Hole relaxation process in quantum wells

Superlattices

155

Vol. 17, No. 2, 1992

and Microstructures,

HOLE RELAXATION PROCESS IN QUANTUM WELLS T. Uenoyama*t

and L.J. Sham?

*Central Research Laboratories, Matsushita Electric Industrial Co., Ltd. Moriguchi, Osaka 570, Japan ?Department of Physics University of California, San Diego La Jolla, CA 92093 (Received 19 May 1991)

Hole momentum relaxation times for two channels: namely, conserving and reversing parity of hole subband states has been evaluated, taking into account the deformation potential and piezoelectric scattering by acoustic The quantitative evaluation, in which parity-conserving hole phonons. relaxation is dominant, is consistent with the empirical value of the relaxation time ratio, deduced in our theory of luminescence polarization.

1. Introduction

In this paper, are evaluated

We have minescence doped

recently

quantum

of quantum

So far,

sumption

is the

of the complete

excited

holes,

tum number

since

sponds

namely

crucial

point

process

tum

the usual

of reversing complete explanation quantum

parity, times,

has been

spin relaxation,

the subindex

of the wavefunctions.

wells.

band

To charof

In the hole and of pre-

states.

The in-

< 1, has led to the in the n-doped

state

subband

is characterized

index

V, in-plane

Hole relaxation or reversing scattering culations transitions, are labeled

proceeds

the parity processes

tion-potential

p refers

$02.00/O

is

to even

($1)

and odd

(-1)

Thus,

each

valence

sub-

uniquely

via two channels,

of the subband

and piezoelectric

with

Here, two by deformaand the calfor three

1. The subbands

h or e, depending

hole levels at the r valley, is an intra-subband

states.

scattering,

p.

conserving

times are performed

in Fig.

of the

kll and parity

of holes are considered

of the relaxation as shown

in terms

wave vector

in Fig.

on the heavy

1

or light

A (lh + lh) B (I! + Ih) and

respectively.

transition,

C (2h -+ le) are inter-subband

+03

where

(J = !, m = &i,&f),

(2.3)

parity

one, for the scattering

well [2].

0749-6036/92/020155

degeneracy

where

the two phenomenologi-

reversal

Time

states at the rs valley,

subband

is fourfold

investigated

version

i.e. 7+/r_

The valence there

spin relaxation

is for the scattering

of the polarization

Relaxation

2.

L

is an extended

of hole Subband

the

in Ref. 1.

The

7+ and T_ have been adopted,

the parity

confirming

time ratio

hole

a new quan-

and the latter

phonons,

corre-

relaxation.

has been introduced.

time

of the relaxation

quan-

uniquely,

process,

relaxation

the parity

states

as-

and heavy

the incomplete

p, which

relaxation

cal relaxation the former serving

subbands

by acoustic

value

the hole relaxation

holes in quantum

parity

space

momentum

is that

scattering

in the valence

essentially

to the hole momentum

the hole subband

number

wave vector light

times and piezo

of the photo-

is not a good

between

it has derived

role

quantum

common

spin relaxation

in our theory

of the photo-excited acterize

photo-

Hole spin relaxation

in the valence

and that

In particular,

and

electric empirical

relaxation

of deformation-potential

led to spectra

natural

at finite in-plane

states.

has

polarization

in n-doped

the hole spin

due to the mixture

subband

theory

and un-

plays a very important

polarization

there

of the lu-

n-doped

this

wells.

hole spin relaxation

to the luminescence

band

[l] and

of the experimental

[2] in all type

wells.

a new theory

for p-doped,

wells

an explanation excited

proposed

polarization

the hole momentum

in terms

and

transitions.

The relax-

0 1992 Academic Press Limited

156

Superlattices and Microstructures,

which,

being

a long-range

agonal

terms

of the interaction

factor

lh

potential,

ei4 is the appropriate

both

plane

1 shows

piezoelectric

wave of the

the

occurs

calculated

lattice values

transitions

of the hole

r* is given

by Fermi’s

The

coefficient.

displacement. of the

Alh --t Ih

and inter-subband

Ei,,t.

In

are only considered

Table 1. The ratio of hole relaxation

Fig. 1. Intra

in the di-

Hamiltonian

qz dependencies

calculations,

in the

Vol. 11, No. 2, 1992

Table

ratio

r+/r_

for

times. T+/T.

Bli! -+ lh

C2h + lt

deformation

potential

0.080

0.73

0.52

piezoelectric

coupling

0.005

0.14

0.45

relaxation. ation

times

Golden

of the two channels

the three

transitions.

calculation,

Rule:

the ratio

the weak J(%(kll)

the scattering

either using

Hamiltonian,

acoustic

the initial

to be at the I valley

obtained,

the axial

state

replacements

subband

approximation Luttinger

the

state

is

states

are

by the fol-

Hamiltonian

in

holes

in both

scattering.

gives This

,

DdTrs’

=

smaller

means

relaxation

transition

into account potential

the empirical

where

D, are the deformation

strain

tensor,

to cancel

the ratio

in

T+/T_

as 0.62 [6], taking for the deformationis smaller deduced

in Ref. 1. Therefore,

than

ours,

by fitting

to

it is confirmed

value of the relaxation

time ratio,

we

reasonable.

being

related

mode

the assumption

of acoustic

tion confirms

displacement.

which

phonon.

Here,

we

in the subband terms

between

calculation, the matrix

el-

of Hint in Eq. (2.4).

For the piexoelectric gives rise to an electric

scattering, potential

the lattice

vibration q:

phonons.

the stronger

is consistent

luminescence

and piezoelectric This quantitative

parity

conserving

with our proposed

and conscat-

calcularelaxation,

theory

[l] of the

polarization.

Acknowledgements This DMR

[5] at wave vector

the ratio between parity-conserving hole momentum relaxation times,

by acoustic

and si,j is the

of D, = DL, which is the same

approximation

the interference

potentials to the lattice

Conclusion

the deformation-potential

terings

ements

Recently,

value

elements

of the two channels

+ &,“, - 2&,X,) ,

sidering

as the axial

the off-diagonal

This

[l], is quite

r+/r_

that

close to the value, data

heavy

The piezoelecof the ratio

B was calculated

scattering.

is rather

lh sub-

-;

the deformation-potential

the full qr dependence

We evaluate parity-reversing

the

values

process.

hole

: and

than

the

3.

adopted

=

1.0, due to

in the

Hamiltonians.

the mixture

that

-+,(E:,

X indicates

between

in the Hint enhance

adopted &

in the

= 2.1,~~

is much less than hole mixture

coupling

transitions

experimental

=

7+/r_

interaction

scattering

for the three

but

Eq. (2.1).

P

we used

yi = 6.85,~~

and heavy

and no direct

in the

141.

Hi,,t is given

in the

wave

To simplify

light

band tric

describing

or the final

and both

(2.4)

dimensional

phonon.

For deformation-potential, lowing

+ 911) - fiw(q)),

and q is a three

process

of the emitted

calculation, taken

- E&l1

Htnt is the interaction

vector

parameters,

2.9, Dd = 4.8 eV and D, = 3.8 eV. In the intra-subband transition,

where

The

are as follows:

work

is partially

88-15068.

discussions. S. Hatta,

supported

We thank

One of us (T.U.) and H. Kotera

Dr.

by NSF U. tissler

is grateful

for their

Grant

No.

for helpful

to Y. Fujiwara,

encouragement.

Superlattices

157

Vol. 11, No. 2, 1992

and Microstructures,

References

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and

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ConThes-