## Monique Combescot and Shiue-Yuan Shiau

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DOI: 10.1093/acprof:oso/9780198753735.001.0001

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# Links Between Cooper Pairs and Excitons

Chapter:
(p.270) 11 Links Between Cooper Pairs and Excitons
Source:
Excitons and Cooper Pairs
Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198753735.003.0011

# Abstract and Keywords

Chapter 11 compares Wannier excitons, Frenkel excitons, and Cooper pairs, with respect to their potentials, particle degrees of freedom, ground-state energies, wave functions, and many-body parameters. Shiva diagrams for composite boson many-body effects are used to visualize the physics. For these composite particles, it is possible to assign the same physical meaning to the mathematical parameter appearing in the density expansion of their many-body effects. This many-body parameter is N/Nmax, where Nmax is the maximum number of composite bosons the sample can accommodate. Characteristic lengths of Cooper pair wave functions for a single pair and a dense regime of pairs are also discussed. Finally, this chapter discusses the density regimes for excitons and Cooper pairs: excitons exist in the dilute regime, since at high density they dissociate into an electron-hole plasma. By contrast, Cooper pairs can strongly overlap without breaking, owing to the peculiar form of the reduced BCS potential.

In this last chapter on Cooper pairs, we come back to some fundamental properties of Wannier and Frenkel excitons to make links with similar properties of Cooper pairs. This will allow us to pin down the key parameters ruling composite bosons, to stress the consequences of the degrees of freedom these composite bosons have, and to identify the precise role played by interactions between fermionic components. We will successively list each of these properties for Cooper pairs, Frenkel excitons, and Wannier excitons, in order to better see similarities and differences between these three composite bosons, of major importance in two major fields of condensed matter physics: superconductors and semiconductors.

# 11.1 Degrees of freedom

Let us start with some characteristics shared by excitons and Cooper pairs that are associated with their degrees of freedom, as these explain the similarities between Frenkel excitons and Cooper pairs.

## 11.1.1 Wannier excitons

A Wannier exciton is constructed from one free conduction electron and one free valence electron absence, which is called hole, this hole behaving like a particle with positive charge and positive mass. The positively charged hole attracts the negatively charged conduction electron through direct Coulomb processes similar to the ones that exist between the proton and the electron in a hydrogen atom. So, starting from two free particles, that is, an electron in a plane-wave state ke, and a hole in a plane-wave state kh, (p.273) we end with a correlated state i called Wannier exciton, whose center of mass is a plane wave $Qi=ke+kh$. The creation operator of the resulting Wannier exciton reads

(11.1)
$Display mathematics$

The Wannier exciton state index i has two degrees of freedom, like the electron-hole pairs $(ke,kh)$ from which Wannier excitons are constructed: in addition to a center-of-mass momentum Qi, a second quantum index νi characterizes the relative motion of the electron-hole pair in the exciton. As with a hydrogen atom, this relative motion index differentiates exciton bound states from extended (unbound) states. The double-index character of Wannier excitons is necessary in order to write closure relations in the one-electron-hole subspace, either in terms of free electron-hole pairs or in terms of Wannier excitons:

(11.2)
$Display mathematics$

## 11.1.2 Frenkel excitons

In the case of Frenkel excitons, we start with electron-hole pairs bound at lattice sites, as a result of the tight-binding approximation, which is valid in semiconductors hosting Frenkel excitons. These on-site excitations are delocalized over the whole sample by intersite interatomic-level Coulomb processes. The creation operator of the resulting exciton reads

(11.3)
$Display mathematics$

where Ns is the number of lattice sites located at the Rn position. The operator $an†$ creates an electron at site n, while $bn†$ creates a hole; so, $an†bn†$ creates an atomic excitation at lattice site n.

## 11.1.3 Cooper pairs

Cooper pairs can seem at first very different from excitons because they are not constructed from electron-hole pairs but from up-spin and down-spin electron pairs. However, this is not the major difference between these composite bosons; rather, what distinguishes excitons from Cooper pairs is the nature of the attractive force which binds their fermionic components. In order to overcome the strong Coulomb repulsion between two negatively charged electrons making a Cooper pair, one has to bring in an additional process, that is, the ion motion, to end up with an effective attraction between electrons. This attraction is very different from the Coulomb attraction existing between electrons and holes: in the reduced BCS potential, the attraction responsible for Cooper (p.274) pair formation occurs between electrons having opposite spins and opposite momenta; so, one up-spin electron can interact with one down-spin electron only. By contrast, in the case of Wannier excitons, a ke electron can interact with anykh hole.

The creation operator of a single Cooper pair, as obtained by Léon Cooper, is given by

(11.4)
$Display mathematics$

with E1 solution of

(11.5)
$Display mathematics$

wk is a sharp cutoff equal to 1 in the energy layer in which the effective attractive potential acts, that is, over a phonon energy omegac on both sides of the normal electron Fermi level, and $wk=0$ otherwise. For a pair of free electrons added to the frozen Fermi sea $|F0〉$, the lowest E1 falls slightly below the frozen Fermi level. The energy gain resulting from the attractive BCS potential is equal to $εc≃4ωce−2/ρV$, where $V$ is the (small) potential scattering, and ρ‎ is the density of states taken as constant in the energy layer where the potential acts.

However, this single Cooper pair is not relevant for BCS superconductivity, which fundamentally is a collective many-body phenomenon occurring in a dense system. Indeed, in the BCS ansatz $(B†)N|F0〉$, the pair creation operator is given by

(11.6)
$Display mathematics$

the “wave function” $vk/uk$ being drastically different from the single Cooper pair wave function appearing in Eq. (11.4). We will come back to this important difference.

## 11.1.4 Discussion

Since the electron pairs used to construct Cooper pairs depend on k only, Cooper pairs are single-index composite bosons. This characteristic is shared by Frenkel excitons, which are made of electron-hole pairs characterized by a single index, the lattice site n.

By contrast, Wannier excitons, in addition to having a center-of-mass momentum Q, have a quantum index ν‎, which characterizes the relative motion of the correlated electron-hole pair. Yet, this relative motion index is often forgotten, because most people have in mind Wannier excitons that are in their ground state. This second degree of freedom ν‎ makes the many-body physics of Wannier excitons quite rich but far more complex than that of Frenkel excitons or Cooper pairs. This fact explains the absence of compact analytical results for N Wannier excitons even though such results exist for Frenkel excitons and Cooper pairs.

# (p.275) 11.2 Potentials

The formation of composite bosons is caused by the potential felt by their fermionic components. In the case of Cooper pairs, this potential originates from the ion motion, which, in materials in which superconductivity exists, overcomes the natural Coulomb repulsion between two electrons. In the case of excitons, the potential responsible for their formation is a part of the Coulomb potential which exists between electrons, this part being not the same for Wannier excitons and Frenkel excitons. Moreover, this potential plays a different role: in the case of Cooper pairs and Wannier excitons, the relevant potential is attractive and binds fermions in pairs. In the case of Frenkel excitons, the relevant potential delocalizes electron-hole pairs already bound at lattice sites into a coherent excitation extending over the whole sample.

A precise understanding of composite bosons starts with comprehending what the potential does.

## 11.2.1 Cooper pairs

In BCS superconductivity, the effective potential between the electrons forming Cooper pairs is highly simplified. It acts between electrons with opposite spins and opposite momenta only. Moreover, to allow analytical resolution, the potential scattering is taken as constant and separable. So, the following very simple form, called the “reduced BCS potential,” is taken (p.276)

(11.7)
$Display mathematics$

where $Bk†=ak↑†a−k↓†$, while $wk=1$ for $εF0≤εk≤εF0+Ω$, and $wk=0$ otherwise. This potential, represented in Fig. 11.1(a), is attractive, $V$ being positive. Its repeated action on one up-spin electron and one down-spin electron leads to the formation of a single Cooper pair.

Figure 11.1 (a) Attractive BCS process between up-spin and down-spin electrons with opposite momenta (see Eq. (11.7)), leading to the formation of a Cooper pair. (b) Transfer Coulomb process between atomic excitations at different lattice sites $n2≠n1$, leading to coherent excitation extending over the whole sample; this excitation is known as the Frenkel exciton. (c) Attractive direct Coulomb process between a free electron ke and a free hole kh, leading to the formation of a Wannier exciton. Solid and dashed lines represent up-spin and down-spin electrons, respectively, in the case of Cooper pairs (a), while they represent electrons and holes in the case of excitons (b, c).

## 11.2.2 Frenkel excitons

To get Frenkel excitons, we start with electrons bound at lattice sites. Coulomb interaction enables transitions between atomic levels located at the same lattice site as well as between sites, and plays multiple roles in Frenkel exciton physics. Let us recall them:

1. (i) First, we consider the intrasite direct and exchange Coulomb processes shown in Fig. 11.2. They read in terms of the operator $aνn†$ that creates an electron at the atomic level $ν=(0,1)$ of lattice site n as

(11.8)
$Display mathematics$

Figure 11.2 Direct (a) and exchange (b) Coulomb processes between ground (ν‎ = 0) and excited (ν‎ = 1) atomic levels at the same lattice site.

The elementary scattering for electrons that are at lattice sites n1 and n2, separated by $R=Rn1−Rn2$ (possibly equal to zero), and changing from atomic level ν‎ to ν' (possibly equal to ν‎) is given, in terms of atomic wave function $〈r|aνn†|0〉=φν(r−Rn)=〈r−Rn|ν〉$, by

(11.9)
$Display mathematics$

By writing $a0n†a1n†a1na0n$ as $a1n†(1−a0na0n†)a1n$ and by turning to electron and hole operators $a1n†=an†$ and $a0n†=bn$, the two potentials in Eq. (11.8) give rise to a two-body interaction which reads (p.277)

(11.10)
$Display mathematics$
δ‎, equal to $VR=0(1100)−VR=0(0110)$, is positive because, atomic levels ν‎ = 0 and ν‎ = 1 being orthogonal, the direct and exchange Coulomb scatterings in Eq. (11.9) would reduce to 1 and 0, respectively, in the absence of the Coulomb factor. Moreover, δ‎ is large compared to other Coulomb scatterings because it involves same-site scatterings. The fact that δ‎ is positive and large forces the electron and the hole to be at the same lattice site. As a result, at the very first stage of the Frenkel exciton problem, we already have bound electron-hole pairs since the electron and the hole are attached to the same atom. This is in contrast to Cooper pairs and Wannier excitons, which are made from free fermion pairs.

2. (ii) Next, we delocalize the electron-hole pair at site n through the intersite interlevel Coulomb process shown in Fig. 11.3(a): an electron-hole pair recombines at site n2 while a pair is created at a different site n1. Starting from the interlevel potential for electrons on different lattice sites,

(11.11)
$Display mathematics$

Figure 11.3 (a) Interatomic-level transfer process between different lattice sites, responsible for Frenkel exciton formation. (b) Interband Coulomb process between valence and conduction bands, usually neglected in the case of Wannier excitons. The main effect of these interband processes is to dress direct Coulomb scatterings through a dielectric constant. They also produce a small energy splitting between bright and dark excitons when electron spin and orbital degrees of freedom are taken into account.

and writing $a1n1†a0n2†a1n2a0n1$ as $a1n1†a0n1a0n2†a1n2$, since $n1≠n2$, we generate a transfer potential between lattice sites which reads in terms of electron and hole operators as

(11.12)
$Display mathematics$

where $Bn†=an†bn†$ creates an electron-hole excitation at lattice site n. This potential is represented by the Feynman diagram in Fig. 11.4(a) (see also Fig. 11.1(b)). Its repeated (p.278) action on a pair initially located at site n delocalizes this pair into a Frenkel exciton. Note that this transfer potential does not have to be attractive, because the electron-hole pair is already bound at the lattice site by Vneutral: the pair just has to be delocalized to form a Frenkel exciton.

Figure 11.4 Feynman diagrams for Coulomb processes (a) between lattice sites, as shown in Fig. 11.3(a), and (b) between valence and conduction bands, as shown in Fig. 11.3(b).

3. (iii) In addition to Vneutral and Vtransf, atomic excitations also suffer direct Coulomb processes that take place between different lattice sites but in which each excitation stays at its respective site (see Fig. 11.5). Such processes produce vcoul

(11.13)
$Display mathematics$

Figure 11.5 Direct Coulomb processes between different lattice sites, the electrons staying in their atomic levels.

As the diagonal scatterings are positive, interactions between two electrons or between two holes are repulsive, while direct interactions between one ground level and one excited level turn attractive when written in terms of electrons and holes, as physically expected. The vcoul potential, shown in the three diagrams in Fig. 11.6, requires two pairs to act; so, it does not contribute to the formation of a single Frenkel exciton but instead plays a role in Frenkel exciton many-body effects.

Figure 11.6 Feynman diagrams for the vcoul potential of Frenkel excitons. It consists of electron-electron, hole-hole, and electron-hole interactions at different lattice sites. This potential requires two Frenkel excitons to enter into play.

## (p.279) 11.2.3 Wannier excitons

To get a Wannier exciton, we start with two free fermions, namely, one conduction electron and one valence electron absence, with arbitrary momenta, the valence electron absence behaving as a positively charged fermion.

1. (i) Intraband Coulomb scatterings, that is, scatterings in which each carrier stays in its band (see Fig. 11.7), reduce for small momentum transfer to the scatterings of two free charges in a medium having a dielectric constant εsc. As a result, the potential between one electron and one hole, as shown in Fig. 11.1(c), reads

(11.14)
$Display mathematics$

Figure 11.7 Intraband Coulomb process between one conduction electron and one valence electron absence, that is, one hole, responsible for Wannier exciton formation.

(p.280) with $Vq=4πe2/ϵscL3q2$ in 3D. The repeated action of this attractive interaction between one free electron and one free hole leads to the formation of a Wannier exciton, its center-of-mass momentum being equal to the center-of-mass momentum $Q=ke+kh$ of the electron-hole pair which is kept constant along these repeated Coulomb processes. In order to compare this electron-hole potential with the reduced BCS potential of Cooper pairs (Eq. (11.7)) and the transfer potential of Frenkel excitons (Eq. (11.12)), let us introduce a creation operator for a free pair with center-of-mass momentum Q and relative motion momentum p, namely, $BQ,p†=ap+γeQ†b−p+γhQ†$, with $γe=1−γh=me/(me+mh)$ in order for the electron and hole kinetic energies $(p+γeQ)2/2me+(−p+γhQ)2/2mh$ to split as a center-of-mass energy plus a relative motion energy, $Q2/2MX+p2/2μX$, with $MX=me+mh$, and $μX−1=me−1+mh−1$. The electron-hole potential in Eq. (11.14) can be written in terms of these free pair operators as

(11.15)
$Display mathematics$

It then becomes obvious that the three potentials responsible for composite boson formation (Eqs. (11.7, 11.12, 11.15)), are very similar; they are shown in Fig. 11.8.

Figure 11.8 (a) Attractive BCS scattering inside one electron pair $(k1↑,−k1↓)$, leading to the formation of a Cooper pair. (b) Transfer scattering inside one atomic excitation, leading to the formation of a Frenkel exciton. (c) Attractive direct Coulomb scattering inside one free electron-hole pair, leading to the formation of a Wannier exciton.

2. (ii) In addition to the intraband electron-hole Coulomb processes shown in Fig. 11.1(c) and 11.7, conduction electrons and valence holes also undergo interband Coulomb processes (see Figs. 11.3(b) and 11.4(b)) in which one electron-hole pair recombines while another pair is created. Although usually forgotten, these interband Coulomb processes account for the semiconductor dielectric constant appearing in the intraband Coulomb scatterings. They also bring a small energy splitting between bright and dark excitons when carrier spin and orbital degrees of freedom are included in the problem. It is (p.281) worth noting that their equivalents in the case of Frenkel excitons produce the excitation transfer from site to site; this transfer is the key mechanism for Frenkel exciton formation.

3. (iii) Finally, a repulsion also exists between conduction electrons and between valence holes, the associated potential between two electrons reading

(11.16)
$Display mathematics$

with a similar expression for the Vhh potential between two holes, ak being replaced with bk.

## 11.2.4 Discussion

In the BCS problem, we start with pairs of free up-spin and free down-spin electrons having a center-of-mass momentum equal to zero. These pairs suffer the attractive potential, VBCS. The pair center-of-mass momentum stays equal to zero throughout the repeated action of VBCS, which binds one up-spin and one down-spin electron into Cooper pair. It is worth noting that Cooper pairs do not directly interact through the reduced BCS potential, as shown in Fig. 11.9.

Figure 11.9 Two Cooper pairs cannot directly interact through the reduced BCS potential, because this would impose $k1=k2$. The two free electron pairs would then have the same momentum, which is prohibited by the Pauli exclusion principle.

In the case of Frenkel excitons, we start with electron-hole pairs bound at a lattice site by the intrasite attractive potential. These bound pairs are delocalized into a plane wave Q with the help of the intersite interlevel Coulomb potential, Vtransf. The equivalent (p.282) potential in the case of Wannier excitons only brings a small energy splitting between dark and bright excitons, when carrier spin and orbital degrees of freedom are taken into account.

To construct Wannier excitons, we start, as with Cooper pairs, with two free fermions, this fermion pair however having an arbitrary center-of-mass momentum. Since Coulomb interactions conserve momentum, the intraband electron-hole Coulomb attraction, Veh, leads to the formation of a bound or unbound Wannier exciton with the same center-of-mass momentum as the free pairs from which it is constructed. Its equivalent in the case of Frenkel excitons is negligible within the tight-binding approximation.

Finally, Frenkel and Wannier excitons also have repulsive direct Coulomb processes between electrons or between holes. However, these processes only act at the many-body level.

The reduced BCS potential has similarities with the intraband Coulomb potential Veh given in Eq. (11.15): being both attractive, they are responsible for binding fermion pairs into Cooper pairs and Wannier excitons. The reduced BCS potential also has similarities with the transfer potential Vtransf given in Eq. (11.12) and responsible for Frenkel exciton formation through delocalizing on-site atomic excitations: VBCS and Vtransf both act on single-index pairs, in contrast to Veh, which acts on double-index pairs. So, the resulting correlated states, either Cooper pairs or Frenkel excitons, are characterized by a single index, while Wannier excitons are characterized by two indices: a center-of-mass momentum Q and a relative motion index ν‎.

The Cooper pair problem is notably simpler than the Wannier exciton or Frenkel exciton problem because of the constant and separable form taken for the scattering between single-index pairs in the reduced BCS potential. This fact explains why exact eigenstates can be obtained in all three problems for a single pair but only in the case of Cooper pairs for more than one pair, thanks to the Richardson-Gaudin procedure.

# 11.3 One composite boson

We now recall how the composite bosons of interest—Cooper pairs, Frenkel excitons, and Wannier excitons—are formed through the potentials described in the preceding section. Let us start with one fermion pair, to better see how these potentials act.

## 11.3.1 One Cooper pair

We first consider the BCS Hamiltonian $HBCS′=H0′+VBCS$ for electrons in the energy layer where the potential acts, $H0′$ being the free electron Hamiltonian $∑wkεkaks†aks$. When the Hamiltonian $HBCS′$ acts on one pair of up-spin and down-spin electrons with zero center-of-mass momentum added to the frozen Fermi sea $|F0〉$, it gives

(11.17)
$Display mathematics$

(p.283) where $|Vk1〉=VBCSBk1†|F0〉$, shown in Fig. 11.8(a), reads

(11.18)
$Display mathematics$

After some algebra, it is possible to show that $B†(E1)|F0〉$, with

(11.19)
$Display mathematics$

is eigenstate for a single Cooper pair, $(HBCS′−E1)B†(E1)|F0〉=0$, provided that E1 fulfills

(11.20)
$Display mathematics$

This equation has a unique bound state with an energy below the lowest free pair energy $2εF0$. This energy is given by $E1=2εF0−εc$, with $εc=2Ωσ/(1−σ)$, where $σ=e−2/ρV$.

## 11.3.2 One Frenkel exciton

In the case of Frenkel excitons, we split the system Hamiltonian as

(11.21)
$Display mathematics$

with $He=εe∑an†an$, and $Hh=εh∑bn†bn$. As vcoul, defined in Eq. (11.13), requires carriers at different lattice sites to act, we find, for Vneutral given in Eq. (11.10),

(11.22)
$Display mathematics$

where $|Vn1〉=VtransfBn1†|0〉$, shown in Fig. 11.8(b), reads

(11.23)
$Display mathematics$

After some algebra, it is possible to show that $BQ†|0〉$, with

(11.24)
$Display mathematics$

(p.284) is $H(F)$ eigenstate for a single electron-hole pair, $(H(F)−EQ)BQ†|0〉=0$, with energy

(11.25)
$Display mathematics$

Frenkel excitons are characterized by a single index Q only, since they are made of single-index electron-hole pairs, this pair index being the lattice site n.

## 11.3.3 One Wannier exciton

In the case of Wannier excitons, Coulomb interaction is reduced to intraband processes; this reduction amounts to taking the system Hamiltonian as

(11.26)
$Display mathematics$

with $He=∑εk(e)ak†ak$, and $Hh=∑εk(h)bk†bk$. Since Vee and Vhh require two electron-hole pairs at least to act, we find that

(11.27)
$Display mathematics$

where $|VQ1p1〉=VehBQ1,p1†|0〉$, shown in Fig. 11.8(c), is given by

(11.28)
$Display mathematics$

The electron-hole pair keeps its center-of-mass momentum Q1 in the series of repeated Coulomb scatterings making the Wannier exciton. Although more demanding than for one Cooper pair or one Frenkel exciton, it is possible to show that the single-pair eigenstate of H reads $BQν†|0〉$, with

(11.29)
$Display mathematics$

In addition to the center-of-mass momentum Q, Wannier excitons have the quantum index ν‎, which characterizes the relative motion of the pair, $〈p|ν〉$ being the Wannier exciton relative motion wave function in momentum space.

The two degrees of freedom $(Qi,νi)$ of the i exciton can be traced back to the two degrees of freedom $(ke,kh)$ or (Q, p) of the free electron-hole pairs from which Wannier excitons are constructed,

(11.30)
$Display mathematics$

(p.285) The eigenenergy of the (Q, ν‎) exciton is equal to $Egap+εν+Q2/2MX$, the relative motion energy εν and wave function $〈p|ν〉$ being such that

(11.31)
$Display mathematics$

This equation is similar to the Schrödinger equation for a hydrogen atom, its reduced mass being $μX=memh/(me+mh)$. The relative motion energy εν comes from intraband Coulomb processes between one electron and one hole.

## 11.3.4 Discussion

Wannier excitons have two degrees of freedom, (Q, ν‎), which originate from the two degrees of freedom, $(ke,kh)$ or (Q, p), of the free electron-hole pairs from which they are constructed. Frenkel excitons and Cooper pairs are constructed on single-index pairs: site n electron-hole excitations in the case of Frenkel excitons, and free up-spin and down-spin electron pairs with opposite momenta, $(p,−p)$, in the case of Cooper pairs. So, they are characterized by a single index only. For Frenkel excitons, this index is the center-of-mass momentum Q. For Cooper pairs, since their center-of-mass momentum is equal to zero by construction, the index can only be associated with their energy. For the reduced BCS potential, this energy either is the (single) bound level E1 or the various unbound levels close to $2εk$. These unbound levels are not considered in BCS superconductivity, which fundamentally deals with a volume-linear number of Cooper pairs and with excited states in which a bound pair is broken and one electron of the broken pair is removed from the system.

It is worth noting that the way the center-of-mass momentum Q appears in Wannier excitons is completely different from the way it appears in Frenkel excitons. For Wannier excitons and for Cooper pairs, the center-of-mass momentum of the pair has a well-defined value right from the beginning, this value being, by construction, zero for Cooper pairs. The attractive BCS potential brings two opposite-momentum electrons close enough to form a bound state. In the case of Wannier excitons, the attractive potential between one free electron and one free hole produces correlated states which can be either bound or unbound. For Frenkel excitons, this is somewhat the opposite: we start with an electron-hole pair already bound at a lattice site, as a result of intrasite direct and exchange Coulomb processes. This pair is then delocalized into a plane-wave state with momentum Q by the intersite interlevel potential. These interlevel Coulomb processes are mandatory for Frenkel exciton formation, while their equivalents in the case of Wannier excitons, shown in Fig. 11.3(b), play no role in the Wannier exciton formation.

# 11.4 Two composite bosons

The precise study of two pairs is quite useful for understanding the interplay between fermion-fermion interaction and fermion exchange in composite boson systems, because (p.286) the Pauli exclusion principle enters into play at the two-pair level. This is why it is rewarding to begin the study of composite boson many-body effects with just two composite bosons.

## 11.4.1 Two Cooper pairs

The situation is quite peculiar in the case of Cooper pairs because, in the absence of Pauli blocking, there would be no interaction at all between two Cooper pairs, a point rarely mentioned. Indeed, two zero-momentum pairs do not interact through the reduced BCS potential because, for such interaction between $(k1,−k1)$ and $(k2,−k2)$ pairs to exist, we should have $k1=k2$; so, the two fermion pairs would be identical (see Fig. 11.9), which is prohibited by the Pauli exclusion principle.

This feature greatly simplifies Cooper pair many-body effects. In the case of two free pairs, the interaction part $|Vk1k2〉,$ visualized in the two diagrams of Fig. 11.10, follows from

(11.32)
$Display mathematics$

Figure 11.10 When the reduced BCS potential acts on two free electron pairs $(k1,k2)$, one of the pair can be transformed into a k pair, as in (a), or an electron exchange between the two pairs can first occur, as in (b), thus inducing an exchange interaction scattering between the two pairs—direct interaction between the two pairs through the reduced BCS potential being prohibited by the Pauli exclusion principle, as shown in Fig. 11.9.

As two Cooper pairs cannot directly interact through the reduced BCS potential, the only way they can feel each other is through fermion exchange resulting from the Pauli exclusion principle, as shown in Fig. 11.10(b).

Using Eq. (11.32), we can show that $B†(R1)B†(R2)|F0〉$, with $B†(R)$ given in Eq. (11.19), is the exact two-pair eigenstate of $HBCS′$, provided that $(R1,R2)$ fulfill

(11.33)
$Display mathematics$

the two-pair energy being $E2=R1+R2$. These two coupled equations can be analytically solved, the two-pair ground-state energy reading in the large sample limit as

(11.34)
$Display mathematics$

(p.287) where ρ‎ is the density of states, $εc=2Ωσ/(1−σ)$ is the binding energy of a single Cooper pair with $σ=e−2/ρV$, and NΩ‎ is equal to ρ‎Ω‎, where $Ω=2ωc$ is the energy extension of the potential layer.

## 11.4.2 Two Frenkel excitons

Atomic excitations result from (i) the intrasite potential Vneutral, which forces the electron and the hole to be at the same lattice site, (ii) the transfer potential Vtransf, which delocalizes the bound pair over the whole sample, and (iii) the direct Coulomb potential vcoul, which acts on electrons and holes at different lattice sites and therefore requires at least two Frenkel excitons to enter into play.

The direct Coulomb potential vcoul induces four scattering processes between two Frenkel excitons, as shown in Fig. 11.11(a). These direct scatterings are very similar to the direct Coulomb scattering $ξnjmi$ of two Wannier excitons. Direct processes can also be mixed with carrier exchanges to generate the exchange Coulomb scatterings shown in Fig. 11.11(b). However, unlike in the case of Wannier excitons, for which these exchange Coulomb processes are dominant for small momentum transfer, it is possible to show that exchange Coulomb scatterings between two Frenkel excitons reduce to zero.

Figure 11.11 (a) Direct and (b) exchange Coulomb scatterings between two Frenkel excitons, Q1 and Q2. While exchange Coulomb processes control the interaction between two Wannier excitons for small momentum transfer, similar exchange Coulomb scatterings between two Frenkel excitons, shown in (b), reduce to zero.

In addition to these rather standard scatterings, there also exist intersite interlevel Coulomb processes (see Fig. 11.4(a)) that come from the Vtransf potential responsible for Frenkel exciton formation. They produce scatterings between two Frenkel excitons through a mixing with carrier exchange, as shown in Fig. 11.12. Since their equivalents in the case of Wannier excitons, namely, the interband valence-conduction scatterings, (p.288) are commonly neglected, such exchange scatterings never appear in usual Wannier exciton many-body effects. By contrast, their equivalent in the case of Cooper pairs, shown in Fig. 11.10(b), rules the entire many-body physics of these composite bosons.

Figure 11.12 Exchange transfer scattering between two Frenkel excitons.

## 11.4.3 Two Wannier excitons

Wannier excitons interact through direct Coulomb processes between two electrons, between two holes, and between one electron and one hole. While it is rather easy to handle correlations between one free electron and one free hole to form a single exciton, the situation is far more complicated when turning to two pairs, as can be seen from the diagram in Fig. 11.13, which shows possible intraband Coulomb interactions between two free electrons and two free holes. The difficulty comes from the exciton composite nature, more precisely, the impossibility of assigning one electron to a specific hole in order to form a well-defined exciton and then constructing an interaction potential between two excitons.

Figure 11.13 Direct Coulomb processes between two free electrons and two free holes in the case of two Wannier excitons.

To overcome this difficulty and derive the direct Coulomb scattering $ξnjmi$ shown in Fig. 11.14(a) in a clean way, we have constructed an operator formalism based on (p.289) the semiconductor Hamiltonian H written in terms of electron and hole operators. This formalism generates direct Coulomb scatterings through two commutators, namely,

(11.35)
$Display mathematics$
(11.36)
$Display mathematics$

Figure 11.14 (a) Direct Coulomb scattering $ξnjmi$ between two Wannier excitons. (b) Pauli scattering $λnjmi$ for carrier exchange between these excitons.

The two possible ways to associate two electrons and two holes into two Wannier excitons appear through

(11.37)
$Display mathematics$

where $λnjmi$ is the Pauli scattering for carrier exchange shown in Fig. 11.14(b). Inserting Eq. (11.37) into the RHS of Eq. (11.36) readily shows that it is possible to substitute the direct Coulomb scattering $ξnjmi$ in Eq. (11.36) with $−ξinnjmi$, the “in” exchange Coulomb scattering shown in Fig. 11.15(a) being given by

(11.38)
$Display mathematics$

Figure 11.15 (a) “In” exchange Coulomb scattering $ξin(njmi)$ between two Wannier excitons, as defined in Eq. (11.38). (b) “Out” exchange Coulomb scattering $ξout(njmi)$. These two exchange Coulomb scatterings are linked to the Pauli scattering $λ(njmi)$ for carrier exchange through Eq. (11.39).

or with any linear combination of the two scatterings, $aξ(njmi)−bξin(njmi)$, with $a+b=1$. Physical results do not depend on the (a,b) indetermination—a fact that can be used to check their correctness.

Another energy-like exchange Coulomb scattering exists, called $ξout(njmi)$, in which Coulomb interactions take place among the “out” states $(m,n)$, as shown in Fig. 11.15(b). It is possible to show that these two exchange Coulomb scatterings are linked to the (p.290) dimensionless Pauli scattering $λ(njmi)$ through

(11.39)
$Display mathematics$

So, these two exchange Coulomb scatterings are equal for energy-conserving processes, as required by time-reversal symmetry. Note that in these three energy-like scatterings, ξ‎, $ξin$, and $ξout$, the electron-hole Coulomb attraction responsible for exciton formation appears along with electron-electron and hole-hole repulsions.

The two commutators given in Eqs. (11.35, 11.36) provide a clean mathematical way to derive the physically intuitive scatterings taking place between two Wannier excitons, with “in” and “out” excitons possibly constructed on different pairs. Yet, the exact eigenstates of two electron-hole pairs, resulting from these repeated scatterings, are not analytically known. The two-pair eigenstates are called biexcitons. These four-fermion structures have bound states analogous to hydrogen molecules when the two electrons, as well as the two holes, are in a spin-singlet state. As the effective exciton-exciton attraction resembles the attraction between two dipoles, the resulting biexciton binding energy is very small compared to the exciton binding energy. Still, even for very small binding, an exact treatment is required to obtain the energy poles associated with biexciton bound states. We will come back to biexcitons in Part III, which deals with quantum composite particles related to excitons.

## 11.4.4 Discussion

Since Cooper pairs do not interact directly through the reduced BCS potential, energy-like scatterings between two Cooper pairs result from electron exchange induced by the Pauli exclusion principle between up-spin electrons and between down-spin electrons. Similar scatterings assisted by carrier exchange also exist for Frenkel excitons, the exchange being mixed with the scattering coming from the transfer potential responsible for Frenkel exciton formation. Two Frenkel excitons also have direct Coulomb scatterings between electrons and holes, which are similar to the scatterings existing for two Wannier excitons. So, similarities also exist between the many-body physics of Frenkel excitons and that of Wannier excitons. Although not often studied, Frenkel excitons definitely are quite interesting quantum objects, because they bridge BCS superconductors having Cooper pairs and semiconductors having Wannier excitons.

It is worth noting that, while two Cooper pairs have analytically known eigenstates, two excitons have not. This is due to the highly simplified reduced BCS potential, as opposed to the complex long-range Coulomb potential.

# 11.5N composite bosons

The many-body physics of N composite bosons directly follows from what we have learned from N = 2 because, with just two composite quantum particles, we already face Pauli blocking between two identical fermions, and interaction between composite (p.291) bosons induced by fermion-fermion potential. Considering N composite bosons, however, helps elucidate the importance of the particle degrees of freedom as well as the consequences of the potential at hand.

## 11.5.1N Cooper pairs

Let us first consider N Cooper pairs. Because of the very simple form of the reduced BCS potential, their exact eigenstates can be analytically obtained. To get these eigenstates, we first calculate the Hamiltonian acting on N zero-momentum pairs:

(11.40)
$Display mathematics$

The interacting part $|Vk1k2…kN〉$ is visualized in the diagrams in Fig. 11.16. The remarkable feature of this interacting part is that one or two pairs at most are involved in the interaction processes.

Figure 11.16 The reduced BCS potential acting on N up-spin and down-spin electron pairs, $(k1,−k1)$, $(k2,−k2)$, …, $(kN,−kN)$, leads either to a scattering inside one pair (a) or to such a scattering mixed with electron exchange between two pairs (b), leaving the other pairs unaffected. The fact that at most two pairs interact is directly related to the fact that the N-pair energy has no term in $N(N−1)(N−2)$ and higher.

From the above equation, it is possible to show that the N-pair eigenstates read

(11.41)
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where the operator $B†(R)$ is the one already appearing in the one-pair eigenstate (see Eq. (11.19)), the Ri’s being linked by N nonlinear equations:

(11.42)
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(p.292) The N-pair energy being $EN=R1+R2+⋯+RN$, the analytical resolution of these equations gives the ground-state energy in the large sample limit as

(11.43)
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with $σ=e−2/ρV$, the single Cooper pair energy being E1. We note that EN has no term in $N(N−1)(N−2)$ and higher, although such higher-order terms usually are present in many-body effects involving N particles. This rather surprising result is linked to the fact that only two Cooper pairs among N are involved in interaction processes, as seen from Fig. 11.16. A similar higher-order cancellation is found when calculating the Hamiltonian mean value in the N–identical Frenkel exciton state $B0†N|0〉$, but owing to a subtle cancellation.

## 11.5.2N Frenkel excitons

As for two excitons, the eigenstates of N Frenkel excitons are not analytically known because of the complexity of the various Coulomb processes existing between atomic levels. We can estimate the ground-state energy of N Frenkel excitons through the mean value of the Hamiltonian $H(F)=H0(F)+VCoul$ in the state , the exact N–Frenkel exciton ground state being expected to be close to this state. This mean value reduces to only two terms:

(11.44)
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where Ns is the number of lattice sites, while $ξ(0000)$ contains all direct and exchange Coulomb scatterings existing between different lattice sites

(11.45)
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We note that, like the EN energy for N Cooper pairs, this Hamiltonian mean value has no term in $N(N−1)(N−2)$ and higher, because of a subtle cancellation between exchange terms appearing in the numerator and the denominator of the Hamiltonian mean value.

## 11.5.3N Wannier excitons

As for Frenkel excitons, the N–Wannier exciton eigenstates are not analytically known. We can estimate the ground-state energy of N Wannier excitons through the (p.293) Hamiltonian mean value in the $B0†N|0〉$ state, where $B0†$ creates a ground-state exciton with center-of-mass momentum Q = 0 and relative motion ground-state index ν0, this state being close to the N–Wannier exciton ground state. We find

(11.46)
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The direct Coulomb scattering $ξ(0000)$ cancels because of the fact that electron-electron and hole-hole repulsions are as large as electron-hole attraction. So, correction to the energy of N noninteracting ground-state excitons comes from exchange Coulomb scattering only. The Hamiltonian mean value is found to expand in 3D as

(11.47)
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where $η=N(aX/L)3$ is the dimensionless parameter associated with Wannier exciton many-body effects, and RX is the 3D exciton Rydberg. We note that the first-order correction to the energy $−NRX$ of N independent excitons is positive, as is necessary to avoid density collapse of the exciton gas. This positive correction also shows that the average exciton binding energy decreases with exciton number as a result of the “moth-eaten effect” induced by Pauli blocking. A similar binding energy decrease with pair number exists for Cooper pairs, for the same physical reason.

We wish to stress that the Wannier excitons considered here have same carrier spin; so, they suffer Pauli blocking and do not form molecular biexcitons (see Part III on biexcitons).

## 11.5.4 Discussion

Although not commonly mentioned, the many-body physics of Cooper pairs within the reduced BCS potential is entirely controlled by the Pauli exclusion principle because the up-spin electron k can only interact with the down-spin electron (−k). Moreover, as interaction among N zero-momentum pairs leaves at least (N − 2) pairs unchanged, the exact N–Cooper pair energy, known by now, has no terms in $N(N−1)(N−2)$ and higher.

Absence of such higher-order terms also occurs in the Hamiltonian mean value of N Frenkel excitons, but because of a subtle cancellation between carrier exchanges occurring in the numerator and denominator of this mean value. Such a cancellation does not occur in the Hamiltonian mean value calculated in the N–Wannier exciton state $B0†N|0〉$, the many-body physics of these excitons being definitely far more complex.

# (p.294) 11.6 Many-body parameters

Many-body effects among excitons or among Cooper pairs are controlled by a dimensionless parameter η‎ that, when understood properly, demonstrates the fact that the many-body physics of these composite bosons is controlled by the Pauli exclusion principle. Indeed, for Wannier excitons, Frenkel excitons, and Cooper pairs, it is possible to assign the same physical meaning to the mathematical parameters appearing in the density expansion of relevant quantities such as the N-pair energy. We can cast the parameter attached to many-body effects as

(11.48)
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where Nmax is the maximum number of composite bosons the sample can accommodate. In the case of Cooper pairs, Nmax is the number of k states available for pairing in the BCS potential layer while, for Frenkel excitons, it is the number of lattice sites. This maximum number is less obvious for Wannier excitons. However, by noting that, at large density, Wannier excitons suffer a Mott dissociation into an electron-hole plasma, this maximum number corresponds to the number at which Wannier exciton wave functions start to overlap.

## 11.6.1 Cooper pairs

In textbooks, Cooper pairs are commonly studied in two cases: (i) when a single up-spin and down-spin electron pair is added to the energy layer where the reduced BCS potential acts, and (ii) when this layer is half-filled. In these two cases, the electrons which feel the potential have energies extending over an energy range Ω‎ above the frozen Fermi sea $|F0〉$. These two configurations correspond to N = 1 pairs and $N=ρΩ/2$ pairs, respectively, in the potential layer having a constant density of states ρ‎. With just two N numbers, it is difficult to pin down the physical parameter ruling many-body effects between Cooper pairs, and, more so, to understand that these many-body effects only come from the Pauli exclusion principle.

By analytically solving the Richardson-Gaudin equations for arbitrary N, we find that the N-pair ground-state energy reads in the large sample limit as

(11.49)
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where $E1=2εF0−εc$, with $εc=2Ωσ/(1−σ)$, is the single Cooper pair energy. This gives the condensation energy for N pairs resulting from the reduced BCS potential as

(11.50)
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where $Nmax=ρΩ+1$ is the number of the k states from which Cooper pairs can be constructed, that is, the number of k states with energy $εF0≤εk≤εF0+Ω$ when (p.295) the density of states is taken as constant and equal to ρ‎. Owing to the Pauli exclusion principle, this number also is the maximum number of Cooper pairs the sample can accommodate.

It is interesting to note that one can recover this condensation energy for arbitrary N in the dense limit by using the BCS ansatz in the situation where the attractive potential between up-spin and down-spin electrons does not extend symmetrically on both sides of the normal electron Fermi level. This can be done by taking the chemical potential introduced in the grand canonical ensemble approach to the BCS problem not exactly in the middle of the potential layer.

The condensation energy given in Eq. (11.50) shows that the average pair binding energy linearly decreases when the pair number increases, through an effect that, in the exciton context, we called the “moth-eaten effect”: each added electron pair removes one available state for pairing because of Pauli blocking, as if a little moth had eaten it. Equation (11.50) moreover shows that the average pair binding energy exactly cancels for $N=Nmax$. This is physically reasonable because, when the potential layer is completely full, the system has lost any flexibility to gain energy from the potential.

The average pair binding energy given in Eq. (11.50) furthermore shows that, for N and Nmax both large—as in the physical configuration—the binding energy decrease is ruled by the dimensionless parameter η‎ given in Eq. (11.48). This parameter is equal to 1/2 for a BCS potential extending symmetrically on both sides of the normal electron Fermi level, leading to a decrease by half of the single-pair binding energy.

## 11.6.2 Frenkel excitons

The many-body parameter ruling Frenkel excitons is also derived from quantities involving N excitons. One of these quantities is the normalization factor of N identical Frenkel exciton states, $〈0|BQNBQ†N|0〉=N!FN$. The Pauli exclusion principle between electron-hole pairs located at lattice sites and available for making Frenkel excitons leads to a FN decrease exactly given by

(11.51)
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where Ns is the number of lattice sites; Ns obviously is the maximum number of excitations in the absence of spin and orbital degrees of freedom and thus is the maximum number of Frenkel excitons the sample can accommodate. Consequently, $N/Ns$ corresponds to $N/Nmax$, a result that supports the form of the many-body parameter given in Eq. (11.48).

We can also consider the Hamiltonian mean value in the N–Frenkel exciton state $B0†N|0〉$. It reads

(11.52)
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(p.296) The effective scattering $VR(eff)$ comes from all possible Coulomb processes between ground and excited atomic levels at lattice sites separated by R, namely,

(11.53)
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In the tight-binding approximation, $VR(eff)$ scales as $e2/R3$, leading to a sum over R which is sample volume free. So, the Hamiltonian mean value quoted in Eq. (11.52) also provides support for a Frenkel exciton many-body parameter reading $N/Ns$, that is, $N/Nmax$.

## 11.6.3 Wannier excitons

In the case of Wannier excitons, the many-body parameter can be extracted from the normalization factor of N ground-state excitons and also from the Hamiltonian mean value calculated in this state; $〈0|B0NB0†N|0〉$ is equal to $N!FN$ with, in 3D,

(11.54)
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while the Hamiltonian mean value is given by

(11.55)
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Into these two calculations performed for excitons in ground state $0=(Q=0,ν0)$ enters the exciton ground-state wave function $〈r|ν0〉$. Its spatial extension aX appears through η‎ given by

(11.56)
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A simple way to physically understand this quantity is to note that the relative motion of composite excitons extends over a $aX3$ volume, while their center of mass is delocalized over the sample volume L3. They have a chance to interact through fermion exchange when they overlap. So, the chance for N excitons to have a fermion exchange is equal to the exciton volume multiplied by the number of excitons that can overlap and divided by the sample volume in which they move: this just is the parameter η‎ given in Eq. (11.56).

Yet, this physical understanding of the Wannier exciton many-body parameter in terms of exciton relative motion extension cannot be extended to Frenkel excitons, because the spatial extension of electron-hole pairs from which Frenkel excitons are constructed reduces to zero within the tight-binding approximation. By contrast, it is easy (p.297) to extend to Wannier excitons the physical understanding of the Cooper pair many-body parameter or the Frenkel exciton many-body parameter in terms of the maximum number of composite bosons the sample can accommodate. Indeed, when the Wannier exciton density is large enough to have excitons overlapping, electron-hole pairs are known to dissociate into an electron-hole plasma. The number of excitons at which overlap starts in a sample volume L3 is this L3 volume divided by the exciton volume $aX3$. So, the maximum number of Wannier excitons that a L3 volume can accommodate without dissociation is of the order of

(11.57)
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in agreement with the Wannier exciton many-body parameter $η=NaX3/L3=N/Nmax$. It is interesting to note that this $L3/aX3$ upper bound for the Wannier exciton number, which is fundamentally associated with Mott dissociation, appears in a natural way. Algebraic calculations on Wannier exciton many-body effects do reflect this underlying physics.

## 11.6.4 Discussion

Through the N dependence of quantities involving N Cooper pairs, N Frenkel excitons, and N Wannier excitons, we have been able to cast the dimensionless parameter ruling their many-body effects as $η=N/Nmax$, where Nmax is the maximum number of composite bosons the sample can accommodate. However, the ways Nmax appears for single-index pairs like Cooper pairs or Frenkel excitons and for double-index pairs like Wannier excitons are rather different. For single-index pairs, Nmax simply appears as the number of states available for pairing, that is, the number of k states in the potential layer in the case of Cooper pairs, and the number of lattice sites in the case of Frenkel excitons. The way Nmax appears in the case of Wannier excitons is more subtle. The numbers of $(ke,kh)$ states available for pairing in the conduction and valence bands are huge; these numbers do not play a role in the Wannier exciton many-body parameter. Instead, through $η=NaX3/L3$, the many-body parameter is associated with the fact that, above a certain number, Wannier excitons overlap and dissociate into an electron-hole plasma. This overlap is directly related to the finite extension of the Wannier exciton relative motion and to the fact that Wannier excitons have, in addition to a center-of-mass momentum Q, a quantum index ν‎ which characterizes this relative motion. We wish to recall that in the physical BCS configuration Cooper pairs strongly overlap but do not dissociate because they feel each other by Pauli blocking only. This is not so for Wannier excitons because they interact through long-range Coulomb forces.

# 11.7 Wave functions

Composite boson wave function is commonly thought to be given by the prefactor in the expansion of their creation operator. It also is commonly agreed that, for fermion (p.298) pair bound states, the wave function spatial extension aB is related to the binding energy $εB$ through a simple dimensional argument, namely, $εB≃1/2μaB2$, where μ‎ is the pair reduced mass. By considering Wannier excitons, Frenkel excitons, a single Cooper pair, and a dense system of Cooper pairs, we will see that the answer is more subtle.

## 11.7.1 Wannier excitons

The creation operator of a Wannier exciton with center-of-mass momentum Q and relative motion index ν‎ reads

(11.58)
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where $BQ,p†=ap+γeQ†b−p+γhQ†$, with $γe=1−γh=me/(me+mh)$, creates a free electron-hole pair with center-of-mass momentum Q and relative motion momentum p. These free pair states are eigenstates of the free electron and free hole Hamiltonians $He=∑kεk(e)ak†ak$, and $Hh=∑kεk(h)bk†bk$

(11.59)
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where $εp=p2/2μX$, and $EQ=Q2/2MX$, with $μX−1=me−1+mh−1$, and $MX=me+mh$. By writing the Coulomb potential between electrons and holes as

(11.60)
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(11.61)
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So, $BQν†|0〉$ is the one-pair eigenstate of the interacting electron-hole Hamiltonian, with energy $Egap+EQ+εν$, provided that the above bracket satisfies

(11.62)
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(p.299) This result shows that the prefactor of a Wannier exciton creation operator expanded in terms of free electron-hole pair operators is indeed the wave function associated with the exciton relative motion taken in momentum space.

In 3D, that is, for $Vq=4πe2/ϵscL3q2$, the solution of the above equation leads to a ground-state binding energy $εν0=1/2μXaX2$, and a ground-state wave function, normalized in a sample volume $L3,$ as

(11.63)
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So, the $|ν0〉$ wave function extension in momentum space is $1/aX$ (see Fig. 11.17(a)).

Figure 11.17 Extension of the Wannier exciton ground-state wave function $1/aX$ in momentum space (a), and aX in real space (b). The exciton Bohr radius aX is related to the ground-state relative motion energy through $εν0=1/2μXaX2$.

To get this extension in real space, we use

(11.64)
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This shows that $〈r|ν0〉$ stays finite in the $r=0$ limit, while it becomes very small for large r due to rapid oscillations of $sinpr$ over the p extension of $〈p|ν0〉$. These oscillations lead to a spatial extension of $〈r|ν0〉$ that scales as aX. It is actually possible to calculate the above integral analytically. By transforming the integral of $sinpr$ from 0 to ∞ into the integral of $eipr$ from $−∞$ to $+∞$, the residue theorem leads to

(11.65)
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(p.300) This result demonstrates that the spatial extension of the exciton ground state is indeed aX (see Fig. 11.17(b)), this extension being related to the ground-state Wannier exciton binding energy through $εν0=1/2μXaX2$.

## 11.7.2 Frenkel excitons

We now consider the Frenkel exciton creation operator characterized by the single index Q. This operator reads

(11.66)
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where $Bn†=an†bn†$ creates an electron-hole excitation at lattice site n. From the excitation transfer potential given by

(11.67)
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we find that the Frenkel exciton Hamiltonian acting on a single correlated pair yields

(11.68)
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It is then easy to check that the Hamiltonian eigenstate which satisfies

(11.69)
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reads $〈n|Q〉=eiQ⋅Rn/Ns$, in agreement with Eq. (11.66), the associated Frenkel exciton energy being given by

(11.70)
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owing to translational invariance which makes the sum over $n′≠n$ in Eq. (11.69) independent of n. So, as in the case of Wannier excitons, the prefactor of the Frenkel exciton creation operator in terms of creation operators for site n excitations is the exciton wave function, solution of the corresponding Schrödinger equation.

(p.301) Equation (11.66) shows that the Frenkel exciton wave function, $〈n|Q〉=eiQ⋅Rn/Ns$, is just a phase; so, it is flat in real space and infinitely narrow in momentum space (see Fig. 11.18):

(11.71)
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Figure 11.18 The Frenkel exciton wave function is just a phase; so, $|〈n|Q〉|$ is flat in real space (a) and infinitely narrow in momentum space (b).

## 11.7.3 Cooper pairs

Cooper pairs are commonly studied in two cases: as a single Cooper pair, and as a dense volume-linear regime of Cooper pairs.

### Single Cooper pair

Cooper showed that the eigenstate for one up-spin and one down-spin electron in the BCS potential layer reads

(11.72)
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where $|F0〉$ is the “frozen Fermi sea,” $Bk†=ak↑†a−k↓†$ is the creation operator of a zero-momentum electron pair, and E1 is the single Cooper pair energy. The function $ωk$, equal to 1 for $εF0≤εk≤εF0+Ω$, and 0 otherwise, characterizes the potential layer. By analogy with the exciton, we can rewrite this single Cooper pair creation operator as

(11.73)
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For $VBCS=−V∑k′kωk′ωkBk′†Bk$, the Hamiltonian $HBCS′$ acting in the potential layer gives

(11.74)
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(p.302) So, $BE1†|F0〉$ is eigenstate of the BCS Hamiltonian with energy E1, provided that

(11.75)
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This equation gives the single Cooper pair wave function in momentum space as $〈k|E1〉=ωk/(2εk−E1)$, in agreement with Eq. (11.72), the single-pair energy E1 following from

(11.76)
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The solution of this equation yields a single bound state with energy $E1=2εF0−εc$, where $εc=2Ωσ/(1−σ)$, and $σ=e−2/ρV$ for a constant density of states ρ‎.

By noting that

(11.77)
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the normalized single Cooper pair wave function $|ϕ1〉$ appears in momentum space as

(11.78)
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So, $ρΩ〈k|ϕ1〉$ varies from $(2Ω+εc)/εc1/2$ to $(2Ω+εc)/εc−1/2$ when εk varies from εF0 to $εF0+Ω$. This result also shows that the energy extension of the $〈k|ϕ1〉$ wave function above εF0 scales as εc (see Fig. 11.19). To get the $〈k|ϕ1〉$ extension in momentum space, (p.303) we write εc as $1/2μcac2$, with $1/μc=2/m$. So, an energy extension from $εF0=kF02/2m$ to $εF0+εc$ corresponds to a $|k|$ extension above $kF0$ that scales as

(11.79)
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Figure 11.19 Normalized single Cooper pair wave function $〈k|ϕ1〉$ given in Eq. (11.78), as a function of $εk=k2/2m$.

This extension depends not only on ac but also on the minimum momentum $kF0$ of electrons that make Cooper pairs.

When turning to real space, a calculation similar to the one performed for Wannier excitons (see Eq. (11.64)) but with a constant density of states ρ‎ gives, for $k=2mεk$,

(11.80)
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This result shows that $〈r|ϕ1〉$ stays finite for $r=0$, while it is very small for r large. Indeed, the sizable part of the integral controlled by the factor $1/(2ε−2εF0+εc)$ extends from εF0 to $εF0+εc/2$, a range which corresponds to $kF0. For r much larger than $kF0ac2$, the integral goes to zero, because $sinkr$ has many oscillations in this k range. So, the extensions of the single Cooper pair wave functions in r and k spaces scale as $kF0ac2$ and $1/kF0ac2$, respectively (see Fig. 11.20). These extensions are different from the naive scales, ac and $1/ac$, that are associated with the single Cooper pair binding energy $εc=1/mac2$. The difference fundamentally comes from the fact that, while its energy extension is equal to the pair binding energy εc, the single Cooper pair wave function is (p.304) made of k states, with $|k|$ larger than a (large) finite value $kF0$, in contrast with Wannier excitons, which are made of pairs with momentum starting from $|k|=0$.

Figure 11.20 Normalized single Cooper pair wave function in momentum space (a) and in real space (b). Note that the spatial extension is not $ac=1/mεc$ but $kF0ac2$.

### Dense regime of Cooper pairs

The physically relevant regime for BCS superconductivity does not correspond to a single Cooper pair but to a volume-linear number of pairs feeling the attractive BCS potential. There are essentially two ways to approach this regime: either via the BCS ansatz, which is mathematically supported by the Bogoliubov procedure, or via the exact Richardson-Gaudin equations.

1. (i)The BCS ansatz: Let us start with the BCS ansatz for condensed pairs. This ansatz can be written as (P. Anderson 1958)

(11.81)
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with $Bk†=ak↑†a−k↓†$ and

(11.82)
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Since, in the following discussion, we are not concerned about the Josephson effect (Josephson 1962, 1965; Tinkham 2004), we take uk and vk as real. The projection of the grand canonical state $|ψBCS〉$ onto the N-pair subspace, namely, $B†N|F0〉$, corresponds to having all pairs in the same B state. By writing the creation operator B as

(11.83)
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we find the pair wave function as

(11.84)
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where $Ek=ξk2+|Δ|2$, and $ξk=εk−μ$. In the physically relevant configuration where the attractive potential extends equally on both sides of the normal electron Fermi level, the chemical potential μ‎ lies in the middle of the potential layer, that is, $μ=εF0+Ω/2$. (p.305) We then find $|Δ|≃Ωe−1/ρV$. This gives $〈k|φ〉≃−2ξk/|Δ|$ for $ξk<−|Δ|$, and $〈k|φ〉≃|Δ|/2ξk$ for $ξk>|Δ|$. As

(11.85)
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with $NΩ=ρΩ$, the normalized pair wave function

(11.86)
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of the BCS condensate exhibits three different scales (Zhu et al. 2012), as shown in Fig. 11.21:

(11.87)
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(11.88)
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(11.89)
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Figure 11.21 The normalized wave function $〈k|ϕ〉$ for condensed Cooper pairs in the half-filling configuration, as deduced from the BCS ansatz, is sizable for $εF0≲εk≲εF0+Ω/2=εF$ only. Inset: $Fk=ukvk$, commonly called “pair wave function.”

This shows that the sizable part of the pair wave function in the BCS condensate is a linearly decreasing function of εk between εF0 and $εF0+Ω/2$. Since the phonon energy $ωc≃Ω/2$ is small compared to the Fermi energy εF0 of the frozen Fermi sea, this regime corresponds to k momentum extending from $kF0$ to $2m(εF0+Ω/2)≃kF0+1/2kF0aΩ2$, with $aΩ=1/mΩ$ (see Fig. 11.22). A procedure analogous to Eq. (11.79) then gives the (p.306) spatial extension of the condensed pair wave function as

(11.90)
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Figure 11.22$〈k|ϕ〉$ and Fk as functions of k. The Fermi momentum for normal electrons is given by $kF=2m(εF0+Ω/2)≃kF0+1/2kF0aΩ2$, with $aΩ=1/mΩ$.

Here also, this spatial extension is not related to the average Cooper pair binding energy, which, in the BCS configuration, that is, for half-filling, is equal to half the single-pair binding energy, $εc/2$. The Cooper pair wave function is usually not associated with $〈k|φ〉=vk/uk$ appearing in the creation operator B when $N=NΩ/2$ pairs feel the BCS potential, but with

(11.91)
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This quantity, peaked on $εk=εF0+Ω/2$, spreads over an energy extension $|Δ|$ (see inset in Fig. 11.21). The reason for associating Fk with the pair wave function most probably originates from the incorrect understanding of the BCS condensation energy $ρ|Δ|2/2$ as resulting from $ρ|Δ|$ pairs having an average binding energy $|Δ|$, instead of $ρΩ/2$ pairs having an average binding energy $εc/2$. With this incorrect understanding, one is naturally led to look for a quantity having an energy extension equal to the “binding energy” $|Δ|$, a possible quantity being $Fk=ukvk$. Since $|Δ|$ is not the pair binding energy in the condensed configuration, there is no particular reason to consider Fk as a pair wave function. If we still want to stay with this idea, we are led to introduce three length scales, namely, aΔ‎, ac, and aΩ‎, with $|Δ|≃Ωe−1/ρV$ set equal to $1/maΔ2$, $εc≃2Ωe−2/ρV$ set equal to $1/mac2$, and Ω‎ set equal to $1/maΩ2$. These three lengths are linked by $aΔ≃aΩac$. Since the Fk function has an energy extension $|Δ|$ around the normal electron Fermi energy, (p.307) $εF=εF0+Ω/2=kF2/2m,$ its k extension around kF scales as $1/kF0aΔ2$. This would lead to a pair wave function extension in r space that scales as

(11.92)
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which is far larger than the spatial extension of the $〈k|φ〉$ wave function given in Eq. (11.84). The above length scale actually corresponds to the Pippard coherence length $ξ=kF0/πm|Δ|$ (Pippard 1953). Even if Fk is not the Cooper pair wave function in the BCS regime, the $Fk≠0$ domain has a physical significance: it corresponds to the energy range in which condensed electron pairs differ from normal electrons. Indeed, normal electrons correspond to ($vk=1$, $uk=0$) for εk below εF, and to ($vk=0$, $uk=1$) above εF. So, $Fk=ukvk$ reduces to zero for all normal electrons; the energy domain in which Fk differs from zero pins down the effect of the BCS potential on normal electrons. Since physical effects come from this difference, it is natural to think that Fk plays a role in BCS superconductivity, even if the true BCS pair wave function is not $Fk=ukvk$ but $〈k|φ〉=vk/uk$.

2. (ii)The Richardson-Gaudin procedure: Richardson-Gaudin approach to the BCS problem allows us to obtain the exact wave function for N Cooper pairs added to the frozen Fermi sea $|F0〉$ as

(11.93)
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with $B†(Ri)=∑kωkBk†/(2εk−Ri)$, the Ri’s being the solutions of N coupled equations

(11.94)
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From the last term of the above equation, it is clear that the Ri’s must have different values. As a result, the exact ground state is a product of N pairs that are not all in the same state. The idea of having the same wave function for all pairs comes from elementary boson condensation, while we here deal with an N-pair system in which Pauli blocking plays a key role. Indeed, we cannot even say that the eigenstate corresponds to N pairs, each being in a state with creation operator $B†(Ri)$, because each time we add one more $B†(Ri)$, the Pauli exclusion principle removes one pair state $(k,−k)$ from the available k states through the moth-eaten effect. What in the end remains in the exact eigenstate given in Eq. (11.93) is a highly intricate many-body state. This state has to be very close to the BCS ansatz, as indicated by comparing the k state occupation numbers in the BCS ground state and in the Bogoliubov ground state.

## (p.308) 11.7.4 Discussion

The wave function of paired fermions is unambiguous in the case of a single pair but cannot be cleanly defined in the case of N pairs. Indeed, even if we accept the idea that the BCS condensate corresponds to a product of N identical operators, as in the BCS ansatz, these N pairs end up being in different states because of the “moth-eaten effect” induced by the Pauli exclusion principle acting on the B operators of the $B†N|F0〉$ condensed state.

According to Richardson and Gaudin, the exact N–Cooper pair state corresponds to a product of N different operators, these operators also suffering the “moth-eaten effect.” The study of the k state occupation number indicates that the BCS ansatz and the exact N-pair state must be very close after the action of the Pauli exclusion principle.

The exciton wave function is known in the case of a single electron-hole pair. In the case of N excitons, we can approach the N-exciton ground state by expanding it on the overcomplete N-free-exciton basis. However, here again, Pauli blocking acts on the product of free exciton operators. To know what the ground state of N excitons precisely looks like is beyond present capacity. This information would, however, be highly valuable for understanding the difference between Bose-Einstein condensation of elementary bosons, and “bosonic condensation” occurring in composite boson systems. The knowledge of the exact N–Cooper pair ground state, as obtained from the Richardson-Gaudin procedure, is a first strong clue toward this microscopic understanding.

# 11.8 Density regimes

Wannier excitons exist in the dilute regime; when the pair density increases beyond overlap, excitons dissociate into an electron-hole plasma. By contrast, Cooper pairs can strongly overlap without dissociating. These two different behaviors, quite puzzling at first, result from the potential that exists between the fermions making these composite bosons. Let us reconsider this question.

## 11.8.1 Wannier excitons

A single Wannier exciton is characterized by a center-of-mass momentum Q and a relative motion index ν‎ that differentiates the various bound and extended states. The ground state ν0 has a spatial extension of the order of the exciton Bohr radius aX. When many excitons are created in a semiconductor sample, they interact via Coulomb forces between their carriers. They can also change states through carrier exchanges because, being indistinguishable quantum particles, there is no way to pair one particular hole with a particular electron. The effects of these interactions increase with density. At low density, the N-exciton ground state is expected to be close to $B0†N|0〉$, where $B0†$ creates a ground-state exciton ν0 with center-of-mass momentum Q = 0. At large density, when excitons overlap, the Coulomb interaction between two excitons becomes as strong as the electron-hole attraction inside one exciton (see Fig. 11.23); excitons then dissociate, and the N-pair state changes drastically: it turns from a set of (weakly) interacting excitons, (p.309) which are neutral objects, to a two-component fermionic gas made of electrons and holes which conduct electric current.

The physically relevant regime for studying Wannier excitons is the dilute regime; it corresponds to

(11.95)
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Figure 11.23 In the dilute limit, electron-hole pairs form an exciton gas (a) which dissociates into a two-component fermionic gas (b) when the density increases above exciton overlap.

In this regime, the N-exciton ground state is close to $B0†N|0〉$. Corrections to this $B0†N|0〉$ state can be expanded in terms of exciton states $Bi1†Bi2†⋯BiN†|0〉$, the first-order correction in Coulomb interaction reading as a sum of $BQ,ν0†B−Q,ν0†B0†N−2|0〉$.

## 11.8.2 Frenkel excitons

Within the tight-binding approximation, the spatial extension of the electron-hole pair in a Frenkel exciton reduces to zero while the sample can have as many excitations as the number of lattice sites it contains. As a result, the dimensionless parameter associated with Frenkel exciton density is certainly less than 1 but not necessarily much smaller

(11.96)
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## 11.8.3 Cooper pairs

Although the study of a single electron pair added to a frozen Fermi sea nicely demonstrates the fact that the BCS potential contains enough physics to bind a pair even for an extremely small amplitude, the physically relevant density regime for BCS superconductivity is dense. This regime is commonly studied in the thermodynamic limit, with a volume-linear number of pairs.

The dimensionless parameter ruling the many-body physics of N Cooper pairs also is smaller than 1 by construction since it reads

(11.97)
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(p.310) where $Nmax=ρΩ+1$ is the number of up-spin or down-spin k electrons feeling the BCS potential. The BCS configuration in which the potential layer is half-filled corresponds to

(11.98)
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This regime definitely excludes a low-density approach similar to the one performed in the case of Wannier excitons. Fortunately, the BCS problem has an exact analytical solution, which allows the Cooper pair study from the dilute to the dense regime.

## 11.8.4 Discussion

The physically relevant regime for Wannier excitons is dilute; so, to study a Wannier exciton gas, we can perform a density expansion in the dimensionless parameter η‎, which stays much smaller than 1; otherwise, the exciton system would drastically change toward an electron-hole plasma.

By contrast, in the physically relevant regime for BCS superconductivity, the dimensionless parameter η‎ is not small compared to 1 but equal to 1/2. So, we have to resort to a nonperturbative procedure—either through the BCS ansatz followed by a variational or mean-field calculation, or through the Richardson-Gaudin exact procedure.