Intersubband collective modes of a multiwire superlattice

~

Solid State Communications, Vol. 77, No. 5, pp. 331-335, 1991. Printed in Great Britain.

0038-1098/9153.00+ .00 Pergamon Press plc

lntersubband Collective Modes of a Multiwire Superlattice X.J. Lu, H.L. Cui and N.J.M. Horing Department of Physics and Engineering Physics Stevens Institute of Technology Hoboken, NJ 07030, USA

(Received 19 July 1990 by A.A. Maradudin) (Revised copy Received 29 October 1990 by A.A. Maradudin) The intersubband plasma excitation spectrum of a multiwire supedattice with finite potential barrier height is investigated theoretically within the random phase approximation. In this we consider a semiconductor quantum-wire supedattice composed of one-dimensional wires periodically repeated in two perpendicular directions. We examine the role of tunneling associated with incomplete confinement of the electrons in regard to their collective behavior, particularly in conjunction with intersubband transitions. The weak tunneling of electrons between adjacent quantum wires is treated employing a tight-binding scheme to approximate the single-electron wavefunction. It is shown that the intersubband plasmon modes depend critically on the symmetry properties of the subband wavefunctinns.

In this paper we extend our previous study [1] of col-

space ( a matrix of periodically aligned wires ) have been

lective excitations in a multiwire superlattice of finite

investigated by several authors both experimentally [2][3]

barrier height to incorporate the role of its intersubband

[4] and theoretically [51161171[8][91 [10111111151. However, the corresponding problem of a superlattice array of 1D

modes. A multiwire superlattice is an array of wires in whi~ch electrons move freely along the wire direction while being confined within the wire in other directions by a po-

quantum wires in 3D space with wave function overlap has not yet been treated, and it forms the subject of this paper. In the following we will first outline the formulation

tential barrier created by band offset of a different material surrounding the wire. If the barrier is of finite height then the electrons are not totally confined and are per-

of density-density correlation function for this multisub-

mitted to tunnel to adjacent wires. There are two most obvious features of this system contrasting with those of a similar system but with infinite barrier height. Firstly, for finite barrier height, the energy spectrum of electrons

band multiwire superlattice system with a finite barrier height including the two features we mentioned earlier and then we present illustrative calculations of tile dispersive intersubband plasmon spectrum. In our model we assume that the electrons are free to

in the direction perpendicular to the wires is dispersive, associated with the formation of minibands which replace the flat sublevels of the infinite barrier case. Secondly the electrons which are largely localized in one wire may in fact be scattered t o another quantum wire via mu-

move along the z direction and are more strongly confirmd in the directions transverse to z so that the system is highly modulated in density in tile xy-planc. A tight binding wave function is chosen to describe the motion of the electrons in the xy-plane

tual Coulomb interaction. These features were incorporated in our earlier calculation of intrasubband collective modes. A prominent characteristic of the calculated plasmon dispersion is that in its long-wavelength limit the plasmon frequency approaches the nonvanishing subband width. Beyond our earlier considerations, however, there are in fact a number of subbands in the quantum wires, and correspondingly there also exist intersubband collective excitations in this system. These intersubband

~'K,m(X) =

~ 1K

eik:EeikaCm(r-R)

(1)

R

Here, x = ( r, z), K = ( k, k2), R = (t,3:+v~)d, a n d / t , v = integers indicating the locations of the quantum wires at the sites of the 2D square lattice of lattice constant d. ¢,~(r - R) is the Wannier wave function centered at the wire position and it also bears a subband index m, N is the total number of wires, and L is the total length of the wire. The energy spectrum of the ruth subband is

plasmons are caused by Coulomb excitations of virtual intersubband transitions. The intersubband plasmon dispersion relation of a 1D multiwire superlattice array on a plane (a row of periodically aligned wires) and in 3D 331

Vol. 77, No. 5

INTERSUBBAND COLLECTIVE MODES OF A MULTIWIRE SUPERLATT[CE

332

k2 -" + E,,,(k) EKm = Eo,m + 2m

(2)

with the notation

where Era(k) is the transverse energy in the xy-plane sub-

E

jected to miniband formation[13] corresponding to the nian for the system is

(11)

where II is related to the elements l~l(i'j) of the I~Imatrix as 1

Km

shown in gq. 10. Moreover, FI° is related to the elements of the 1'I° in a similar way,

K, K', Q m , m l , n , n ' , .t t , , K , K )CK+QnCK,_QmC K m CKn'

Vn,nqm,,m(q,

lln,n,,pl ,pi ~:pl~:)p2,p~llOp{212p~,)m,m, E ~°"''') ^' A

Pl ,P~ ,P2,P~ l! ,12

wave function of Eq. 1. The second quantized Hamilto-

(3)

11°(Q,w)

=

~

~

~

f~i}~,(q + G)

n , n ' , m , m t i,j G,G'

where the Coulomb matrix elements are given by

15I°(i,J) ¢'(J) I n _ G'), n,n*,rn,rnJ J m~,m \ "~

V.,.,,m, re(Q, K , K') = e2

[J daxdax'¢I*~+q"¢t*c'q"~~ I x - x' i q'~''~'~bK"'' (4)

(12)

and finally 11o is defined in terms of noninteractive density density correlation function

where ~¢ is the background dielectric constant. In terms of the matrix notation we used previously [14] involving

Eeik.Ri_ik.R J f(EK+Q,.) -- f(EK,n')

V.,,v,=,,=(Q, K, K') = T(k)Q.,.,,,~,,,~(Q)Tt(k ' - q)

K

(5) where T ( k ) = (1,e ik'R' .... ) is a l x 5 matrix.

In this

Ri designates the 4 nearest wire sites positions Ri = 4-d&,:l:d~) for i = 1,2,3,4(i = 0 indicates the primary

EK+q,,~ -- EK,,,, -- ~ -- i(~

6.,m~5.,,,,vf ~ . *0) , , . . ( ' ~" "

-

G').

notation 1] as

fl = flo(1 + ~fl)

(14)

fl = (1 - flo?)-,flo.

(15)

or

trix with ( i , j ) element

47re2

¢(i) In

*0)

~ I q + G [='"'" " ~ + G)f'~"m(q + G) (6)

where G is a reciprocal lattice vector conjugate to R,

The collective modes of the interacting system are decided by the singularities of the l~I(Q,tv), which are the roots of the secular equation

and

det I 1 - l~I°? ]= 0,

f,(i),n'~'~ in

+

G)

=

(13)

The RPA equation for 11 may be reexpressed in matrix

wire position). Furthermore V.,.,,m,,m(Q) is a 5 x 5 ma-

V(/'J) (O) = ~ '"'*" . . . . .

i,j G,G'

n,rd,m,m'

intersubband superlattice phenomena, we have

/ darei(q+°)'r¢*n(r)¢n,(r - Ri)

(7)

is the Fourier transform of the product of wavefunctions The electron density operator can be written in the same fashion as the Coulomb interaction

(16)

These roots contain both intrasubband and intersubband modes which are coupled since the off-diagonal elements of Eq.(14) are generally not zero. Novertheless we make a diagonal approximation similar to that of Tselis and Quinn [12] in their calculation of the 2D electron gas su-

p(Q) = ~ ~ ,.,-','~¢(') r ~ + G)p~I.,(Q) n,n I

(8)

i,(]

perlattice system, neglecting all the off-diagonal elements in the matrix by setting

here p~!.,(Q) = ~ K T(O(k)C~+Q,,~CK,,~' The density-density correlation function as defined by II(q,w)

- i f dte i~'t < [p(Q,t),pt(Q,O)] > (9)

satisfies the algebraic-matrix equation in the randomphase approximation (RPA)

n(q,~)

=

E n,n',m,rn'

6o(i,J)

0 (if(n,n') # ( m , m ' ) )

(17)

Thus the intersubband mode corresponding to the subband transition n ~ rn is given by the solution of the equation det [ 1 - ,¢~0o~c;,J) ~.. -,,,,~, . . . . , l= o [(n,~') = (m,m')].

(lS)

Considering a square lattice multiwire superlattice

+ ~ j - . . , . , . . . . ',~'(J) =',,.~"~ E E E f~,.,(Q (') ~,~",J) ,,, - a') n , n t , m , m s ~,j G,G' ~-

(H°V) {i'a) -- --/n,nt,rndrd

n,nl ~' ~ ~ + G) E E d'(`) i,j G,G'

I r'0) , ' n }n,n~,rn,mtjdm,,ml"~ -- G t ) ,

t6oO~o~(i,J}

.tXn,nt,m,mt DV [at r x x

(10)

and taking account only the nearest neighbors, this involves a 10 x 10 matrix whose indices refer jointly to site and subband designations. We make a further approximation noting that those terms in the determinant involving V21,~{),n,,v with i # 0 and j # 0 at the same time

Vol. 77, No. 5

are proportional to the second order of overlap and are negligible in the case of weak electron tunneling under

t...~t

° - n , m , n j n

" n,m,n,m

--

~..~i

p,t~

.t,,2,,

'~ I Q 12qA4-a

~ )exp(--~) (23)

consideration. Hence we are left with 1 V'. If0(0,0 V(i,o) 5-" H0(o, I) --

333

INTERSUBBAND COLLECTIVE MODES OF A M U L T I W ERE S U P E R L A T T I C E

where we have dropped the terms in the summation ovcr

v(i, o)

-'n,m,n,m

" n,m,m,n

reciprocal lattice vector G in the Coulomb intcraction - - g " . l] 0(°'I) •--~t

--m,njn,n

x 1-[(1 - 11 °(i'°) --n,m,n,rrl

V (i'°) •

7rt,n~n,m

V" n ,(°'i) tn,n,rn!

i¢0

l

V (i'°)

g-' Iq 0(°'I) --

z-..,i

"-Tn,t~,m,n

~ ]"[¢! - H °(i'°) --m,*1,~,~ l l X -

matrix elements V °'i etc. as the terms with G = 0 arc

" m,n,mj~

V• thin,re,n/ (°'i) ~ = 0

finite at long wave length limit and so are dominant.

i#0

(19) and the last two appearances of V (°'i) at the end of the left hand side of Eq. 19 may be neglected since they occur under products excluding i = 0, and thus are proportional to overlap which indicates proximity to single particle excitations and damping. Finally the basic information concerning collective modes are contained in the determinantal equation of (Eq. 19), whose derivation was not committed to specific Wannier wave-functions nor to detailed subband structures. In the following we adopt a quasi-parabolic potential for

We assume that the 2p subband has a band structure E2v(k) = A(1 - cos k=d) and the ls band is simply flat as compared to 2p. The secular equation becomes 1 - ~ ,/(o.0

2 2

v(r) = 0

elsewhere

w~p,,,

=

8re 2 q~ (E0,2p - E0,,8) 2 + - [ Q 12 d 2 4a TM -

(20)

For this potential, we take the simple harmonic oscillator wave function as the approximate Wannier wave function era(r), noting that care must be taken to orthogonalize the wave function on different site since the Wannier wave functions should be orthogonal from site to site and from subband to subband. Even with these simplifications it

•

0

(21) The Fourier transform for the wave function product can be performed in closed form to give ( define a = (1/2)mw0) =

J~s,sp(q) . =

iq= e x p ( - q~_~2) 2al/2 [~-~ -- ~ ,iq.R exp( ~

ls,2s,ls,SsXLls,2s,ls,Ss

~ --

(/(O,i)

ads

e x p ( - 9--~-025)

fl(i,0)

v2s,ls,28,lsLaSS,Is,Ss,ls

~

i

~

O.

i

(26) The Fourier transforms of the ls and 2s wave-function products are fl°.s,(q)

_), [ q8a [2 e x p ( - _q~8~

_

(

I q Is

iq2-'Ri)

8a

exp(iq_.Ri2

ls ~ =

i

floan(q)

2R~)Ant

a I Ri Is 2

I q IS~ (27) 8a "'

However, for the collective mode corresponding to the

~/(o,0 rl{i,o) i/(o,i) rl(i,o) "l.,sn,x.,sp'q,,,sp,,,,,2~, - ~ "sp,,.,spj.,"s~,,,.ap,,, i

2 _-1

~i(i,O)

1 - ~ f/(o,0

long-wave length limit (I Q I--+ 0) the dispersion relation

1- ~

4re2

,~ I q 12 dsq=(c~ -

Turning attention to the ls ~ 2s intersnbband mode (i.e. Eq. 19 with n = ls, m = 2s) we have the dispersion relation

is still difficult to derive a general explicit expression for the plasmon frequency. Here we only attempt to derive the plasmon frequency at long wave length limit for the ls --* 2p~ transition. In the following we always imply 2p~ when we say 2p unless explicitly noted. In the (Eq. 19)is simplified to

K

Solving this equation we obtain the local ls ---* 2p plasmon frequency as (hi is the electron density per unit length in the wire)

(1=1 < d/2, Ivl < d/2)

v(r) = ~m,~0~,

~ eik.n,

E0,2p - Eo,1, + A(1 -- cos k=d) f(EKa')w2 :~o,-~p:-~o,~--~&~- cos k=d)] 2 = 0. (24)

the q u a n t u m wire in the transverse direction 1

'1

. 2p.l~N2d2 L

i

(1 -- eiq'R' )]

Ot I ~ I s 2

._:lql~

2s transition the situation is different fi'om the

ls ~ 2p counterpart discussed above, as we can see from Eqs. 27 for Iql --' 0, since now f°;is, ~1 q Is ~ 0 while from Eqs. 22 we see f°;isp ~ q= approaches zero more slowly; This occurs because the ls and 2s wave-function share the same rotational symmetry property. The consequence is that their Coulomb coupling is weak since it is mainly quadratic coupling (as compared with the dipole coupling fo the ls - 2p transition) and favoring Umklapp processes with G # 0 in (Eq. 6). In conjunc-

(22)

tion with the weakness of the Coukunh intcractio,, we also bear in mind that the 2s miniband width is finite, so

The corresponding Coulomb matrix elements in the long wavelength limit are given as

the elements of the density correlation matrix I'I° remain finite even in the limit [ q ] ~ 0 and w approaches the

8a "

334

Vol. 77, No. 5

I N T E R S U B B ~ D COLLECTIVE MODES OF A MULTIWIRE SUPERLATTiCE

band gap E0,2~ - E0.1~. All these considerations suggest that the products of V and l~I in the ls --* 2s secular equation Eq. 19 may remain less than l so there will be no plasmon excitations for ls --* 2s transition (if the 2s miniband width AT, is finite). However, if both the ls and 2s minibands are flat, the elements of Yl° will be-

8.U.

1.5

~

~/d

1

come infinite as w approaches E0.2~- E0.1~, and this rapid change will induce the presence of another root into the plasmon dispersion relation. Physically this corresponds to electron movements which are strongly in phase with each other throughout the whole system. We have calculated this branch of the plasmon spectrum numerically from Eq. 19, and the result is shown in Fig.1. In the calculation we intentionally used a small subband width to insure the existence of the ls ---*2s plasmon mode. In Fig.2 we show the numerical result for the l s ---*2p~ plasmon branch which is also obtained from Eq. 19. from Fig.2 as well as Eq. 25 we see that due to strong dipole coupling, the plasmon exists through a large range of wavelengths and wider subband width. Moreover, the dispersion relation is anisotropic; Along the y direction a.u, 0.06

qx='rr/d \ 0.04 qx=O

0,02 -

o

I 4

I 8

I 12 qz

I 16

20

"10"5glo

F i g u r e 1. The ls --4 2s intersubband plasmon dispersion relation. There is no plasmon excitation as q~ becomes greater than about 10-4ao. Parameters used in calculation are: c~ = .002~ -2, E2~ - E I , = . 9 7 e V , t~ = 13.0, d = 75,~i, A2s = .01eV, nt = 0.55 x 10rcm - l . a0 is the Bohr radius.

0.5

qx = o 01

l

0.01

qz

I 0.02

a: I

0.03

F i g u r e 2. The l s ---* 2p intersubband plasmon dispersion relation. The parameters used are: a = .001~1-2, E 2 p - E~s = . 5 e V , t; = 13.0, d = 75)1, A 2 t, = . l e V , nr = 0.55 x t0rcm - l . the Coulomb coupling becomes mainly through the Umklapp process , which is much stronger than that of the ls ---* 2s case. An important aspect of anisotropy in the present case is that there are two plasmon branches corresponding to the to tile ls ---+ 2px and ls --~ 2pu transitions respectively. In our calculation, they are degenerate in the diagonal approximation which excludes the coupling between these two modes. This degeneracy may be lifted by including the off-diagonal terms of the secular equation. In conclusion, we have examined the plasmon excitations of a multiwire superlattice having finite barrier height, focusing particularly on the inter-subband modes. In our calculation, we considered both the subband dispersion in the transverse direction and the wave function overlap features. We also explicitly showed how the symmetry of the Wannier wave functions will affect tile corresponding plasmon modes. Such symmetry properties of the intersubband plasmons will undoubtedly be manifested in experimental studies as some of the modes have much weaker oscillator strengths since they involve higher multipole transitions.

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[3] W. Hansen, M. Jorst, J.P. Kotthaus, U. Merkt, Ch. Sikorski and K. Ploog, Phys. Rev. Lett. 58

2586(1987), [2] U. Mackens, D. Heitmann, L. Prager, J.P. Kottahaus and W. Beinbvogl, Phys. Rev. Lett. 53 1485(1984),

[4] T. Demel, D. Heitmann, P. Grambow and K. Ploog, Phys. Rev. B38 12732(1988),

Vol. 77, No. 5

INTERSUBBAND COLLECTIVE MODES OF A MULTIWIRE

SUPERLATTICE

335

[5] W.M. Que and G. Kirczenow, Phys. Rev. B37 7153(1988),

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Phys. Rev. B36

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