Current driven plasma instability in quantum wires

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Solid State Communications, Vol. 76, No. 6, pp. 835-837, 1990. Printed in Great Britain.

C U R R E N T DRIVEN PLASMA INSTABILITY IN Q U A N T U M WIRES P. Bakshi, J. Cen and K. Kempa Department of Physics, Boston College, Chestnut Hill, MA 02167, USA

(Received 19 June 1990 by J. Tauc) We investigate current driven plasma instabilities in quantum wires. We find that such instabilities might be observed in such systems for experimentally achievable drift velocities. We also find, that the threshold drift velocity for this instability is proportional to the Fermi velocity, as was the case in higher dimensional systems. Lowering the Fermi energy of the plasma is thus advantageous for generating these instabilities and possible device applications. C U R R E N T driven plasma instabilities occur when a sufficiently fast current passes through a plasma. Plasma waves can then grow at the expense of the energy of the current. This phenomenon has a potential for device applications, since with proper couplers these plasma waves can produce an electromagnetic radiation. While the current driven plasma instabilities are well known in gaseous plasmas, they have not received similar attention in solid state systems. The main reason for lack of experimental realization of this effect in solid state systems, so far, is the need for extremely high critical threshold drifts to drive such plasma modes in these systems. In our previous work we have developed a systematic approach for the study of such instabilities in solid state systems. We established the feasibility of amplification of acoustic plasma modes in type I [1, 3] and type II [2, 3] superlattices, and in superconductor systems [4]. Recently we have also studied semiconductor heterojunctions [5], and found that allowing multi-subband occupancy significantly reduces the threshold drift required to achieve instability. In this communication we study the possibility of current driven plasma instabilities in quasi one-dimensional (1D) systems: quantum-wires, and quantum-wire superlattices. Such systems can currently be fabricated [6] and several theoretical attempts have already been made to study the electromagnetic properties of such lowdimensional systems [7, 8]. We adopt here the geometry which was used in [8]: the superlattice consists of a periodic, in the y-direction, arrangement (with period d) of parallel, infinitely long wires. Each individual wire is an effectively 2D-strip of electron plasma of Width L = ~ i.e. it contains electrons confined in the y-direction, to the lowest subband, by a parabolic potential ~.-,* . . . ,2,2 0 y , (m* is the effective mass

of an electron), and the finite thickness, z0, (allowing only for a single-subband occupancy) of the electron gas is defined through a model wave function G0(z) = ( l / x / ~ z e -z/2z°. There is no confinement in the x-direction. We investigate the electromagnetic response of this system in the presence of a constant current driven by an external electric field, parallel to the wires (x-direction). As in our previous treatments [1-3, 5], we obtain the distribution function in the presence of an external electric field by solving the constant collision model Boltzmann equation, leading to f(p)

=

f

d x e -x

{ ~lfo~ ( m P* - o ~ iax ---fi--, r )

T,

0

+ ~hf~q P - - (1 + a,x)----ff--, /'2

,

(1)

Then using the random phase approximation formula we evaluate the susceptibility, which for 1D case is given by

f(P + qx) " f('P) 2m* [(p + qx)2 - p2] _ ho9

1-100(qx, o9) = 2 f (~-~n) h 2

(2) Above, vdr is the drift velocity, re_~ and Ve_phare the electron-electron and the electron-phonon scattering frequencies [9] respectively, T~ is the lattice temperature and T2 is the electron temperature. We assume here that only a single subband of the wire is occupied. The equilibrium distribution function is given by 1

f~q(p, T ) =

835

F(h2p2 exp L\2--~ - - #(T)

)/ T] k

(3) + 1

836

C U R R E N T DRIVEN PLASMA INSTABILITY

where /~(T) is the chemical potential, and k is the Boltzmann constant. The procedures for obtaining T: and/~(T) are given in [2]. In equation (2), p and qx are the electron and plasmon momenta respectively. As shown in [8], the dispersion relation o f plasma waves in this system, is given by V0(q~, q,)Hoo(q~, to) =

1

(4)

where

+ nG)A*(qy +nG)

Vo(q~, qy) = ~ B ( q , ) A o o ( q y

(5) 27t

G = ---~, q, = x / ~ + (qy + n G ) 2,

B(q) = f dz' dze-ql'-~'Jl~o(Z')121~o(Z)l 2, Aoo(qj, ) = f dye-iq: dp*( y)dpo ( y). qy is the y-component of the photon wave vector, n is an integer, and ~b0(z) is the ground state harmonic oscillator wave function. The mode structure of the system is given by equation (4), and modes with Im (to) > 0 are unstable, i.e. their amplitudes grow in time. Here, to obtain the instability condition, we use a simpler method, and look for real frequency (Im (to) = 0) solutions of equation (4), which describes the situation for which growth just compensates 1 . 2 0

~

'

,

,

i

. . . .

1.00

i

. . . .

i

. . . .

i

. . . .

i

. . . .

i

. . . .

i

. . . .

b .~.//~ c

~0.60

e~

Vol. 76, No. 6

absorption, i.e. the threshold condition [2]. The threshold drift velocity depends on q. Threshold boundary curves are shown in Fig. 1 for a variety of physical situations. The electron plasma is unstable in the domain to the right of a given curve. The corresponding dispersion curves to = to(q) are given in Fig. 2. We have considered GaAs quantum wires with m* = 0.0665me and e = 13.1. The line (a) is for a single wire of z0 = 160/~, with L = 100 A and the line density n~ = 1.0 x 105 cm -~ . Other lines are for the superlattices with different parameters: (b) for nl = 1.0 x 10Scm -1, d = 400A, L = 100A, z0 = 160 A; (c) same as (b) but with d = 500 A; (d) is same as (c) but with nl = 3.2 x 105 cm-~; (e) is for nt = 1.6 x 1 0 5 c m - l , d = 500A, L = 160A,,z 0 = 160/~. The curves (a) to (c), for the single wire and the two superlattices with the same density, have the same drift threshold in the long wavelength limit (qx = 0). This is similar to our earlier results in 2D systems, that drift threshold for q = 0 is essentially determined by the single layer physics. A superlattice arrangement basically enhances the range o f unstable q-modes, here as well as in 2D systems. Case (c), with the larger separation between wires, is closer to the single wire than case (b); this is easily understood, since a single wire may be viewed as the d ~ oo limit of a superlattice arrangement. Increasing density, ease (d), leads to a corresponding increase o f the drift threshold. This can be understood as follows. The ratio o~" ]OFis relatively insensitive to the variation of parameters. The Fermi velocity, on the other hand, is proportional to n~, therefore o~in is approximately proportional to nt. The dispersion curves in Fig. 2 show, that this is an acoustic mode, and the phase velocity essentially scales as n~ or OF. mi, is proportional to OF, it can therefore be Since Oar reduced (for given aerial density n,) by reducing L.

0.40 a

0.40

d

'

'

'

f-,

,

,

i

,

,

,

i

,

,

,

i

,

,

,

I

,

,

,

0.20 0.00

0.30 0.0

0.5

1.0

1.5 2.0 2.5 ~k (107 cm/sec)

3.0

3.5

Fig. 1. Threshold boundary curves for GaAs (m* = 0 0665me, 8 = 13 1) quantum wires, for q. = 0, and R = V~]V,._ph = 10. Electron plasma is unstable m the domain to the right of a given curve. The line (a) is for a single wire o f z0 = 160A, with L = 100A and the line density nt = 1.0 × 105 cm -l . Other lines are for the superlattices with different parameters: (b) for nt=ol.O x 105cm -I, d = 400A, L = 100A, z0 = 160 A; (c) same as (b) but with d = 500/~; (d) is same as (c) but with n~ = 3.2 x 105cm-~; ~e) is for nj 1.6 x 1 0 S c m - ~ , d = 500A, L = 160A, z0 = 160A. •

.

•

Y

d,

b

4.0 0.20

.

O.lO 0.00 0.0

'

0.2

0.4

~

0.6 0.8 q (10Scm-I)

'

1.0

1.2

Fig. 2. Dispersion relations of the current induced plasma mode. Parameters and notation as in Fig. 1.

Vol. 76, No. 6

CURRENT DRIVEN PLASMA INSTABILITY

However, as shown in [10], this will also reduce the experimentally achievable maximum of the drift velocity o~xp, for L < 160 A, due to electron-boundary collisions. Thus further reduction of L will not necessarily lead to further significant reduction of the ratio rain / e x p . odr /odr • the case (e) represents the situation with the same surface density as in case (c), but with L = 160 A. While o~n is lower for the case (c), where L = 100A, udr .,exp.~ ;o also similarly reduced [10]. Further reduction of o~n would also be possible by reducing ns. However, values of ns less than 10H cm -z are currently difficult to achieve. All the cases described so far were with qy = O, where electrons in all wires oscillate in phase in the min y-direction. Other modes are also possible, but the od~ increases with qy, reaching its maximum for qy = r~]d when electrons in alternate wires are out of phase. In conclusion, we have investigated the current driven plasma instabilities in quantum wires. As in all cases of higher dimensionality [1-3, 5] studied so far, the amplifiable plasma mode in quantum wires is an acoustic one, tied to the edge of the single particle continuum, and therefore unobservable until a threshold drift velocity of carriers is reached. We find, that this critical velocity is experimentally achievable, and therefore this mode, and its instability, might be experimentally observed. We also find that the basic properties of this instability, such as scaling of threshold drift velocity with the Fermi velocity, effect of the superlattice arrangement, etc., are identical in the cases studied before. We conclude therefore, that there is a negligible effect of the dimensionality itself on the current driven instabilities in semiconductor systems. These instabilities would be easiest to observe in high mobility systems, v~ith low Fermi velocity. In fact, not wires, but the wide parabolic

837

quantum wells in which the quasi-3D, high mobility electron gas exists, seem to be most advantageous [5]. The 3D electron density is easily tunable in these systems, and therefore the Fermi velocity can be significantly lowered in a controllable manner. Acknowledgements - This work was supported in part by the U.S. Army Research Office. REFERENCES 1. 2. 3. 4. 5. 6.

7. 8. 9.

10.

J. Cen, K. Kempa & P. Bakshi, Phys. Rev. B38, 10051 (1988). P. Bakshi, J. Cen & K. Kempa, J. Appl. Phys. 64, 2243 (1988). K. Kempa, P. Bakshi & J. Cen, Proceedings of SPIE, 945, 62 (1988). K. Kempa, J. Cen & P. Bakshi, Phys. Rev. B39, 2852 (1989). J. Cen, K. Kempa & P. Bakshi, submitted for publication. see for example: T. Demel, D. Heitmann, P. Grambow & K. Ploog, Phys. Rev. B38, 12732 (1988), and references therein. We consider in the following discussion a somewhat more idealized system than can be fabricated at present time; transport properties of presently available samples are known to be affected by the technological nonuniformity of the wire width. Qiang Li & Das Sarma, Phys. Rev. B40, 5860 (1989). Wei-ming Que & G. Kirczenow, Phys. Rev. B37, 7153 (1988). We assume here that the phenomenological constant-collision frequency model is still valid in this effectively 1D case. The Monte Carlo simulation of [10] seem to support this assumption for wires of thickness > 160A. D. Chattopadhyay & A. Bhattacharya, Phys. Rev. B37, 7105 (1988).