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Physica E 34 (2006) 417–420

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Spectra of standing and traveling plasma waves in two-dimensionalelectron channels

Akira Satoua,, Alexander Chaplikb, Victor Ryzhiia, Michael S. Shurc

aCSSP Laboratory, University of Aizu, Aizu–Wakamatsu 965–8580, JapanbInstitute of Semiconductor Physics, RAS, Novosibirsk 630090, Russia

cDepartment of ECSE, Rensselaer Polytechnic Institute, Troy, MI 12180, USA

Available online 24 April 2006

Abstract

We calculate spectra of standing and traveling plasma waves in two-dimensional electron channels with strip-like side contacts by

solving the linearized hydrodynamic equations coupled with the Poisson equation. We show that the obtained spectra of plasma waves

can be substantially different from those calculated using an approach invoking ‘‘quantization’’ rules.

r 2006 Elsevier B.V. All rights reserved.

PACS: 73.20.Mf

Keywords: Plasma wave; Two-dimensional electron channel; Slot diode

1. Introduction

Plasma oscillations in devices similar to field-effecttransistors with two-dimensional (2D) electron channelscan be used for generation and detection of terahertz (THz)radiation [1–3], as recent experiments have demonstrated[4,5]. With sufficiently high electron mobility, the channelsof such devices can serve as resonant cavities for plasmawaves. The plasma wave spectra in different 2D electronsystems were considered in a number of papers (forexample, Refs. [6–9]). In our previous works, we calculatedspectra of standing plasma waves in a slot diode with semi-infinite side contacts [10,11], using a hydrodynamic electrontransport model for a 2D electron channel coupled with thePoisson equation. In this paper, we generalize the problemby considering (a) a slot diode with the side contacts offinite width and (b) traveling plasma waves in a slot diode.

2. Standing plasma waves

Schematic view of the structure under consideration isshown in Fig. 1.

e front matter r 2006 Elsevier B.V. All rights reserved.

yse.2006.03.101

ing author. Tel.: +81242 37 2563.

ess: d8071201@u-aizu.ac.jp (A. Satou).

The linearized version of the hydrodynamic equationscoupled with the Poisson equation give the followingequation:

q2jo

qx2þ

q2jo

qz2¼

4piæ o

qqx

soqjo

qx

dðzÞ, (1)

where jo is the AC component of the potential, æ is thedielectric constant, o is the frequency of plasma waves,so ¼ is0;chnch=ðoþ inchÞ for jxjplch, so ¼ is0;scnsc=ðoþinscÞ for lchpjxjplch þ lsc, and so ¼ 0 for jxjXlch þ lscare the AC conductivities, nch and nsc are the electroncollision frequencies, s0;ch ¼ S0;che2=mnch and s0;sc ¼S0;sce

2=mnsc are the DC conductivities, S0;ch and S0;sc

are the DC electron sheet concentrations, and e and m arethe electron charge and effective mass. The fact thatthe dielectric region in z ¼ 0 plane can be treated in theproblem by setting DC (and hence AC) conductivity zerohas been confirmed in Ref. [11]. Since we focus on theplasma oscillations with the electric field concentratedmainly in the channel, it is natural to assume that theelectric potential vanishes far from the channel center. Sothat the boundary conditions for the AC potential arechosen to be the following: jojjxj¼1 ¼ 0. Then, Eq. (1) is

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Fig. 1. Schematic view of a slot diode.

10-2 10-1 1 10 102

σ0,sc/σ0,ch

-0.45

-0.40

-0.35

-0.30

Im(ω

/2π)

, TH

z

0.75

0.80

0.85

0.90

Re(

ω/2

π), T

Hz

lsc/lch = 10

=1

= 10-1

= 10-2

10-3 10-1 10

lsc/lch

0.7

0.8

0.9

Fig. 2. Real and imaginary (damping) parts of plasma frequency vs the

ratio of DC conductivities s0;sc=s0;ch calculated with different lsc=lch. The

inset is the dependence of Reðo=2pÞ on lsc=lch with s0;sc=s0;ch ¼ 2000.

A. Satou et al. / Physica E 34 (2006) 417–420418

reduced to

Z 11

eikx dk

jkj

Z 11

Eoðx0Þ eikx0 dx0

¼ 4p2iæ o

soðxÞEoðxÞ, ð2Þ

where Eo is the electric field at z ¼ 0 plane. We replace jkj

by

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2þ p2

qin Eq. (2), where p is a parameter which we

tend to zero, and obtain the following integral equation:Z 11

dx0Eoðlchx0ÞK0ðplchjx x0jÞ

¼ 2p2i

æ olchsoðlchxÞEoðlchxÞ. ð3Þ

The replacement of jkj by

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2þ p2

qcan be validated by

checking the convergence of the solution of Eq. (3), i.e. thevalue of o, when we take the limit p! 0. Replacing theintegral in Eq. (3) with the finite Riemann sum with N

terms, we can approximately reduce it to the followingeigenvalue problem:

XN

m¼1

AnmEm ¼ RnðoÞEn, (4)

where n ¼ 1; 2; . . . ;N, Em ¼ EoðxmÞ, Anm ¼ K0ðpljxn

xmjÞDm, Dm ¼ jxm xm1j, RnðoÞ ¼ ð2p2i=æ oÞsoðxnÞ,and xn is some value of x and x0 which we use to discretizeEq. (3). The singularity in Ann can be avoided by replacingK0ð0Þ by K0ðÞ, where is an appropriately small number.In matrix representation, Eq. (4) can be written as

ðo2Aþ ioBA CÞE ¼ 0, where B and C are somematrices, which is, in turn, reduced to the usual eigenvalueproblem:

0 I

A1C iA1BC

E

E0

¼ o

E

E0

. (5)

The plasma frequencies o can then be found as theeigenvalues whose corresponding electric field distributionsare concentrated mainly in the channel.

Fig. 2 shows the real and imaginary parts of plasmafrequency vs the ratio of DC conductivities s0;sc=s0;ch withdifferent ratios of channel to side contact width lsc=lch. We

assume nch ¼ 4 1012 s1; nsc ¼ 5nch, S0;ch ¼ 1012 cm2,

mch ¼ 6 1029 g, æ ¼ 12; lch ¼ 1:6 mm. For these values,the characteristic frequency of plasma waves in the channel

Och=2p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis0;chnch=2æ lch

pis approximately equal to

1THz. It can be seen from Fig. 2 that the plasma frequencyand the damping are almost constant when s0;sc=s0;ch51

and s0;sc=s0;chb1. However, they drastically changearound some value of s0;sc=s0;ch which depends on lsc=lch.

This value is almost constant when lsc=lch41 and decreaseswith decrease of lsc=lch. For example, for lsc=lch ¼ 1 and 10

this value is 1, and for lsc=lch ¼ 101 it is 101.The dependence of the plasma frequency on lsc=lch is

shown in the inset of Fig. 2. It looks similar to thedependence on s0;sc=s0;ch with lsc=lchb1, and indeed theplasma frequencies for limiting cases lsc=lch ! 0 and!1coincide with s0;sc=s0;ch! 0 and !1, respectively, asexpected from the equivalence of the problem for theselimiting cases. It is evident from the inset of Fig. 2 that,even if the side contacts can be considered to be highlyconducting, the plasma frequency strongly depends onthe width of the side contacts; in realistic cases where themagnitude of lsc=lch is likely to be between 101 and 1,the plasma frequencies vary between 0:76 and 0.85THz.

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0 5 10 15

qylch

0.0

0.5

1.0

1.5

2.0

ω/Ω

ch

Fig. 3. The dispersion relation of traveling plasma waves in a slot diode

for first/second symmetric/antisymmetric modes (solid lines). The dotted

line the dispersion relation of infinite 2D plasmons.

−1 −0.5 0 0.5 1

00.050.10.15

0.2

0

0.2

0.4

x/lchz/lch

00.2

0.4

0.6

Pote

ntia

l, a.

u.

0

0.5

1

1.5

qylch = 30

qylch = 10

qylch = 3

Fig. 4. The potential distribution of first symmetric mode with different

qylch.

A. Satou et al. / Physica E 34 (2006) 417–420 419

3. Traveling plasma waves

We now consider the traveling waves with wave numberqy along y-axis, which can be expressed in terms of thepotential as joðx; y; zÞ ¼ joðx; zÞ expðiqyyÞ. Then Eq. (1) isgeneralized as follows:

q2jo

qx2þ

q2jo

qz2 q2

yjo

¼4pi

æ oqqx

soqjo

qx

soq2

yjo

. ð6Þ

To treat the problem analytically, we assume here thatlscblch; q1y and s0;scbs0;ch. These assumptions lead to thefollowing boundary conditions: jojjxjXlch ¼ 0. Using theGreen function method, Eq. (6) is reduced to the following:

joðx; zÞ ¼ 2lch

plo

Z lch

lch

dx0Gðx; z; x0; 0Þ

q2

qx02 q2

y

joðx

0; 0Þ, ð7Þ

where lo ¼ oðoþ inchÞ=O2ch and G is the Green function of

the system under consideration. For jxjplch and z ¼ 0, Eq.(7) gives the following integral equation:

joðxÞ ¼ 2lch

plo

Z lch

lch

dx0Gðx;x0Þq2

qx02 q2

y

joðx

0Þ, (8)

where Gðx;x0Þ ¼ Gðx; 0; x0; 0Þ and joðxÞ ¼ joðx; 0Þ. TheGreen function of the system under consideration is givenby

Gðx; z; x0; z0Þ ¼1

2p

X1

n¼1

K0 qylch

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðy y0 2npÞ2 þ ðc c0Þ2

q "

X1

n¼1

K0 qylch

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðyþ y0 2npÞ2 þ ðc c0Þ2

q #, ð9Þ

where yþ ic¼ cos1½ðxþ izÞ=lch and y0 þ ic0 ¼ cos1½ðx0þiz0Þ=lch. Eq. (8) is reduced to the eigenvalue problem byexpanding jo over the following series:

joðxÞ ¼X1k¼1

ck cos½ð2k 1Þpx=2lch (10)

for symmetric modes of the potential distribution and

joðxÞ ¼X1k¼1

ck sinðkpx=lchÞ (11)

for antisymmetric modes. Calculating the eigenvalueslo ¼ loðqyÞ, we obtain the dispersion relation of plasmawaves o ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiloðqyÞ

pOch. We have neglected nch here;

retaining it naturally leads to some modification of theplasma wave dispersion relation [10] and results in somedamping. Fig. 3 shows the dispersion relation for somemodes with solid lines. The dotted line is the dispersion

relation of infinite 2D plasmons which is given by

op ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqylch=p

qOch. It can be seen from Fig. 3 that at

sufficiently large qylch the plasma frequencies of all modes

are lower that those for infinite 2D plasmons. Such modesare identified as strip modes [9]. The potential distributions,and hence the electron concentrations, of such modes arelocalized near both edges of the 2D electron channel, asseen from Fig. 4.

4. Conclusions

Spectra of standing and traveling plasma waves in 2Delectron channels with strip-like side contacts were

ARTICLE IN PRESSA. Satou et al. / Physica E 34 (2006) 417–420420

calculated using the linearized hydrodynamic equationscoupled with the Poisson equation. The dependence of theplasma frequency and damping of standing waves oncontact width was investigated. We calculated the disper-sion relation and the potential distribution of travelingwaves and found that modes localized to the edges of the2D electron channel can exist.

Acknowledgment

The work at RPI has been partially supported by theNational Science Foundation (Project Monitor Dr. JamesMink).

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