Electronic raman scattering in quantum wells: Coupled levels in tilted magnetic fields

Superlattices

and Microstructures,

ELECTRONIC

493

Vol. 3, No. 5, 1987

RAMAN

SCATTERING IN TILTED R.

Department Ann

IN QUANTUM WELLS: MAGNETIC FIELDS

Borroff

Research

University of Physics, Louisiana 70148, Orleans,

Laboratory,

of

S.

Michigan,

A.

Greene

J. Naval

LEVELS

Merlin

The University of Physics, Arbor, MI 48109-1120, U. R.L.

Department New

R.

and

COUPLED

of New U. S.

Orleans, A.

Comas Washington,

(Received

29 June

D.C.

20375,

U.

S.

A.

1987)

We report on a magneto-Raman scattering investigation of free and donor-bound electrons in GaAs-AlxGal_ As quantum wells. For fields perpendicular to the layers, tie spectra show ls*ls’ intersubband transitions of photoexcited electrons and donor excifations. Tilted fields lead to subband-Landau level and ls’-2p coupling. Experimental results for the latter case agree very well with variational calculations. Data on combined intersubband-cyclotron resonances at arbitrary tilt angles are accurately described by expressions valid for parabolic wells. The parabolic approach is shown to provide a good approximation in situations where coupling to higher subbands can be neglected.

In quasi two-dimensional (2D) electron systems the motions in the confinement plane (x,y) and perpendicular to it are coupled for magnetic fieldsle2at angles 0 + 0 with respect to z. 9 This leads to excitations of mixed character, such as combined intersubband-cyclotron resonances, exhibiting the well-known 2-7 anticrossing behavior near degeneracy. Subband-Landau level coupling has been extensively studied for 20 systegs7formed at semiconductor heter junctions Si-accumulation layers ’ using the tetE:ique of cyclotron resonance. For small 6’s these measurements provide a determination of the energies of intersubband ns which are forbidden at Small tilt angles were also used in the far infrare g (FIR) experiments of Jarotik et al. to study crossing of donor levels in quantum-well structures (QWS’s). In this work, we report on a Raman scattering (RS) investigation of tilted-field-induced mixing in GaAs-AlxGa _xAs QWS for both free and donor-boun d electron states. Our results

0749-6036/87/050493+04$02.00/0

complement t ose obtained from FIR-ex3-b periments; transitions that are weak or forbidden in FIR dominate the Raman spectra. For donor excitations, we find good agreement between the experimental results and those of calcu ations based B on a variational approach. The data on free electrons reveal coupling between the cyclotron mode and a, + E, transitions associated with the ground and first excited subband states. This coupling has been investigated in the range of parameters where perturbation theory applies, and beyond that range. Analytical ex ressions derived for parabolic wellsl’ were found to describe extremely well the latter results, for arbitrary 8. This unlikely situation is explained by the close similarity between the coupledmode equations for parrbolic and square wells under conditions where the coupling to higher subbands can be ignored. A QWS grown by molecular beam expi taxy on (001) GaAs was studied. It consisted of thirty uncoupled GaAs-wells of thickness L w 460 A, with 125-A-thick

0 1987

Academic

Press Limited

494

Superlattices

and Microstructures,

Vol 3, No. 5, 1987

A'0

dop6

;;“=“;~;,fA”;;-Y*ithThea

concentration width of the

donor

spike was 1: L/3. Free electron excitations were investigated on the same QWS. either by photoCarriers w re generated fl excitation or by thermal ionization of donors. The laser beam used to obtain RS data also served to photopump the sample. Power densities were in the range P = 2 - 10 Wcme2, giving an vt RS cm‘“9W. free electron density P - 10 experiments were performed at fields 8 C 7 T (the maximum field provided by our split-coil superconducting magnet) and at angles G between O" and 80°. The laser energy hwL was tuned to resonate with the gap derived from the Eo+AO gap of GaAs. At this resonance, the scatconductering involving states from th f3 Data tion band is strongly enhanced. were recorded in the z(x',x')? and )i backscattering configurations, z(x',y' with x' and y' denoting the [llO] and [lTO] directions and with z normal to the The former geometry allows scatlayers. tering by charge-density fluctuations while spin-density f~yc;~a;;;;:,;:;,;llowed in the latter. differences were found between spectra in the two configurations, indicating that depolarization effects are negligible. This is consistent with our estimate for a small value of P. In Fig. 1, we show Raman spectra of the QWS for different values of T, 8, and B. Features labeled D are donor-related, which is clear from their rapid quenching with increasing temperature due to impurity-ionization (data-for G = 300). The In the spectra strongest line at 88 cm for 0 = 30° is mostly due to the transidenotes the lowest donor tion Is+ls' (1s' state associated with the first-excited subband while the weaker structures at respec132 cm -1, and 137 cm-' are due 't transito the ls+2s and Is+2p tively, The latter excitations shift to tions. higher energies with increasing 0 (constant 8) while ls+ls’ moves in the opposite direction. The assignment of these transitions is based on the comparison with calculations considered below (for As opposed to see also Ref. 12). ls+ls’, the ls+ls' and the RS-allowed ls+Zs transitions, the ls+2p+ transition cannot be seen for fields perpendicular to the layers (G = 0). The 2p+-level transforms like {x,y) and its Raman activity at 0 $: 0 is derived from the coupling with which transforms like Is', j2;I.r";:;;;;" speaking, neither the ls+Zp transitions are allowed at 4 = 0 because the states have different parities ($ is the scattering wavevector). However, it

40

0

80

160

120

RAMAN SHIFT (cm-l) Fig.

1:

Raman spectra of the structure with L = 460 A as a function of T,G, and B. Features labeled D are donor-related. Dashed lines indicate the positions of the combined intersubband-cyclotron resonances. The scattering ge"j metry is z(x',y')i, P a 5 Wcm and hwL = 1.875 eV. 0 is the angle between the magnetic field and the z-axis of the QWS.

is a matter of fact that (z)-symmetry excitations (e.g., e,+e, intersubband transitions) can be obs rved in Raman 15 spectra for 4 along z. Mechanisms I"volving ;-dependent RS are requred to 13 account for these observations. The

dashed

combined nances.

lines

in

Fig.

1

intersubband-cyclotron At

2

K,

the

Intensity

lndlcate reso-

of

the

two

,

Superlattices

and Microstructures,

Vol. 3, No. 5, 1987

7-

E

-

0

>

i2

s W

c

01234567

4

1 0 EXPERIMENT

1

. THEORY

151(T) 3:

Fig.

The

energy

as

a

Q

is

2-4

6

8

2:

Comparison

between

calculated

values

function

of

angle.

25,

1s'

at

field measured

the

center

P60

A

GaAs-AlO

The

solid

tne

eye.

are

of

for

donor

a

of

the

a

QWS. guide

to

nearly due

with to

= 300, -1 cm IS

while the

An

=

cotip!

posltlon

not

tion

rusrng

at

=

=

slrghtly

It not

53

level

transition. by

?he

line

ascrlbed

c-loiely w!th

m

f requency aair.).

The

iarger

tr,e

ma55

f,>?r eie-t;r,ns

thr

GaAc

conouctlor,

to

=

O.G7 mj

tnsn

m at

band.

=

0

which

at

a

cor-

Eol.

explain

near

are as

is

the

the

free

,:alue

of

O.C,665

We

as

a

of

results

these

m

mQ,

the

bottom

of

The

Raman

sh:ft

is

very

In

FIR

edges

15

In

= of 9,

reason

which

our 1s Of

between

to

what

has

show

in

shown

feature

srmilar

for

plotted

transltions

coupling

we

any

calculations upon

trlted

3,

scat-

Al so

($1.

the

Fig.

of

are

8"

by

8

show at

rnterestrng

wells.

8

unlike

donor-related

0

Induced

the

is

that

for

drfference.

donor

the

out h0,

do

know

Ref.

measurements of

at

mode

this

the

most

data

point

varlational

of

2p '

reasonably

which

of

III

The

in

experimental

This

function

::1cLXt

al.

peak

not

at

described

and

do

rjansi-

ontinuities 14 predicts

to a

posrtions

assignment

the

the

cyclotron

measured 2

is

layers.

that

e,+e,

peI;.":

et

QWS

might

the

dis

important

he

Fig.

where

m,(nc

measured

shows

vs.

E

on

which

observe the

The

to

Miller

,

terlng by 0 _ ,.15,:k

based.

3s

inset T,

=

is

with

is

to

peaks,

our

fo,ilnws and

b -r

modulation-doped

For

from

:C"),

did

normal

transltions

!;

4.8

band-gap

agreement

value.

supported

angles

from

parabolic

The

hn,

much

the cm

we

at

derives

56

li-

'"2 cAn ue neglected except i,4 Fljr small angles, the

eiiB;mc,

eli=ctrcn

c,+ts, '1

cm 1s

5mai;

ms:de

cyciotron

1 -

component

to 98

=

G's,

depend

of

forL~c~;tering

Inter-Landau

.lf

cyclotron

1;

due

peak

roughly

carriers.

mostly I

intensity

expected

;ow-energy

i0r

degeneracy.

h$;,

as

rdentlflcatlon

results the

P

the

the

the

increases

pnotrexclted

G

This

to

phononsj

(81

assigned

determined

(normairzed

optical

10).

to

small

does

FOI good peaks

line at

tion,

are

obtained

The a

24GaQ.76As

lines

of

theore-

2pt

energies.

data

field.

The

splitting

responds

and

ls+2p-,

magnetic

for

(Ref.

inter-

excitations

where

coupled-mode

transition

theoretical

tilt

expressions

wells

of

coupled

level

curves

the

ISI (T) Fig.

the

the

tical

0

of

subband-Landau

donors the

1s'

field. been

This

reported at

imeasured

the

496

Superlattices

energies of the coupled intersubbandcf.clotron modes for two different tilt Level-repulsion is evident in the angles. figure. 8 = 80 is within the range where perturbati n theory can be applied. The q,4 of a splitting prediction proportional to 8 for hgc = E Ihas been confirmed by inset). Deparour experiments ? see the tures from results of perturbative calculations can be seen in the data of the inset at large angles and in the results for 8 = 60°. In particular, we find that the lower branch of the coupled modes for large B's, but does not approach E considerab 9'y from that value. deviates The theoretical curves in Fig. 3 were valid for paraexpressions obtained fro TO The agreement between bolic wells. theory and experiment is quite remarkabis also the case for the asymple. This of the lower branch which :;t;;,;e;;:vns to tend to EOtcos8 at high fields. The reason why a square well can be approximated by a parabolic one is basically due to the fact that the corresponding ground and first-excited subband states are very similar. The matrix elements involved in the coupled-mode problem for the two cases differ by less one can expect that parathan 2%. Hence, bolic solutions will apply to square wells if the coupling to higher subbands A detailed analysis of is not important. the coupled equations indicates that this condition is well-fullfilled for the range of parameters in our experiments.

2.

3. 4.

5.

6.

7.

8.

9. 10. 11.

12.

13.

14.

This work was supported by the U.S. Army Research Office under Contract No. DAAG29-85-K-0175 and the U.S. Office of Naval Research.

15.

16. REFERENCES 1.

F. Stern 163, 816

and W.E. (1967).

Howard,

Phys.

Rev.

and Microstructures,

Vol. 3, No. 5, 7987

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