Electron-LO-Phonon Quantum Kinetics in quantum Dots

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Electron-LO-Phonon Quantum Kinetics in quantum Dots

K. Král, Z. Khás

To cite this version:

K. Král, Z. Khás. Electron-LO-Phonon Quantum Kinetics in quantum Dots. Journal de Physique I,

EDP Sciences, 1997, 7 (11), pp.1431-1443. �10.1051/jp1:1997139�. �jpa-00247461�

Electron-LO-Phonon Quantum Kinetics in Quantum Dots

K. Kr61

(*)

^{and Z. Khfis}

Institute of

Physics, Academy

of Sciences of Czech

Republic,

Na Slovance 2, 180 40

Prague

8, Czech

Republic

(Received

10 March 1997, revised 20

May 1997, accepted

15

July 1997)

PACS.72. Electronic transport in condensed matter PACS.73.40.-c Electronic transport in interface structures

PACS.72.10.-d

Theory

of electronic transport;

scattering

mechanisms

Abstract. Quantum kinetic equations _are solved numerically for the _process of the relax-

ation of _an excited electronic system ⁱⁿ a cubic quantum dot in _a

GaAs/ALGaI-~As

structure with the

electron-LO-phonon

interaction included and with the carrier-carrier interaction _ne-

glected.

The

theory

is

developed

within the equation of motion method for the reduced

density

matrices. It is shown that _near the low electron and hole

density

limit _a

relatively

broad inter- val of the dot lateral size may exist in which the relaxation of the electrons _may develop at _a

picosecond

time scale.

1. Introduction

Recently,

the

improvement

in the

technology

has led to _a further reduction of the dimensional-

ity

of the seIniconductor _structures

having,

besides the fundaInental

significance,

_a

large iInpact

on the

expectation

of

applications

in

signal processing.

There is _a

large

interest in understand-

ing

the

physical properties

of the heterostructures in which the electronic _wave functions _are

confined in three diInensions

(quantuIn dots).

These seIniconductor structures _are

proInissing

due to the discrete electronic

density

^of

states,

^{which makes their candidates for}

high efficiency

lasers

ill.

The

perforInance

of the semiconductor lasers is

expected

to be

seriously

affected

by

the process of the relaxation of the excited electrons to the

ground

state, which is

supposed

to be critical in

quantum

^dots

having

lateral dimensions less than 50 _nm

[2-5j.

In

particular,

earlier theoretical estimates show that unless the

separation

of the electronic energy levels is close to the energy of the

Longitudinal Optical (LO) phonon

at the _center of the Brillouin _zone

(ltuJo),

the radiative

efficiency

of the

quantuIn

^{dot would be weak} [3j ^{at least in} the

approxiInation

at which other interaction of electrons than the Fl0hlich

coupling

to the

polar optical phonons

are

neglected. Although

these conclusions appear ^to be

supported by

_soIne

experiInents [6-8],

a luminescence _was

reported [9-llj

in

samples,

in which the effect of the

phonon

bottleneck

on the luminescence should be noticeable.

Recently, photoluminescence

measurements have been

reported

in GaAs

quantum

^dots hav-

ing

lateral dimension of about 450nm

[12j.

^The

scattering

of electrons with the acoustic

(*)

Author for correspondence

(e-mail: kraltlfzu.cz)

@

Les

#ditions

_de

_Physique

₁₉₉₇

phonons

_was found _too slow to

explain

the small value of the relaxation time

(not larger

than

10ps)

observed.

Any strong

^{threshold behaviour in the}

photoluIninescence spectra, expected

to _{occur on} the basis of the reference

[3],

^{when the excitation} energy scans

through

the

optical

phonon

_energy

hula,

_was

reported

to be absent in the

experimental

data.

The

high

value of the electronic

cooling rate,

^{obtained in the Born}

approximation

_[3] under the condition of the near-resonance between the electronic energy level

separation

and the

optical phonon

_energy

hwo, brings

about _a

question

about the role of the electron _energy collision

broadening

in the effect under consideration. The electronic relaxation in quantum dots _was studied

recently

in

coInprehensive

_papers

[13,14].

The authors took into account the

electron-energy

collision

broadening

in a self-consistent _Inanner and included into consideration the electron-hole electrostatic

coupling. They

find that in

dependence

_on the

density

^of

injected

carriers the electronic relaxation time _can reach hundreds of

picoseconds

in _a broad _range of the dot size and carrier densities. This rather

rapid

electronic relaxation is found to be influenced in _a fundamental _manner

by

the

rapid

relaxation of the hole

subsystem having

rather

high density

of states.

The electronic relaxation channel due to the Fl0hlich

coupling

to the LO

phonons

is _usu-

ally

the most intensive way of the _energy relaxation in

polar

semiconductors and _even in the

quantum

^{dots leads} to _an intensive relaxation

[3],

^{at least in the Born}

approximation

to the

scattering rate, though

under _some rather restrictive conditions

imposed

on the relation be-

tween the electronic

energy-level spacing

and the

optical phonon

_energy. With the collision

broadening

included

[13,14]

^it

might

offer _a window of reasonable width in the electronic energy

scale,

_or in the scale of the dot

size,

in which the electronic relaxation

might

be intensive under suitable circumstances. Besides the collision

broadening,

indicated

by

the Born

approximation

value of the relaxation rate, ^the

rapid

electronic relaxation _may have _some additional quantum effects. As such the

electron-LO-phonon

relaxation channel alone deserves _{a more} detailed

attention.

The _purpose of this work is to

analyze

the

electron-LO-phonon

relaxation channel in _a

greater

^detail

going beyond

the

approach

of _papers

[13,14].

For this purpose the other energy relaxation channels _are

neglected

here

together

with the

coupling

of electrons to the holes in the valence band states. A

simple

model of _a cubic quantum ^{dot with}

only

two

lowest-energy

states is considered. In difference to the Monte Carlo

approach

of references

[13,14]

^the present

theory

is

developed

in the frame of the

quantum transport equations [15,16] using

^{the reduced-}

density

matrix formalism. This

approach

Inakes it

possible

to treat

siInultaneously

the effect of the electron energy collision

broadening

and the effect of coherent

exchange

of energy between the electron and

phonon subsysteIns.

In Section 2 the

quantuIn

^kinetic

equations, giving

the

electronic relaxation in

quantuIn dot,

_are forInulated within the second Born

approximation (see

Ref.

[17]), including

the self-consistent collision

broadening

contribution to the electron

energies.

The results of the nuInerical solution of the kinetic

equations,

for the _case of _a

quantuIn ^{dot with}

only

two electron energy

levels,

and _a

single

electron in the

quantuIn dot,

are

presented

in Section 3.

2. Kinetic

Equations

2,I. HAMILTONIAN. A

siInple

_case of the

quantuIn

dot will be

assuIned, naInely,

the GaAs

quantuIn

^{dot eInbedded} in the

A[Gai-~As crystal,

in which the

single-electron

states _are

represented by

their

energies

^and wavefunctions. The Inodel electronic

potential

of the

quantuIn

dot and the

corresponding

electronic wavefunctions and

energies

^{will be}

specified

in _a later section. The effect of the holes in the valence band states and the electrostatic

coupling

_aInong

electrons will be

neglected.

In this

respect

^{the model used}

corresponds

to the liInit of low

density

of electrons and holes in the

saInple

with

quantuIn

dots. The

following

_process will be considered here: _at the tiIne t

= 0 the electronic

subsysteIn

is

prepared

at _a state described

by

the

single-electron

distribution function. The electronic

systeIn

relaxes _{at t >}

0, interacting only

with the

LO-phonons.

which _are

kept

at

equilibriuIn

with the ambient lattice

throughout

the _process. The effects connected with the finite duration of the

optical

excitation

pulse

and

the coherent

effects,

which may be induced

by

the effect of the

pulse

in the electron-hole

system

in the _course of the excitation process

[15,16],

are thus

neglected

here.

The

unperturbed

electronic states, localized within the

dot,

form the orthonormal set

(in jr )

of orbital _wave

functions,

with the

corresponding energies En. Neglecting

the effect of _con- fined and interface

optical phonons,

the

system

of the

unperturbed

lattice vibrations will be

approximated by

the

LO-phonon

modes of bulk GaAs

crystal interacting

with the electrons via the Fl0hlich

coupling _[18].

The electronic

screening

of this

coupling

will be

neglected.

The

Hamiltonian of the whole

system

then is:

H

=

£ _Enc(~cn,~

₊

£ _ltuJqb(bq

₊

£ £ £ _Aq4l(n,

_m,

q) _(bq b+~) c(~cm,~, (l)

n,«

'

q m,n q «

'

where the three _{terms on} the rhs of

(I)

_are,

respectively,

the Hamiltonian of free

electrons,

the Hamiltonian of free LO

phonons

with the _energy

ltuJq

^{and the Fr6hlich}

coupling.

In

(I)

_cn,~

and

bq

are,

respectively,

the annihilation

operators

of the electron in the n-th orbital

state,

with

spin

_a, and the annihilation operator ^of the

LO-phonon

with the momentum q. The

quantity 4l(n,

_m,

q)

is the forIn-factor of the

quantuIn dot,

defined _as

4l(n,

_m,

q)

=

d~r ~fi((r)e~~~~fim(r), (2)

having

the

property

4~~ln,

_m,

qj

"

~jm,n, -qj. j3)

The

coupling

constant

Aq [18]

^is

ie

~i _~-i

~q

_"

~j _2eoV

^~~° ° '

(4)

where _q

=

)q),

_-e is electronic

charge,

_so is

perInittivity

of free space and _~c~ and _{~co are}

high frequency

and static dielectric constants, ltuJo ^is the

LO-phonon

_energy at the r

point

of the

Brillouin _zone. V is the voluIne of the

saInple.

2.2.

QUANTUM-KINETIC EQUATIONS.

The effect of the

nonequilibrium

of the

phonon

_sys-

te1n will be

neglected

here.

Thus,

the basic

quantity.

for which the quantuIn ^kinetic

equations

will be

forInulated,

will be the

density fn,~

of electrons in the state

in, a),

defined _as

fn,~

~

(c$ ~cn,~)

~

Ti(c( ~cn,~p), (5)

p

being

the statistical

operator

^{of the whole}

systeIn, Trip)

= I.

Starting

with the

Heisenberg equation

for the tiIne evolution of the operator

c(~cn,~

and

using

the

property (3)

of the forIn

factor,

_one finds ^'

j

=

-) _{L L} lAqlRel4lln>1> q)llbq b±q)Ct,«Ci,«11.

16) At the

right

hand side of this

equation

the _Inean values _are found of the

product

of _one

phonon-

and _two

electron-particle

_operators. This

equation

is _a

beginning

of _an infinite series

of

equations.

^As

usual,

the latter set of

equations

_can be solved

approxiInately by factorizing

the _mean values at the

right

hand sides in such _{a way} that _a closed set of

equations

for _a restricted set of reduced

density

matrices is obtained. In the lowest

order,

_a factorization of the

phonon-assisted

matrices

present

at the rhs of

equation (6)

can be

performed, resulting

in the

product

of the _mean value

(bq)

^{and the} mean value

(c(~ci,«). Assuming

the

diagonal approximation, namely, (cl _~ci,«)

~

6n,1(c$,~cn,~),

^the

right (and

_{side of}

_{equation (6) gives}

zero. In this

approximatiin,

the _nonzero value of the rhs of

equation (6)

would thus be

connected with the _presence of coherent

phonons ((bq))

and

by

the

mixing

of the states n

and due to the

electron-LO-phonon coupling,

manifested

by

_nonzero

quantities (c(~ci,«)

with _n

~

l. We shall

neglect

these effects and

proceed

further to consider the

higher'order

equations.

The

coInplete equation

for the

phonon-assisted density

Inatrix

(bqc( ~cn,~)

reads:

~~~~~'"~~'"~ lt

~~ ~"

_{~ ~~°~~}

~~~~~>""'"~

)lA~l4~li>'~>

~q) lfn,~(i fl,~)

+

"qlfn,~ fl,~)l

+) L lAql4~ln',m',-q)61C$,«Ci,«C$,,«,Cm,,«<I

w,~,,~,

+) ^£ lA~'l4~li'~"q')61b~q'b~~$,~°°'>"l

m',q'

£ _{jAq, j4l(n',}

n,

q')6(b+~, bqc$,

ci

al

11

~, ~,

'" '

~

£ _{jAq, 4l(1,} m', q') _(bq, bqc$

_~cm,

~)

h

, ,

' '

m ,q

+

~j _{)Aq, )4l(n',}

n,

q') _(bq, bqci

~

ci

al (7)

11

~, ~,

"

' '

The second terIn on rhs of this

equation

was obtained

by factorizing two-particle density

Inatrices into

products

of electronic and

phonon

distribution functions

fn,~

and uq =

(b(bq) (coInpare

with the

analogical

derivation in bulk

Inaterials,

Ref.

[16]).

^The

neglection

of the

last five terIns in

(7)

would represent ^the second Born

(28) approxiInation [17]

to the quantuIn

kinetic

equation

for

fn,~.

It is well known [17] that 28

approxiInation

is

unstable, giving solutions,

in which the electronic distribution function leaves the interval

(0,1)

in the _course of the time evolution. The inclusion of the last five terms allows _one to Inake the

present

level of

approxiInation

consistent. The three terIns

containing

the

syInbol ii...)

_are the

two-particle

correlation functions defined _as the difference between the

two-particle density

Inatrix and the factorized forIn of the latter. For

instance,

61b+q,bqc$,,«Ci,«I

=

16+~,bqc(,,~ci,«1- 6-q,,q61,n,uqfi,« 18)

and

61C$,«Ci,«c$~,«~Cm~,«~) ⁼

ic$,«ci,«c$~,«,cm~.«,) 6n,m~61,n~6«,«~fn,«li fi,«).

₁₉₎

The last two terIns in

equation (7)

contain the reduced

density

Inatrices which

give

_{zero con-}

tribution in the factorized form.

They

^do not contribute to the second term _on rhs of

equation

(7).

^This is due to the _zero values attributed to the _terms

(bqbq,).

It is well known that the above mentioned

instability

of 28

approximation

_can be removed upon

introducing iInaginary coInponents

to the electronic

energies

in the

equation (7),

or the collision

broadening

of the

single-electron

_energy levels

[16].

^{The Train}

steps

^of the derivation

are

following.

The

equations

of1notion _can be written down for the five

two-particle

correlation functions of

equation (7).

These

equations

_express the tiIne derivatives of the

two-particle

correlation functions in terms of _mean values of

products

of five

particle operators.

^{The latter}

five-particle

operator mean values _can be factorized to the

products

of the

phonon-assisted density

matrix

(bqc$ ~ci,«)

^and

single-particle

distribution functions. The

equations

for the five

correlation

functioni _present

_in

_{equation (7)}

can be solved under the adiabatic and Markov

approximation _[15].

The details of the derivation _are

given

in the

appendix. Substituting

the result into

(7),

the

equation

for the

phonon-assisted density

matrix

(bqc$~ci,«),

with the

energy

broadening

of the electronic

energies included,

reads: ^'

~~~~~>""'"~ h l~ ~" ~

~~~'"

~

~"'"~

_~

~~~

~~~~~>""'"~

+) _lAq jail'll'~q)lfn,~li fl,~)

+

"q(fn,~ fl,~)l' lio)

In this

equation

the

damping

factor

r~,~

was determined in the second order

approximation

to the

electron-phonon coupling

and in adiabatic and Markov

approximation (see appendix).

It

reads

r~>"

~

)LlAql~l@(~>3,qll~

q>3

~

llf3>"

+

"~q)61E~ E3

+

h~-q)

₊

_l~

_{+ "q}

f3.")61Ei EJ h~q)I' l~~)

The

singularity

ofthe

density

of states ofthe

quantum

dot leads to the

necessity

to calculate the

damping rate,

or the collision

broadening,in

_a self-consistent _manner

[13,14].

For this purpose, the formula

ill)

_can be

generalized by substituting

the Lorentzian function with a finite width

instead of the

energy-conserving

delta function. Let _us note that in references

[13,14]

the

energy-conserving

delta function is substituted

by

_a Gaussian line. In the _case of the

presently

used Lorentzian function _one obtains self-consistent

equations

^for the

damping

rate which _can be shown to

correspond

to _a self-consistent Tamm-Dancoff

[19] approximation

to the electronic

self-energy represented by

the

imaginary

part of the

latter,

taken in the

pole approximation.

Other differences between the Gaussian and Lorentzian version of the substitution of the delta function _are not considered here. It is obtained that

C~

=

~~£jAqj~j4l(I,j,q)j~

' lt

q,J

x

j /~,~

+

v-~) ' ~~>"~~ )j"~

7r

(E~ Ej

+

ltuJ-q)

+

~(r~,~

+

rj,~)

~

+(l

+ uq

fj,~)

^~

^~~~'"

^~

~l'"~ j. ⁽¹²⁾

~r

lE~ Ej

_hU~q)~ + _m

lrz,«

₊

rj,«)~

The latter forInula allows _one to calculate

self-consistently

the

damping

rates

r~,~

^{for all}

I,

solving iteratively

the set of

equations (12),

at _a

given

state of the

system

of the

quantum well,

given by

the

single-particle

distribution functions.

The

equations (6, 10, 12)

^form

together

_a closed set of

equations

for the

quantum-kinetic

determination of the time

development

of the electronic distribution

function,

the

phonon-

assisted distribution functions and the self-consistent collision

broadening

of electronic

energies, ltr~,~,

_or the

damping

factor

r~,~.

This set of

equations

will be solved

numerically

in Section 3.

2.3. SEMICLASSICAL APPROXIMATION. As it is well known

[16],

^the semiclassical

approxi-

mation _can be obtained

by solving formally

the

equation (10)

with

applying

the adiabatic and Markov

approximation.

In the

present

case these

steps

lead to the Boltzmann

equation

for

In,~

with the collision

broadening:

~

l'~

"

-'f($~~fn,« +'i($~ ^{Ii fn,«),} (13)

where

~/1[l~~ =

LLLlAql~l4~ln,1,qjj~

+>- q I#n

rn,~

₊

Ti,«

(I fi,«) "+q

~ ~

~

~~~~

x

~~ ~~

~ /~~~~~j2 ⁺

((rn,«

₊

_Fi,«)~

and

~li',f

=

LLLlAql~l4~ln,1,q)l~

+>~ q I#n

~

(Ei En

+

lt~j$~~)(rn,~

₊

_ri~«)~ ^{~'~ ~~~}

^{~ ~ ~~~~}

As it _was shown earlier

[17]

^{for the} case of the

quantum-kinetic equations

ⁱⁿ the bulk electron-

phonon system,

the _sum ~y($~~ +

~y(# gives

the

daInping

factor of the _energy level

En,

'fn,a ~'fn,a

~

~n,a. (16)

The latter forInula is

easily

verified for the

present

_case

too, coInparing

the

equations (14, 15)

with the forInula

(12).

The

equations (13-16) provide

the relation between the collision broad-

ening

^and the relaxation rate

given by

the Boltzlnann

equation. Taking

the liInits of

Tn,~

and

ri,« going

to _zero at the

right

hand side of

equations (14, 15),

the Boltzlnann

equation (13)

is obtained with the collision

integral

taken into account in Born

approxiInation,

as it _was used in the reference

[3],

^when

estiInating

the

efficiency

of the

electron-LO-phonon scattering

channel in the lowest order.

In the

special

_case of

fn,~

=

l,

with all the other _states

eInpty,

^y~,$~ is _zero. The

daInping

factor

rn,~, given by

the

equation (16),

then

gives

the initial

It

=

0)

relaxation rate,

naInely, rn,~

=

id fn,~ /dtj.

The

equation (13)

_can be used in seIniclassical calculations of the electronic

relaxation in

quantuIn

dots with electron energy collision

broadening

included.

3. Numerical Evaluations

For the purpose

of1nini1nizing

the

coInputer

^deInands several

approximations

_are Trade. The

quantuIn

^dot is

supposed

to have the

shape

of _a cube with the GaAs material inside. The electronic

potential

is

supposed

to be infinite outside the dot.

Therefore,

the

energies

of _an electron _are

~2fi2

~n

"

l~nin2n3 @

^{~~~ ~}

^7l(

^{~ 7l(~}

^(~~)

while the

corresponding

orbital _wave functions _are:

~bn>n~n~

ir) 11)

~/~

sin

i~i~ (xi iii

_sin

i~i~

x~ i)i

sin

i~i~

x~ iii

,

i18)

where _r

=

(xi,

_x2,

x3)

is the electron coordinate and _m is the effective _mass. The indices _n~, I =

1, 2, 3,.

, are

positive integers.

The

composed

index

(ni,

_n2,

n3)

_now

plays

the role of the index _n in the

equation (I). Eiii

is the electron

ground

state energy. The first electronic

excited state is

degenerate, E211

_"

E121

_"

Ei12.

The kinetic

equations

will be solved

taking

into the consideration

only

two orbital

states,

(I, I, I)

and

(2,1,1),

out of the whole _space of

possible

orbital _states of _an electron in the

dot, assuIning

that this restriction does not _cause any serious _error in the Train conclusions of this work. We _assuIne that there is _a

single

electron with

spin

_a in the dot. At the initial

tiIne oft

= 0 the

population f211«

of the orbital state

(2,1,1),

with the

spin

_a, ^is assuIned to be

I,

while the electronic

population

in the lowest _energy orbital state

(I, I, I)

with

spin

a,

fill«,

is taken to be _zero. At t > 0 the electrons _are

being

transferred to the

ground

state

by

the relaxation _process. The

optical-phonon systeIn

^{is assumed} to be

constantly

at

equilibrium

at the temperature of the ambient

lattice, TL

"

10K, ignoring

in this _way the

effect of hot

phonons.

The distribution function uq is

given by

the Bose-Einstein distribution function.

Consistently

with the

assumption

of

TL"

10K of the ambient

lattice,

the processes of the transfer of the electron from the

ground

state to the orbital states

(1, 2,1)

and

(I,1, 2),

in the process of

absorption

of _an

optical phonon,

will be

ignored.

Therefore _{we assuIne}

f211«

+

fill«

" I in the _course of the relaxation. In the

following

_we

keep

the index _a in

fn~n~n~~,

to

keep

record of the

fact,

that

fn~n~n~~

e

(0,1).

The

dispersion hwq

of the LO

phonon

is assuIned

isotropic, given by

the forInula

h~°q

"

Fo

+ AF _cos

l~~

~B ^'

(~~)

approxiInating

the Brillouin _zone

by

the

sphere

in the _q-space, with the radius

qB " 1.09 _x

10~°1n-~

We take

Fo

" 5.219 _x

10~~~

J and AF

= 5.79 _x

lo-~~ J,

to

approximate

the well-known

dispersion dependence

of the

LO-phonons _[20].

The Inaterial

paraIneters

^of

GaAs _are taken _over froIn reference

[21].

The

probleIn

under consideration

requires

that the

daInping

factors

r~,~

^be deterInined froIn the self-consistent

equations (12),

rather than froIn the first-order

expression (11).

The

iInpor-

tance of the

higher

orders of the

perturbation expansion

of

r~,~

ⁱⁿ powers of the electron-LO-

phonon coupling,

_can be

clearly

_seen in the process of the iterative solution of the self-consistent

equations (12). Therefore,

the nuInerical results

presented

in this paper are

coInputed

with the self-consistent solution for the

daInping

^factor

r~,~

included.

The seIniclassical relaxation rate d

f211«/dt, given by

the

equation (13), coInputed

at the initial instant of the

tiIne,

t ₌

0,

is

presented

in the

Figure

I. This rate is

displayed

_{as a}

function of the lateral size d. The Inaximuln of the rate is found at about d

= 21.7nm. At

this distance the electronic

energy-level separation E211 El

ii

equals

the zone-center value of the

LO-phonon

_energy

hwo.

The value of the relaxation rate is of the order of 10~~

s~~

in the

range of d from about 20nm to about 23nm. This range of d

corresponds approxiInately

to the _{energy range} froIn 32 to 411neV of the

energy-level separation E211 El

_ii

(see Fig. I).

Let

us reInark at this

point,

that the

scattering

rate calculation

[13,14],

based _{on a} self-consistent determination of the

electron-energy

collision

broadening

and the electron-hole

scattering

_gave the relaxation times of the order of hundreds of

picoseconds

in _a broad _range of the dot

size,

_

0

v~

~

_{41 mev 36 mev 32 mev}

~ Q~

li

~ 5

_8

li

~

Q~

0C

§0

21 22 23

Lateral size

(nm)

Fig,

I. Relaxation rate

-df211«/dt,

_{as a} function of the lateral size d, computed from the semi-

classical equation

(13),

at the state with f211«

= 1 and

fill«

= 0, with the

damping

factor determined from the self-consistent equation. The lattice temperature TL is lo K. The values of the

energy-level

separation E211

Eiii, corresponding

to three selected values of the lateral size

(given by

the

arrows),

are presented in the

graph.

which differs from the

present

result. This difference is ascribed to the

neglection

of the electron-hole

coupling [13,14]

^{in the}

present

^{model of the} dot. The

shape

of the relaxation rate d

f211«/dt (Fig. 1),

^{which extends} _{over a} rather wide range of

d,

is in _a

partial agreement

with the

experimental

observation _on the

photoluminescence

in

quantum

^dots

[12]. Namely,

in reference

[12]

an absence is

reported

of _any threshold behaviour of the

photoluIninescence

response, when the energy of the

exciting photon

scans

through

the

optical phonon

_energy

ltwo.

Such _a threshold behaviour _can be

expected

_on the

grounds

of the earlier theoretical

estiInates

[3],

^which gave a very

sharp dependence

of the electronic relaxation rate on the electron energy level

separation.

It has to be realized

however,that

_a

coInparison

^ofthe

present

result with the

experiIneni _[12]

should be Trade with care, because in the

experiInent

[12] ^the nuInber of electrons _per

single

dot Tray be

larger

than I and the processes of the

exchange

of energy between electron and

holes1nay play

_a role _as

pointed

out in

[13,14].

^The present ^result should rather be

coInpared

with data

corresponding

to the low electronic and hole

density

liInit.

The

quantuIn

kinetic

equations (6, 10),

with the

equation (12)

for

r~,~,

_were solved

by Runge-Kuttalnethod

of the nuInerical solution of _a set of differential

equations.

The tiIne

step

was 0.25fs. This way of

solving

the

equations

was rather

deInanding

_{on Inelnory} and CPU time. The nuInerical solution of the quantum kinetic

equations

is

presented

in

Figure

2. In this

figure,

the electronic

population f21la

is

displayed

_{as a} function of tiIne t and the

quantuIn

^dot lateral size d. In the interval of the lateral size d froIn about 20 _nIn to about 23 _nm the electronic

population

decreases

monotonously

with time. This range of d agrees with the range, inside which the semiclassical relaxation rate d

f211«/dt,

of

Figure _I,

is _nonzero. In this semiclassical

region

of d in

Figure 2,

the characteristic time of the relaxation

corresponds

to the semiclassical values of the relaxation rate of

Figure

I. Let _us

note,

that at the very small

times,

under about 30

fs,

the relaxation rate in the semiclassical _range of d is rather

weak,

which _can be

regarded

as a manifestation of the

energy-nonconservation

effect due _to the time

inhomogeneity

at t ₌ 0 in the

presently

considered

"thought" experiment,

in which the

system

^{is excited at} t ₌ 0

by

an excitation

pulse

with the

infinitely

short duration. The solution of the

quantum

^kinetic

equations

can be

expected

to agree in the semiclassical

region

^{with the numerical results based}

8~

~

l~

I

~

t

Fig.

2. Time evolution of the

population

f211«, _{as a} function of the lateral size d and time t,

in the cubic quantum dot with two energy levels taken into account,

computed

from the quantum kinetic equations. TL

= 10K. The oscillations of the

population

at

large

and small d demonstrate the reversible transfer of energy between the electronic and

phonon

subsystems.

.oo

c

o a

= m

~ o.95

if

f

~ ^m

d =19 _nm

o.90~

_{~~~ loco}

Time

Ifs)

Fig.

3. Time evolution of the

population f211«,

for the selected value of the quantum dot lateral size d. The numerical data show that the

population oscillates,

due to the reversible transfer of _energy between electrons and phonons, between the lower limit

fm (dotted line)

and I. The

averaged

value of the population is

fa (dashed line).

on the semiclassical formula

(13).

In the semiclassical _range of the lattice size the solution

presented

in

Figure

2

supports

the above

given interpretation

of the absence of _a threshold in the

photoluminescence experimental

data

[12].

^{It has to be noted}

again,

the numerical results

should be better

compared

with

experiments performed

_near the low carrier

density

limit.

In the range of d above about 23nm and below about 20nm in

Figure 2,

the electronic

population

is not constant and

equal

to

I,

_as it would be

expected

_on the

grounds

of the semiclassical

approach,

but it oscillates with time between I and

fm, fm being

_a number between _zero and _one. This time

dependence

of

f211«

is

displayed

in1nore detail in

Figure

3 for _a selected value of d. F1o1n

Figure

2 it is observed that the

Inagnitude

of

fm increases,

approaching

the value of I, when either d increases above about

23nm,

or d decreases below

about 20 _nm.

Similarly,

the

frequency

of the oscillations increases with

increasing

the

separation

of d from the semiclassical _range of d values. In other

words,

the

frequency

of the oscillation of

f211« along

the time axis increases with the difference between the electronic

energy-level

separation E211 Eiii

and the energy of the LO

phonons.

Let _us denote the

region

of d

values,

which _are outside the semiclassical

region,

_as the

quantum region

of d. The

inspection

of the numerical data

shows,

that the electronic

population, averaged

in time _{over one}

period

of the

oscillation,

is about

la

=

(I

+

fm)/2,

with

fm depending

_on d in the way

given

above. It is essential to see,

that,

the electronic

population, averaged

_{over one}

period,

is lower than I in the

quantum region.

The _more the lateral size d

approaches

the semiclassical range of the dot

size,

the

stronger

the decrease of the _average

population

is. The numerical calculation shows that in the

vicinity

of the lateral size of about 20nm _or about 23nm _a transition

region

_appears

to occur, in which the collision

broadening

is still _nonzero, but small

enough

for the reversible oscillations to

prevail.

The oscillations deInonstrate the

exchange

of energy between the electronic and vibrational

subsysteIns.

This

exchange

is reversible and is _an

iInplication

of the

fact,

that the states of the whole

systeIn,

at which the electronic

subsystem

is in the excited

state,

are not

eigenstates

of the whole

systeIn. Relying fully

_on the

plausibility

of the

approxiInation

to the

quantuIn

kinetic

equations, accepted

in this

work,

_one should

expect

the oscillations of the electronic

population

to be

observable, providing

the initial state of the

electron-phonon systeIn

^{of the} dot is realized.

It should be

expected,

that the behaviour of

f211«,

deInonstrated in the

quantuIn region

of

d,

is influenced

by

the

quantuIn

^{effect of} the energy nonconservation due to the time in-

homogeneity,

in the scenario considered. In _a real

experiment,

the manifestation of the time

inhomogeneity

would be

different,

in

dependence

_on the _process of excitation of the electronic

systeIn. Nevertheless,

in the _case of

exciting

the electronic

systeIn

ⁱⁿ

quantuIn

^{dots with}

help

of _a femtosecond laser

pulse,

the conditions of the

experiment

would be close to the scenario under consideration.

The effect of the decrease of the electronic

population f211«

in the

quantuIn region

at short

tiInes,

as it contributes to the overall status of the

systeIn resulting

froIn the decrease of the electronic

population

in the whole range of

d,

is therefore

coInplelnentary

to the seIniclassi- cal relaxation of

Figure

I. In contrast to the Born

approxiInation

result of reference

[3],

^the

d-dependence

of the electronic relaxation is then

relatively

broad,

including,

besides the semi- classical

region,

the

quantum region

as well. The electronic relaxation of

Figure

2 thus does not contain any

strong

threshold feature to be observed in

photoluIninescence experiInents.

With

taking

into consideration the quantuIn

region

of the decrease of

f211«,

the range of

d,

at which _a finite aInount of the electronic

population

is relaxed at a

given

instant of

tiIne,

is therefore broader than it results froIn the seIniclassical

approach only.

In the

quantum region

of d, the average of the

population fill«

of the electronic

ground

state has thus _{a nonzero}

value at t > 0. This fact should have _an

iInpact

on the _processes, not included in the

present

paper, like the luminescence connected with transitions froIn the electronic

ground

state to the valence band states.

The results

presented

are based _on the

analysis

of the dot with two electron energy levels

only. Quantitatively,

these results _can be Inodified to an extent when

considering

not

only

an

exciting optical pulse

of _a finite

width,

but also

quantum

dots with _a

large

nuInber of electron energy levels and _many electrons.

As it has been mentioned

already

in the Section

2.2,

the numerical solution of the _quan- tum kinetic

equations,

^similar to those considered

above,

_can

display

an

unphysical

property of

f~ii~ leaving

^{the interval}

(0,1),

in the _case, when the

damping

factors

r~,~

are

approxi-

mated

by

_zero. ^{This effect} was

analyzed recently

in _a

quantum

^dot

system [17].

^It was found

that the effect is the

property

^{of the}

approxiInation,

^{rather than} a nuInerical

inaccuracy.

It has been

shown,

that the

unphysical

behaviour of the electronic distribution function _can be

reInoved,

if the

electron-energy

collision

broadening

is introduced

[16].

^As

expected,

the _nu- 1nerical solution for

f211«,

of the

quantuIn

^kinetic

equations (6, 10),

with

(12),

in both the

seIniclassical and

quantuIn regions,

is found within the interval

(0, 1).

Let _us reInark that the nuInerical results _are

coInputed

for the

quantuIn

dot with

infinitely deep potential

well.

Taking

into account _{a Inore} realistic

shape

of the

confining potential

_Tray

influence the

quantitative

results to an extent.

4.

Summary

The hot electron relaxation _was studied in

polar

semiconductor

quantum

dots in the _approx- imation of two electronic _energy levels

coupled by

the interaction of electrons with

only

LO

phonons.

The relaxation _was studied within the method of

equations

^of motion for the reduced

density matrices, applying

the

approximations

^used

recently

in studies of the femtosecond

phe-

nomena in bulk materials. The

emphasis

_was

put

on

quantitative

estimates of the effects

given

by

the semiclassical and

quantum

^kinetic

equations.

The solution of the

quantum

^kinetic

equations displays

two main

regions

of the quantum ^dot size

(or

electronic

energy) according

to the behaviour of the electronic relaxation. In the interval

from about 20nm _to about 23 _nm of the lateral size of the cubic dot the electronic relaxation time is found at the

picosecond

time scale and the electronic

population

decreases

monotonously

as it would be

expected

when

using

^the irreversible semiclassical

approximation

to the _process of relaxation with the electron energy collision

broadening

included. In the

quantum region

of the lateral size above about 23nm and below about 20nm the relaxation _{process appears} to be reversible in the

present model,

with the excited-state electronic

population oscillating

around _{an average} value of the

population.

The electronic

population

starts to oscillate around this average value at

early

times which appear ^to be not

larger

than the relaxation time in the semiclassical

region.

The numerical results appear ^to indicate the presence of _narrow

transition

regions

between the two main

regions,

in which the

damping

factors _{are nonzero} but small

enough

to lead to a non-monotonous decrease of the electronic

population.

Both the

quantum

^{and the transition}

regions might

contribute to the relaxation from the excited

electronic level in _cases when the electron has _a finite lifetime in the

ground

state.

Although

the range of the electron

energy-level separation,

at which the relaxation is irre- versible and is _at the

picosecond

time

scale,

is

only

about _a

one-quarter

^of the

optical phonon

energy, it

corresponds

to a rather broad interval of the lateral dot size. The width of this dot size interval _may be

interesting

from the

point

of view of the current

technology

of

quantum

dots.

According

to the

present results,

the values of the electronic energy level

separations,

at which the electronic relaxation is very

fast,

_are not restricted to _a very narrow

vicinity

of

the _zone center LO

phonon

_{energy, as} indicated

by

the Born

approximation. Providing

that

quantum

dots

having

the size of tens of nanometers _are

prepared

with the

energy-level

_sep-

aration

being

within the energy window

given

in

Figure I,

the relaxation time of the

polar

semiconductor

quantum

^dot laser structures _can be at the

picosecond

time scale _even at low

injection

levels of electronic

density.

Appendix

Let _us write down the

equation

for the correlation function

6(b+~,bqc(, ~ci,«)

^of

^equation

ii)

and

demonstrate,

on this

example,

the

approximations applied. Using'the

definition

(8),

we obtain in _a

straightforward

_way

j6(b+~,bqc(,

^~ci

^al

^"

^(Ei

⁺

^{ltuJq En,} ltuJ-q,)(b+~,bqc(,

~ci

al IA. I)

t ^' i '

+

£ _{)Aq, j4l(ni,}

ml,

-q') _(bqc+,

_ci

«c(

~ cm~ _~~

~

ni,mi,t

" '~ ~' ~ ~

i

~i

q(~ ^'~~" '~~~'~~'~"~l~~"~~q~~~CS~,«Cmi,«i

+

L _{iAq,, i#ml, n~,}

_q~~i

_{ibq,, b+~,} bqcsi,«ci,«1

~,,,~,

+) L lAql4~lni,mi,-qllb+q,C$,,«Ci,«c$i,«icmi,«il

ni,mi,«i

+

L _{lAq,, la fl,}

_mi,

q"I16+q, b+q,, bqc$,,«Cmi,«I

q",mi

L _lAq" la Ini, n', q")16+q, b+q,, bqc$i,«Ci,«1

~,,,~~

+

)6-q,,q61,n,v~ L L _{iAq,, iReiaii,}

r,

q~~)iibq,, b+~,, )c[~cr«i I

q,, r#1

The reduced

density

matrices _on the rhs of this

equation, containing

five

particle operators each,

can be factorized _to the

products

of two mean

values,

_one of which is of the

type (bqcf~cj,~

and the other is _a

single-particle

distribution function. After

doing

the

factorization, seieral

_terms

on the rhs of

equation (20)

are

found,

which have the form of _{a sum over}

q",

of terms

containing expressions

like

(bq,,cf~cj,~).

These _{terms are} eliminated _on the

grounds

of the Random-

Phase-Approximation irgument (RPA) (compare

Ref.

[16]). Namely,

the latter _mean values

are

regarded

_as

changing rapidly

in

phase

in the Brillouin _{zone, over} which the summation extends.

Solne of the

terIns,

which still reInain in

(20)

after the RPA reduction of this

equation,

are

found _to be

proportional

to the

quantity (bqc$,~ci,«) only

upon

assuIning

that n'

= n in the

correlation function

(b+~,bqc(, ~ci,«).

^Such terms contribute

only

to the _term with n'

= n in

the _{sum over} n' _in the fifth

term

on the rhs of

equation iii. Realizing

that

according

to the

equation (6)

the electronic states with the orbital indexes _n and I _are

coupled

via emission _or

absorption

of the

phonon,

the terms like

6(b+~,bqc$~ci,«)

are not

expected

to contribute _on the basis of the energy conservation

argument

^and

tiey

are therefore

neglected.

In the

approximation specified above,

it is obtained that

~~~~~q'~~~$,al>") j _~~

~

~~q ~'l' ~~-q') ~~~q'~qc~',a~l>")

+) lAq'l@l~>11'> ~q'll~

_{+ "-q'}

fn',al lbq~t,a~l,al. jA.21

Let the letter t denote the time

dependence

of the

phonon-assisted density matrices, writing

(bqc$~ci,«)t.

In the zero-order

approximation (in _Aq)

(bqc$,~ci,«)t-t~

=

(bqc$,«Cl>«)t ~~P~)~~~'~' ^~~

^~~

where

ER

"

Ei En

+

ltuJq. Taking

this

rapid part

of the

time-dependence

into account, the

equation (21)

can be

integrated formally. Applying

the adiabatic and Markov

approximation

[15] ^to the formal solution of

equation (21),

_one obtains

6(b+~,bqc(,

~ci

al

"

ijAq, j4l(n, ^n', -q')(I

_{+ u-q,}

fn, ~)(bqc(~ci al

,

(A.4)

' ' '

ER

ltuJ + is

where _e

=

0+

^{and ltuJ} ₌

Ei

+

ltuJq En, ltuJ-q,.

This

equation provides

the

self-energy

contribution to

equation iii.

In this _paper the real part of the

self-energy, giving

the

polaron

shifts of the electronic

energies,

will be

neglected.

Analogically

the other correlation functions in

equation iii

_are treated.

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