Surface thermoplasma waves and their interaction with flow of charged particles at a semiconductor-metal interface

Solid-State ElectronicsVol. 39, No. 4, pp. 611-614, 1996

Perg.mo.

0038-1101(95)00150-6

Copyright © 1996ElsevierScienceLtd Printed in Great Britain.All rights reserved 0038-1101/96 $15.00+0.00

SURFACE THERMOPLASMA WAVES AND THEIR INTERACTION WITH FLOW OF CHARGED PARTICLES AT A SEMICONDUCTOR-METAL INTERFACE V. L. F A L K O , S. I. K H A N K I N A and V. M. Y A K O V E N K O Institute of Radiophysics and Electronics of National Academy of Sciences, 12 Akad. Proscura Str., 310085 Kharkov, Ukraine (Received 8 October 1994; in revised form 14 June 1995)

Abstract--lt is shown that slow surface thermoplasma waves exist at an interface of a semiconductor with an ideal metal if thermal motion of conduction electrons causes the spatial dispersion of the dielectric permeability in the semiconductor. The spectrum and the damping of these waves are obtained. The coUisionless damping is due to the interaction of the surface thermoplasma waves with the conduction electrons which move along a normal to the boundary without collisions through the region of the surface wave localization. The Landau damping is exponentially small in comparison with the damping. Amplification of the surface thermoplasma wave can be realized by using a flow of charged particles injected into the semiconductor from the metal.

I. INTRODUCTION It is known that surface waves can propagate along a boundary of a conducting medium (piezosemiconductor, plasma, etc.) with an ideal metal if a spatial dispersion of the dielectric permeability E exists in the conducting medium. This results from the fact that a new branch of eigen oscillations appears besides the ordinary electromagnetic waves with the dispersion law q2 = O)2E/C 2 (q is the wave vector, 09 is the wave frequency) which are due to piezoeffect (electro-acoustic waves) or to thermal motion of charged particles (concentration waves or spatial charge waves). The surface waves come into existence in the conducting medium as a result of the superposition of these waves at the boundary with the metal. The present paper is devoted to an investigation of the slow surface thermoplasma waves which exist at the boundary of a semiconductor with a metal when the spatial dispersion of the dielectric permeability E is connected with the thermal motion of conduction electrons in the semiconductor. We have studied a possibility of this wave amplification by a flow of charged particles which cross the interface.

2. THE SPECTRUM OF THE SURFACE THERMOPLASMA WAVES T o determine the spectrum of the surface thermo-

plasmas in the semiconductor one needs to solve the

system of the Maxwell equations: 1 3H

rot E = c--~- ; ~0 dE 4n rot H = c - ~ - + --c J '

(1)

accounting the boundary conditions at the interface of the semiconductor with the ideal metal (the plane y = 0) and the conditions of irradiations for electromagnetic waves far from the boundary as y ~ oo. The space y > 0 is occupied by the semiconductor, the metal is in the region y < 0. The boundary conditions on the plane y = 0 are the conditions of equality to zero of the tangential components of the electric field Et ly= 0 = 0 because of the large electrical conduction of the metal. In addition, it is suggested that the boundary is an infinitely high potential barrier for the conduction electrons in the semiconductor; a particle reflects from this and moves in an opposite direction. Consequently, the following boundary condition for the normal component of the current density Jy is considered to hold: J , I~ = o = o.

(2)

The current density J(r, t) is determined by using the density matrix p(r, t). In a linear approximation: O) .(0) J -= S p ( ~ ) = ~ {P.,..I.,,, ~o).~,~ + P.m.l,..}"

(3)

ngn

The indexes "0" and "1" concern the equilibrium or perturbation conditions, correspondingly; ,.,~t°) =,m pt0) S . I,,to) is the equilibrium distribution function of n ~nm~ ~n 611

612

V. L• Falko et al.

the charged particles, a form of which depends on the state of the semiconductor electron gas. Here, we suppose the electron gas is nondegenerate and its distribution function is given by the Maxwell function:

in the quasi-static approximation, the wave of two types propagates in unbounded semiconductors: the electrostatic wave for which q 2 = 0 and the wave of the spatial charge with the dispersion law:

• (2nh) 3 t/ p~, "~ P(,,°) = Iv0 (2~m ~__.~_~,3'2 T) ~ exp~ - ~m--T).

q: = 09(09 + iv) - 090 v2

2

(4)

In eqn (3), the indexes " n " and " m " determine the states of the electron with the momentum p. or p,., The matrix elements of the current operators ](o) and ]") are equal to: i(o) = ieh {~lO)(r)V~).(r ) - ~ , , ,(o).(r)V~..(0,(r)}, (5) Jmn 2m

co~ = 4ne2No/mEo is the electron plasma frequency; v

is the effective collision frequency. The inclusion of the boundary conditions at the interface between the semiconductor and the metal leads to the existence of surface thermoplasma waves as the result of the superposition of the abovementioned partial waves in the frequency range:

e2

j~.~ = - - -

mc

A(r, t)~,~)(r)~O~')*(r).

cop > 09 > qvx > v.

(6)

Here, A is the vector potential: E = ~ A / c O t ; ~,~)(r) is the wave function of a stationary state of an unperturbed system of the conduction electrons (not including the time factor); e, m, No and T are the charge, the effective mass, the equilibrium concentration and the temperature (in the energetic units) of the conduction electrons, respectively, and e0 is the lattice dielectric constant. The perturbation density matrix is determined by the equation:

(13)

(14)

The dispersion law of these surface thermoplasmons has the form: ~o = (cooqvv)l/2 _ iv/2.

(15)

(The axis x coincides with the direction of the surface wave propagation: ql[x.) The electric field in eqn (I 5) has the following components: E~(~o, q) = E 1{e Iqly- -

ei(q. . . . . );

e-lqlly}

Ey(og, q) = iE~ {Iq[ e-lql.'-lql[ e :q,L,.}e,(q...... ), (16) q where

ih op(I) =/~(0,fi,) _/Sll)B~o) + BIIlfi(0~ _ fi(o)Bo) ' (7)

q~ =

Ot

where ieh

/~¢0)=~m;

/~¢1)= 2mc { V A ( r , t ) + A ( r , t ) V } . (8)

~(1)

~ t

(1)

,

H,,,(t ) e

konm t'

t.

dt ,

.... ) HO)(t = f ~k(,°)*(r)/~(l)(t, r)~k~)(r) dr;

~o..,=(p2.-p2..)/2mh.

(9) (10) (11)

In the unbounded isotropic semiconductor, all the functions of the set (1) can be represented as Fourier transforms on the coordinates and time. In this situation, the current density J, calculated from eqns (3)-(11), agrees with the known one, found previously by using the kinetic equations[l]. In the case of the weak spatial dispersion, the components of the current density have the form: J, = iN°e2 mfD

(17)

S,: is the area of the specimen surface on the plane y=0.

n

×

2 - - q2,

w0 = X, zE0~ IE, 12 4nlqlco2

1 e i°'"~t/~(0)--p(m0)) ip

2

and [q~ [ > [q[. The energy of the wave in eqns (15) and (16) is equal to:

The solution of eqn (7) has the form: P nm = h

2

( C O p __ CO ) / U T

I + O~i

Ei(o) >>qVT).

(12)

Here the index " i " identifies the longitudinal ("long") and transverse ("tr") components of the current J and the field E; V2T= 3 T / m ; the constants ~ are equal: ~zo.8 = 1; ~,, = 1/3. It follows from system (1) that

3. COLLISIONLESS DAMPING OF SURFACE THERMOPLASMONS

The localization depth of both the components of the field of the surface thermoplasma wave of eqn (16), l/Iql and l/Iqj I, and its wave length 2 are less than the length of the electron free path l that arises from the inequalities (14). In other words, the ballistic (without collisions) motion of the conduction electrons through the region of the localization of the surface wave is realized. That is why, along with the collision damping of the wave (15))'¢ol = - v / 2 , it is necessary to obtain the collisionless damping ),, which is due to interaction of the conduction electrons with the surface thermoplasma wave• One can determine the damping )' starting from the expression for the change of the wave energy and assuming that the wave amplitude varies slowly with time ([(I/E) c~E/~3t[<
Re[wL0 0tw ).' 0w ~ ----- f ( J E )

dr.

(18)

Surface thermoplasma waves The integration in eqn (18) is carried out over the whole region of the surface wave localization, the angular brackets stand for averaging over the oscillation period. The current density J is a sum of currents induced by the electrons, which move boundarywards (J+), and the reflected ones (J_). These currents J± are described by the expressions (3)-(11), which contain the wave functions of the falling particles ~) (r) and the reflected ones ~k{,°2.Let us suppose that in the semiconductor the conduction electrons, which fall and reflect from the boundary y = 0, are not correlated to one other. Then: .t, t0) (r)

"t---

Wn-+

1

(19)

e iknx x + iknz z ± kn "y

--%//V

"

.

(kn; = 2rm/L;; n = 1, 2 ...; V is the sample volume; L; is its size along the /-axis). The choice of the wave functions in eqn (19) ensures the fulfillment of the boundary condition (2). The first term of expression (3) for the current density J determines the current due to collective behavior of the electrons in the electric field of the wave• The dissipative current, connected with inelastic scattering of the electrons on the surface thermoplasmons, is described by the second term in eqn (8). The collisionless damping ~ is due to this current. The matrix elements p ~ contain the function l/(gOnm- O9) as a multiplier, which is the result of integrating over time in eqn (9). Since the damping is much smaller than the frequency (Im o9 <
1

1 in6(x) + B - -

-

A~0 X

~

=

(20)

X "

The symbol P means that one must understand the peculiarity at x = 0 as a main value of an integration. In our case, this describes the imaginary part of the energy change ~ W / ~ t of eqn (18) and determines the small change of the surface wave frequency, which is not under discussion in this paper. We have the following expression of the collisionless damping y after the simple calculations: 7 "~" "~rio9 'W ' 0- ' , p

~

[p~O)

-pm]ln,q,~(go)] (0) (1) 2 ~(o9.,~--o9); (21)

• m~

P~: = P,:;

P,,x = P ~ - hq.

H(0 nqm is the Fourier transform of the matrix element (10): H(,) 2

h2elE,{ 2

2

,qm = 16--~m2~lg.qm[ •

(22)

The quadrate of the dimensionless matrix element IB.qml 2 is:

IW.~ 12 = q~[(P,.~ + p~x)hq +

2

_2 ~,2

(Pny -- V m y ].l

q 2[(pray_ p~y)2 + h 2q 2] [(p,.y _ p~y)2 + h 2q 2,]"

(23)

Equation (21) describes the picture of the physical processes of the energy exchange between the charged

613

particles and the surface thermoplasma wave: the first term corresponds to the transition of the electrons from the n-state into the m-state with radiation of the quantum of the field. The second term is due to the transition from m-state into m-state with its absorption. As the electrons interact with the surface wave, the conservation laws for tangential components of momentum of the system are fulfilled: Pnx = P,,x + hq; Pn: =P,,z. Because of the medium inhomogeneity along the y-axis, the conservation law for the normal component of momentum of the system is not fulfilled. It is convenient to find the damping coefficient T of eqn (18) in the following way: in the sum over p,,y we pick out the terms with P,v ~- Pn>. and transform the sum over p, into the integrals ( Z p . . . V(2~h) -3 S d p . . . ) . As a result, ~ is a sum of two terms 7 = 7, + ~2. The term ~ is determined by the contribution to the absorption from processes of interaction between the wave and the electrons, which move synphase-wise and interact with the wave efficiently: p,x = mo9/q. The normal components of momentum of these electrons change negligibly: Lpny-p,,y[<
gO 2.~-~

" v

() gO

3

exp

- - 2---'~T

.

(24)

The damping 72 is formed by the conduction electrons, for which the change of the normal component of momentum must satisfy the inequality: IP~>p,t~.l>>hq, hqp~x/pn~, and the value of p~.,. is near zero. This damping is due to the current J, and is: T2 "" - 0.42qvr.

(25)

Thus, the Landau damping (24) is exponentially small in comparison with the damping V2 (25). 4. A M P L I F I C A T I O N

OF SURFACE THERMOPLASMONS

Let us consider the interaction of the surface thermoplasmons with the flow of the charged particles, which are injected into the semiconductor from the metal (for example, by d.c. electric field). This flow moves with constant velocity v0 along the axis y. It is suggested that the particles fly the whole region of the surface wave localization without collisions (the ballistic fly). We shall only investigate the individual interaction of the injected particles with the surface thermoplasmons. Such interactions with waves of different nature have been studied in Refs [2-4]. As the charges fall at the interface, the field produced by these partially transforms into surface oscillations and the electric energy of these oscillations changes the field of the charged particles. The change of the surface thermoplasmon energy associated with its interaction with particles is described by eqns (3)-(11) and (18)-(21), where P,,.mand m are the

614

V.L. Falko et al.

tunnel through it. (The condition of the specular reflection is always fulfilled for the principal electrons in the semiconductor: the conduction electrons.) The distribution of the normal components of the momentum of the injected electrons is localized near t9(n o)= n0f(2xh)33 (P,x)J (Pny- m0v0)3(p,~), the central value P0 = movo and has a f-function n0f is its equilibrium concentration, m0 is the free character. As the injected electrons enter the semielectron mass. conductor, they have a kinetic energy move~2 which In the classical approximation, when m0v0~>>hco; is much more than the conduction electron energy. The injected electrons move through the whole of the movo>>hq, 7 is a positive value given by: space of their interactions with the wave field without 2 2 4 F = cof°~pm°V° collisions (ballistic transport). Their drift velocity v0 2 2 2 2~:0hVT ((.O2 + q2v02)(fO2 + (-OpU0/VT) is determined by the applied voltage eEod ~ m0v02/2 and has to reach ~ 108 cm s -l. The small penetration x {(m~v~ --2mohog)-1:2 depth of the surface wave allows the use of semi-(m2ov~+ 2mohco)-I/2} conductor materials where the ballistic charge transV0 (O 2 (.OpO) 2 port is only realized within the narrow layer near the 2 2 2 ' (26) surface at a distance of the wave penetration depth at EoVT(CO~ + q2V2o)(Co2+ %Vo/VO the asot or room temperatures (v <
F>72>o/2,

or

2 2 VT (.O~ > fD" E0 V0

> q v r > o . (27)

COp COp

These inequalities could be fulfilled for co = 10~2s in the semiconductors with the following parameters: E "~ 16, N0= 10t6cm-3, m = 10-29 g, VT~ 108cm s -j, v = 10II s -1 if ~of~>1013s-~. 5. C O M M E N T S

The proposed mechanism of surface thermoplasmon instability can be used, in principle, for making solid state amplifiers or generators of electromagnetic waves in the submillimetcr band, which find wide application in different fields of science and engineering. For making such devices, one should use a structure consisting of plane-parallel metallic and semiconductor plates, separated by an insulator of thickness d, which is small in comparison with the penetration depth of the surface wave in the semiconductor (MIS-structure). If a voltage U = eEod is applied between the metal and the semiconductor (E 0 is the d.c. field), the charge carrier drift from the metal to the semiconductor can be realized. As this takes place, the voltage falls off on the clearance. The injected electrons pass above the potential barrier or

6. C O N C L U S I O N

Thus, at the interface between the semiconductor and the ideal metal, the superposition of the electrostatic wave and the wave of the spatial charge (existing in the unbounded semiconductors) gives rise to the slow surface thermoplasma wave with the dispersion law (15). The interaction of this surface wave with the conduction electrons causes its collisionless damping (25) which is due to the ballistic character of the electron motion along the normal to the boundary in the region of the surface wave localization. The amplification of the thermoplasma wave (26) appears when it interacts with the flow of the charged particles injected into the semiconductor from the metal. REFERENCES

1. A. I. Akhiezer (Ed.), Electrodynamics of Plasma, p. 184. Moscow, Nauka (1974) (in Russian). 2. S. I. Khankina, V. M. Yakovenko and Izv. Vuz'ov, Radiofizika 34, 436 (1991). 3. V. M. Yakovenko and I. V. Yakovenko, UFZh. 29, 1930 (1984). 4. M. V. Byrtyka, O. V. Glukhov and V. M. Yakovenko, Solid-St. Electron. 33, 1339 (1990); 34, 559 (1991).