Links Between Cooper Pairs and Excitons
Links Between Cooper Pairs and Excitons
Abstract and Keywords
Chapter 11 compares Wannier excitons, Frenkel excitons, and Cooper pairs, with respect to their potentials, particle degrees of freedom, groundstate energies, wave functions, and manybody parameters. Shiva diagrams for composite boson manybody 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 manybody effects. This manybody parameter is N/N_{max}, where N_{max} 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 electronhole plasma. By contrast, Cooper pairs can strongly overlap without breaking, owing to the peculiar form of the reduced BCS potential.
Keywords: manybody parameter, Cooper pair, Wannier exciton, Frenkel exciton, Shiva diagram, pair wave function, potential, degree of freedom, groundstate energy
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 planewave state k_{e}, and a hole in a planewave state k_{h}, (p.273) we end with a correlated state i called Wannier exciton, whose center of mass is a plane wave ${\mathbf{\text{Q}}}_{i}={\mathbf{\text{k}}}_{e}+{\mathbf{\text{k}}}_{h}$. The creation operator of the resulting Wannier exciton reads
The Wannier exciton state index i has two degrees of freedom, like the electronhole pairs $({\mathbf{\text{k}}}_{e},{\mathbf{\text{k}}}_{h})$ from which Wannier excitons are constructed: in addition to a centerofmass momentum Q_{i}, a second quantum index ν_{i} characterizes the relative motion of the electronhole pair in the exciton. As with a hydrogen atom, this relative motion index differentiates exciton bound states from extended (unbound) states. The doubleindex character of Wannier excitons is necessary in order to write closure relations in the oneelectronhole subspace, either in terms of free electronhole pairs or in terms of Wannier excitons:
11.1.2 Frenkel excitons
In the case of Frenkel excitons, we start with electronhole pairs bound at lattice sites, as a result of the tightbinding approximation, which is valid in semiconductors hosting Frenkel excitons. These onsite excitations are delocalized over the whole sample by intersite interatomiclevel Coulomb processes. The creation operator of the resulting exciton reads
where N_{s} is the number of lattice sites located at the R_{n} position. The operator ${a}_{n}^{\u2020}$ creates an electron at site n, while ${b}_{n}^{\u2020}$ creates a hole; so, ${a}_{n}^{\u2020}{b}_{n}^{\u2020}$ 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 electronhole pairs but from upspin and downspin 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 upspin electron can interact with one downspin electron only. By contrast, in the case of Wannier excitons, a k_{e} electron can interact with anyk_{h} hole.
The creation operator of a single Cooper pair, as obtained by Léon Cooper, is given by
with E_{1} solution of
w_{k} is a sharp cutoff equal to 1 in the energy layer in which the effective attractive potential acts, that is, over a phonon energy omega_{c} on both sides of the normal electron Fermi level, and ${w}_{\mathbf{\text{k}}}=0$ otherwise. For a pair of free electrons added to the frozen Fermi sea ${F}_{0}\u3009$, the lowest E_{1} falls slightly below the frozen Fermi level. The energy gain resulting from the attractive BCS potential is equal to ${\epsilon}_{c}\simeq 4{\omega}_{c}{e}^{2/\rho \mathcal{V}}$, where $\mathcal{V}$ 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 manybody phenomenon occurring in a dense system. Indeed, in the BCS ansatz $({B}^{\u2020}{)}^{N}{F}_{0}\u3009$, the pair creation operator is given by
the “wave function” ${v}_{\mathbf{\text{k}}}/{u}_{\mathbf{\text{k}}}$ 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 singleindex composite bosons. This characteristic is shared by Frenkel excitons, which are made of electronhole pairs characterized by a single index, the lattice site n.
By contrast, Wannier excitons, in addition to having a centerofmass momentum Q, have a quantum index ν, which characterizes the relative motion of the correlated electronhole 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 manybody 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 electronhole 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)
where ${B}_{\mathbf{\text{k}}}^{\u2020}={a}_{\mathbf{\text{k}}\uparrow}^{\u2020}{a}_{\mathbf{\text{k}}\downarrow}^{\u2020}$, while ${w}_{\mathbf{\text{k}}}=1$ for ${\epsilon}_{{F}_{0}}\le {\epsilon}_{\mathbf{\text{k}}}\le {\epsilon}_{{F}_{0}}+\mathrm{\Omega}$, and ${w}_{\mathbf{\text{k}}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}0$ otherwise. This potential, represented in Fig. 11.1(a), is attractive, $\mathcal{V}$ being positive. Its repeated action on one upspin electron and one downspin electron leads to the formation of a single Cooper pair.
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:
(i) First, we consider the intrasite direct and exchange Coulomb processes shown in Fig. 11.2. They read in terms of the operator ${a}_{\nu n}^{\u2020}$ that creates an electron at the atomic level $\nu =(0,1)$ of lattice site n as
(11.8)$${V}_{\mathbf{\text{R}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\mathbf{\text{0}}}\left(\begin{array}{cc}1& 1\\ 0& 0\end{array}\right){\displaystyle \sum _{n}}{a}_{0n}^{\u2020}{a}_{1n}^{\u2020}{a}_{1n}{a}_{0n}+{V}_{\mathbf{\text{R}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\mathbf{\text{0}}}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right){\displaystyle \sum _{n}}{a}_{1n}^{\u2020}{a}_{0n}^{\u2020}{a}_{1n}{a}_{0n}.$$The elementary scattering for electrons that are at lattice sites n_{1} and n_{2}, separated by $\mathbf{\text{R}}={\mathbf{\text{R}}}_{{n}_{1}}{\mathbf{\text{R}}}_{{n}_{2}}$ (possibly equal to zero), and changing from atomic level ν to ν^{'} (possibly equal to ν) is given, in terms of atomic wave function $\u3008\mathbf{\text{r}}{a}_{\nu n}^{\u2020}0\u3009\phantom{\rule{thinmathspace}{0ex}}={\phi}_{\nu}(\mathbf{\text{r}}{\mathbf{\text{R}}}_{n})=\u3008\mathbf{\text{r}}{\mathbf{\text{R}}}_{n}\nu \u3009$, by
(11.9)$${V}_{\mathbf{\text{R}}}\left(\begin{array}{cc}{\nu}_{2}^{\mathrm{\prime}}& {\nu}_{2}\\ {\nu}_{1}^{\mathrm{\prime}}& {\nu}_{1}\end{array}\right)=\int d{\mathbf{\text{r}}}_{1}d{\mathbf{\text{r}}}_{2}\phantom{\rule{thinmathspace}{0ex}}\u3008{\nu}_{1}^{\mathrm{\prime}}{\mathbf{\text{r}}}_{1}\u3009\u3008{\nu}_{2}^{\mathrm{\prime}}{\mathbf{\text{r}}}_{2}\u3009\frac{{e}^{2}}{{\mathbf{\text{r}}}_{1}{\mathbf{\text{r}}}_{2}+\mathbf{\text{R}}}\u3008{\mathbf{\text{r}}}_{2}{\nu}_{2}\u3009\u3008{\mathbf{\text{r}}}_{1}{\nu}_{1}\u3009.$$By writing ${a}_{0n}^{\u2020}{a}_{1n}^{\u2020}{a}_{1n}{a}_{0n}$ as ${a}_{1n}^{\u2020}(1{a}_{0n}{a}_{0n}^{\u2020}){a}_{1n}$ and by turning to electron and hole operators ${a}_{1n}^{\u2020}={a}_{n}^{\u2020}$ and ${a}_{0n}^{\u2020}={b}_{n}$, the two potentials in Eq. (11.8) give rise to a twobody interaction which reads (p.277)
(11.10)δ, equal to ${V}_{R=0}\left(\begin{array}{cc}1& 1\\ 0& 0\end{array}\right){V}_{R=0}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, 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 samesite 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 electronhole 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.$${V}_{neutral}=\delta {\displaystyle \sum _{n}}{a}_{n}^{\u2020}{b}_{n}^{\u2020}{b}_{n}{a}_{n}.$$
(ii) Next, we delocalize the electronhole pair at site n through the intersite interlevel Coulomb process shown in Fig. 11.3(a): an electronhole pair recombines at site n_{2} while a pair is created at a different site n_{1}. Starting from the interlevel potential for electrons on different lattice sites,
(11.11)$$\sum _{{n}_{1}\ne {n}_{2}}}{V}_{{\mathbf{\text{R}}}_{{n}_{1}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathbf{\text{R}}}_{{n}_{2}}}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right){a}_{1{n}_{1}}^{\u2020}{a}_{0{n}_{2}}^{\u2020}{a}_{1{n}_{2}}{a}_{0{n}_{1}}\phantom{\rule{1.2pt}{0ex}},$$and writing ${a}_{1{n}_{1}}^{\u2020}{a}_{0{n}_{2}}^{\u2020}{a}_{1{n}_{2}}{a}_{0{n}_{1}}$ as ${a}_{1{n}_{1}}^{\u2020}{a}_{0{n}_{1}}{a}_{0{n}_{2}}^{\u2020}{a}_{1{n}_{2}}$, since ${n}_{1}\ne {n}_{2}$, we generate a transfer potential between lattice sites which reads in terms of electron and hole operators as
(11.12)$${V}_{transf}={\displaystyle \sum _{{n}_{1}\ne {n}_{2}}}{V}_{{\mathbf{\text{R}}}_{{n}_{1}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathbf{\text{R}}}_{{n}_{2}}}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right){B}_{{n}_{1}}^{\u2020}{B}_{{n}_{2}},$$where ${B}_{n}^{\u2020}={a}_{n}^{\u2020}{b}_{n}^{\u2020}$ creates an electronhole 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 electronhole pair is already bound at the lattice site by V_{neutral}: the pair just has to be delocalized to form a Frenkel exciton.

(iii) In addition to V_{neutral} and V_{transf}, 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 v_{coul}
(11.13)$${V}_{Coul}={V}_{ee}+{V}_{hh}+{V}_{eh}.$$As the diagonal scatterings ${V}_{\mathbf{\text{R}}}\left({}_{{\nu}_{1}\text{}{\nu}_{1}}^{{\nu}_{2}\text{}{\nu}_{2}}\right)$ 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 v_{coul} 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 manybody effects.
(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.
(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 ε_{sc}. As a result, the potential between one electron and one hole, as shown in Fig. 11.1(c), reads
(11.14)$${V}_{eh}={\displaystyle \sum _{\mathbf{\text{q}}\ne \mathbf{\text{0}}}}{V}_{\mathbf{\text{q}}}{\displaystyle \sum _{{\mathbf{\text{k}}}_{e}{\mathbf{\text{k}}}_{h}}}{a}_{{\mathbf{\text{k}}}_{e}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\mathbf{\text{q}}}^{\u2020}{b}_{{\mathbf{\text{k}}}_{h}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathbf{\text{q}}}^{\u2020}{b}_{{\mathbf{\text{k}}}_{h}}{a}_{{\mathbf{\text{k}}}_{e}},$$(p.280) with ${V}_{\mathbf{\text{q}}}=4\pi {e}^{2}/{\u03f5}_{sc}{L}^{3}{q}^{2}$ 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 centerofmass momentum being equal to the centerofmass momentum $\mathbf{\text{Q}}={\mathbf{\text{k}}}_{e}+{\mathbf{\text{k}}}_{h}$ of the electronhole pair which is kept constant along these repeated Coulomb processes. In order to compare this electronhole 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 centerofmass momentum Q and relative motion momentum p, namely, ${B}_{\mathbf{\text{Q}},\mathbf{\text{p}}}^{\u2020}={a}_{\mathbf{\text{p}}+{\gamma}_{e}\mathbf{\text{Q}}}^{\u2020}{b}_{\mathbf{\text{p}}+{\gamma}_{h}\mathbf{\text{Q}}}^{\u2020}$, with ${\gamma}_{e}=1{\gamma}_{h}={m}_{e}/({m}_{e}+{m}_{h})$ in order for the electron and hole kinetic energies $(\mathbf{\text{p}}+{\gamma}_{e}\mathbf{\text{Q}}{)}^{2}/2{m}_{e}+(\mathbf{\text{p}}+{\gamma}_{h}\mathbf{\text{Q}}{)}^{2}/2{m}_{h}$ to split as a centerofmass energy plus a relative motion energy, ${\mathbf{\text{Q}}}^{2}/2{M}_{X}+{\mathbf{\text{p}}}^{2}/2{\mu}_{X}$, with ${M}_{X}={m}_{e}+{m}_{h}$, and ${\mu}_{X}^{1}={m}_{e}^{1}+{m}_{h}^{1}$. The electronhole potential in Eq. (11.14) can be written in terms of these free pair operators as
(11.15)$${V}_{eh}={\displaystyle \sum _{\mathbf{\text{Q}}}}{\displaystyle \sum _{\mathbf{\text{p}}{\mathbf{\text{p}}}^{\prime}}}{V}_{{\mathbf{\text{p}}}^{\prime}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathbf{\text{p}}}{B}_{\mathbf{\text{Q}},{\mathbf{\text{p}}}^{\prime}}^{\u2020}{B}_{\mathbf{\text{Q}},\mathbf{\text{p}}}.$$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.

(ii) In addition to the intraband electronhole 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 electronhole 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.

(iii) Finally, a repulsion also exists between conduction electrons and between valence holes, the associated potential between two electrons reading
(11.16)$${V}_{ee}=\frac{1}{2}{\displaystyle \sum _{\mathbf{\text{q}}\ne \mathbf{\text{0}}}}{V}_{\mathbf{\text{q}}}{\displaystyle \sum _{{\mathbf{\text{k}}}_{1}{\mathbf{\text{k}}}_{2}}}{a}_{{\mathbf{\text{k}}}_{1}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\mathbf{\text{q}}}^{\u2020}{a}_{{\mathbf{\text{k}}}_{2}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathbf{\text{q}}}^{\u2020}{a}_{{\mathbf{\text{k}}}_{2}}{a}_{{\mathbf{\text{k}}}_{1}},$$with a similar expression for the V_{hh} potential between two holes, a_{k} being replaced with b_{k}.
11.2.4 Discussion
In the BCS problem, we start with pairs of free upspin and free downspin electrons having a centerofmass momentum equal to zero. These pairs suffer the attractive potential, V_{BCS}. The pair centerofmass momentum stays equal to zero throughout the repeated action of V_{BCS}, which binds one upspin and one downspin 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.
In the case of Frenkel excitons, we start with electronhole 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, V_{transf}. 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 centerofmass momentum. Since Coulomb interactions conserve momentum, the intraband electronhole Coulomb attraction, V_{eh}, leads to the formation of a bound or unbound Wannier exciton with the same centerofmass momentum as the free pairs from which it is constructed. Its equivalent in the case of Frenkel excitons is negligible within the tightbinding approximation.
Finally, Frenkel and Wannier excitons also have repulsive direct Coulomb processes between electrons or between holes. However, these processes only act at the manybody level.
The reduced BCS potential has similarities with the intraband Coulomb potential V_{eh} 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 V_{transf} given in Eq. (11.12) and responsible for Frenkel exciton formation through delocalizing onsite atomic excitations: V_{BCS} and V_{transf} both act on singleindex pairs, in contrast to V_{eh}, which acts on doubleindex 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 centerofmass 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 singleindex 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 RichardsonGaudin 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 ${H}_{BCS}^{\mathrm{\prime}}={H}_{0}^{\mathrm{\prime}}+{V}_{BCS}$ for electrons in the energy layer where the potential acts, ${H}_{0}^{\mathrm{\prime}}$ being the free electron Hamiltonian $\sum {w}_{\mathbf{\text{k}}}{\epsilon}_{\mathbf{\text{k}}}{a}_{\mathbf{\text{k}}s}^{\u2020}{a}_{\mathbf{\text{k}}s}$. When the Hamiltonian ${H}_{BCS}^{\mathrm{\prime}}$ acts on one pair of upspin and downspin electrons with zero centerofmass momentum added to the frozen Fermi sea ${F}_{0}\u3009$, it gives
(p.283) where ${V}_{{\mathbf{\text{k}}}_{1}}\u3009={V}_{BCS}{B}_{{\mathbf{\text{k}}}_{1}}^{\u2020}{F}_{0}\u3009$, shown in Fig. 11.8(a), reads
After some algebra, it is possible to show that ${B}^{\u2020}({E}_{1}){F}_{0}\u3009$, with
is eigenstate for a single Cooper pair, $({H}_{BCS}^{\mathrm{\prime}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{E}_{1}){B}^{\u2020}({E}_{1}){F}_{0}\u3009=0$, provided that E_{1} fulfills
This equation has a unique bound state with an energy below the lowest free pair energy $2{\epsilon}_{{F}_{0}}$. This energy is given by ${E}_{1}=2{\epsilon}_{{F}_{0}}{\epsilon}_{c}$, with ${\epsilon}_{c}=2\mathrm{\Omega}\sigma /(1\sigma )$, where $\sigma ={e}^{2/\rho \mathcal{V}}$.
11.3.2 One Frenkel exciton
In the case of Frenkel excitons, we split the system Hamiltonian as
with ${H}_{e}={\epsilon}_{e}\sum {a}_{n}^{\u2020}{a}_{n}$, and ${H}_{h}={\epsilon}_{h}\sum {b}_{n}^{\u2020}{b}_{n}$. As v_{coul}, defined in Eq. (11.13), requires carriers at different lattice sites to act, we find, for V_{neutral} given in Eq. (11.10),
where ${V}_{{n}_{1}}\u3009={V}_{transf}{B}_{{n}_{1}}^{\u2020}0\u3009$, shown in Fig. 11.8(b), reads
After some algebra, it is possible to show that ${B}_{\mathbf{\text{Q}}}^{\u2020}0\u3009$, with
(p.284) is ${H}^{(F)}$ eigenstate for a single electronhole pair, $({H}^{(F)}{E}_{\mathbf{\text{Q}}}){B}_{\mathbf{\text{Q}}}^{\u2020}0\u3009=0$, with energy
Frenkel excitons are characterized by a single index Q only, since they are made of singleindex electronhole 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
with ${H}_{e}=\sum {\epsilon}_{\mathbf{\text{k}}}^{(e)}{a}_{\mathbf{\text{k}}}^{\u2020}{a}_{\mathbf{\text{k}}}$, and ${H}_{h}=\sum {\epsilon}_{\mathbf{\text{k}}}^{(h)}{b}_{\mathbf{\text{k}}}^{\u2020}{b}_{\mathbf{\text{k}}}$. Since V_{ee} and V_{hh} require two electronhole pairs at least to act, we find that
where ${V}_{{\mathbf{\text{Q}}}_{1}{\mathbf{\text{p}}}_{1}}\u3009={V}_{eh}{B}_{{\mathbf{\text{Q}}}_{1},{\mathbf{\text{p}}}_{1}}^{\u2020}0\u3009$, shown in Fig. 11.8(c), is given by
The electronhole pair keeps its centerofmass momentum Q_{1} 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 singlepair eigenstate of H reads ${B}_{\mathbf{\text{Q}}\nu}^{\u2020}0\u3009$, with
In addition to the centerofmass momentum Q, Wannier excitons have the quantum index ν, which characterizes the relative motion of the pair, $\u3008\mathbf{\text{p}}\nu \u3009$ being the Wannier exciton relative motion wave function in momentum space.
The two degrees of freedom $({\mathbf{\text{Q}}}_{i},{\nu}_{i})$ of the i exciton can be traced back to the two degrees of freedom $({\mathbf{\text{k}}}_{e},{\mathbf{\text{k}}}_{h})$ or (Q, p) of the free electronhole pairs from which Wannier excitons are constructed,
(p.285) The eigenenergy of the (Q, ν) exciton is equal to ${E}_{gap}+{\epsilon}_{\nu}+{\mathbf{\text{Q}}}^{2}/2{M}_{X}$, the relative motion energy ε_{ν} and wave function $\u3008\mathbf{\text{p}}\nu \u3009$ being such that
This equation is similar to the Schrödinger equation for a hydrogen atom, its reduced mass being ${\mu}_{X}={m}_{e}{m}_{h}/({m}_{e}+{m}_{h})$. The relative motion energy ε_{ν} 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, $({\mathbf{\text{k}}}_{e},{\mathbf{\text{k}}}_{h})$ or (Q, p), of the free electronhole pairs from which they are constructed. Frenkel excitons and Cooper pairs are constructed on singleindex pairs: site n electronhole excitations in the case of Frenkel excitons, and free upspin and downspin electron pairs with opposite momenta, $(\mathbf{\text{p}},\mathbf{\text{p}})$, in the case of Cooper pairs. So, they are characterized by a single index only. For Frenkel excitons, this index is the centerofmass momentum Q. For Cooper pairs, since their centerofmass 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 E_{1} or the various unbound levels close to $2{\epsilon}_{\mathbf{\text{k}}}$. These unbound levels are not considered in BCS superconductivity, which fundamentally deals with a volumelinear 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 centerofmass 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 centerofmass momentum of the pair has a welldefined value right from the beginning, this value being, by construction, zero for Cooper pairs. The attractive BCS potential brings two oppositemomentum 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 electronhole pair already bound at a lattice site, as a result of intrasite direct and exchange Coulomb processes. This pair is then delocalized into a planewave 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 fermionfermion interaction and fermion exchange in composite boson systems, because (p.286) the Pauli exclusion principle enters into play at the twopair level. This is why it is rewarding to begin the study of composite boson manybody 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 zeromomentum pairs do not interact through the reduced BCS potential because, for such interaction between $({\mathbf{\text{k}}}_{1},{\mathbf{\text{k}}}_{1})$ and $({\mathbf{\text{k}}}_{2},{\mathbf{\text{k}}}_{2})$ pairs to exist, we should have ${\mathbf{\text{k}}}_{1}={\mathbf{\text{k}}}_{2}$; 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 manybody effects. In the case of two free pairs, the interaction part ${V}_{{\mathbf{\text{k}}}_{1}{\mathbf{\text{k}}}_{2}}\u3009,$ visualized in the two diagrams of Fig. 11.10, follows from
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 ${B}^{\u2020}({R}_{1}){B}^{\u2020}({R}_{2}){F}_{0}\u3009$, with ${B}^{\u2020}(R)$ given in Eq. (11.19), is the exact twopair eigenstate of ${H}_{BCS}^{\mathrm{\prime}}$, provided that $({R}_{1},{R}_{2})$ fulfill
the twopair energy being ${E}_{2}={R}_{1}+{R}_{2}$. These two coupled equations can be analytically solved, the twopair groundstate energy reading in the large sample limit as
(p.287) where ρ is the density of states, ${\epsilon}_{c}=2\mathrm{\Omega}\sigma /(1\sigma )$ is the binding energy of a single Cooper pair with $\sigma ={e}^{2/\rho \mathcal{V}}$, and N_{Ω} is equal to ρΩ, where $\mathrm{\Omega}=2{\omega}_{c}$ is the energy extension of the potential layer.
11.4.2 Two Frenkel excitons
Atomic excitations result from (i) the intrasite potential V_{neutral}, which forces the electron and the hole to be at the same lattice site, (ii) the transfer potential V_{transf}, which delocalizes the bound pair over the whole sample, and (iii) the direct Coulomb potential v_{coul}, 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 v_{coul} 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 $\xi \left(\begin{array}{cc}n& j\\ m& i\end{array}\right)$ 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.
In addition to these rather standard scatterings, there also exist intersite interlevel Coulomb processes (see Fig. 11.4(a)) that come from the V_{transf} 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 valenceconduction scatterings, (p.288) are commonly neglected, such exchange scatterings never appear in usual Wannier exciton manybody effects. By contrast, their equivalent in the case of Cooper pairs, shown in Fig. 11.10(b), rules the entire manybody physics of these composite bosons.
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 welldefined exciton and then constructing an interaction potential between two excitons.
To overcome this difficulty and derive the direct Coulomb scattering $\xi \left(\begin{array}{cc}n& j\\ m& i\end{array}\right)$ 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,
The two possible ways to associate two electrons and two holes into two Wannier excitons appear through
where $\mathrm{\lambda}\left(\begin{array}{cc}n& j\\ m& i\end{array}\right)$ 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 $\xi \left(\begin{array}{cc}n& j\\ m& i\end{array}\right)$ in Eq. (11.36) with ${\xi}^{in}\left(\begin{array}{cc}n& j\\ m& i\end{array}\right)$, the “in” exchange Coulomb scattering shown in Fig. 11.15(a) being given by
or with any linear combination of the two scatterings, $a\xi \left(\begin{array}{cc}n& j\\ m& i\end{array}\right)b{\xi}^{in}\left(\begin{array}{cc}n& j\\ m& i\end{array}\right)$, 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 energylike exchange Coulomb scattering exists, called ${\xi}^{out}\left(\begin{array}{cc}n& j\\ m& i\end{array}\right)$, 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 $\text{\lambda}\left(\begin{array}{cc}n& j\\ m& i\end{array}\right)$ through
So, these two exchange Coulomb scatterings are equal for energyconserving processes, as required by timereversal symmetry. Note that in these three energylike scatterings, ξ, ${\xi}^{in}$, and ${\xi}^{out}$, the electronhole Coulomb attraction responsible for exciton formation appears along with electronelectron and holehole 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 electronhole pairs, resulting from these repeated scatterings, are not analytically known. The twopair eigenstates are called biexcitons. These fourfermion structures have bound states analogous to hydrogen molecules when the two electrons, as well as the two holes, are in a spinsinglet state. As the effective excitonexciton 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, energylike scatterings between two Cooper pairs result from electron exchange induced by the Pauli exclusion principle between upspin electrons and between downspin 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 manybody 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 longrange Coulomb potential.
11.5N composite bosons
The manybody 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 fermionfermion 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 zeromomentum pairs:
The interacting part ${V}_{{\mathbf{\text{k}}}_{1}{\mathbf{\text{k}}}_{2}\dots {\mathbf{\text{k}}}_{N}}\u3009$ 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.
From the above equation, it is possible to show that the Npair eigenstates read
where the operator ${B}^{\u2020}(R)$ is the one already appearing in the onepair eigenstate (see Eq. (11.19)), the R_{i}’s being linked by N nonlinear equations:
(p.292) The Npair energy being ${E}_{N}={R}_{1}+{R}_{2}+\cdots +{R}_{N}$, the analytical resolution of these equations gives the groundstate energy in the large sample limit as
with $\sigma ={e}^{2/\rho \mathcal{V}}$, the single Cooper pair energy being E_{1}. We note that E_{N} has no term in $N(N1)(N2)$ and higher, although such higherorder terms usually are present in manybody 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 higherorder cancellation is found when calculating the Hamiltonian mean value in the N–identical Frenkel exciton state ${B}_{\mathbf{\text{0}}}^{\u2020N}0\u3009$, 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 groundstate energy of N Frenkel excitons through the mean value of the Hamiltonian ${H}^{(F)}={H}_{0}^{(F)}+{V}_{Coul}$ in the state ${B}_{\mathbf{\text{Q}}=\mathbf{\text{0}}}^{\u2020\text{}N}0\u3009$, the exact N–Frenkel exciton ground state being expected to be close to this state. This mean value reduces to only two terms:
where N_{s} is the number of lattice sites, while $\xi \left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$ contains all direct and exchange Coulomb scatterings existing between different lattice sites
We note that, like the E_{N} energy for N Cooper pairs, this Hamiltonian mean value has no term in $N(N1)(N2)$ 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 groundstate energy of N Wannier excitons through the (p.293) Hamiltonian mean value in the ${B}_{0}^{\u2020N}0\u3009$ state, where ${B}_{0}^{\u2020}$ creates a groundstate exciton with centerofmass momentum Q = 0 and relative motion groundstate index ν_{0}, this state being close to the N–Wannier exciton ground state. We find
The direct Coulomb scattering $\xi \left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$ cancels because of the fact that electronelectron and holehole repulsions are as large as electronhole attraction. So, correction to the energy of N noninteracting groundstate excitons comes from exchange Coulomb scattering only. The Hamiltonian mean value is found to expand in 3D as
where $\eta =N({a}_{X}/L{)}^{3}$ is the dimensionless parameter associated with Wannier exciton manybody effects, and R_{X} is the 3D exciton Rydberg. We note that the firstorder correction to the energy $N{R}_{X}$ 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 “motheaten 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 manybody physics of Cooper pairs within the reduced BCS potential is entirely controlled by the Pauli exclusion principle because the upspin electron k can only interact with the downspin electron (−k). Moreover, as interaction among N zeromomentum pairs leaves at least (N − 2) pairs unchanged, the exact N–Cooper pair energy, known by now, has no terms in $N(N1)(N2)$ and higher.
Absence of such higherorder 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 ${B}_{0}^{\u2020N}0\u3009$, the manybody physics of these excitons being definitely far more complex.
(p.294) 11.6 Manybody parameters
Manybody effects among excitons or among Cooper pairs are controlled by a dimensionless parameter η that, when understood properly, demonstrates the fact that the manybody 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 Npair energy. We can cast the parameter attached to manybody effects as
where N_{max} is the maximum number of composite bosons the sample can accommodate. In the case of Cooper pairs, N_{max} 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 electronhole 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 upspin and downspin electron pair is added to the energy layer where the reduced BCS potential acts, and (ii) when this layer is halffilled. In these two cases, the electrons which feel the potential have energies extending over an energy range Ω above the frozen Fermi sea ${F}_{0}\u3009$. These two configurations correspond to N = 1 pairs and $N=\rho \mathrm{\Omega}/2$ 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 manybody effects between Cooper pairs, and, more so, to understand that these manybody effects only come from the Pauli exclusion principle.
By analytically solving the RichardsonGaudin equations for arbitrary N, we find that the Npair groundstate energy reads in the large sample limit as
where ${E}_{1}=2{\epsilon}_{{F}_{0}}{\epsilon}_{c}$, with ${\epsilon}_{c}=2\mathrm{\Omega}\sigma /(1\sigma )$, is the single Cooper pair energy. This gives the condensation energy for N pairs resulting from the reduced BCS potential as
where ${N}_{max}=\rho \mathrm{\Omega}+1$ is the number of the k states from which Cooper pairs can be constructed, that is, the number of k states with energy ${\epsilon}_{{F}_{0}}\le {\epsilon}_{\mathbf{\text{k}}}\le {\epsilon}_{{F}_{0}}+\mathrm{\Omega}$ 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 upspin and downspin 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 “motheaten 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 $N={N}_{max}$. 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 N_{max} 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 singlepair binding energy.
11.6.2 Frenkel excitons
The manybody 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, $\u30080{B}_{\mathbf{\text{Q}}}^{N}{B}_{\mathbf{\text{Q}}}^{\u2020N}0\u3009=N!{F}_{N}$. The Pauli exclusion principle between electronhole pairs located at lattice sites and available for making Frenkel excitons leads to a F_{N} decrease exactly given by
where N_{s} is the number of lattice sites; N_{s} 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, $N/{N}_{s}$ corresponds to $N/{N}_{max}$, a result that supports the form of the manybody parameter given in Eq. (11.48).
We can also consider the Hamiltonian mean value in the N–Frenkel exciton state ${B}_{\mathbf{\text{0}}}^{\u2020N}0\u3009$. It reads
(p.296) The effective scattering ${V}_{\mathbf{\text{R}}}^{(eff)}$ comes from all possible Coulomb processes between ground and excited atomic levels at lattice sites separated by R, namely,
In the tightbinding approximation, ${V}_{\mathbf{\text{R}}}^{(eff)}$ scales as ${e}^{2}/{R}^{3}$, 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 manybody parameter reading $N/{N}_{s}$, that is, $N/{N}_{max}$.
11.6.3 Wannier excitons
In the case of Wannier excitons, the manybody parameter can be extracted from the normalization factor of N groundstate excitons and also from the Hamiltonian mean value calculated in this state; $\u30080{B}_{0}^{N}{B}_{0}^{\u2020N}0\u3009$ is equal to $N!{F}_{N}$ with, in 3D,
while the Hamiltonian mean value is given by
Into these two calculations performed for excitons in ground state $0=(\mathbf{\text{Q}}=\mathbf{0},{\nu}_{0})$ enters the exciton groundstate wave function $\u3008\mathbf{\text{r}}{\nu}_{0}\u3009$. Its spatial extension a_{X} appears through η given by
A simple way to physically understand this quantity is to note that the relative motion of composite excitons extends over a ${a}_{X}^{3}$ volume, while their center of mass is delocalized over the sample volume L^{3}. 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 manybody parameter in terms of exciton relative motion extension cannot be extended to Frenkel excitons, because the spatial extension of electronhole pairs from which Frenkel excitons are constructed reduces to zero within the tightbinding approximation. By contrast, it is easy (p.297) to extend to Wannier excitons the physical understanding of the Cooper pair manybody parameter or the Frenkel exciton manybody 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, electronhole pairs are known to dissociate into an electronhole plasma. The number of excitons at which overlap starts in a sample volume L^{3} is this L^{3} volume divided by the exciton volume ${a}_{X}^{3}$. So, the maximum number of Wannier excitons that a L^{3} volume can accommodate without dissociation is of the order of
in agreement with the Wannier exciton manybody parameter $\eta =N{a}_{X}^{3}/{L}^{3}=N/{N}_{max}$. It is interesting to note that this ${L}^{3}/{a}_{X}^{3}$ upper bound for the Wannier exciton number, which is fundamentally associated with Mott dissociation, appears in a natural way. Algebraic calculations on Wannier exciton manybody 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 manybody effects as $\eta =N/{N}_{max}$, where N_{max} is the maximum number of composite bosons the sample can accommodate. However, the ways N_{max} appears for singleindex pairs like Cooper pairs or Frenkel excitons and for doubleindex pairs like Wannier excitons are rather different. For singleindex pairs, N_{max} 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 N_{max} appears in the case of Wannier excitons is more subtle. The numbers of $({\mathbf{\text{k}}}_{e},{\mathbf{\text{k}}}_{h})$ states available for pairing in the conduction and valence bands are huge; these numbers do not play a role in the Wannier exciton manybody parameter. Instead, through $\eta =N{a}_{X}^{3}/{L}^{3}$, the manybody parameter is associated with the fact that, above a certain number, Wannier excitons overlap and dissociate into an electronhole 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 centerofmass 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 longrange 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 a_{B} is related to the binding energy ${\epsilon}_{B}$ through a simple dimensional argument, namely, ${\epsilon}_{B}\simeq 1/2\mu {a}_{B}^{2}$, 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 centerofmass momentum Q and relative motion index ν reads
where ${B}_{\mathbf{\text{Q}},\mathbf{\text{p}}}^{\u2020}={a}_{\mathbf{\text{p}}+{\gamma}_{e}\mathbf{\text{Q}}}^{\u2020}{b}_{\mathbf{\text{p}}+{\gamma}_{h}\mathbf{\text{Q}}}^{\u2020}$, with ${\gamma}_{e}=1{\gamma}_{h}={m}_{e}/({m}_{e}+{m}_{h})$, creates a free electronhole pair with centerofmass momentum Q and relative motion momentum p. These free pair states are eigenstates of the free electron and free hole Hamiltonians ${H}_{e}=\sum _{\mathbf{\text{k}}}{\epsilon}_{\mathbf{\text{k}}}^{(e)}{a}_{\mathbf{\text{k}}}^{\u2020}{a}_{\mathbf{\text{k}}}$, and ${H}_{h}=\sum _{\mathbf{\text{k}}}{\epsilon}_{\mathbf{\text{k}}}^{(h)}{b}_{\mathbf{\text{k}}}^{\u2020}{b}_{\mathbf{\text{k}}}$
where ${\epsilon}_{\mathbf{\text{p}}}={\mathbf{\text{p}}}^{2}/2{\mu}_{X}$, and ${E}_{\mathbf{\text{Q}}}={\mathbf{\text{Q}}}^{2}/2{M}_{X}$, with ${\mu}_{X}^{1}={m}_{e}^{1}+{m}_{h}^{1}$, and ${M}_{X}={m}_{e}+{m}_{h}$. By writing the Coulomb potential between electrons and holes as
we readily find
So, ${B}_{\mathbf{\text{Q}}\nu}^{\u2020}0\u3009$ is the onepair eigenstate of the interacting electronhole Hamiltonian, with energy ${E}_{gap}+{E}_{\mathbf{\text{Q}}}+{\epsilon}_{\nu}$, provided that the above bracket satisfies
(p.299) This result shows that the prefactor of a Wannier exciton creation operator expanded in terms of free electronhole pair operators is indeed the wave function associated with the exciton relative motion taken in momentum space.
In 3D, that is, for ${V}_{\mathbf{\text{q}}}=4\pi {e}^{2}/{\u03f5}_{sc}{L}^{3}{q}^{2}$, the solution of the above equation leads to a groundstate binding energy ${\epsilon}_{{\nu}_{0}}=1/2{\mu}_{X}{a}_{X}^{2}$, and a groundstate wave function, normalized in a sample volume ${L}^{3},$ as
So, the ${\nu}_{0}\u3009$ wave function extension in momentum space is $1/{a}_{X}$ (see Fig. 11.17(a)).
To get this extension in real space, we use
This shows that $\u3008\mathbf{\text{r}}{\nu}_{0}\u3009$ 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 $\u3008\mathbf{\text{p}}{\nu}_{0}\u3009$. These oscillations lead to a spatial extension of $\u3008\mathbf{\text{r}}{\nu}_{0}\u3009$ that scales as a_{X}. It is actually possible to calculate the above integral analytically. By transforming the integral of $sinpr$ from 0 to ∞ into the integral of ${e}^{ipr}$ from $\mathrm{\infty}$ to $+\mathrm{\infty}$, the residue theorem leads to
(p.300) This result demonstrates that the spatial extension of the exciton ground state is indeed a_{X} (see Fig. 11.17(b)), this extension being related to the groundstate Wannier exciton binding energy through ${\epsilon}_{{\nu}_{0}}=1/2{\mu}_{X}{a}_{X}^{2}$.
11.7.2 Frenkel excitons
We now consider the Frenkel exciton creation operator characterized by the single index Q. This operator reads
where ${B}_{n}^{\u2020}={a}_{n}^{\u2020}{b}_{n}^{\u2020}$ creates an electronhole excitation at lattice site n. From the excitation transfer potential given by
we find that the Frenkel exciton Hamiltonian acting on a single correlated pair yields
It is then easy to check that the Hamiltonian eigenstate which satisfies
reads $\u3008n\mathbf{\text{Q}}\u3009={e}^{i\mathbf{\text{Q}}\cdot {\mathbf{\text{R}}}_{n}}/\sqrt{{N}_{s}}$, in agreement with Eq. (11.66), the associated Frenkel exciton energy being given by
owing to translational invariance which makes the sum over ${n}^{\prime}\ne n$ 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, $\u3008n\mathbf{\text{Q}}\u3009={e}^{i\mathbf{\text{Q}}\cdot {\mathbf{\text{R}}}_{n}}/\sqrt{{N}_{s}}$, is just a phase; so, it is flat in real space and infinitely narrow in momentum space (see Fig. 11.18):
11.7.3 Cooper pairs
Cooper pairs are commonly studied in two cases: as a single Cooper pair, and as a dense volumelinear regime of Cooper pairs.
Single Cooper pair
Cooper showed that the eigenstate for one upspin and one downspin electron in the BCS potential layer reads
where ${F}_{0}\u3009$ is the “frozen Fermi sea,” ${B}_{\mathbf{\text{k}}}^{\u2020}={a}_{\mathbf{\text{k}}\uparrow}^{\u2020}{a}_{\mathbf{\text{k}}\downarrow}^{\u2020}$ is the creation operator of a zeromomentum electron pair, and E_{1} is the single Cooper pair energy. The function ${\omega}_{\mathbf{\text{k}}}$, equal to 1 for ${\epsilon}_{{F}_{0}}\le {\epsilon}_{\mathbf{\text{k}}}\le {\epsilon}_{{F}_{0}}+\mathrm{\Omega}$, and 0 otherwise, characterizes the potential layer. By analogy with the exciton, we can rewrite this single Cooper pair creation operator as
For ${V}_{BCS}=\mathcal{V}\sum _{{\mathbf{\text{k}}}^{\prime}\mathbf{\text{k}}}{\omega}_{{\mathbf{\text{k}}}^{\prime}}{\omega}_{\mathbf{\text{k}}}{B}_{{\mathbf{\text{k}}}^{\prime}}^{\u2020}{B}_{\mathbf{\text{k}}}$, the Hamiltonian ${H}_{BCS}^{\mathrm{\prime}}$ acting in the potential layer gives
(p.302) So, ${B}_{{E}_{1}}^{\u2020}{F}_{0}\u3009$ is eigenstate of the BCS Hamiltonian with energy E_{1}, provided that
This equation gives the single Cooper pair wave function in momentum space as $\u3008\mathbf{\text{k}}{E}_{1}\u3009={\omega}_{\mathbf{\text{k}}}/(2{\epsilon}_{\mathbf{\text{k}}}{E}_{1})$, in agreement with Eq. (11.72), the singlepair energy E_{1} following from
The solution of this equation yields a single bound state with energy ${E}_{1}=2{\epsilon}_{{F}_{0}}{\epsilon}_{c}$, where ${\epsilon}_{c}=2\mathrm{\Omega}\sigma /(1\sigma )$, and $\sigma ={e}^{2/\rho \mathcal{V}}$ for a constant density of states ρ.
By noting that
the normalized single Cooper pair wave function ${\varphi}_{1}\u3009$ appears in momentum space as
So, $\sqrt{\rho \mathrm{\Omega}}\u3008\mathbf{\text{k}}{\varphi}_{1}\u3009$ varies from ${\left[(2\mathrm{\Omega}+{\epsilon}_{c})/{\epsilon}_{c}\right]}^{1/2}$ to ${\left[(2\mathrm{\Omega}+{\epsilon}_{c})/{\epsilon}_{c}\right]}^{1/2}$ when ε_{k} varies from ε_{F0} to ${\epsilon}_{{F}_{0}}+\mathrm{\Omega}$. This result also shows that the energy extension of the $\u3008\mathbf{\text{k}}{\varphi}_{1}\u3009$ wave function above ε_{F0} scales as ε_{c} (see Fig. 11.19). To get the $\u3008\mathbf{\text{k}}{\varphi}_{1}\u3009$ extension in momentum space, (p.303) we write ε_{c} as $1/2{\mu}_{c}{a}_{c}^{2}$, with $1/{\mu}_{c}=2/m$. So, an energy extension from ${\epsilon}_{{F}_{0}}={k}_{{F}_{0}}^{2}/2m$ to ${\epsilon}_{{F}_{0}}+{\epsilon}_{c}$ corresponds to a $\mathbf{\text{k}}$ extension above ${k}_{{F}_{0}}$ that scales as
This extension depends not only on a_{c} but also on the minimum momentum ${k}_{{F}_{0}}$ 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 $k=\sqrt{2m{\epsilon}_{\mathbf{\text{k}}}}$,
This result shows that $\u3008\mathbf{\text{r}}{\varphi}_{1}\u3009$ stays finite for $r=0$, while it is very small for r large. Indeed, the sizable part of the integral controlled by the factor $1/(2\epsilon 2{\epsilon}_{{F}_{0}}+{\epsilon}_{c})$ extends from ε_{F0} to ${\epsilon}_{{F}_{0}}+{\epsilon}_{c}/2$, a range which corresponds to ${k}_{{F}_{0}}<k\lesssim {k}_{{F}_{0}}+1/2{k}_{{F}_{0}}{a}_{c}^{2}$. For r much larger than ${k}_{{F}_{0}}{a}_{c}^{2}$, 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 ${k}_{{F}_{0}}{a}_{c}^{2}$ and $1/{k}_{{F}_{0}}{a}_{c}^{2}$, respectively (see Fig. 11.20). These extensions are different from the naive scales, a_{c} and $1/{a}_{c}$, that are associated with the single Cooper pair binding energy ${\epsilon}_{c}=1/m{a}_{c}^{2}$. The difference fundamentally comes from the fact that, while its energy extension is equal to the pair binding energy ε_{c}, the single Cooper pair wave function is (p.304) made of k states, with $\mathbf{\text{k}}$ larger than a (large) finite value ${k}_{{F}_{0}}$, in contrast with Wannier excitons, which are made of pairs with momentum starting from $\mathbf{\text{k}}=0$.
Dense regime of Cooper pairs
The physically relevant regime for BCS superconductivity does not correspond to a single Cooper pair but to a volumelinear 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 RichardsonGaudin equations.
(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)$$\begin{array}{rcl}{\psi}_{BCS}\u3009& =& {\displaystyle \prod _{{w}_{\mathbf{\text{k}}}=1}}\left({u}_{\mathbf{\text{k}}}+{v}_{\mathbf{\text{k}}}{B}_{\mathbf{\text{k}}}^{\u2020}\right){F}_{0}\u3009\\ & =& \left({\displaystyle \prod _{{w}_{\mathbf{\text{k}}}=1}}{u}_{\mathbf{\text{k}}}\right){\displaystyle \sum _{N=0}^{\mathrm{\infty}}}\frac{1}{N!}{B}^{\u2020N}{F}_{0}\u3009,\end{array}$$with ${B}_{\mathbf{\text{k}}}^{\u2020}={a}_{\mathbf{\text{k}}\uparrow}^{\u2020}{a}_{\mathbf{\text{k}}\downarrow}^{\u2020}$ and
(11.82)$${B}^{\u2020}={\displaystyle \sum _{\mathbf{\text{k}}}}{w}_{\mathbf{\text{k}}}{B}_{\mathbf{\text{k}}}^{\u2020}\frac{{v}_{\mathbf{\text{k}}}}{{u}_{\mathbf{\text{k}}}}.$$Since, in the following discussion, we are not concerned about the Josephson effect (Josephson 1962, 1965; Tinkham 2004), we take u_{k} and v_{k} as real. The projection of the grand canonical state ${\psi}_{BCS}\u3009$ onto the Npair subspace, namely, ${B}^{\u2020N}{F}_{0}\u3009$, corresponds to having all pairs in the same B^{†} state. By writing the creation operator B^{†} as
(11.83)$${B}^{\u2020}={\displaystyle \sum _{\mathbf{\text{k}}}}{\omega}_{\mathbf{\text{k}}}{B}_{\mathbf{\text{k}}}^{\u2020}\u3008\mathbf{\text{k}}\phi \u3009,$$we find the pair wave function as
(11.84)$$\u3008\mathbf{\text{k}}\phi \u3009=\frac{{v}_{\mathbf{\text{k}}}}{{u}_{\mathbf{\text{k}}}}=\sqrt{\frac{{E}_{\mathbf{\text{k}}}{\xi}_{\mathbf{\text{k}}}}{{E}_{\mathbf{\text{k}}}+{\xi}_{\mathbf{\text{k}}}}},$$where ${E}_{\mathbf{\text{k}}}=\sqrt{{\xi}_{\mathbf{\text{k}}}^{2}+\mathrm{\Delta}{}^{2}}$, and ${\xi}_{\mathbf{\text{k}}}={\epsilon}_{\mathbf{\text{k}}}\mu $. 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, $\mu ={\epsilon}_{{F}_{0}}+\mathrm{\Omega}/2$. (p.305) We then find $\mathrm{\Delta}\simeq \mathrm{\Omega}{e}^{1/\rho \mathcal{V}}$. This gives $\u3008\mathbf{\text{k}}\phi \u3009\simeq 2{\xi}_{\mathbf{\text{k}}}/\mathrm{\Delta}$ for ${\xi}_{\mathbf{\text{k}}}<\mathrm{\Delta}$, and $\u3008\mathbf{\text{k}}\phi \u3009\simeq \mathrm{\Delta}/2{\xi}_{\mathbf{\text{k}}}$ for ${\xi}_{\mathbf{\text{k}}}>\mathrm{\Delta}$. As
(11.85)$$\u3008\phi \phi \u3009={\displaystyle \sum _{\mathbf{\text{k}}}}\u3008\mathbf{\text{k}}\phi \u3009{}^{2}={N}_{\mathrm{\Omega}}\left(1+\frac{{\mathrm{\Omega}}^{2}}{6\mathrm{\Delta}{}^{2}}\right)\simeq \frac{{N}_{\mathrm{\Omega}}}{6}\phantom{\rule{thinmathspace}{0ex}}{e}^{2/\rho \mathcal{V}}$$with ${N}_{\mathrm{\Omega}}=\rho \mathrm{\Omega}$, the normalized pair wave function
(11.86)$$\u3008\mathbf{\text{k}}\varphi \u3009=\frac{\u3008\mathbf{\text{k}}\phi \u3009}{\sqrt{\u3008\phi \phi \u3009}}$$of the BCS condensate exhibits three different scales (Zhu et al. 2012), as shown in Fig. 11.21:
(11.87)$$\begin{array}{rcl}\u3008\mathbf{\text{k}}{\varphi}^{(1)}\u3009& \simeq & \frac{1}{\sqrt{{N}_{\mathrm{\Omega}}}}\frac{\mu {\epsilon}_{\mathbf{\text{k}}}}{\mathrm{\Omega}},\phantom{\rule{1em}{0ex}}\mathrm{f}\mathrm{o}\mathrm{r}\phantom{\rule{1em}{0ex}}0\le {\epsilon}_{\mathbf{\text{k}}}{\epsilon}_{{F}_{0}}\lesssim \frac{\mathrm{\Omega}\mathrm{\Delta}}{2},\end{array}$$(11.88)$$\begin{array}{rcl}\u3008\mathbf{\text{k}}{\varphi}^{(2)}\u3009& \simeq & \frac{{e}^{1/\rho \mathcal{V}}}{\sqrt{{N}_{\mathrm{\Omega}}}},\phantom{\rule{6pt}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\mathrm{f}\mathrm{o}\mathrm{r}\phantom{\rule{1em}{0ex}}\frac{\mathrm{\Omega}\mathrm{\Delta}}{2}\lesssim {\epsilon}_{\mathbf{\text{k}}}{\epsilon}_{{F}_{0}}\lesssim \frac{\mathrm{\Omega}+\mathrm{\Delta}}{2},\end{array}$$(11.89)$$\begin{array}{rcl}\u3008\mathbf{\text{k}}{\varphi}^{(3)}\u3009& \simeq & \frac{{e}^{2/\rho \mathcal{V}}}{\sqrt{{N}_{\mathrm{\Omega}}}}\frac{\mathrm{\Omega}}{{\epsilon}_{\mathbf{\text{k}}}\mu},\phantom{\rule{1.2pt}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{f}\mathrm{o}\mathrm{r}\phantom{\rule{1em}{0ex}}\frac{\mathrm{\Omega}+\mathrm{\Delta}}{2}\lesssim {\epsilon}_{\mathbf{\text{k}}}{\epsilon}_{{F}_{0}}\le \mathrm{\Omega}.\end{array}$$This shows that the sizable part of the pair wave function in the BCS condensate is a linearly decreasing function of ε_{k} between ε_{F0} and ${\epsilon}_{{F}_{0}}+\mathrm{\Omega}/2$. Since the phonon energy ${\omega}_{c}\simeq \mathrm{\Omega}/2$ is small compared to the Fermi energy ε_{F0} of the frozen Fermi sea, this regime corresponds to k momentum extending from ${k}_{{F}_{0}}$ to $\sqrt{2m({\epsilon}_{{F}_{0}}+\mathrm{\Omega}/2)}\simeq {k}_{{F}_{0}}+1/2{k}_{{F}_{0}}{a}_{\mathrm{\Omega}}^{2}$, with ${a}_{\mathrm{\Omega}}=1/\sqrt{m\mathrm{\Omega}}$ (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)$${k}_{{F}_{0}}{a}_{\mathrm{\Omega}}^{2}={a}_{\mathrm{\Omega}}\sqrt{\frac{2{\epsilon}_{{F}_{0}}}{\mathrm{\Omega}}}.$$Here also, this spatial extension is not related to the average Cooper pair binding energy, which, in the BCS configuration, that is, for halffilling, is equal to half the singlepair binding energy, ${\epsilon}_{c}/2$. The Cooper pair wave function is usually not associated with $\u3008\mathbf{\text{k}}\phi \u3009={v}_{\mathbf{\text{k}}}/{u}_{\mathbf{\text{k}}}$ appearing in the creation operator B^{†} when $N={N}_{\mathrm{\Omega}}/2$ pairs feel the BCS potential, but with
(11.91)$$\begin{array}{rcl}{F}_{\mathbf{\text{k}}}& =& {u}_{\mathbf{\text{k}}}{v}_{\mathbf{\text{k}}}=\frac{\u3008\mathbf{\text{k}}\phi \u3009}{1+\u3008\mathbf{\text{k}}\phi \u3009{}^{2}}\\ & =& \frac{\mathrm{\Delta}}{2{E}_{\mathbf{\text{k}}}}.\end{array}$$This quantity, peaked on ${\epsilon}_{\mathbf{\text{k}}}={\epsilon}_{{F}_{0}}+\mathrm{\Omega}/2$, spreads over an energy extension $\mathrm{\Delta}$ (see inset in Fig. 11.21). The reason for associating F_{k} with the pair wave function most probably originates from the incorrect understanding of the BCS condensation energy $\rho \mathrm{\Delta}{}^{2}/2$ as resulting from $\rho \mathrm{\Delta}$ pairs having an average binding energy $\mathrm{\Delta}$, instead of $\rho \mathrm{\Omega}/2$ pairs having an average binding energy ${\epsilon}_{c}/2$. With this incorrect understanding, one is naturally led to look for a quantity having an energy extension equal to the “binding energy” $\mathrm{\Delta}$, a possible quantity being ${F}_{\mathbf{\text{k}}}={u}_{\mathbf{\text{k}}}{v}_{\mathbf{\text{k}}}$. Since $\mathrm{\Delta}$ is not the pair binding energy in the condensed configuration, there is no particular reason to consider F_{k} as a pair wave function. If we still want to stay with this idea, we are led to introduce three length scales, namely, a_{Δ}, a_{c}, and a_{Ω}, with $\mathrm{\Delta}\simeq \mathrm{\Omega}{e}^{1/\rho \mathcal{V}}$ set equal to $1/m{a}_{\mathrm{\Delta}}^{2}$, ${\epsilon}_{c}\simeq 2\mathrm{\Omega}{e}^{2/\rho \mathcal{V}}$ set equal to $1/m{a}_{c}^{2}$, and Ω set equal to $1/m{a}_{\mathrm{\Omega}}^{2}$. These three lengths are linked by ${a}_{\mathrm{\Delta}}\simeq \sqrt{{a}_{\mathrm{\Omega}}{a}_{c}}$. Since the F_{k} function has an energy extension $\mathrm{\Delta}$ around the normal electron Fermi energy, (p.307) ${\epsilon}_{F}={\epsilon}_{{F}_{0}}+\mathrm{\Omega}/2={k}_{F}^{2}/2m,$ its k extension around k_{F} scales as $1/{k}_{{F}_{0}}{a}_{\mathrm{\Delta}}^{2}$. This would lead to a pair wave function extension in r space that scales as
(11.92)$${k}_{{F}_{0}}{a}_{\mathrm{\Delta}}^{2}\simeq {k}_{{F}_{0}}{a}_{c}{a}_{\mathrm{\Omega}}\simeq {a}_{\mathrm{\Omega}}\sqrt{\frac{2{\epsilon}_{{F}_{0}}}{{\epsilon}_{c}}},$$which is far larger than the spatial extension of the $\u3008\mathbf{\text{k}}\phi \u3009$ wave function given in Eq. (11.84). The above length scale actually corresponds to the Pippard coherence length $\xi ={k}_{{F}_{0}}/\pi m\mathrm{\Delta}$ (Pippard 1953). Even if F_{k} is not the Cooper pair wave function in the BCS regime, the ${F}_{\mathbf{\text{k}}}\ne 0$ 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 (${v}_{\mathbf{\text{k}}}=1$, ${u}_{\mathbf{\text{k}}}=0$) for ε_{k} below ε_{F}, and to (${v}_{\mathbf{\text{k}}}=0$, ${u}_{\mathbf{\text{k}}}=1$) above ε_{F}. So, ${F}_{\mathbf{\text{k}}}={u}_{\mathbf{\text{k}}}{v}_{\mathbf{\text{k}}}$ reduces to zero for all normal electrons; the energy domain in which F_{k} 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 F_{k} plays a role in BCS superconductivity, even if the true BCS pair wave function is not ${F}_{\mathbf{\text{k}}}={u}_{\mathbf{\text{k}}}{v}_{\mathbf{\text{k}}}$ but $\u3008\mathbf{\text{k}}\phi \u3009={v}_{\mathbf{\text{k}}}/{u}_{\mathbf{\text{k}}}$.

(ii)The RichardsonGaudin procedure: RichardsonGaudin approach to the BCS problem allows us to obtain the exact wave function for N Cooper pairs added to the frozen Fermi sea ${F}_{0}\u3009$ as
(11.93)$${B}^{\u2020}({R}_{1}){B}^{\u2020}({R}_{2})\cdots {B}^{\u2020}({R}_{N}){F}_{0}\u3009,$$with ${B}^{\u2020}({R}_{i})=\sum _{\mathbf{\text{k}}}{\omega}_{\mathbf{\text{k}}}{B}_{\mathbf{\text{k}}}^{\u2020}/(2{\epsilon}_{\mathbf{\text{k}}}{R}_{i})$, the R_{i}’s being the solutions of N coupled equations
(11.94)$$\frac{1}{\mathcal{V}}={\displaystyle \sum _{\mathbf{\text{k}}}}\frac{{\omega}_{\mathbf{\text{k}}}}{2{\epsilon}_{\mathbf{\text{k}}}{R}_{i}}+{\displaystyle \sum _{j\ne i}}\frac{2}{{R}_{i}{R}_{j}}.$$From the last term of the above equation, it is clear that the R_{i}’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 Npair 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 ${B}^{\u2020}({R}_{i})$, because each time we add one more ${B}^{\u2020}({R}_{i})$, the Pauli exclusion principle removes one pair state $(\mathbf{\text{k}},\mathbf{\text{k}})$ from the available k states through the motheaten effect. What in the end remains in the exact eigenstate given in Eq. (11.93) is a highly intricate manybody 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 “motheaten effect” induced by the Pauli exclusion principle acting on the B^{†} operators of the ${B}^{\u2020N}{F}_{0}\u3009$ 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 “motheaten effect.” The study of the k state occupation number indicates that the BCS ansatz and the exact Npair 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 electronhole pair. In the case of N excitons, we can approach the Nexciton ground state by expanding it on the overcomplete Nfreeexciton 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 BoseEinstein 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 RichardsonGaudin 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 electronhole 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 centerofmass momentum Q and a relative motion index ν that differentiates the various bound and extended states. The ground state ν_{0} has a spatial extension of the order of the exciton Bohr radius a_{X}. 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 Nexciton ground state is expected to be close to ${B}_{0}^{\u2020N}0\u3009$, where ${B}_{0}^{\u2020}$ creates a groundstate exciton ν_{0} with centerofmass momentum Q = 0. At large density, when excitons overlap, the Coulomb interaction between two excitons becomes as strong as the electronhole attraction inside one exciton (see Fig. 11.23); excitons then dissociate, and the Npair state changes drastically: it turns from a set of (weakly) interacting excitons, (p.309) which are neutral objects, to a twocomponent 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
In this regime, the Nexciton ground state is close to ${B}_{0}^{\u2020N}0\u3009$. Corrections to this ${B}_{0}^{\u2020N}0\u3009$ state can be expanded in terms of exciton states ${B}_{{i}_{1}}^{\u2020}{B}_{{i}_{2}}^{\u2020}\cdots {B}_{{i}_{N}}^{\u2020}0\u3009$, the firstorder correction in Coulomb interaction reading as a sum of ${B}_{\mathbf{\text{Q}},{\nu}_{0}}^{\u2020}{B}_{\mathbf{\text{Q}},{\nu}_{0}}^{\u2020}{B}_{0}^{\u2020N2}0\u3009$.
11.8.2 Frenkel excitons
Within the tightbinding approximation, the spatial extension of the electronhole 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
11.8.3 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 volumelinear number of pairs.
The dimensionless parameter ruling the manybody physics of N Cooper pairs also is smaller than 1 by construction since it reads
(p.310) where ${N}_{max}=\rho \mathrm{\Omega}+1$ is the number of upspin or downspin k electrons feeling the BCS potential. The BCS configuration in which the potential layer is halffilled corresponds to
This regime definitely excludes a lowdensity 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 electronhole 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 meanfield calculation, or through the RichardsonGaudin exact procedure.