Appendix F Direct Coulomb Scatterings for Wannier Excitons - Oxford Scholarship Jump to ContentJump to Main Navigation
Excitons and Cooper PairsTwo Composite Bosons in Many-Body Physics$

Monique Combescot and Shiue-Yuan Shiau

Print publication date: 2015

Print ISBN-13: 9780198753735

Published to Oxford Scholarship Online: March 2016

DOI: 10.1093/acprof:oso/9780198753735.001.0001

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(p.493) Appendix F Direct Coulomb Scatterings for Wannier Excitons

(p.493) Appendix F Direct Coulomb Scatterings for Wannier Excitons

Source:
Excitons and Cooper Pairs
Author(s):

Monique Combescot

Shiue-Yuan Shiau

Publisher:
Oxford University Press

F.1 Creation potential

Because of the exciton composite nature, there is no way to define an interaction potential between two Wannier excitons. It is, however, possible to obtain Coulomb scatterings between two excitons that account for the Coulomb forces between their carriers, through the creation potential Vi associated with the i exciton, defined as (M. Combescot, Betbeder-Matibet, and Combescot 2007b)

(F.1)
H,Bi=EiBi+Vi,

where Bi is the i exciton creation operator

(F.2)
Bi=pephapebphph,pe|i

while H is the electron-hole Hamiltonian defined as

(F.3)
H=He+Hh+Vee+Vhh+Veh.

The kinetic part of this Hamiltonian is given by

(F.4)
He+Hh=kεk(e)akak+kεk(h)bkbk.

The repulsive Coulomb interaction between electrons reads

(F.5)
Vee=12q0Vqk1k2ak1+qak2qak2ak1,

(p.495) with the Coulomb scattering Vq equal to 4πe2/ϵscL3q2 in 3D. The operators a are replaced by b for the repulsive Coulomb interaction between holes, while the attractive Coulomb interaction between electrons and holes is given by

(F.6)
Veh=q0Vqkekhake+qbkhqbkhake.
Appendix F Direct Coulomb Scatterings for Wannier Excitons

Figure F.1 The electron-hole potential participates in the Wannier exciton formation. It also participates in exciton-exciton Coulomb scatterings.

This electron-hole attraction is responsible for the formation of Wannier excitons. It also participates in the interaction between two excitons, as seen in Fig. F.1. To demonstrate this fact, we calculate the commutator

(F.7)
[Veh,Bi]=q0Vqkekhpephph,pe|i[ake+qbkhqbkhake,apebph]=q0Vqkekhpephph,pe|iake+qbkhq×(δkepeδkhphδkepebphbkhδkhphapeake).

From

(F.8)
[He,Bi]=kεk(e)pephph,pe|i[akak,apebph]=pephεpe(e)ph,pe|iapebph

and a similar result for Hh, we end up with

(F.9)
He+Hh+Veh,Bi=Ai+Ah+Ae,

where Ai contains all terms in ab, Ah contains all terms in abbb, and Ae contains all terms in abaa.

  1. (p.496) (i)Ai comes from [He+Hh,Bi] and the first term of [Veh,Bi] in Eq. (F.7). It precisely reads

    (F.10)
    Ai=pephεpe(e)+εph(h)ph,pe|iapebphq0Vqpephph,pe|iape+qbphq.

    By replacing (pe+q,phq) by (pe,ph), we can rewrite these two terms as

    (F.11)
    Ai=pephapebph{εpe(e)+εph(h)ph,pe|iq0Vqph+q,peq|i}.

    To calculate the term in curly brackets, we note that the single exciton state |i is eigenstate of H with energy Ei. So,

    (F.12)
    0=ph,pe|HEi|i=ph,pe|He+Hh+VehEi|i=(εpe(e)+εph(h)Ei)ph,pe|iq0Vqphq,pe+q|i.

    As Vq=Vq, the term in curly brackets of Eq. (F.11) thus reduces to Eiph,pe|i, which yields

    (F.13)
    Ai=EiBi.

    As expected, the formation of a Wannier exciton from free electron-hole pairs is entirely due to electron-hole Coulomb attraction.

  2. (ii)Ah comes from the second term in Eq. (F.7), and can be rewritten as

    (F.14)
    Ah=q0Vqpephph,pe|iape+qbphkhbkhqbkh.

    Ae comes from the third term in Eq. (F.7), and can be rewritten as

    (F.15)
    Ae=q0Vqpephph,pe|iapebphqkeake+qake.

We now turn to the electron-electron and hole-hole potentials given in Eq. (F.5). Their commutators read

(F.16)
[Vee,Bi]=12q0Vqk1k2pephph,pe|i[ak1+qak2qak2ak1,apebph]=122q0Vqpephph,pe|iape+qbphkakqak

(p.497) and, since Vq=Vq,

(F.17)
Vhh,Bi=q0Vqpephph,pe|iapebphqkbk+qbk.

By collecting the above results, we end up with

(F.18)
H,Bi=EiBi+Wh+We,

where Wh, which comes from Ah and [Vee,Bi], reads

(F.19)
Wh=q0Vqpephph,pe|iape+qbphk(akqakbkqbk),
Appendix F Direct Coulomb Scatterings for Wannier Excitons

Figure F.2 In Wmi(h) given in Eq. (F.22), the excitons m and i have the same ph hole, while their electrons change momentum because of Coulomb scattering with a k electron, as in (a), or a k hole, as in (b).

with a similar expression for We, which comes from Ae and [Vhh,Bi].

The next step is to transform free electron-hole pair operators into exciton operators. This is done using

(F.20)
apebph=mBmm|pe,ph.

The creation potential Vi defined in Eq. (F.1) then reads

(F.21)
Vi=Wh+We=mBmWmi(h)+Wmi(e),

with

(F.22)
Wmi(h)=q0Vqpephm|pe+q,phph,pe|ik(akqakbkqbk),

as visualized in Fig. F.2: the “in” exciton i and “out” exciton m have the same ph hole, while the pe electron of the i exciton interacts either with an electron, as in Fig. F.2(a), or with a hole, as in (p.498) Fig. F.2(b), before forming the m exciton. Note that the electron-electron Coulomb interaction is repulsive, while the electron-hole Coulomb interaction is attractive, which explains the sign difference in Eq. (F.22).

In the same way, in Wmi(e) visualized in Fig. F.3, the excitons m and i have the same pe electron, while the ph hole of the i exciton interacts either with a hole, as in (a), or with an electron, as in (b), before forming the m exciton. The Wmi(e) operator precisely reads

(F.23)
Wmi(e)=q0Vqpephm|pe,ph+qph,pe|ik(bkqbkakqak).
Appendix F Direct Coulomb Scatterings for Wannier Excitons

Figure F.3 In Wmi(e) given in Eq. (F.23), the excitons m and i have the same pe electron, while their holes change momentum after Coulomb scattering with a k hole, as in (a), or a k electron, as in (b).

F.2 Direct Coulomb scatterings

The direct Coulomb scatterings ξnjmi between two excitons follow from one more commutator, namely,

(F.24)
Vi,Bj=mnξnjmiBmBn.

F.2.1 In momentum space

Using Vi given in Eq. (F.21), and the fact that [Bm,Bj]=0, we find that [Wmi(h)+Wmi(e),Bj] can be written as nBnξnjmi. Indeed,

(F.25)
[Wmi(h),Bj]=q0Vqpephpephm|pe+q,phph,pe|iph,pe|j×[k(akqakbkqbk),apebph].

(p.499) Equation (F.20) gives the commutator on the RHS in terms of excitons as

(F.26)
apeqbphapebphq=nBn(n|peq,phn|pe,phq).
Appendix F Direct Coulomb Scatterings for Wannier Excitons

Figure F.4 Shiva diagrams in momentum space for the four Coulomb processes contained in ξnjmi, given in Eq. (F.27). These processes take place between two Wannier excitons (i,j), the “out” states (m,n) being constructed on the same electron-hole pairs as the (i,j) states are.

Combining with a similar result for [Wmi(e),Bj] leads to

(F.27)
ξ(njmi)=q0Vqpephpeph×(m|pe+q,phn|peq,ph+m|pe,ph+qn|pe,phqm|pe+q,phn|pe,phqm|pe,ph+qn|peq,ph)×ph,pe|iph,pe|j.

This ξnjmi scattering, which looks rather complicated at first, can be directly read from the four Shiva diagrams shown in Fig. F.4, which represents the four possible Coulomb processes between the two electrons and the two holes of excitons i and j.

F.2.2 In real space

We can rewrite the direct Coulomb scattering ξnjmi in real space using Fourier transform

(F.28)
ph,pe|i=L3d3red3rhph|rhpe|rerh,re|i,

(p.500) with r|p=eipr/L3/2. The first term of ξnjmi in Eq. (F.27), represented in momentum space by the Shiva diagram in Fig. F.4(a), transforms into

(F.29)
q0Vqpephpephd{r}m|re,rhre|pe+qrh|ph×n|re,rhre|peqrh|ph×ph|rhpe|rerh,re|j×ph|rhpe|rerh,re|i.

To calculate the above quantity, we first use the closure relation for free particle states, namely,

(F.30)
phrh′′|phph|rh=rh′′|rh=δ(rh′′rh),

which imposes rh′′=rh. In the same way,

(F.31)
pere′′|pe+qpe|re=peeipe(re′′re)L3eiqre′′=eiqre′′pere′′|pepe|re=eiqre′′re′′|re

imposes re′′=re. So, after having eliminated the four p sums, Eq. (F.29) reduces to

(F.32)
d{r}m|re,rhn|re,rhrh,re|jrh,re|iq0Vqeiq(rere),
Appendix F Direct Coulomb Scatterings for Wannier Excitons

Figure F.5 Shiva diagram in real space for the four direct Coulomb processes contained in ξnjmi, given in Eq. (F.33), which take place between the “in” excitons (i,j), the “out” states (m,n) being constructed on the same electron-hole pairs as (i,j) are.

the sum over q giving e2/ϵsc|rere|.

The four terms of ξnjmi in Eq. (F.27), calculated in the same way, give the direct Coulomb scattering in real space as

(F.33)
ξnjmi=d{r}m|re,rhn|re,rhrh,re|jrh,re|i(Vee+VhhVehVeh),

(p.501) with Vee=V(rere), and V(r)=e2/ϵscr. The above expression for ξnjmi is just what we read from the Shiva diagram in Fig. F.5.

F.3 Symmetry properties

The expression of ξnjmi in real space readily shows that

(F.34)
ξnjmi=ξminj=ξjnim.

This can also be obtained through the expression of ξnjmi in momentum space, as given in Eq. (F.27), by setting pe+q=ke, ph=kh, peq=ke, and ph=kh in the first term of this equation, and similarly for the three other terms.