Pergamon Solid State Communications, Vol. 94, No. 5, pp. 397400, 1995 Elsevier Science Ltd

printed in Great Britain

0038-1098(95)00061-5 0038-1098/95 $9.50+.00

INTERSUBBAND EXCITATIONS OF QUASI-ONE-DIMENSIONAL ELECTRON SYSTEMS

IN A I~IAGNETIC FIELD

Seiya Kumada, Seiichiro Suga and Ayao Okiji

Department of Applied Physics, Osaka University, Suita Osaka 565

(Received 24 October 1994 by H. Kamimura)

Intersubband excitation spectra of quasi-one-dimensional electron systems in quan- tum wires are calculated by the perturbation expansion with respect to the Coulmb

interaction between electrons in the presence of perpendicular magnetic fields. We

find the peak structure at the upper edge of spectra of the single-particle excite

tion, in addition to the main peak corresponding to the magnetoplasmon. With

increasing the magnitude of magnetic fields, this additional peak separates from the

spectra of the single-particle excitation and the intensity of this peak is gradually

decreased. We consider that the additional peak is connected with the attractive

interaction between electrons which appears only in the presence of magnetic fields.

Keywords: A. nanostructure, A. semiconductors, D. electron-electorn interactions,

E. inelastic light scattering

Recent progress in semiconductor fabrication enables

the realization of quantum wires in which free electrons

are confined to quasi-one-dimension (QlD). Intersub-

band excitation spectra of QlD electron systems in GaAs

/ AlGaAs heterostructures are measured by resonant

inelastic light scattering experiments, [I, 21 which have

been used for studies of the spectra in quantum well.

[3 - 51 From these experiments, the wave vector disper-

sions of single-particle excitations (SPE) and collective

excitations are investigated. The theoretical study for

the intersubband excitation spectra of the QlD elec-

tron systems in magnetic fields predicted the existance

of the magnetoroton, [S] which have already been found

in the two-dimensional (2D) electron systems in mag-

netic fields. [7 - 91 The above theoretical study [6] also

brought about the many extra collective modes in addi-

tion to the main peak of the magnetoplasmon (MP). In

this letter, we calculate the density correlation function

in the QlD electron system by the perturbation expau-

tion with respect to the Coulomb interaction between

electrons in the lowest and the first excited subbands

for the various values of fields. We report that one ad-

ditional peak is obtained at the upper edge of the SPE

spectra besides the main peak corresponding to the MP.

To describe electron states of the quantum wire, we

consider a 2D electron system on zy-plane with a con-

fining potential V(y) = m’gy’f2. The Landau gauge

(-By, 0,O) is used in the case where the external mag-

netic fields are applied to the z direction. Eigenstates

and eigenenergies in this system are given by

kk(~, Y) = -&+ xn(y - Yk) ,

where Yk = wc12kli12, fl = dm, 1 = ,/m., . . X = (sl/w,)l, L, is the length of the electron system

along the r direction, w, the cyclotron frequency given

by eBfm’, m* the effective electron mass, and x,, the

wave function of a harmonic oscillator. Using the above

basis, the Hamiltonian for the electron system of the

‘quantum wire is written as

+v 1 -

x+&L,n,(k, qtrr k - qX,,,Jk’,qy, k’ + qz)

@!, (k ub,?,(k’, u’)Gs(k’ + qz, a’)a,,(k - qs, 0) ,

(3)

J,&k, qu, k’) = /_I x..(?/ - yk) eigyp xm(?/ - yk’) dy , (4)

397

398 QUASI-ONE-DIMENSIONAL ELECTRON SYSTEMS Vol. 94, No.5

where u(q) = e’/(2&(=), S is the area of the sys-

tem, E the background dielectric constant, and oi(L,u) t ( ai (k,o) ) the annihilation (creation) operator of the

electron in i-th subband with the wave vector k and

the spin u. The second term of eq.(3) expresses the in-

teraction between electrons. We focus our attention on

elementary excitation spectra from the lowest subband

to the first excited one without spin-flip. In this case,

the effect of the Zeeman splitting need not be taken

into account. We assume that only the lowest subband

is occupied by electrons. Spectral intensities (transition

probabilities) are given as the product of the Raman

tensor and the imaginary part of the retarded density

correlation function. We do not consider the wave vec-

tor dependence of the Raman tensor, since the transition

is expected to occur mainly in the vicinity of the l? point.

Accordingly, spectral intensities are given by

I(fiw) a -Im c Gol:o,o~(k, k’ : iw -+ fiw + is), (5) k,k’,o,o’

where Tw is excitation energies, 6 the linewidth. G,,l:,_l

(k, k’ : iw) stands for the Fourier transform of two-

particle Green’s function which is written by,

Go+/(k, k’ : T) = - < T, c&k, u : +l(k + &,a : T)

x&k’ + Q, o’)ao(k’, a’) > , (6)

where < a-- > means the thermal average, ai(k,u :

7) (i = 0,l) is the Heisenberg representation of the an-

nihilation operator, T, the time ordering operator, Q

the scattering wave vector, and the suffix O(1) stands

for the lowest (first excited) subband. We assume in

this letter that QID electron systems in quantum wires

are discribed by the normal Fermi liquid and evaluate

eq.(6) by the perturbation expansion with respect to the

interaction part of eq.(3). The equations obtained are

as follows

c Gol:,,,~(k, k’ : iw) k,k’,m,r’

= c GfL,(k : iw) A~l,ti(k : iw) , (7)

&.,(k : iw) = 1 + c V(Q : k - k‘) do~~p,(k’ : iw) k’,r’

x hol:,~(k’ : iw) , (8)

x ;( J+12Q2 + qy”) cos('?vyk), (9)

where X’ = w&l, G$iC(k : iw) = f(i!&k)/(iw + && -

&,k+Q), f(&,k) is the Fermi distribution, and hol,,(k :

iw) the vertex function. It is noted that in the absence

of magnetic fields, the series obtained by eqs.(7)-(9) give

those in the usual random phase approximation. The

interaction shown in eq.(9) depends not only on Q but

also on k in magnetic fields, because the wavefunction

along the y direction is shifted to the edge of the wire

by magnetic fields. In order to treat the k-dependence

of the interaction, V(Q : k), precisely, we solve eq.(8)

numerically. Substituting Asr,Jk : iw) into eq.(7) and

using eq.(5), we can calculate the spectral intensity.

The numerical results are shown in Fig.1. The exci-

tation energy, tiw, is plotted by using trw, as the unit,

where fiw, is the excitation energy between the lowest

and the first excited subbands in the absence of mag-

netic fields. We set n = 3.78 x 107m-‘, tW, = 2.5meV,

c = 1360, m* = O.O67m,, and Ef = 2.OmeV, where n

is the electron density, co the dielectric constant of the

vacuum, m, the free electron mass, and Ef the Fermi en-

ergy measured from the bottom of the lowest subband.

In the absence of magnetic fields(Fig.l(a)), we can see

the broad peak and the sharp peak in the spectra, which

correspond to the SPE and the MP respectively. In low

fields(Fig.l(b)), a sharp peak other than the MP comes

out at the upper edge of the SPE spectra. As the mag-

nitude of magnetic fields is increased(Fig.l(c)-(d)), this

peak separates from the SPE spectra. For B > l.OT, its

Fig.

2 3

(hdhoo)

0 1 2 3 (Who)

(ho/boo)

1. The dependence of the additional peak struc-

ture upon magnetic fields is shown. B=OT(a),

2.5T(d). tiw, = 2.5meV, I, =

x 10s8m, C/b,, = 0.01, and

temperature Tis OK.

Vol. 94. No.5 QUASI-ONE-DIMENSIONAL ELECTRON SYSTEMS

intensity decreases . In order to interpret these behav-

iors, we separate the interaction as

V(Q:~)=v,(Q)+v,(Q:~), (10)

v,(Q) =

X

v,(Q : k)

(11)

x f( X”Q2 + ‘$) [1 - c-(&,yk)l, (12)

where V,(Q) is the piece of the interaction which does

not depend on the wave vector k. It is to be noted that

the term, V,(Q : k), has a finite and negative value only

in magnetic fields. Using eq.(lO), eqs.(7), (8) and (9)

are rewritten as follows

c Gol:,,,O, k’ : id k,k’,o,o’

= E Gic(k : iw) &,,(k : iw) , (13)

A&(k : iw) = 1 + c V,(Q : k - k’) Ggc,(k’ : iw)

x iizot(k’ : iw), (14)

Giz(k : iw) = &Lb(k : iw)

1 - V,(Q) Ckl,_t d$,(k’ : iw) . (15)

As the second term of eq.(14) vanishes in the absence

of magnetic fields, the spectra without magnetic fields

is given by -Im zk,o Ggz(k : hu + ia). From eq.(lS),

it is easy to understand that the spectra in this case

consist of the SPE and the collective mode only. In the

presence of magnetic fields, we must consider not only

the k-independent interaction, V,(Q), but also the k-

dependent interaction, Vl(Q : k). Then, eq.(14) means

in the presence of magnetic fields adding the effect of the

interaction, V,(Q : k), to the spectra of the SPE and the

main collective mode given by the term, Gg:‘,(k : iw),

in eq.( 15). Therefore, the interaction, V,(Q : k), is con-

sidered to cause the additional peak. The interaction,

V,(Q : k), acts as an attractive one between the elec-

tron in the lowest subband and the electron in the first

excited subband, hence the peak structure is considered

to be connected with the attractive interaction between

electrons in magnetic fields. On the other hand, the

interaction, V,(Q) gives the spectra of the MP corre-

:b) B = l.OT

Qlo = 0.5

Fig. 2. The behaviors of the additional peak for the

various scattering wave vectors are shown. B=OT(a),

l.OT(b), 2.OT(c). trW, = 2.5meV.

sponding to the collective mode of the electron which is

shifted to the edge of the wire due to magnetic fields.

The behavior of this peak for various scattering wave

vectors is shown in Fig.2. The region of the SPE, which

is enclosed with dashed lines in Fig.2, becomes smaller

for high fileds. With increasing the magnitude of mag-

netic fields, the wave vector dependence of each sub-

band tends to be weak and subbands show the Landau-

level-like behaviors in 2D electron systems in magnetic

fields. Therefore, it is considered that the additional

peak and the SPE spectra vanishes gradually and that

at last there remains the sharp peak corresponding to

the inter-Landau-level excitation without spin-flip. The

additional peak predicted here is expected to be mea-

sured by the experiments.

Acknowledgement - We would like to thank Professor K.

Cho for useful comments and fruitful discussions.

400 QUASI-ONB-DIMENSIONAL ELECTRON SYSTEMS

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Vol. 94, No.5

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