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Solid State Communications, Vol. 78, No. 5, pp. 433-437, 1991. Printed in Great Britain.

A M P L I F I C A T I O N OF PLASMA MODES IN S E M I C O N D U C T O R H E T E R O S T R U C T U R E S J. Cen, K. Kempa and P. Bakshi Department of Physics, Boston College, Chestnut Hill, MA 02167, USA

(Received 9 November 1990 by J. Taut) We investigate current driven instabilities of plasma modes in semiconductor heterostructures. Amplification of the plasma modes becomes possible when electrons are driven parallel to the interface by a sufficiently large electric field. Effects of electron-electron and electronphonon scatterings are included. We find that the idealized, strictly two-dimensional treatment of the charge carriers used in our previous studies is an excellent approximation if only one subband is occupied at T = 0, and the frequency of the generated oscillation is much less than the intersubband separation. We also find that if more subbands are occupied, the threshold drift velocity for this instability can be significantly reduced, making it more practical for device applications. IN OUR previous studies the possibility of current driven instabilities in layered solid state systems has been established and potential device applications of this phenomenon have been noted [1-4]. For semiconductor systems, we employed a model in which layers of mobile charges were modelled as strictly twodimensional (2D) planes of free charges. In reality, of course, the carrier layers have a finite thickness and for such situations, an extension to a more general quasi-2D treatment is desired. This paper provides such an extension, using the formalism of [5] which allows for an investigation of the electromagnetic response of semiconductor heterostructures. We begin with a system consisting of a slab of electrons confined by barrier potentials. Plasma modes of the system are given in the non-retarded (c ---, or) and in the long wavelength limits (q --* 0)

by [5] = ~

d,,(q, 09) - dp(q, 09) ,

(1)

where 09 is the mode frequency, q is the wave vector in the plane parallel to the slab. d,(q, 09) and dp(q, 09) are the normal and parallel surface response functions [5]. Equation (1) can be further re-written to bring out the resemblance to the formulation used in [i], 21re----~2D(q, 09) q

e. + eo ~

i + qd.(q, 09) "~, 2-~ )

(2)

where e and ~o are the dielectric constants inside and outside the slab, respectively. The susceptibility 433

D(q, 09) in the random phase approximation (RPA) is given by [6] D(q, 09)

23~ dp f ( p + q) h f ( p ) (2~)2 ( 2 p . q + q 2 ) ~ _ h09

(3)

where m* is the effective electron mass, hp is the electron momentum in the plane of the slab and f(p) is the distribution function of electrons. In principle co should be replaced by 09 + iv in the presence of dissipative collisions with effective frequency v; we have omitted this term since in the systems of interest the single particle absorption, inherent in equation (3), is the dominant effect. The dispersion relation, equation (2), contains all possible plasma modes of the system (in the nonretarded and long wavelength limits), for any distribution function. In the absence of a driving current, both the intra- and the inter-subband plasma modes would be recovered [5]. Introducing a current (which modifies the distribution function) will alter the mode structure for these modes and, above a threshold, also lead to new (acoustic) modes, with phase velocity of the order of the Fermi velocity [1]. We are interested here in such amplifiable acoustic modes [l]. Since their frequencies are (for small q) much below the intersubband mode frequencies, the function dn(q, 09), which is significant only in the vicinity of the intersubband transition frequencies, is vanishingly small in the range of frequencies of interest here [5], and can be taken to be zero in the following. In the absence of an external electrical field (E = 0), the distribution functionf(p) has the simple

434

A M P L I F I C A T I O N OF PLASMA MODES

Vol. 78, No. 5

equation (7) with (3), we obtain

form, feq(P, T) =

1

2 .=, exp [(E~ - #(T))/kT] + 1'

(4)

where E2 = E. + h2p2/2m *, E. is the electron energy in the n-th subband, and k is the Boltzmann constant. The chemical potential/1 = ~(T) is calculated from the electron number conservation condition (n, is the total surface electron density),

D(q, o9) =

(8) where D~o~d(q, ¢o, Vd,)

n~.

=

(Deq(q,

w -

q " Vd,~jX,

Tj)),

Dhot(q, ~o, Vdr) =

2 / (~-~~)2f~q(P T) =

~lDcold(q, W, Vdr) + ~2Dhot(q, ¢0, Vat),

(Deq(q , o) -- q" Vdr(1 + ~lX), T2)),

(5)

( f ( x ) ) --= f dx e Xf(x). It is clear from equation (4) that f(p) is the sum of electron distributions from all available subbands. For T = 0, this sum is confined to occupied subbands (En < ~F), where the Fermi energy eV is determined from the total surface density and the subband energies through the relation

[

eF =

]/

#(0) + ~ E,

rt=l

l,

(6)

I being the number of occupied subbands. Taking E~ = 0 as the reference energy, #(0) = h"(2rm,)/2m* is the 2D Fermi energy. In the presence of a constant electric field E parallel to the slab, we determinef(p) using the formalism of [l]. The distorted f(p) is obtained by solving the Boltzmann equation in the constant collision frequency model [7-9]. The field strength is restricted to the linear mobility domain; effects such as intervalley transfers (Gunn effect) and generation of optical phonons do not come into play here. The final result for f(p) is given by f(p)

=

0

d x e x elf~q

[

P -- alx--fi---, T,

m*Vdr ]} ,

+ ~2feq p -- (l + ~,X)----~, T2

(7)

where al = %_ph/(V¢~ + %-oh), aZ = 1 -- ~l, V~_~ the electron-electron and re_oh the electron-phonon collision frequencies, respectively, vd~ = eE/m*v~-ph is the electron drift velocity in the heterostructure under E, Tj is the lattice temperature, Tz is the effective electron temperature determined from the kinetic energy conservation condition [1, 2] and equation (5). It is clear from equation (7) that the velocity distribution consists of a "cold" component and drifting "hot" component, governed respectively by the low lattice temperature Tt and the high effective electron temperature 7",, with their relative populations in the ratio R = ot2/uf = %~/v~_r,h. The collision averaging does not alter their relative strengths. Combining

0

T) is the equilibrium susceptibility obtained whenfeq(p, T) given by equation (4) is substituted in equation (3). The total susceptibility D(q, o9), as seen from equation (8), also separates into "cold" (Tl) and "hot" (T2) components, which are just the collision averaged equilibrium susceptibilities. Equation (2) with D(q, ~o) given by (8) constitutes the dispersion relation determining the plasma modes of the current driven system. We are interested in modes for which Im (~o) > 0; these are the unstable modes, i.e. their amplitudes increase in time. Now we examine the dependence of the plasma mode properties on the subband energy structure. If there is effectively only one subband in the system, equation (2) reduces, as expected, to the surface eigenmode condition for a 2D system [1, 4]. This would be the case, for example, when the separation between the first and second subbands is much larger than kT2. For finite thickness layers, however, subband separations can be comparable to kT2 and the effects of subband structure will become important. In general, two distinct situations are possible where at T~ ,~ 0 and Vdr = 0, (a) only the lowest subband is occupied (l = 1) or, (b) two or more subbands are occupied (t >/ 2). In the first case, the cold component of equation (8) which is at the lattice temperature Tt (taken to be near 0 K) includes only the lowest subband contributions, so it is identical to the corresponding 2D cold component [1]. On the other hand, in the hot (T2) component one must include contributions from all subbands because of the thermal redistribution of electrons amongst the subbands. Since the required instability threshold drift velocity is expected to be large (of the order of 2vv) [1], the effective electron temperature T2 is also quite high [kT 2 ~ (V~r/ V~)eF] and the equilibrium distribution function then approaches a Maxwellian fM(P) = F exp [-h2p2/ 2 m * k ~ ] with the pre-factor F = Y~exp [(/~(~) E~)/k~] ,~ exp [#(0)/k~] - I. The corresponding 9 e q ( q , tO,

Vol. 78, No. 5

A M P L I F I C A T I O N OF P L A S M A M O D E S

435

1.0 .~t" 0.9

..i"

0.8 "•

. l i "A

0.7

O

"O i •°

O

0.6 •.-"O O O

0.5 04.

°''

0.4

I

,

0.5

q

i

J

I

i

i

0.6

i

i

I

~

J

0.7

i

L

I

0.8

L

i

,

,

i

0.9

. . . .

1.0

a9F [ a3O0 Fig. I. Change in threshold drift velocity for various subband structures. The horizontal axis is the Fermi rain ratio. Other parameters are: total surface velocity VF ratio, the vertical axis the threshold drift velocity Var electron density n, = 10~cm 2, effective electron mass m * = 0.0665me, dielectric constants ~ = 13.1, e0 = !.0, and collision frequency ratio R = Ve~/V~_ph = 10. D~q(q, co, T2) is given by, Deq(q, co, T2) ..~ 2 [

./

dp fM (P + q) - fM (P) , (2r0 z h 2 2m* [(P + q)2 _ p2] _ hco

(9)

which is identical to the corresponding 2D expression [1]. Thus the hot component of equation (8) is also identical to the corresponding 2D 'result. Therefore, we conclude that for multi-subband heterostructures in which only the lowest subband is occupied at T = 0 and E = O, the amplification threshold o f the acoustic plasma mode (for small q) is the same as that obtained in a single subband treatment. Note that this is not a trivial result (in contrast to the current-less case). Once the current is switched on, electrons are scattered into higher subbands, leading to multi-subband occupancy. It requires a certain amount of algebra, as shown above, to establish that the 2D-limit is still valid. In the case of multi-subband occupancy, keeping the same ns, we can expect a reduction of the threshold min• It was shown for a 2D system in [1] drift velocity vat min that vat oc vv. It is clear from equation (6) that the Fermi energy ev, and thus the vv, for the system will be reduced by multi-subband occupancy. I f the proportionality of Vdr with VF still prevails, then multisubband occupancy would lead to lower threshold drift velocities. Our detailed calculations show that this is indeed

the case; this is displayed in Fig. 1 which shows the change in threshold drift velocity for different subband configurations, for a given total surface density ns = 10~cm 2 which corresponds to / a ( 0 ) = 3.6meV, and wave number q = 5.0 × 102cm t. The horizontal axis is vv/vvo = ~ and the vertical axis gives the corresponding threshold drift velocity ratio Vdr min min min /V~o, where va~0 is the threshold drift corremin sponding to one subband occupancy (l = I), and vat is the threshold drift corresponding to a reduced Fermi energy eF" The triangles represent a series of two subband configurations with AE _= E , - E~ varying from a value above/l(0) to a vanishingly small interval. The triangle in the right hand corner at (I, 1) represents the case when AE > p(0), so that l = 1. As pointed out above, this coincides with the threshold condition for a single subband treatment, i.e. the idealized 2D plasma of [1]. As the subband structure is changed so as to reduce AE, the same point also represents the case of marginal second subband occupancy (AE = #(0), l = 2). As AE is further reduced successively to 3.0, 2.0, 1.0, 0.5 and 0.2meV, ev is reduced according to equation (6) and we find a corremin sponding reduction in the threshold drift vdr as well. We indeed see that an approximately linear min and VF exists. In fact, this relationship between vat rain and vv can be formally proportionality between vat shown to hold exactly for the limiting case of AE = 0, represented by the last triangle at (1/x/-2, l/,,f2). The circles represent a multi-subband situation

436

A M P L I F I C A T I O N OF P L A S M A M O D E S

Vo'l. 78, No. 5

where more levels (l > 2) are occupied, with equal eV = 2.05meV. The middle curve represents 1 = 3, spacing AE between the levels. Again, the approxi- AE = 1.0 meV with ev = 2.2 meV. The extreme right mate linearity ofvam~"vs VF is maintained. This process curve corresponds to eF = /~(0) = 3.6 meV and coinmin (i.e. increasing l) can push the vdr to even lower cides with the boundary curve from our previous calvalues, although inclusion of intersubband effects culation [1] for a 2D layer, for corresponding physical would become necessary beyond a certain point, if and parameters. This confirms our conclusion that the 2D when the intersubband separations become comparable result is reproduced when electrons occupy only the to the mode frequencies. Note also that increasing l, lowest subband at T = 0 and E = 0. We have dis(which implies increasing the width of the sample) played only a limited range of q-values in Fig. 2; for while keeping n~ constant, amounts to a reduction much large q's, the curves bend to the right as in [1]. in the effective three-dimensional electron density. In conclusion, we have investigated current driven Even for moderate l the Fermi velocity begins to be instabilities in quasi-two-dimensional semiconductor governed by the bulk density rib, rather than ns, as can heterostructures, where multi-subband occupancy be shown from equation (6). Further decrease in vv can occur. We found, that the two-dimensional treatcan be ascribed to the reduction of the bulk density ment of the electron gas used in our previous studies with increasing size (for thick slabs VF ~ n~/3). The is an excellent approximation, and remains valid for lowest achievable n~ therefore determines the lowest VF finite thickness heterostructures if at E = 0 and in a practical system. T = 0 only one subband is occupied, and the freFigure 2 shows the boundary curves of insta- quency of the generated oscillation is much less than bility for various subband configurations. The hori- the intersubband separation. We also find that if more zontal axis is the drift velocity v~'", the vertical axis is subbands are occupied (at E = 0 and T = 0), the wave number q. To the left of a given curve, the threshold drift velocity for instability can be signifisurface plasma mode is damped due to single particle cantly reduced. A procedure to construct growth absorptions, while to the right, this mode becomes potentials which lead to various desirable subband unstable and its amplitude can grow in time. The structures has been described elsewhere [10]. This boundary curves describe steady situations where the can be used to advantage in fabricating heterostrucenergy transferred from the driving electric field to the tures which offer a reduction in the threshold drift plasma mode is balanced by the energy loss to single velocity. Since increasing the subband occupancy (which particle absorptions. The extreme left curve describes the case of two subbands with AE = 0.5meV and reduces the drift thresholds) requires increased sample 0.8

0.6 I

¢.)

0.4

0.2

I

I

0,0

2.0

2.5

i

i

r

3.0 ~) min(107 dr -

I

3.5

4.0

cm/sec)

Fig. 2. The instability boundary curves for heterostructures with different subband configurations. The horizontal axis is the drift velocity, the vertical axis the wave number. For the curve on the left: AE = 0.5 meV, r,v = 2.05meV, l = 2; curve in the middle: AE = 1.0meV, ev = 2.2meV, l = 3; curve on the right: AE = p(0) = 3.6 meV. Other parameters are as in Fig. 1.

Vol. 78, No. 5

AMPLIFICATION OF PLASMA MODES

size, it implies that for a given n , shifting the narrow quasi-2D system towards a wider configuration offers a practical advantage. Another way to reduce the drift threshold would be to reduce n~, keeping the width of the sample constant. This also amounts to reducing the bulk density n~. Both these approaches suggest that systems with lower nh are the better candidates for device applications. However, since the surface scattering effects in narrow samples reduce the maximum achievable drift velocities [11] and also increase the effective damping which has to be overcome to achieve instability, the wider configurations are preferable from the practical point of view. In addition, they would sustain higher currents for the same nh, leading to a larger transfer of energy from the current to the plasma wave. Thus we may expect best results (lowest drift thresholds and higher power output) in threedimensional geometries. In semiconductor systems, a high mobility, effectively three-dimensional electron gas can be obtained in wide parabolic quantum wells [12], they may turn out to be the best candidates for device applications. In fact we have initiated a study of the properties of current-driven modes in such systems, and in a 3D electron gas. Even though the mode interactions are no longer negligible (i.e. one must retain the full effect of d, in equation (2)), the basic results confirm the conclusion of the present paper: the threshold drift velocity for instability remains tied to Vv with about the same constant of proportionality, and thus seeking systems with lower v~. is advan-

437

tageous. Superlattice arrangements of such wells might offer further advantages [i-3].

Acknowledgement - This work has been supported by the U.S. Army Research Office. REFERENCES I. 2. 3. 4.

5. 6. 7. 8. 9. 10. 11. 12.

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, J. Cen & P. Bakshi, Phys. Rev. B39, 2852 (1989). K. Kempa, P. Bakshi & J. Cen, in Proceedings of the Conference on Advanced Processing of Semiconductors (H), 1988, (Edited by H. Craighead & J. Narayan), pp. 62-67, SPIE Conf. Proc. 945, (1988). K. Kempa, D.A. Broido, C. Beckwith & J. Cen, Phys. Rev. !t40, 8385 (1989). F. Stern, Phys. Rev. Lett. 18, 546 (1967). P.M. Bakshi & E.P. Gross, Annals of Physics 49, 513 (1968). N.S. Wingreen, C.J. Stanton & J.W. Wilkins, Phys. Rev. Lett. 57, 1084 (1986). K. Kempa, Appl. Phys. Lett. 50, 1185 (1987). D.A. Broido, P. Bakshi & K. Kempa, SolidState Commun. 76, 6 i 3 (1990). D. Chattopadhyay & A. Bhattacharya, Phys. Rev. B37, 7105 (1988). See, for example, T. Sajoto, J. Jo, M. Santos & M. Shayegan, Appl. Phys. Lett. 55, 1430 (1989).

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