. 4. Electron density eigenfunction !left" and elec-tric field !right" for #=10, $=1, R=1, and %i20.87%p !arbitrary scale" at the first unstable eigen-mode m=1.



Moreover, this critical mode can be written in terms of theplasma frequency as

mc =#R

4'!%p""2 . !24"

It can also be written as

mc =1

u!' H

4'"( . !25"

As an example, choosing #=10, $=1, and R=1, we getmc20.8, therefore m.1 guarantees an unstable regime. Asis shown in Fig. 3, which describes the evolution of electrondensity and electric field through the first period of oscilla-

t = 0

0

0.5

1

0

0.2

0.4

0.6

0.8

1

0n′

xy0 0.5 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

t = (π/2)/ωi

0

0.5

1

0

0.2

0.4

0.6

0.8

1

0n′

xy0 0.5 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

t = (π)/ωi

0

0.5

1

0

0.2

0.4

0.6

0.8

1

0n′

xy0 0.5 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

t = (3π/2)/ωi

0

0.5

1

0

0.2

0.4

0.6

0.8

1

0n′

xy0 0.5 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

t = (2π)/ωi

0

0.5

1

0

0.2

0.4

0.6

0.8

1

0n′

xy0 0.5 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

FIG. 5. Electron density eigenfunction !left" and elec-tric field !right" for #=10, $=1, R=2, and %i20.87%p !arbitrary scale" at the first unstable eigen-mode m=2.



tions in time for a configuration with R=0.5, the first un-stable mode is m=1. When R=1, the first unstable mode ism=1 and the evolution of electron density and electric fieldare shown in Fig. 4. Otherwise, when R=2 the first unstablemode is m=2 as is shown in Fig. 5. It illustrates how the firstunstable mode is determined by R. The magnitude of theoscillatory components of the spectrum for ky,m$0 is!&# /%p"&%p

2 −1 /4"2. It depends only on #, $, and " andtends to the plasma frequency %p in the ballistic limit "→(. This limit is reached when there is very low electron-impurity scattering.

C. Dispersion relation

By definition, the dispersion relation provides a relation-ship between the oscillatory component of the temporalmodes, %i, and the oscillatory component of the spatialmodes, ki

x and kiy. Many situations of physical interest may

have multiple and discrete roots of %i. In the proposed prob-lem, the relation can be obtained from Eq. !18".

For 4$2+1, the dispersion relation takes the form

#

2$20−2$%i

&#& &4$2 − 11 − ki

x = 0. !26"

This linear relationship between %i and kix is two parallel

lines as shown in Fig. 6. The values of %i are determined byEq. !21" and the corresponding values in Table I. From this,the phase and group velocities can be deduced to be

cp =$&#

2$%i

2$%i + &#!4$2 − 1", cg =

$&#

, !27"

respectively.For 4$2*1, the expression for the dispersion relation is

&#%i

$+ ki

x = 0. !28"

The relation is linear as in the previous case, but it is just onestraight line crossing the origin, as shown in Fig. 7. Thisexpression is valid only for ky,m=ky,0=0 since %i$0, other-wise for ky,m$0, %i=0. From this, the phase and group ve-locities can be deduced to be

cp = cg =$&#

, !29"

respectively. It can be noticed that the steady state electronvelocity in Eq. !7" and the group velocity of the instabilitywaves are equal.

V. CONCLUSIONS

The instabilities in the hydrodynamic model of a two-dimensional electron flow in ungated semiconductors areanalyzed. Analytical expressions for the spatial and temporalplasma oscillation modes are derived. The spectrum of tem-poral modes shows a predominant unstable region, whichdepends strongly in the applied voltage through the semicon-ductor. As the applied voltage decreases, the spectrum ismore stable. Also, the aspect ratio determines how unstablethe temporal modes can be. As the aspect ratio decreases, theunstable modes become more unstable. The unstable region,which means temporal modes with positive real part, hasoscillatory components able to describe terahertz frequenciesunder specific parameter values.

In summary, the required operating condition to supportsemiconductors in a two-dimensional configuration !such asin a HEMT structure" as a radiative source can be obtainedunder the right set of parameters such as applied voltage,aspect ratio and doping density, among others. The theoreti-cal formalism presented extends the earlier works on the sub-ject by revealing a spectrum of both stable and unstablemodes for plasma-mode oscillations and presents a directmethod for analyzing their dependence on the material andgeometrical parameters of the device. This will prove to bevaluable in the design process of compact electronic tera-hertz sources of the future.

ACKNOWLEDGMENTS

W.C.-M. is grateful for a scholarship from CONICYT-Chile and the Universidad de Chile.

γ√αb

− γ√αb

ωi

−b bkxi

0

0

FIG. 6. Dispersion relation for 4$2+1.

ωi

kxi

0

0

FIG. 7. Dispersion relation for 4$2*1.



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