Ex itons

Negatively hargedele tronsinthe ondu tionbandandpositively hargedholesin

thevalen ebandaresubje ttoCoulombattra tion. Thisele tron-holeintera tiongivesrise

toboundstates, alledex itons. Insemi ondu torswiththemorphologiesweare onsidering

valen eband. Inthis asethenetwavefun tionsoftheboundele tronandholeareextended

overseveral rystalunit ells, and the statesare regarded asWannier-Mott ex itons. This

situationisopposedtothatfoundinioni rystals, inwhi hele tronsandholesarestrongly

bound together and their wavefun tions lo alized in the same or nearest unit ell. In this

aseex itons are saidto be of theFrenkel type.

To model the ex itons within the ee tive mass approximation, let us just

on-sider that we have in the rystal an ex ited ele tron with momentum

k e

^{and a hole with}

momentum

k _h

^{. The time}independent S hr

¨

^{odinger equation wouldbeinthis ase:}

 p ² _e

2m ^∗ _e + p ² _h

2m ^∗ _h + e ² ǫ |r ^e − r ^h |



ψ (r e , r h ) = (E − E ^gap ) ψ (r e , r h )

^(2.35)

where

p _e

⁽

p _h

^{) and}

m ^∗ _e(h)

^are,respe tively,the momentumand ee tivemassoftheele tron (hole),

r _e

^and

r _h

^{aretheele tronandholepositions,}

E _gap

^isthesemi ondu torfundamental gap, and

ǫ

^{isthe diele tri fun tion of thematerial. In the ase of themassof thehole, in}

Eq. 2.35we have onsidered a meanhole mass

m ^∗ _h

^{asdis ussed inSe . 2.1.4 asa results of}

thespheri al band approximation.

In the onsidered pi ture of a single ele tron-hole pair ex itation, the ee t of

the rest of the valen e ele trons present in the rystal is a ounted by the ee tive mass

approximationandbythediele tri onstant,thatin ludestheba kgroundintera tionwith

thoseele trons,andresultsinanee tives reeningoftheele tron-holeCoulombintera tion.

Noti ethat Eq. 2.35 is analogous to that des ribing the hydrogen atom in atomi physi s.

AstheCoulombintera tion dependsonly ontherelativedistan ebetween theele tronand

the hole, it is onvenient to dene a new set of oordinates given by the position of the

enterof massofthe ele tron and hole(

R

^{)and its relative distan e(}

r

^):

R = m ^∗ _e r _e + m ^∗ _h r _h

m ^∗ _e + m ^∗ _h , r = r e − r ^h .

^(2.36)

Withthat hange of variables, Eq.2.35be omes:

P ²

2 m ^∗ _e + m ^∗ _h
+ p ² 2µ _eh + e ²

ǫr

!

ψ (R, r) = (E − E gap ) ψ (R, r) ,

^(2.37)

where

P = −i~∇ ^R

^,

p = −i~∇ ^r

^and

µ ⁻¹ _eh = (m ^∗ _e ) ⁻¹ + (m ^∗ _h ) ⁻¹

^{is the redu ed mass of the}

ele tron-hole pair. Thesolutions to 2.37 anbe fa torizedinto

ψ (R, r) = ϕ (R) ϑ (r)

^{. The}

equation orrespondingto

ϕ (R)

orrespondstothatofafreeparti leofmass

m ^∗ _e +m ^∗ _h

^,while

ϑ (r)

theprodu t of Laguerre polynomials [

L _nl (r)

^{℄and spheri al harmoni fun tions [}

Y _lm (θ, φ)

^℄

withthe eigenenergies givenby:

E _n ^r = − E _b

n ² , n = 1, 2, 3, . . . E _b = µ _eh e ²

2ǫ ² ~ ² .

^(2.38)

Therefore,the total energy and eigenfun tionsof theex itonare:

E _nlm = E gap + ~ ² K ²

2 m ^∗ _e + m ^∗ _h
− E _b

n ² ,

^(2.39)

ψ _nlm (R, r) = e ^iK·R L _nl (r) Y _lm (θ, φ) ,

^(2.40)

where

n, l, m

^{are the prin ipal, angular an magneti quantum numbers,} respe tively,

K

^is

the ex iton enter of massmomentum and

E _b

^{is the binding energy. The ex iton ground}

state at restis givenbythe

n = 1, l = 0, m = 0

wavefun tion with

K = 0

^:

ψ ₁₀₀ (R, r) = 2 q

a ³ _B

e ^{−r/a B} ,

^(2.41)

where

a _B = ~ ² ǫ/µ _eh e ²

^{is the ex iton Bohr radius and} hara terizes the spatial extension of the ex itoni omplex. In GaAs the heavy-hole ex iton (formed from the binding of a

ondu tion ele tron and a heavy-hole) has a Bohr radius at low temperature of 11.4 nm,

while its binding energy is of 4.2 meV, four orders of magnitude smaller than the binding

energy of an ele tron to a proton inthe hydrogen atom. If the ex itonis formed from the

binding of an ele tron with a light-hole we will refer to it as a light-hole ex iton. Note

that so far we have onsidered just a mean hole mass for the ex itons. This would mean

that both heavy- and light-hole ex itons have the same binding energy and Bohr radius.

Nonetheless we an still distinguish and des ribe them independently be ause ea h

hole-type wavefun tion has dierent angular momenta. When we redu e thedimensionality of

thesystem,the breakdownofthesymmetrywillresultintheinhibitionofthemassreversal

ee t and heavy-and light-hole ex itons will present dierent redu ed masses and binding

energies (seeSe . 3.1.1).

Equations 2.39 and 2.40 tell us some important properties of the ex itons. One

of them is that ex itons an move freely within the rystal. Nonetheless, intera tion with

J ^z

^hh

+3/2

^hh

−3/2

^lh

+1/2

^lh

−1/2

+1/2 |+2i |−1i |+1i |0i

−1/2 |+1i |−2i |0i |−1i

Table 2.2: Ex iton spin state as a fun tion of the third omponent of the total angular

momentumof its onstituent ele tron (left olumn)andhole (toprow).

in dissipation. We an also see that the ground state energy of the ex iton (before the

onstituentele tronandholepairsre ombine)islo atedatanenergy

E _b

^{belowthebandgap}

energy,and that the ex iton possessesits ownparaboli band. Therefore,thefundamental

ex itationinthe systemisnot onformedbyanele tronandaholeattheirbandedges, but

by theformationof a1s ex itonat rest.

So far we have not onsidered the spin of the ex itons. As we have seen in the

previous se tions, the onstituent parti les of the ex iton may have a well dened total

angular momentum. For the sake of simpli ity, given the ompli ated band mixing ee ts

in the heavy and light hole bands far from the

k = 0

^{states, lets restri t the analysis to}

ex itons formed from ele trons and holes with

k = 0

^{. In this ase both parti les have a}

well dened total (

J

^{) and third omponent (}

J ^z

^{) of the angular momentum, as depi ted in}

Eqs.2.22-2.25. Theresultingex itonwillalsohaveawelldenedtotalandthird omponent

of the angular momentum (

J _X

^,

J _X ^z

^{). From now on we will refer to}

J _X ^z

^{as the spin of the}

ex iton. Table2.2 presents all the possible spin statesof ex itons formed fromthe binding

of an ele tron with a heavy or a light hole. Ex itons with

J _X ^z = 0, 1 , 2

^{an be formed,}

however, as we will des ribe later, only ex itons with

J _X ^z = 1

^{,0 an ouple to light due to}

momentum onservationrules.

Notealso that though the ex iton is onformed by two fermions, an ele tron and

a hole, its total angular momentum is integer. For this reason, in the dilute limit regime,

in whi h interex itoni intera tion and s reening is not very important, ex itons an be

des ribed as bosons. Parti les of this kind, whose onstituents are fermioni but follow

bosoni statisti sarereferredtoas ompositebosons. Therefore,ex itoni intera tionsare

well des ribedbybosoni operators,andsu h anapproa hiswidelyusedwhentreatingthe

photon-ex iton intera tion in the strong oupling regime (see Se . 3.2.2). However, there

have been some re ent theoreti al proposals whi h stress the importan e of the fermioni

The des riptionwe have followed sofar made used of the single ele tron pi ture,

inwhi h the systemHamiltonian 2.5 a ounted for the average ee t of thesea of valen e

ele trons on a parti ular ele tron. Moreover, it assumed that all ele trons where in the

groundstateofthesystem,i.e.,fullyo upyingthevalen eband. Underthese onditionsthe

banddiagramsdepi tedinFig.2.3andtheasso iatedee tivemassesnearthe

Γ

^pointwere

al ulated. Ifwe reate one ex itationinthe system,for instan e, an ex iton, thesituation

remains pretty mu h the same, with the same band gaps, ee tive masses and diele tri

onstants hara terizing the material. However, when many ele trons are promoted from

the valen e to the ondu tion band the situation starts to diersigni antly from the one

ele tronpi ture. Underthese ir umstan eslargepopulationsofex itedele trons andholes

withspreadmomentumdistributions an freelymoveand intera t. Ifthedensitiesarehigh

enough so as to over ome the degeneration limit dened in Fig. 2.5, a rst orre tion to

the energy of the systemwith respe t to the single ele tron pi ture would ome from the

antisymmetrization of the fermioni total ele tron and hole wavefun tions, in a

Hartree-Fo kapproa h. Onthesamegrounds,Coulomb orrelationee ts,andthe arriers reening

asso iatedtothem,areverymu hae tedbythewide spreadinmomentumofthe arriers.

Theseproblems have been thoroughly treated theoreti ally inthe literature.[105,308 , 280 ℄

Oneof thedire t onsequen esof themany-bodyee tsisthe hange oftheband gaps. In

parti ular,thein reaseinthes reeningofthesystem ausedbytheex ited arriersresultsin

theredu tionofthefundamentalgap. Alsotheee tivemassesareae tedandevenexoti

spin-dependent phenomena has re ently been predi ted in this regime.[303℄ Intera tions of

the arrierensemblesareae tedtoo,andtherateofex hangeofenergybetweentheex ited

ele trons and holesand the latti e ( arrier-phononintera tion) isaltered.

As for the ex itons, the in rease in the arrier s reening at high densities when

both ex itons and ele tron-hole plasma are present, results in a net hange (an in rease)

of thediele tri onstant

ǫ

^{thatdes ribesthe}surrounding ex itoni media inEqs.2.39and 2.40. Thus,theex itonbindingenergygetsprogressivelyredu edwiththein reaseof arrier

density,and eventually ex itons disso iate,aswill be thoroughly dis ussedinSe . 5.4.

Anestimationofthedensityoftheseee tstobesigni ant might begivenbythe

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