Magnetoplasmon spectrum of a modulation-doped semiconductor superlattice with periodic defects

Superlattices

and Microstructures,

357

Vol. 10, No. 3, 799 7

MAGNETOPLASMON SPECTRUM OF A MODULATION-DOPED SEMICONDUCTOR SUPEBLATTICE WITH PERIODIC DEFECTS Godfrey Gumbs of Physics, University

Department

Lethbridge,

Alberta TlK

of L&bridge

3iU& Canada

H.L. Cui and N.J.M. Horing Department of Physic8 and Engineering Physics Stevens Institute of Technology, Hoboken, New Jersey 07090

(Received 13 August 1990)

The newly proposed modulation doped superlattice with periodic defects simulated by positive potential barriers in the quantum wells of the superlattice has been shown to feature minibands of controllable widths and separations, by adjusting the position and height of the barriers within the wells. We demonstrate that such interesting properties of the single-particle spectrum are correspondingly reflected in the magnetoplasmon excitations of the same structure. A self-consistent field RPA approach is employed here to examine the inter-subband and intra-subband magnetoplasmon spectra of the modulation doped superlattice, with particular attention given to the effects of the positive potential barrier (defect) on these collective charge density modes in a magnetic field.

In the present paper, we extend this work to incorpo

The striking sensitivity of superlattice energy sub band structure to variations of the geometric parame

rate. the role of a magnetic

ters of barriers in the quantum well regions, as eluci-

izing the study of dielectric response properties,

dated by Beltram and Capasso’ as well as by Pee&s

termining

Vasilopoulos,2

hss prompted

further investigations

rok of defects in superlattices dielectric

response

superlattice within

properties

with rectangular

the random

the plasmon

we demonstrated subbsnd

quantum

toplasmon spectrum, including intra-subband and inter-

of a multi-quantum-well defect barriers carried out

with corresponding Moreover, variations

and intratuned sub well, such in the

40k, an order of magnitude inplasma frequency,

changes in the intra-subband of the collective excitation

cies follow closely the electron energy subband

07494036/91/070357+04

mode. frequen-

structures

via the defect barrier.

$02.00/O

barrier, as

ekctric response propertiea in this situation have been

by varying the defect barrier

and are therefore equally tunable

on features

In that work,

its height, and its position

crease is obtained in the inter-subband

we focus attention

associated with the positive defect potential they are modified

of the barrier in the quantum

width from 0 to about

Specifically,

perlattice.

we examined’

that both the inter-subband

well. For example,

and de-

field on the magne-

subband magnetoplasmons of the modulation doped su-

for such a system.

ss the barrier width,

of the magnetic

of the

plasma frequencies can be dlectively

ject to variations

the e&&s

general-

In a recent study of the

phase approximation,

spectrum

and

field, appropriately

by a magnetic

field probe.

The di-

formally examined in the literature in the presence of a lattice/supe.rlattice6~11, and -separatelyof

a

in the presence

magnetic field.12 The problem at hand involves tak-

ing munt complex

of both featurez jointly, unit cell needed

to describe

odically placed in the supczlattia are in turn periodically

and including the ddects

quantum

distributed

the peri-

wells which

in the z-direction.

ThemagneticfiddB=Biiakotakentobeintbe z-direction,

driving electrons into Landau-quantized

cularorbitsinthez-lyplane. eigenstatez,

cir-

Theunperturbaienergy

chosen in the gauge for which

the magnetic

0 1991 Academic Press Limited

358

Superlattices

vectorpotmtid is A

Ike,

&, are the norm&d

fun4n.u

Vol. 10, No. 3, 199 1

M

harmonic ordilator eigen-

having the cycAotron orbit centered at z+, with

tlm cycbtron hgth

= (0, Bz, O), mby be written

and Microstructures,

frequency w,, and L, ia a normdizatiot~

in the plimction.

the pubdic Paey

potdid

Y,c,(s) ir a Blo& function in

model. The bated

*

,b

of the r-direction for the Kronig-

r

electronic energia are

EL,,+. = (n + l/2) hw, + M,,

(2)

Fig.

-1

1

1: Itepatation

of the potential pro6le in

a superlattice containing one defect barrier of width d.

with m am mbband index and -111 < ks c 111. The

The center of the defect h at a distance a from the middle

rupe&ttice

of the quantum well. The width of the quantum well ie

hi

paiod 1 dong the z-direction, L. = Nl

~NirtbenumbaofunitalLJoIyther-direction. rsr,,,,c, P e*+,,,~~(r)

CXp<OW.

ir a Bbch function; u+,(s)

ir pe

rindicaadmaybeexpa&din~Fourieraeriatoobtain

The longitudinal dielectric function of the random

rraprasDktionintemudreciproc.allatticevectomC: UI&)

. ammpdmg 2nr/l

= F c(k, - G) c-~‘,

to the raid

the -ice

pot&M

(n = 0,*1,*2,.-a).

three caqmnaxta

(3)

lattice repraeatation V(z) = &

of

with G =

V’p,

We retain only the Er8t

of the auwlattia

potatial

expa&onbrtheBbchfunction,uinRef. &

phue appmxinution

in the VCJ-

may be written in reciprocal lattice

representation M Rx7 (q,w) =&a~~ -~(e,qz+G)~~+w~),

(5)

where wavevector q = (&qs), and frequency= W, and I( L the background diebetric constant. Also

4,withthe

u (Q,qs + G) % (~*c’/K) [I$ + (qs + G)‘]-1 ,

(I - 2T/l)

and the noninkacting

density-density correlatii

(6) func-

tionia

I&# (q,(Q) =

x&a c h#C*SIC’ XL,*

(k.,q,,w)z~,,,(I;,q,,G)

(k*,q*,G).

(7)

Inthb, d Lrl

Am0,) = [1+ a; (k,)’ + a; (ks)‘)-lD

(ks, qs, 6’) = f ~‘dz~*,(z)e-~~~~r,+r.(E),

and the in&all&ion of the magnetic field yielding

Rxa&&&angulud&ctofwidthdaadheight x&s

v,,amtaalin~quutumweUofwidthloboamledby quare~ofwidthkandlw&htV,(aeeFig.

(k*,e,w)

=

l),

C,,,(#)

= (n!/n’!)za’-ac-c[L~‘-a(z)]~,

where ;t = ~f2m*we, m+rl,

n,) and f(E)

and n’ = mar(nl,na),

n =

b the Fexmi distribution.

Other

359

Superlattices and Microstructures, Vol. 10, No. 3, 1991 u8ehlrep?aaltatioManlldbeobtainedfromamani-

F f

(km)

= [Cn~.l~Cl,

,+1/2, at T = 0 (17a)

feetly gauge invariant treatment employing the magnetic field Green’s function in closed form.1z

= Lk*lliWo

The plasmon dispersion relation is given by

= ;P(-**

aemicluniul zero iield limit

coth (&,/2),

(17b)

nondegeneratelimit (17~)

det 17 (p,w)l = 0. Following our work in Ref. 4, we consider the case in which electron wavefunctions within adjacent quantum

where Eq. (17d) ia the de &as-van Alphen (d&A)

wells overlap substantially, and only a few reciprocal lat-

for ubitruy

tice vectors need be considered. Moreover, our concern

to be added to the aemiclaokal zero fidd limit. Aloo,

here is focused on long wavelengths, corresponding to

[...I,,,.= 1 b the maximum intega function and (n&S 3

q* a 2r/Z and q

( - h$, 18a shifted chemical potential.

a (li/(m*w.)), the

magnetic length.

La&

part

factor g* and temperature,which ir

Under these conditions the modes satisfy 1 - ” (a (I*)@I? (9, w) = 0.

(9)

5

a

This low-wavenumber dispersion relation is restricted to intra-subband plasmons hybridized with the cyclotron frequency in accordancewith

1=&[l+&]8in~e+go8~e,(10) where&$ = 4re’&,/rtml, correapondato an etktive bulk plasmon, for a closed-packed ruperlattice, & ia given by

(12)

J(‘)(m, k.) wherei=O,1,2,.-.

P

f

~‘d.u+,(*)~u;~~(‘ (13))) and J(m, ks) = J(O)(m,k,),

Fig.

2: &gnetopkuna

frequencia aa function

of the defect barrier width d. Magnetic field drcqth

Angle of propagation e = 45*. other pa25OA, b = 5OA, w = 2OOA, s= 0, V, = 313meV, V, = 228meV, and electron sheet density

B = 2T.

rametem are I=

7kj = 1.5 x 1o’%m-1. curve aE high frequent root of

W (m, k,) P 25 (m, k,) J(l) (m, k.) .

(14)

Also, sing = i/ (d + q:)“‘, CCIO 0 = q=/ (9” + qi)“‘, and Gi s 4re’i&/icm.l,

where the dktive

(15)

mama m. in the growth direction of

Eq.(lO); curve b: low frequency root of Eq.(lO); curve c: plasma frequency for B = 0.

To &&rate the depauha

d intruubband magn~+

toplasmo~~on the puametan characterizingthe defect barrier, we exhibit in Fii.2 the two mode frcquenciw UI

the superlattice is defined by

&en

by

the two roOt8 of the long-wavelength&per-

sion rdatiin (Eq.(lO)), M functiona d the defectburim width d. The mic CuryiqoutthemunovuLuxkAev&hEq. ueobtain~allimitiifonrmdi&zwt.

(16), Wehve

(for convenience, in&u #pin splitting = LandAll level 9epU&n)”

field rtrength ia takea to be

B=2T,andthepropagationangkisa,uumaitobe g=45*.

Inti&xlation,quantummagneticfiddd-

fectahavebeenn+ctedinthee&tivemaam,akg thegrowthdiiionuweUuintheFermi~.

Ru-

360

Superla!tm?s

thermore, the pg.- correction ir z&o ignored here. For purpoaa

of compukon,

field pkna width

we abo show in Fig.2 the zero-

frequency

(cum

c).

aa a function of

The introduction

the ddect bar&r

of a magnetic

modifying

and Microstructures,

Furthermore,

additional

Cal magnetoplasmon ometric

(curve a) being eua&Uy

with our analysis

rier

(ree diacukon

chu&tiu

mode (curve b) haa more

Ilpucipum

at d = 38A where

To gti

having

StNCtUH,

Mber

the loweat subband attains

insight into the rok of the magnetic

of ddectr,

we note

dau qua&z&on rod ignoring

ellecb

spectrum

together

These points

spectrum

in the na-

in the nonlo

with defect ge

will be addressed

of the intersubband

jointly

magnotoplasm<+ri

excitation.9 in further work in progress’3.

Acknowledgement

- G.G. was supported

in the prea-

Research Council of Cmsda

in part

Sciences and Engineering and the University

of Leth-

bridge Eeae.arch Fund.

that with the neglect of Lan-

in m. (and in the Fermi enera)

the ,9q,-correction,

Eq.(lO) aao&ted

effects.

by a grunt from the Natural plasmon

resonances

a

width’.

fidd in the 6uperlattice ena

tbc defect bar-

in H.ef.4). The lower

frequency

B maximum

of

plasmon

modea rare to be expected

rplitr this mode into two, with the higher frequency one independent

1.Y9 I

wp through m,, and through the Fermi energy

ture of Bernstein

field

\/ol. IO. No. .I

with Fig.2 hu

the &p&on

Referent

relation

the form

1. F. Ekltram and F. Capawo,

Phys. Rev. B38,356Q

(1988). 2. FM. Lett.

ForO=Othareirjurtonemodew-*(j,,withthektkr giivco by Eq.( 15), and the e&t+ dkctio~

si=

mam along the growth

by ma = C~f(&)l

C~f(&)(@cr.Id%).

amall.

tidd &cta

u

quantum

On the other hand,

mic

6dd c&e&

pakttice

mnle

the kmiliu

4

vault

01~ defect

gannetry. spectrum

nmidusical

the clolropack IIII

with the cyclotron w’ -* q

motion

to yield

+ w:, abnent dependence

For arbitrary

factcam cited act jointly, pa

effecta are negligibly

for 0 = r/2,

act to hybridize

angka,

the vuioua

and there ia dependence

00 both the magnetic

of the

fkld through

hybridktkm,

and QL ddect

null magnetic

fkdd, tbezc ia just one branch of the pke=

llX%BpCt~W’ dance

on

dda~t

L+n2e

geometry

++4de,

width is depicted

umml, the m

mngnetic

and P. Varilopouba,

3. G. Gumbo and A. Salman,

through

whae

(j,‘. For

depen-

in Fig.2, curve c. As field inducen umther

G. Gumbs,

5. H. Ehmnreich

6. D. Fak, Phyu. Rev. 116,105

d which asks

from Q.

This nlrtivdy

low-field

at high

magnetic

B to be

Rev.

115,

(1960).

9. If. Lhida,

J. Phys. Sot. Jpn. 65, 4396 (1986).

10. X.L. Lei, R.Q. Yang, and C.H. Thai, in”P/ayrica of

and Quantum weib”, C.H. Thai et.

Supcrlattica

al. ah.

(World Scientific, Singapore,

11. X.D. Zhu, X. Xia, J.J. Quinn,

1989).

and P. Hawrylak,

Phyr. Rev. B 38, 5617 (1988). Phys.

48

A,

7 (1974);

(N.Y.) 31, 1 (1965); snd M. Yildiz,

ibid 85

Phye.

A, 378 (1981); Phyr.

(N.Y.)

Q7,216 (1976); N.J.M. Horing snd M. Yildiz, Phya. Rev. B 8S, 3895 (19&i).

~rpectrumwilibefurtkenrkhedmd

, cmnphd

Rev.

Phys.

N.J.M. Horing and M. Yildiz, Ann. tiple

B 41 10

8. N. Wiser, Phyn. Rev. 12Q, 62 (1963).

of tbc &

in Fig.2,

Rev.

7. S.L. Adler, Phylr. Rev. 120, 413 (1962).

Lett.

are depicted

Phye.

and M.H. Cohen,

hibruy&e,&.18,andthetwoamci&dbruKbm spectrum

Phyr.

786 (1959).

N.J.M. Horing, hf. Ormq

cws~~dbintheirdepaukceondefectwidth,aU

Phys.

publirhed.

12. N.J.M. Horing, Ann.

motilttothekKdpimmoEl&~iorrnwhfuru-

Appl.

124 (1990). 4. ILL. Cui utd

Thiamodedependaondefectgaunetry,&citdevoidof magnetic

Peeten

IS, 1106 (1989).

6eld rtragths,

LUldulqua&i&~dktaplayanimpo&antrokin

wheze

13. G. Gumhe, liahed.

H.L. Cui, and N.J.M. Horing, unpub-