Formation of an anomalous acoustic plasmon in spatially separated plasmas

Surface Science 98 (1980) 563-570 0 North-Holland Publishing Company and Yamada Science Foundation

FORMATION

OF AN ANOMALOUS

ACOUSTIC PLASMON IN SPATIALLY

SEPARATED PLASMAS *

S. DAS SARMA and Anupam MADHUKAR Departments of Physics and Materials Science, California 90007, USA

**

University of Sou thern California, Los Angeles,

Received 11 July 1979; accepted for publication 4 September 1979

The collective excitation spectrum of a spatially separated, two-component, two-dimensional plasma is investigated within a generalized random-phase approximation. It is shown that in addition to the regular optical and acoustic plasmons, a high frequency mode may exist in such a system provided the separation between the two charge components exceeds a critical length. This mode is undamped in leading order and is linear in wave vector in the long wavelength limit. It may thus be viewed as “acoustic” in nature, and referred to as an Anomalous Acoustic Plasmon. The existence criteria for the regular acoustic plasmon, which is a low frequency damped mode, is also helped by the spatial separation since its phase velocity is enhanced with respect to its damping rate. The regular optical plasmon remains unaffected by the separation in leading order. Finally, the dynamical structure factor of the system is analyzed and the feasibility of observing these modes in InAs/GaSb superlattices or in multi-component inversion layers, is discussed.

The collective excitation spectrum of two-dimensional electronic systems has in the recent past been studied both experimentally and theoretically in the context of semiconducting surface inversion layers [ 1,2]. These studies have brought out the main features of two-dimensional plasmons, even though experimentally it has not yet been possible to obtain the plasma dispersion beyond the leading order wave vector dependent term. There are two special features of two-dimensional plasmon which make it different from the corresponding three-dimensional case: (i) lack of a long-wavelength self-consistent restoring force makes the long wavelength (4 + 0) plasma frequency go to zero in two dimensions (unlike a non-zero value determined by the electron density in three-dimensional case) and, (ii) possibility of a multi-component two-dimensional plasma where the different components are effectively located in different regions of space. In this paper our emphasis is on this second feature and its effect on the collective excitation spectrum of a multi-component two-dimensional plasma. * Work supported by ONR (Contract No. N00014-77-C-0397) AFOSR 78-3530). ** Alfred P. Sloan Foundation Fellow. 563

and by AFOSR (Contract No.

564

S. Das Sarma, il. Madhukar j Anomalous acoustic plasmon

The motivation for this study comes from the experimental realization of such spatially separated multi-component, two-dimensional plasmas in the GaAs/ Al,Gar_,As quantum wells, in InAs/GaSb thin superlattices [3,4] and heterojunction, and in the inversion layers [2]. From the viewpoint of the effects of spatial separation on collective modes, the double (at least) quantum well or superlattice systems may offer greater flexibility since the separation can be controlled. In the InAs/GaSb system the masses of the two carriers involved are also different. Experimental [5,6] and theoretical [7,8] studies within the past year have revealed that the InAs/GaSb superlattice shows “semi-metallic” properties, both parallel and normal to the interface, provided the individual layer thickness is larger than -125 A. This behavior has been reconciled on the basis of the observation that for bulk InAs and GaSb, a line up of the energy bands according to the prevalent electron affinity rule indicates that at the F point the valence band edge of the latter is located above the conduction band of the former. Consequently, approximately 10” electrons per cm3 from the valence bands of GaSb empty into the conduction band of InAs. However, these superlattices also show confined nature of the electrons and the holes, as reflected in the dependence of the SdH behavior on the orientation of the magnetic field. At present, it is not clear whether this behavior is indicative of spatial separation of electrons and holes within the superlattice unit cell or simply a consequence of confinement within the unit cell. However, for the true heterojunction (depletion length > layer thickness), the presence of band bending associated with this charge transfer, will give rise to formation of twodimensional subbands, on both sides .of the interface [5]. As a result one has a spatially separated, two-dimensional and two-component (electron and hole) plasma. In this paper we show that such a system is capable of supporting a mode of collective oscillation which exists in the high frequency regime, is undamped up to a critical wave vector qc, and whose frequency in the long wavelength limit is proportional to the wave vector 4 with a proportionally constant (i.e. phase velocity, cAAP) which is higher than the Fermi velocity of both the components. as well as the sound velocity [9]. We note that this new mode is root the usual Acoustic Plasmon [lo] mode which can exist in the low frequency regime of multi-component plasma, charge separated or not, provided either the effective masses or the Fermi velocities differ, but is Landau-damped. The analogue of such a mode exists [ 1 l] in the present system as well, and is helped by the spatial separation of charge. The existence of this high frequency mode, however, is dependent upon the charge separation exceeding a critical distance. In the following, we will refer to this collective mode as the Anomalous Acoustic Plasmon (AAP) and the usual mode by its prevalent name, Acoustic Plasmon (AP). One of the great difficulties of observing regular AP in ordinary multi-component plasma is their rather large decay rate since their energy lies in the electronhole spectrum of the faster moving electron species. Thus AP decays by creating electron-hole pairs in one of the charge components. On the other hand AAP is

S. Das Sarma, A. Madhukar /Anomalous

acoustic plasmon

Fig. 1. Shows schematically, the wave vector dependence of the collective with respect to the particle-hole continua of the two charge components.

modes

565

(OP and AAP)

undamped in leading order since its frequency is outside the electron-hole regimes of both the components. Thus, just like the regular optical plasmon (OP), AAP is a long lived mode. The separation of the two components also affects the dispersion relation of the AP [ 111. In particular, it increases the phase velocity of the AP from the situation of no separation. In principle, it now becomes possible for the system to support AP even if the masses of the two kinds of charge carriers are the same, provided their separation is greater than a typical screening length of the system. By enhancing the phase velocity, the spatial separation decreases the damping of the regular AP. This makes these systems with spatially separated two-dimensional charged gases a good candidate for observing regular AP in the low frequency regime. The collective modes of such a multicomponent, spatially separated plasma are given by the poles of the inverse dielectric tensor, e-l, of the system [lo]. Neglecting interband polarizability and interband scattering, we can, within the random phase approximation @PA), write the i-j component of the dielectric matrix as, cii(4, W) =

sij

-

vij(cl)

$j(q,

0)

3

(1)

where i and j refer to two-dimensional bands and 4 is a momentum vector parallel to the interface. In eq. (1) all quantities are the retarded functions, i.e. w is assumed to carry a small positive infinitesimal iv. ll$(q, w) is the non-interacting polarizability of the ith band and V,(q) is the Coulomb interaction vertex V’iijj(q) for the interaction between electrons (and/or holes) in the ith and the jth bands with exchange of two-dimensional momentum Q. We choose i = 1,2 to indicate the two-dimensional electron and hole bands. Since the actual bound state wave functions for the quantization of z-motion in this system are not known, we can only make qualitative remarks about the effect of the finite width of the wave functions on the properties of the collective modes being discussed here. One-electron wave functions describing the electrons and the holes can be written as, GiCr)

-

exp(iq

* ru)

Ei@),

where r( is a two-dimensional

(2)

position vector parallel to the interface.

The envelope

fun&ion Q(z) gives the spread of the two-~mensional band in z-space. For algebraic simplicity, we assume l&~(;i(“)l2 to be delta functions located at z = a and b without any loss of generality. Changing [i(z) to more complicated forms (like a Gaussian) does not change the conclusions of this paper in any qualitative fashion. The interaction vertex Vii(q) is given by, I$&4) = CiiiV&:z, .z’)($ ,

i3)

where V(y ; z, 2’) = ~~2~e*~~~~exp(-qiz - z’i) ,

(4) is the two-dimensional Fourier transform of Coulomb interaction. The sign of V in eq. (4) is positive or negative depending on whether one is talking of interaction between same type of charge carrier or not. Since the static, lattice dielectric constants of InAs and GaSb are very close [ 121, we use an average dielectric constant Kin eq. (4) in order to avoid unnecessary ~omplic~tiol~. The band polar~abilit~ within WA, has the following approximate form in the high and low frequency limits : @(4, WI z a&“/d)

+ ~~~~4/#~~,

= -~~~/~~2~{~ + iw/f&

,

for w B @vfc2

f3

for w -@quf, ,

@I

where O!i= mitJ~i/2Th2 &

=

= Nifvni

,

(‘7)

3~~~~~~~~~~ = 3~h~~~~!~rn~ ,

181

Here mi, vff,Ni are the respective band mass, Fermi velocity and particle density for the Tao-d~ensio~~ motion parallel to the interface. The condition for the etistence of a collective mode Is @en by poles of the response function e-r, which is equivalent to vani~in~ of i ~1 for the dielectric tensor defined by eq, (1). Using eqs. (3)-(g) in the determinantal equation,

If&, w>l = l&j - V$$

lIj@, w)l = 0 *

PII

we may seek the high and law frequency collective modes of the system. In the high frequency regime we find the following two branches for the long-wavelength collective modes of the system:

&Q=LJO=Cqf=,

@PI

CJ=~AAP=CAAPq*

011

The coefficients C and CA~a are given by

S. Das Sarma, A. Madhukar / Anomalous acoustic plasmon

567

where

fo = (27re2/Qla - bl ,

(14)

and 4 (a - b I< 1 is assumed. We point out that this approximation is not essential for the existence of a new mode in the system. We make this small 4 expansion only to extract the explicit dispersion relation of the new mode (a*~~) in the long wavelength (4 + 0) limit and to show its “sound-like” character. It should also be remarked that though a new mode (in addition to the OP) may exist in the high frequency regime of spatially separated, two-component, two-dimensional plasma even if one does not retain the 0(q4/04) term in eq. (5) nevertheless in order to obtain the long wavelength dispersion relation for the new mode, one must retain the 0(q4/w4) term for the sake of consistency in solving eq. (9). It is only in the longwavelength limit, that we can identify the new mode as an “anomalous” acoustic mode by noting its linear 4 dispersion in the leading order. In eq. (lo), o. is the optical plasmon of a twocomponent, two-dimensional plasma in which the electrons and the holes oscillate collectively out of phase to each other. w. is simply given by the square root of the sum of the squares of the plasma frequencies of the individual components. The point to note here is that w. is unaffected by the spatial separation between the two components, at least in the leading order in 4. From eq. (14) we find that the existence of AAP is restricted by the necessary condition (2fo~rty2 > Pr + p2) which requires that the separation la - b I satisfy,

(15) Thus, the spatial separation ICI- b 1has to be greater than something like a screening length (6) of the system for AAP to be meaningful. It is remarkable that the AAP exists even if ml = m, and vrr = ur2 as long as Ia -b I > (3/2)u,, where a, = (h2iF/2me2) is the Thomas-Fermi screening length. Employing the values appropriate for the InAs/GaSb and GaAs/AlAs systems, we find that Ia -- b I has to be about 400 and 150 A respectively for the AAP to exist in these systems. In the low frequency regime (qvfr > w > qvfa), the determinantal equation, eq. (lo), has only one solution, given by,

(16) in the long wavelength limit. This is the regular low frequency ity and the damping are given by,

AP. The phase veloc-

(17)

(18)

YA = (m2/4ml)(v?Z/vfd,

Note that as a result of the spatial separation

between the two components,

one can

568

S. Das Sarma, A. Madhukar /Anomalous

acoustic plasmon

have CA > vr2 even when m, = m2 by having (fom2/nh2) > 3. Thus the separation enhances the chance of actual realization of AP in these superlattices by increasing the phase velocity and hence the energy of AP. The damping (eq. 18) remains unaffected by the separation in leading order and hence the condition CA/-y* > 1 can be well satisfied by having a large separation. Some of these results for AP have been obtained by Takada in the context of inversion layers [ 1 I]. The above analysis clearly demonstrates the possible existence of a high frequency collective mode in spatially separated, two-component, two-dimensional plasma. It is well known that OP is the high frequency mode in which the electrons and the holes oscillate out of phase to each other whereas AP is not a true collective mode since the slower moving species lose energy by forming single particle excitations in the faster moving charge component. We believe that AAP is the high frequency collective mode in which the electrons and the holes are moving in phase to each other producing a collective mode parallel to their planes of confinement. Whether such a mode can be observed experimentally or not depends critically on a number of factors - the most important of which is the spectral weight carried by AAP. It is worth noting that even though AP is a theoretically well established mode in three-dimensional multi-component plasma, it has not yet been observed due to the rather large damping compared to its phase velocity, and the small spectral weight. We believe that spatially separated multi-component two-dimensional plasma is a better system for the realization of the AP as well because a large separation reduces the damping considerably. The fact that AAP is undamped in leading order (like the regular optical plasmon) makes it a very attractive experimental possibility. A more detailed examination [ 131 of the existence conditions for the AAP and AP suggests that the AAP is, in fact, the high frequency manifestation of the low frequency AP in the weak coupling regime, and consequently a unique feature of the spatial separation in these systems [9]. To estimate the spectral weight carried by the different modes, we have analyzed the dynamical structure factor [IO] of the system. We find that in the long wavelength limit (4 + 0) the spectral weight is mostly carried by the OP (OP goes as O(q1’2), whereas both the AAP and AP go as O(q)). On the other hand for 4 near qc, the spectral weigth associated with the AAP increases considerably, even though OP is still the dominant one. However, AAP being separated in frequency from the OP, it appears likely that light scattering experiments may be the best available method for the observation of AAP. The AAP being in the two-dimensional plane parallel to the interface, the probe must have an electric field component parallel to the interface. Allowing for a width in the wavefunctions l(z) while maintaining the same average separation, tends to decrease the oscillator strength of the acoustic plasmons (both AP and AAP). In that situation, the best coupling with the external probe is expected to occur for qZ z A-’ where qZ is the z component of the probe wave vector and A is the average spread of the wave functions. It should also be noted that among the two “sound-like” modes in the system, there is a better chance of observing AAP experimentally since it is stable compared to AP.

S. Das Sarma, A. Madhukar /Anomalous

acoustic plasmon

569

An explicit estimate of the effect of the width of the wave functions in the InAs/GaSb system is unfortunately hindered by the unavailability of such wave functions in the appropriate regime of superlattice thickness. For general wave functions describing the two-dimensional confinement in the system (we have assumed the purely two-dimensional limit in our analysis here through the approximation It(z) I2 a S(z)), we can write down the condition for the existence of the AAPto be: Wl2

-f11

-f22)%Q2

>Pl

+02

(19)

9

where, fr I = (2re2/ir)Jh

tT(z)Jdz’

Iz - z’l t’4(z’> ,

(20)

f22 = (2ne2/K)Jdz

EZ,(z)Jdz’ Iz - z’l ga(z’) ,

(21)

f12 = (2ne2/QJdz

g:(z)Jdz’

lz - z’l &z’)

.

(22)

The integrals over z, z’ in eqs. (20) and (21) extend over the region of confinement of the carriers. In conclusion, we have demonstrated the possible existence of a new collective mode in spatially separated multi-component two-dimensional plasma [9]. The new mode is a high-frequency mode and its existence depends critically on having a spatial separation between the different components. Its long wavelength (4 + 0) dispersion is linear in CJand it is a stable (undamped) mode. We also find that the regular low frequency acoustic plasmon can be better realized experimentally in space separated multicomponent plasma. As a suitable means for observing these modes, we suggest light scattering experiments on the InAs/GaSb superlattice system and GaAs/Ga,Alr _,As double quantum wells.

References [l] T.N. Theis, Surface Sci. 98 (1980) 514. [ 21 See, for example, Proc. Intern. Conf. on Electronic Properties of Two-Dimensional Systems, Surface Sci. 73 (1978). [3] H. Sakaki, L.L. Chang, R. Ludeke, Chin-An Chang, G.A. Sai-Halasz and L. Esaki, Appl. Phys. Letters 31 (1977) 211. [4] G.A. Sai-Halasz, L.L. Chang, J.M. Welter, C.A. Chang and L. Esaki, Solid State Commun. 27 (1978) 935. [S] H. Sakaki, L.L. Chang, G.A. Sai-Halasz, C.A. Chang and L. Esaki, Solid State Commun. 26 (1978) 589. [6] L.L. Chang, G.A. Sai-Halasz, N.J. Kawai and L. Esaki, to be published. [7] G.A. Sai-HaIasz, L. Esaki and W.A. Harrison, Phys. Rev. B13 (1978) 2812. (81 R.N. Nucho and A. Madhukar, _I. Vacuum Sci. Technol. 15 (1978) 1530. [9] A. Eguiluz, T.K. Lee, J.J. Quinn and K.W. Chiu, Phys. Rev. Bll (1979) 4989. These authors, in numerically solving the dispersion relation for the collective modes in a

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S. Das Sarma, A. Madhukar /Anomalous

acoustic plasmorz

“double-inversion layer” geometry (in the large separation regime), find a mode that is linear in wave-vector. Our work clearly shows the connection of this mode with the usual acoustic plasmon. [lo] P.M. Platzmann and P.A. Wolff, Waves and Interaction in Solid State Plasmas (Academic Press, New York, 1973). [11] Y. Takada, J. Phys. Sot. Japan43 (1977) 1627. [ 121 AC. Milnes and D.L. Feucht, Heterojunctions and Metal-Semiconductor Junctions (Academic Press, New York, 1972). [ 131 S. Das Sarma and A. Madhukar, to be published.