Electron g factor in quantum wells determined by spin quantum beats

Solid State Communications, Vol. 93, No. 4, pp. 313-317, 1995 Elsevier Science Ltd Printed in Great Britain 003%1098195 $9.50+ .OO 003x- IOYX(Y4)007X4-5

Pergamon

ELECTRON g FACTOR IN QUANTUM R. M. Hannak, Max-Planck-lnstitut,

WELLS DETERMINED

hl. Oestreicht,

A. P. Heberle’

fiir Fcstktirperforschung,

BY SPIN QUANTUM

BEATS

and W. W. Riihle

D-70.509 Stuttgart,

Germany

I<. K6hler

Fraunhofer-Institut (Received

fiir Angewandte

2X September

Spin quantum

beats

toluminescence a function

lYY4. accepted for publication

are detected

netic field perpendicular

Festkiirperphysik,

25 October

lYY4 by J.Kuhl)

quantum

wells in a mag-

in GaAs/Al,Ga,_,As

to the growth

spectroscopy.

D-79108 Freiburg

direction

We directly

by picosecond

determine

the electron

of well width from 1 to 20 nm with unprecedented

quantitative

comparison

Keywords:

A. quantum

with theory

time-resolved

pho-

Landd g factor as

accuracy

which makes

possible.

wells. D. spin dynamics,

E. time-resolved

optical

spectro-

scopies, E. luminescence

1. Introduction

tron wave function

into the barrier which strongly

ifies g:, and this penetration The dependence

of the electron

carrier

confinement

gained

interest

cally3.

The g factor of electrons

quantum

in quantum

experimentally’*’

wells is of special

sign at a well width ods including

electron

and resonant

Raman

perimentally

determine

these methods

of about

as it changes Various

spin resonance4s5,

its

interpretation

of the data is often complicated

by exci-

tonic effects.

Although

the general

widths

< 4.5 nm, although

of special interest

trend

a quantitative

was not possible.

energy

No data

Energy

levels separated

oscillations

states

coherently.

the Larmor angular

light, when both A magnetic

B’ = B e’, applied on a semiconductor ting of the spin states

and at low den-

by an energy difference

of the emitted

are excited

spin quan-

which allows the direct

of g,’ with high accuracy

produces

field

a split-

xzl by AE = tLw~ with WL being

frequency

.

WL = g:pBB/ft

was already

(1)

comparison Excitation

exist for well

these small well widths

since it is the penetration

using electron

has been developed,

AE produce

However, and the

with theory

sitie#:

meth-

Hanle effect’

wells.

a novel method

determination

to high densities

shown by these experiments

tum beats

have been used to ex-

g: in quantum

are restricted

Recently,

in GaAs/Al,Ga,_,As

5.5 nm.

scattering’

recently

as well as theoreti-

interest,

mod-

for small well

widths.

Land6 g factor g: on wells (QWs)

is strongest

with circularly

perpendicular

are

of the elec-

field cre-

ates carriers

with unequal occupation

of the spin states

where z is the excitation

and observation

and x;,

coherent

The spin states superposition

simultaneously

excited

of x:,

relaxation.

Therefore,

$

can be described provided

both states

by a short laser pulse.

tial part of the wavefunction

313

light in the direction

of the magnetic

xt

direction. * Current address: Hitachi Cambridge Laboratory, Madingley Road, Camridge CB3 OHE, United Kingdom ’ Current address: Materials Department, University of California Santa Barbara, CA 93106

polarized

to the direction

as a are

The spa-

suffers a very fast phase

the spin part can be written

sep-

314

ELECTRON

arately.

The time evolution

in z direction

g FACTOR IN QUANTUM

(2)

.

+ i,y~sin(w~t/2)

(3)

and sL = -$.

hence oscillates

These

in a semi-classical

oscillations

picture

the spins around

as the Larmor

by a short

varying circular

emitted

s, = +f

precession

of

stat in a superconducting

laser pulse therefore

samples

a repetition

after ex-

shows a time-

In our experiment bination

the luminescence

of excitons.

The magnetic

using the nonlinear

laser light. spectrometer

interaction,

‘*’ Therefore,

and hole spins.

frequencies

should

be observable:

which couples the two oscillation

spin and another

the hole spins.

However, the spin splitting

one from oscillations

Consequently,

signment

is confirmed

nescence.

of the electron

luminescence

Additionally,

is observed

bination

screened

is observed.

measuring

Electron

spins.

frequencies

density,

lumi-

recom-

therefore,

beat from exciton

the influence this method

GaAs/Al,Gal_,As

quantum-well

samples,

Al-concentration

set of QWs with different

consists

thicknesses,

is dispersed

in a 0.32 m

with a spectral

and temporal by a streak

readout.

1 shows

the temporal

evolution

of the PL

pronounced precession

laser pulse at 733 nm. 400 Won-*.

is about

oscillatious

The oscillation wide

as a consequence

of the electron periods

Excitation

The PL signal shows of the Larmor

spins in the magnetic

are different

due to the depen-

on the well thickness:

wells g; 2: g:(GaAs)

field.

= -0.44.

For very

As the well width

becomes smaller gz increases and finally approaches the

for Al-contents

which is +0.39, +0.46,

x=0.27,

x=0.3,

and x=0.35,

of the excito a set of

1.51

each with a

in particular

also

7 A

c)

1.52

b)

B k I:

1.54

a)

1.56

E

are used in our experiment:

!

I

0

sample

wells with nomi-

20 nm, 10 nm, 5 nm, and 1.8 nm sepof 30 nm Al~.~sGao.ssAs. Sample

of 5 GaAs quantum

hand-

to reflected

of the

in the barrier and each with a

of 4 GaAs single quantum

by barriers

with opposite

of 0.5 nm and 10 ps, respectively,

by a short

intensity

1.58

arated

(PL)

A at 4 Tesla after exci-

and +0.55

recombina-

2. Experiments

A consists

The laser

c) the 10 nm well of sample

s

nal thicknesses

and detected

doubled

b) the 20 nm well, and

z

samples

laser with of the small-

with respect

electron g-factor of Al,GaI-,As,

although

very thin QWs.

Three

direction

The luminescence

dence of the g factor

frequency

i.e. where exci-

We applied

including

for the

and band-to-band

tonic hole is small.

different

sam-

allows the direct determination

g factor gi, because

Ti:Sapphire

P-Barium-Borat.

polarization

The

by picosecond-

from a) the GaAs substrate,

tation

This as-

as for the exciton

This method,

the quantum

tion luminescence,

of the

in p-doped

the same oscillation

at high excitation

tons are completely

direction.

the photoluminescence

with two-dimensional

Figure

of

in our experiment.

by measurements

ples, which show the same oscillation band-to-acceptor

resolution camera

of the heavy

the oscillations

hole spins is too slow to be observable We only see oscillations

direction

field

one from oscillations

of the electron

holes is very small.

in backward

is due to recom-

to break up the exchange

crystal

polarized,

of circular

field is strong enough

electron

to the growth

in growth

wells the laser light is frequency

light is circularly

edness

The magnetic

rate of 80 MHz. For excitation

is detected

polarization.

in a He gas flow cryo-

magnet.

perpendicular are excited

est quantum

field (x-axis).

in z-direction

in Voigt configuration

laser pulses from a mode-locked

can be understood

the axis of the magnetic

The photoluminescence citation

between

by 49.2 nm Al,-,.~~Gtao.~~As. The samples

nm, separated are mounted

is applied The spin orientation

32.8 nm, 16.4 nm, 8.2 nm and 1.8

wells with thicknesses

Y,(r) = (X;eitWL/s)’ + X++e-+~I’)t)/&

Vol. 93, No. 4

Alo.sGao.,As, and sample C consists of 4 GaAs quantum

of the spin state occupation

is then given by6

= x,+cos(w~t/2)

WELLS

wells with thicknesses

nm, 10 nm, 5 nm, 2 nm! and 1 nm, separated

B 20

by 30 nm

FIG. 1. Photoluminescence

500 Time [ps]

1000

of a) the GaAs substrate,

b)

the 20 nm well and c) the 10 nm well of sample A at 4 Tesla after excitation wavelength

by a short

of 733 nm.

laser pulse with an excitation

Vol. 93, No. 4 lo

respectively. material

ELECTRON The sign of the g factor

and in the barrier

consequence,

material

is opposite.

curve)

from the 5 nm QW

and the 10 nm QW of sample

Tesla. The frequencies

of the spin oscillations

the same for both QWs despite

B at 14 are about

the large well-width

dif-

The reason for this is, that the g factor changes

its sign between

these well widths.

For the specific thick-

nesses used in figure 2 the absolute however,

As a

where the

in figure 2, which

shows the decay of the luminescence

ference.

in the well

there must be a well thickness,

g factor is zero. This is demonstrated

(upper

g FACTOR IN QUANTUM

the sign is opposite.

cannot determine

value of g: is equal,

In our experiment,

we

dependence

WELLS of g: is caused by the modification

ergy band-structure effective

masses

by the magnetic

of the en-

field:” energy gaps,

in the well and in the barrier change

with magnetic

field as well as the distribution

wave function.

It is rather difficult to quantify the con-

tributions

of these effects.

In our experiment,

of the

we use

only the data at low magnetic fields to determine Table I compiles

gz.

the various values of g: in the QWs

as obtained

from fits to the linear portion

dependence

of the oscillation

of the data is 5 f0.007

frequency.

of the field The accuracy

for this excitation

which provides oscillations

wavelength,

with large amplitude.

the sign of g:, we hence observe the Penetration

same oscillation frequencies for both well widths. In order to improve accuracy, we determine

of the electron wave function into the

barriers strongly increases for well widths < 5 nm , vigthe os-

orously

affecting

the g factor.

3 In our experimental

cillation period for various magnetic field strengths be-

setup, these thin QWs can only be excited by frequency-

tween 1 and 14 ‘I&la. Figure 3 shows the dependence of

doubling

the oscillation

frequency of the bulk-luminescence

the luminescence

and

of the 20 nm, 10 nm, and 5 nm QW

tation

of the Ti:Sapphire

wavelength

laser light. We use an exci-

of 410 nm and an excitation

density

of about 80 Wcmm2, i.e., carriers are excited in the bar-

of sample B on the applied magnetic field. The dashed

riers. As a consequence,

lines in figure 3 give the field dependence

is very small, which strongly reduces the amplitude

equation

(1) with g: = -0.44,

respectively.

-0.16,

to

and +0.16,

The oscillation frequency increases linearly

for small magnetic fields. spin-splitting

-0.34,

according

This

proves,

that the electron

is even at low magnetic fields always larger

than the exchange coupling between electron and hole spin. Otherwise, a transition between the different coupling regimes should be observable. data at high magnetic

The accuracy of these data is therefore strongly reduced and the experimental

error of the g: value is about

kO.06. The results of these measurements

are summa-

rized in the lower part of table I. 0.08

The experimental

of g:. This field

of

the spin oscillations observed in the photoluminescence.

I

I

I

I

I

I

I

12

14

A Bulk GaAs

fields show a slightly sublinear

behavior, indicating a field dependence

the degree of spin-polarization

0

20nm

Well

4

6

.’

0.06

0.04

0.02

0.00 Magnetic -100

0

100

200

Time

300

400

500

[ps]

FIG. 2. Decay of the luminescence from the 5 nm well (upper curve) and the 10 nm well (lower curve) of sample B at 14 Teela.

FIG. 3. Dependence

6 Field

10

16

[Teslo]

of the spin-oscillation

frequency

on

field for the substrate luminescence and the luminescence from the lowest 3 quantum wells of sample A; dashed line: oscillation frequency dependence expected for field-independent g factors of -0.43 (bulk), -0.34 (20 run well) and f0.16 (10 nm well and 5 nm well). the applied

magnetic

316

ELECTRON g FACTOR IN QUANTUM WELLS

TABLE I. Electron Landk g factor as determined the oscillation frequency of the spin quantum beats.

xezc

Nominal Well Width 20 nm 733 nm 20 nm 16.4 nm 10 nm 10nm 8.2 nm 5 nm 5 nm 410 nm 1.8 nm 2 nm 1.2 nm 1 nm

Sample Al-Content A B C A B C A B C B A B

x=0.35 x=0.3 x=0.27 x=0.35 x=0.3 x=0.27 x=0.35 x=0.3 x=0.27 x=0.3 x=0.35 x=0.3

from 0.6

of an electron

-0.348 rt 0.005 -0.336 zt 0.005 -0.310 f 0.005 -0.186 f 0.005 -0.164 f 0.005 -0.129 f 0.005 0.141 f 0.007 0.164 f 0.007 0.43 i 0.06 0.46 f 0.06 0.485 & 0.06 0.48 f 0.06

in a quantum

width L is in first approximation

c

0

Sample

-

Sample

A B

n

Sample

C

9’

3. Discrlssion

The g factor

Vol. 93, No. 4

(one-band

well of

approxima-

0.2 : t !z 0,

0.0

-0.2

-0.4 I 0

I

1

0

I

J

5

10

15

20

25

Well Width

[nm]

FIG. 4. Dependence of the electron g factor on the well width; dots, triangles, and squares represent the values obtained from the quantum beat frequencies in samples A, B and C, respectively. Dashed line: one-band approximation for x=0.35 ignoring anisotropy. Solid lines: g factor calculated in the Kane model perpendicular and parallel to the sample growth-direction.3

tion) given by3

g’ = g(O)+ Ag

(4) described

above

(one-band

and Kiselev3 applied

with

approximation).

a more sophisticated

on the Kane model to calculate

g(O)= gA

<

@A > +gB < @g >

Ag = h;41i; < @,, > +h;k:,

(5)

< 0~ >

(6)

g(O) is composed

the well material weighted

by the g factors

(gA) and of the barrier

material

(gB),

by < @A > and < 0~ >, the probabilities

finding the electron

in the well or in the barrier,

tively. The second term Ag describes

of

respec-

the contribution

second order terms of the Zeeman interaction

of

of

operator,

Figure 4 shows a comparison the model

calculations

the barrier,

respectively.

The coefficients

hf and hf are

obtained

from k . p theory3s’1*12. This simple approach

neglects

the energetic

and heavy-hole energetic

splitting

tor in direction the growth structures,

between

leads to an anisotropy

parallel

direction.

triangles

give the results ing to eq.(4),

of g: by anisotropy

by the simple approach

our data and

above.

The experi-

respectively.

of the one-band

measure-

calculated

Dashed

approximation

for Al-content

gL to the growth

lated in the framework

3 The

parallel gll

direction

of the three-band

lines

accord-

x=0.35.

solid lines give the g factors in the directions

as calcu-

Kane model

by Ivchenko and Kiselev3. In the experiments

by Dobers et al.’ and Sapega et

et al.’ g1 was determined.

to

between

A, B and C are given in figure 4 as

and squares,

of the g fac-

g1 - gll is not small compared

adequately

dots,

described

al.2 gll was determined,

(gl)

including

from our spin-oscillation

However, the

3 In the case of GaAslAlGaAs

3 The modification

be described

the light-hole

(gll) and perpendicular

the difference

to the g factor. cannot

splitting

ground state of the QWs.

ments from Samples

and perpendicular

where kA and ks denote the wave vectors in the well and

the g factor

anisotropy.

mental data obtained The first term

Ivchenko model based

in the experiments

by Snelling

However, scattering

of these

data is of the order of or even larger than the theoretically expected obtain

splitting

of gll and 91. In our setup we

g1 as the magnetic

lar to the sample

field is applied

growth-direction.

data are in excellent

agreement

perpendicu-

The experimental with the values for g1

Vol. 93, No. 4 as calculated

ELECTRON by the Kane model3 proving

of this model.

The theoretical

lated for an Al-content of g: for x=0.27 imation

x=0.35.

and x=0.3

values

are only calcu-

However,

an estimation

with the one-band

approx-

deviate

from the g factor for x=0.35

than 0.03 for well widths Al-concentrations

increase

> 5 nm. The influence in the barrier

for well widths

of the electron-wave dominant.

< 5 nm,

function

For the smallest

g: should approach

should

well widths

which are 0.39. 0.413, and 0.54 for x=0.27, respectively.

‘O Unfortunately,

thicknesses, of these continuum the accuracy

of the barrier

becomes

material,

0.3, and 0.35,

We have measured

of GaAs/Al,Ga,_,As

nescence widths

magnetic

quantum

of the oscillation

field allowed the accurate g factor as a function

tron g factor was determined where penetration strongly

beats in photolumiwells with well

varying from 1 to 20 nm in magnetic

14 Tesla. The variation

electron

spin quantum

are in excellent from a three-band

agreement

determination

of well width.

with of the

The elec-

for very small well widths,

of the electron

affects the g factor.

fields up to

frequency

wave into the barrier

The experimental with calculations

results obtained

k . p model.

at these small well

Excitation

uncertainties well below

would be necessary

of our measurement,

with the current

4. Summary

we can not see a clear

Al-contents

values.

strongly

the values of

due to the large experimental

particular

of dif-

as the penetration

into the barrier

the g factor of the barrier

trend of g; for different

by less

317

WELLS

the accuracy

shows that the values of the g factor for these

Al-contents

ferent

g FACTOR IN QUANTUM

t,he

to improve

which is not possible

laser system.

Acknowledgements - We would like to thank R. T. Harley and R. H5pfel for helpful discussions, and K. Rother and H. Klann for technical assistance. The financial support of the Bundesministerium fiir Forschung and Technologie, the NATO Science Committee, and the DAAD is gratefully acknowledged.

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