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Exciton Bose condensation : the ground state of an electron-hole gas - II. Spin states, screening and band
structure effects
P. Nozières, C. Comte
To cite this version:
P. Nozières, C. Comte. Exciton Bose condensation : the ground state of an electron-hole gas - II.
Spin states, screening and band structure effects. Journal de Physique, 1982, 43 (7), pp.1083-1098.
�10.1051/jphys:019820043070108300�. �jpa-00209484�
Exciton Bose condensation : the ground state of an electron-hole gas II. Spin states, screening and band structure effects
P. Nozières and C. Comte (*)
Institut Laue-Langevin, BP 156X, 38042 Grenoble Cedex, France
(Reçu le 17 décembre 1981, accepté le 4 mars 1982)
Résumé.
2014Nous généralisons tout d’abord la méthode développée dans l’article précédent en y incluant les degrés
de liberté de spin. Nous classons les états correspondants, et nous discutons brièvement l’effet de l’échange inter-
bande. Nous introduisons ensuite l’effet d’écran, dans le cadre d’une approximation RPA généralisée, incorporant
la condensation de Bose des paires électron-trou. Nous étudions en détail la limite diluée, et nous montrons que les corrections d’écran laissent la compressibilité positive, contrairement a certaines estimations antérieures. Ces corrections RPA ne sont en fait qu’une forme approchée de l’attraction de Van der Waals entre excitons. Aux densités intermédiaires, la RPA fournit une méthode d’interpolation. Nous en proposons plusieurs variantes, qui devraient rendre compte de la transition de Mott, et nous donnons quelques estimations numériques pré- liminaires très grossières. Enfin, nous discutons l’effet d’une dégénérescence des bandes sur l’état fondamental.
Lorsque cette dégénérescence est différente dans les deux bandes, on obtient un plasma normal à haute densité,
alors qu’à basse densité les excitons liés forment un condensat de Bose, avec rupture de leur symétrie interne.
Nous prévoyons une transition du 1er ordre avec séparation liquide gaz.
Abstract. 2014 We first generalize the approach of the previous paper by including spin degrees of freedom. We classify
the various spin states and we discuss the effect of interband exchange interactions. We then introduce screening,
in the framework of a generalized RPA which incorporates Bose condensation of bound electron-hole pairs. We
discuss in detail the low density limit : screening corrections do not change the sign of the compressibility, which
remains positive, in contrast to previous estimates. We show that such RPA corrections reduce to an approximate
form of the Van der Waals attraction between excitons. Viewing this RPA approach as an interpolation procedure
at intermediate densities, we propose several interpolation schemes that should account for the Mott transition,
and we give some preliminary very rough numerical estimates. Finally, we discuss the effect of band degeneracy
on the ground state : different degeneracies in the two bands should lead to a normal plasma at high density while
at low densities bound excitons « Bose condense », with a breakdown of their internal symmetry; we expect a first order transition with a liquid-gas phase separation.
Classification Physics Abstracts
71.35
1. Introduction.
-In a preceding paper [1], we
discussed the ground state of an oversimplified elec-
tron-hole gas : spinless carriers, direct gap semi-
conductor, isotropic non degenerate bands. Using
a simple mean field variational ansatz, equivalent to
the BCS wave function in superconductors, we dis-
cussed the nature of Bose condensation for bound electron-hole pairs as a function of density. In the
present paper, we explore the problem further, and
(*) On leave from Laboratoire de Spectroscopie et d’Optique du Corps Solide (associ6 au C.N.R.S. no 232), 5, rue de l’Universit6, 67000 Strasbourg.
we try to take into account features that were ignored
in I.
First of all, we restore spin degrees of freedom. In
section 2, we show how they can be incorporated as
a 2 x 2 complex matrix A that describes the spin
states of condensed pairs, whether singlet or triplet.
If one neglects interband exchange, which would couple electron and hole spins, the hamiltonian is
separately invariant under a rotation of either the electron spin Se or the hole spin Sh. We show that the matrix A may then be diagonalized, in such a way
as to factorize the ground state wave function. The relevant parameter is not the total spin S
=(Se + Sh)
of condensed pairs (which is not a good quantum
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019820043070108300
number), but rather their « state of polarization »,
which we define precisely. We show that the lowest energy is achieved for an « unpolarized » state, which may correspond to a singlet state, or to a
triplet state with Sz
=0 in some arbitrary direction.
If interband interactions are taken into account within first order perturbation theory, the triplet
state is lowest because of Hund’s rule : we briefly
discuss physical properties of the corresponding
state.
Section 3 is devoted to screening, a crucial feature which is ignored in the mean field approach of I-except
for the ad hoc inclusion of the static dielectric constant
K of the intrinsic material. In the dense plasma phase,
intraband screening is usually treated within the random phase approximation (RPA), suitably modi-
fied by exchange corrections [2]. In the opposite
dilute limit, screening by individual excitons is
implicitly taken into account in the original work of Kjeldysh and Kozlov [3] ; a rough estimate of the
corresponding corrections was made by Anderson,
Brinkman and Chui [4]. In the intermediate density region, many attempts have been made to blend RPA with Bose condensation. The variational approach of
Silin [11] is similar to ours; the self consistent field theoretical formulation of Zimmermann [12] is much
more sophisticated. In principle, all these methods
should agree in the dilute limit, where only second
order corrections are important : screening correc-
tions reduce to a truncated form of the Van der Waals interaction between two excitons, which can
be calculated explicitly. Nevertheless, our conclusion
is opposite to that of [11] and [12] : we find that the exciton repulsion due to the exclusion principle
dominates the Van der Waals attraction at low
density N.
At intermediate densities, one must calculate the dielectric constant E(q, (JJ) self consistently : screening
modifies electron-hole pairing, which in turn modi-
fies s. As screening grows, binding decreases, which
is nothing but the Mott transition. Note that such a
transition is not sharp : in our isotropic model, a
finite gap persists at all densities N - and anyhow
the idea of a sharp bound state is meaningless as
soon as Auger broadening is taken into account [13].
Approximate interpolation schemes were proposed
in [11] and [12]. In this paper, we consider yet another
one, in a simple language which seems easier to
handle. Our variational approach is correct in both
limits of low and high densities : in between it should
provide a reliable interpolation. One may use a variational ansatz more realistic than those used in
[11] and [12], thereby avoiding the spurious instability
as N - 0. Unfortunately, the numerical work needed for that variational calculation was beyond our reach (even though it looks possible). Consequently, after discussing the general formalism, we carried only very rough numerical estimates, hich do not correspond
to a well defined approximation : the quantitative
problem remains open. Taking screening into account
lowers the ground state energy; it seems that the effect might be large enough that it produces a first
order phase separation
-a somewhat unexpected
result in an isotropic band. model, yet consistent with the original picture of Mott. A more reliable
calculation is needed in order to decide whether that guess is correct or not.
Finally, in section 4 we consider briefly how these
conclusions would be modified in a more realistic band structure. Band anisotropies do not modify
the physics much at low densities; they are known to
suppress the excitonic insulator instability at high
densities
-a feature which is apparent in the mean field approach (1). However, Kohn [5] has shown
that the transition was actually quite complicated :
translational symmetry breaks down in successive steps, resulting in a series of nested transitions in the
(n, T) plane. Our variational approach does not
account for that behaviour. We also consider the effect of band degeneracy, whether due to an intrinsic
degeneracy (e.g. a p-band), or to a multivalley structure.
Here again, a different degeneracy in the conduction and valence bands destroys Bose condensation at
high density (because Fermi surfaces do not match).
As a result, the system should return to a normal
plasma state at some critical density n
-a feature
that will enhance first order transitions.
Altogether, we leave many questions open : our
goal is only to stress the importance of Bose conden-
sation in studying the intermediate density regime.
2. The spin structure of condensed particles.
-An extensive discussion of that problem, in the
context of excitonic insulators, may be found in the review article of Halperin and Rice [7]. Here we
limit ourselves to a simple analysis, emphasizing
the role of rotational invariance. We start from the
Kjeldysh wave function, which describes accumula- tion of condensed electron-hole pairs in a single bound
state, with zero total momentum - as for an ideal Bose gas. For spin 1/2 particles, it may be written as
(within a normalization factor). Ok characterizes the internal orbital wave function of the pair. The 2 x 2 complex matrix A fixes the spin state. In the absence of spin orbit coupling, A is k-independent. The nor-
malization of A is unimportant, since an extra factor
can always be absorbed in Ok’ If we discard an overall phase factor which corresponds to the global gauge
(’) The effect of impurities is similar : they do not affect
Bose condensation at low density, while they destroy the
excitonic insulator instability at high density [6] : there
must exist a critical Neat which the ground state returns
to the normal plasma. We did not attempt to describe that
transition.
invariance, we see that the spin state depends on 6 independent parameters. We may quote simple exam- ples :
Case (c) is essentially the one studied in I, in which
all carriers have the same spin direction. We note that cases (a) and (b) both correspond to a factoriza-
tion of the ground state wave function, which may be written as
(the operator Aa involving only electrons and holes with spin Q). In such a state, TT and 11 pairs condense separately, with decoupled order parameters
The only difference between singlet and Sz
=0 triplet excitons lies in the relative phase of xk1 and
xkl
-a kind of « internal gauge symmetry ». In the singlet, they are in phase, in the triplet they are out
of phase.
Singlet and triplet states are only extreme cases
for the wave function of condensed excitons
-any combination of them is also possible. It should be
stressed that the total spin S
=Se + Sh of condensed
excitons is not a good quantum number, despite
rotational invariance of the Hamiltonian. S is of
course a good quantum number for a single exciton;
however, if we take two of them, denoted 1 and 2,
there appears intraband exchange interactions
Set .Se2’ or Sht .Sh2 (for instance the usual Fock
terms). As a result, (Sel + Sh 1 ) is no longer conserved.
Only the total angular momentum of all excitons is well defined, which does not mean that the momen-
tum of a single entity is such. Classifying condensates
as « singlet » or « triplet » is somewhat artificial
-
indeed, we shall see that it is not the relevant ques- tion to ask in order to characterize the ground state.
In the factorized state (2), both the Fock intraband
exchange interaction and the Bogoliubov anomalous
terms couple only particles with parallel spins.
Moreover, the Hartree interaction vanishes because of electrical neutrality. Within a mean field approxima- tion, up and down spins are thus dynamically decou- pled : the ground state energy is a sum (EOT + Eol).
One may view the system as a non interacting mixture of up and down carriers. In a non magnetic system, with N 1
=N , the ratio Eo/N is the same as in a spinless gas : the discussion of I is thereby validated (’).
Such a simple result will break down beyond mean
field approximation, when we take screening into
account.
Let us return to the general wave function (1).
If we neglect interband exchange, we may rotate Se and Sh independently without affecting the Hamilto-
nian. Let U and V be the corresponding unitary
transformations : the spin matrix A is transformed into U Å V + = i. Whatever À., we may choose U and V in such a way that I is diagonal, real and positive :
À.î and A’ are the eigenvalues of AA’, invariant under rotations ( U and Y are respectively the matrices that
diagonalize À.À. + and A’A). The positive numbers A,
and A2 are the significant parameters characterizing
the spin state : they fix the state of polarization of the
condensed exciton (defined independently of any
rotation). An unpolarized exciton corresponds to À.1
=Å.2
=1 (equal weight on the two spin states).
At the other extreme, full polarization corresponds to A2
=0, À.1
=1. The energy depends only on A, and A2, not on the total spin S
=Se + Sh.
Consider for instance an unpolarized state : the
carriers split into two independent groups
-the role of the exclusion principle is minimized. Returning to
the original basis, we see that A
=U + V is a unitary
matrix : within a global phase factor, we can write it
as exp[iQ.S], where G is an arbitrary vector and S
are the Pauli spin matrices. Without any loss of
generality, we may take the z-axis along 0 : A then
takes the diagonal form
Depending on Q, the spin state may be a singlet (0
=0), or an unpolarized S,-
=0 triplet (Q
=2 n) :
as far as the energy is concerned, it makes no difference.
Similarly, a fully polarized state corresponds to a separable spin matrix, Å,aa’
=r:x(1 p(1’ : electrons and
holes each have a single spin state, which makes the
exclusion principle most efficient. The spin states a and may be characterized by the two directions .ne and nh along which the corresponding spin is + 1/2.
If ne and nh are parallel, the total spin is triplet. If they are not, we have an hybridization of singlet and triplet. Once again, it does not matter as far as the energy is concerned.
For an arbitrary polarization, the carriers split into
two independent groups, with respective densities
Within a mean field approximation, the ground state
energy in a volume V is simply (2) The parameter rs being defined in terms of the density
Nt for one spin direction.
where e is the energy per particle in the absence of
spin, calculated in I. In that framework, spin polariza-
tion of the excitons (N 1 :0 N2 ) is equivalent to a liquid-gas phase separation. If the latter is energetically
unfavourable (upward curvature of s(VIN)), polari-
zation will not occur : carriers will occupy the two
spin states equally in order to minimize their kinetic
energy. The gain is of course small at low density,
where the exclusion principle hardly acts ; it rapidly
increases when the excitons start overlapping.
It may happen that a liquid-gas phase separation
is favourable. The resulting equilibrium molar
volumes vi 1 and v2 (v
=VIN) are then obtained by
the usual double tangent construction. At first sight, spin polarization seems to compete with a real separation in two distinct phases. However, we
should realize that spin polarization only involves
one parameter : for an arbitrary density N, it is unlikely
that one could achieve the two optimal values v , and V2 : a real phase separation is thus unavoidable.
In each phase, both up and down carriers will achieve
a molar volume either vi or v2. That is not enough to
fix the amount of polarization, which remains unde- termined within our mean field approximation. In higher orders, however, opposite spins do interact in
a way which favours unpolarized states (see section 3).
The ground state will be two unpolarized phases (3)
with molar volumes vi and v2.
Let us now restore the interband exchange which couples electron and hole spins : it is a weak effect which may be treated within first order perturbation theory. The singlet and triplet levels of an isolated
exciton are now split; because of Hund’s rule, the triplet state is lowest. The primary issue remains the absence of polarization (which controls the large intra-
band exchange). But among the many unpolarized
states, we are free to choose the triplet Sz = 0 state (in an arbitrary direction z), which optimizes the small
interband correction. That state should be the real
ground state throughout the whole density range.
Using the numerical results of I, we may describe qualitatively the evolution with density of the various
spin states : singlet (S), unpolarized triplet (To), fully polarized triplet (T1) (remember they are only extreme cases). In the dilute limit, the effect of the exclusion
principle is small as compared to interband exchange : To and T, are close, well separated from the singlet S.
The reverse holds at high density : the polarization
state is dominant; Sand To are very close, while T,
is way up. Such a behaviour is sketched in figure 1.
Fig. 1.
-A sketch of the energy as a function of density for
various spin states : unpolarized triplet To (full curve), polarized triplet T1 (dashed curve), singlet (dotted curve).
8s and ET are the energies of isolated singlet and triplet
excitons.
In conclusion we note that Bose condensation of
triplet excitons raises a number of interesting physical problems. The triplet state S,,
=0 corresponds to
linear polarization in an arbitrary direction z (the
state T, would instead be circularly polarized).
Rotational symmetry is thus broken; we expect a Goldstone mode corresponding to gradual rotation
of the preferred direction. As a simple model, valid at
low densities, we may consider Bose condensation of
interacting spin 1 bosons [8] : the corresponding
branch of the excitation spectrum is found to be linear, (D
=cp, and doubly degenerate (like the spin
waves in an Heisenberg antiferromagnet). In the dense limit, rotational symmetry breakdown is weaker and
weaker, and such a spin wave mode disappears progressively.
3. The effect of screening on the ground state. -
3 .1 RPA FOR A NORMAL UNCONDENSED PLASMA. -
The standard approximation for a high density one component plasma is the random phase approxima-
tion (RPA), which sums all the ring diagrams in a perturbation expansion. The resulting interaction part of the ground state energy is
vq
=4 ne2/Kq2 is the matrix element of the bare Coulomb interaction (screened by the static dielectric
constant of the intrinsic material). II (q, m) is the free
(3) Such a conclusion may look surprising in view of the well known Stoner criterion for the appearance of ferroma-
gnetism in an electron gas. The difference comes, from the Hartree interaction term, which is not zero in the latter
case
-and which is in fact responsible for the magnetic instability (density changes are anyway precluded by elec-
trical neutrality).
particle polarizability, corresponding to a single loop
in the perturbation series :
(Ek = h2 k2/2 m is the single particle kinetic energy).
If we separate H into its real and imaginary parts, ,71 + in 2’ (4) reduces to
-
E(q, co)
=1 + V q Il is the dynamic dielectric constant of the plasma.
To first order in Vq, Arctg (B2/Bl) reduces to B2 :
the integral in (6) is straightforward, yielding a contri-
bution
Taken together, the two terms in (7) provide the usual
Fock exchange energy. Here, however, the latter appears in two separate pieces : the first term in (7)
involves real transitions across the Fermi surface,
from a filled state k to an empty state (k + q). The
second term of (7) acts to subtract the self interaction of individual electrons, which is unduly introduced
when the Coulomb interaction is written in factorized
form, t Vq Pq p.,.
Higher order corrections to, AEO may be viewed as
a screening of the first order Fock term DEo’ Now, according to (6), it is clear that we should only screen
the first part of (7), leaving the self interaction term untouched. Dividing the whole exchange interaction
by s would be definitely wrong ! That fact is of course
well known - still it has raised some ambiguities in
the past. Since it is crucial in our analysis, especially
at low density, a few comments are in order. Formally, screening corrections can only act on real electron
transitions, allowed by the exclusion principle (put
another way, the perturbation expansion involves
real excited states of the whole plasma, not expecta- tion values within the ground state). As a result, only matrix elements of the form nk(1 - nk+q) can be
subject to screening, as in (7). For instance, the second
order contribution to (6) is easily found to be
It clearly displays virtual excitation of two electron- hole pairs.
In order to achieve a more physical understanding,
let us carry two crude simplifications on (6) :
(i) We ignore the frequency dependence of c 1 (q, ro),
which is replaced by its static limit E 1 (q, 0)
=Eq.
°
(ii) We assume that E2 is small, and we replace Arctg x by x in (6).
Admittedly, the approximations are very bad, espe- cially for small q : we do not claim accuracy, we only
want to stress the underlying physics. The whole
interaction energy (6) thus becomes
which we may rewrite as
The first term in (10) is a screened exchange energy, obtained by the replacement Vq --+ Vq/Bq in the Fock term (7). It is just what common sense would suggest : Coulomb interactions are screened and we divide all
Vq by sq ! That however would miss the second term in (10), which may be viewed as a vacuum polarization
correction : each electron polarizes the surrounding medium, thereby changing its self interaction. Such a
point of view was stressed by Anderson et al. [4].
We now return to the full RPA expression (6). It is
known to be correct at high densities, rs 1. For larger r,, exchange corrections become important. In
second order, for instance, the exchange conjugate graph of figure 2b gives a contribution
which cancels half the direct graph of figure 2a for large values of q. There exist various empirical ways to interpolate between the small and high q limits [9].
The simplest one in principle is that of Hubbard, who
Fig. 2.
-The two second order diagrams in an interacting plasma : (a) Direct RPA term; (b) Exchange conjugate cor-
rection. The spin weight is respectively 4 and 2 for the two
diagrams.
replaces the denominator (1 + V q lI 1 ) in (6) by the empirical form
With such modifications, the RPA is considered
satisfactory at metallic densities, r, - 1.
Such an approach is easily adapted to an electron-
hole gas. If we ignore interband scattering, the basic
vertices are those of figure 3. In the absence of Bose
condensation, the two bands are not hybridized : we
define independent polarizabilities for electrons and holes, na and II6 (see Fig. 4).
Fig. 3.
-The intraband interaction vertices in an electron- hole gas. Indices a )> and « b » refer respectively to the
conduction and to the valence band.
Fig. 4.
-Elementary polarization loops. 17. and II b exist
in a normal, uncondensed plasma. i7 results from hibridiza- tion of the two bands : it appears as a consequence of Bose condensation.
The RPA expression (6) is then replaced by
where 17
=17,, + 7b is the total polarizability, and
N the number of electron-hole pairs. Note that Ila
and lIb depend respectively on the masses me and mh of electrons and holes. As a result, L1Eo depends
on the mass ratio Q
=me ,Inih, in contrast to the Hartree Fock approximation (in which me and mh enter only the kinetic energy, via the reduced mass m*).
At metallic densities, one may account for exchange
corrections by either of the above tricks : one thus obtains an estimate of Eo(N). Such calculations were
carried extensively in the seventies [10], first in a simple isotropic band model, and then in more
realistic band structures. They are supposed to explain
electron-hole droplet formation. It should be realized however that these approaches do not account for
the formation of bound pairs. We saw in I that such
a feature was dominant, even at metallic densities,
within a mean field approach. Sure, screening will
reduce the importance of binding
-nevertheless, we
are led to question RPA-like treatments.
In the preceding discussion, we treated holes as positive particles
-which is perfectly all right. It is
nevertheless instructive to return to the original
valence description. Let nkQ be the distribution of
holes, (1 - nka) that of valence electrons. The real
exchange energy is
The last term + 1 in the bracket of (13) is absorbed in the ground state energy of the intrinsic material, while
the linear terms act to correct the hole energy (thereby renormalizing the energy gap). These contributions
are absorbed in the one hole Hamiltonian, so that
we retain only the hole-hole exchange, nk,,, nk’a. That rather trivial remark is relevant if we include screening.
According to our earlier discussion, the latter acts
only on real transitions : we thus write the hole-hole
exchange as
and we screen only the first term
-which is the same
in a particle and in a hole description. One should
not screen the exchange correction to the energy gap, which involves matrix elements between two filled states. That point is sometimes missed
-hence our
detailed discussion.
3. 2 GENERALIZATION TO A BOSE CONDENSED GAS.
-Bose condensation of excitons implies an hybridiza-
tion of the two bands, leading to « anomalous » propagators
(aZa and bZa refer respectively to the conduction and to the valence band). In a strict perturbation approach,
these propagators must be determined se!f-consis- tently : within a given set of skeleton diagrams for
the ground state energy, the normal and anomalous self energies must be the functional derivatives of
AEO with respect to the corresponding propagators :
If we retain only the first order diagrams, shown on figure 5, equations (14) reduce to
Fig. 5.
-The first order diagram for a condensed system.
The third « anomalous » one is specific of Bose condensation.
The self energies are instantaneous (energy indepen- dent) : we should recover the mean field approxima-
tion of I. In order to see the equivalence in detail, we
note that in a neutral system (4) Ea = Lb
=Z. (15)
then leads to an effective one-electron Hamiltonian
In general, we must allow for different chemical
potentials Jle and in order to ensure Ne
=Nh
=N.
The elementary excitations are obtained in the usual way by writing equations of motion for a* and bra;
the corresponding secular matrix is
For convenience we use the same notations as in I :
The eigenvalues of (17) may then be written as
They describe two types of quasiparticles, one pre-
dominantly electron-like with energy 11:, the other
rather hole-like with energy - ilk
The ground state is the vacuum of these quasi- particles. Its internal structure depends only on the
difference 11: - 11;- = En which controls the eigen-
vectors of (17). (Ek is the energy needed to break a
condensed exciton into a quasiparticle pair with zero total momentum). We thus recover the BCS state (I .12), the parameter vk being given by
(rn* is the reduced mass). If we further note that
we see that the self-consistency equations are identical
with our former equations (I.18) and (I.19). The
variational approach used in I is easily cast in pertur- bative language : minimizing the energy (eqs. (I. 18)
and (1.19)) is equivalent to the self-consistency require-
ment (15).
The parameters p,, and Jlh are chosen at the end in such a way that Ne
=Nh
=N. The situation is
Q
=me/mh
=1. Then the system has full electron- hole symmetry, which implies
In the more general case, Ak 0 0 and fit depend on the
mass ratio (1. Although they are not needed in a first
order calculation, it is easy to calculate the one particle propagators from the effective Hamiltonian (16) :
the corresponding 2 x 2 matrix is simply [Heff - e,] - 1.
Simple algebra yields
(23) may be used for more elaborate higher order
calculations.
In order to account for screening, we must go beyond first order. For instance, we may again sum
the ring diagrams of RPA : the ground state energy is
still given by (6), the only difference being that the (’) Distribution in k-space is the same for electrons and
holes, since they are produced by pairs of equal momenta
out of the original normal plasma ground state.
polarizability H has one more « anomalous » term, shown on figure 4 :
A question then arises : what should we take for G ? In ordinary RPA, one takes the free particle propa- gator : clearly, that would ignore completely the
reaction of binding on the dielectric constant
-a
crucial feature ! Ideally, we should maintain the
exact self-consistency equations (14) : for a given set
of graphs, E and ± are obtained by functional deriva- tion. Such an approach was used by Zimmermann [12] :
the resulting self energies are retarded, and some sort
of approximation is needed to eliminate energy inte-
gration. The calculation is complicated, and it is hard to assess its accuracy. The difficulty may be circumvented at very large or very low densities : when Nag « 1, condensation effects are small, and ordinary RPA should be all right. If instead Na’ 1, screening corrections are weak and a perturbation expansion is possible. The real problem lies at inter-
mediate densities, where a self-consistent treatment of screening and binding is definitely needed. In order to proceed, we must be less ambitious : one way or the
other, we must devise an approximation in which the propagators have the mean field form (23), with effective parameters v,, fit appropriately modified by screening. We shall consider later several possibi-
lities along those lines (see also [11]).
The polarizability I7(q, co) is easily calculated when G has the form (23). Performing the integrations in (24),
we find
The « coherence factor » (uv’ - u’ v)’ is typical of a
BCS ground state. One verifies easily that for a step like Vkl II reduces to the sum (llao + II6o) of normal
electron and hole polarizabilities, each given by (5).
We note that I7(q, m) vanishes when q - 0, which
ensures a finite dielectric constant at long wavelength.
Physically, that feature is a consequence of electron- hole binding, which precludes free carrier Debye screening. In order to clarify orders of magnitude, we
may consider the static polarizability
In the dilute limit Nao 1, we know that
(N (1
=N/2 is the one spin density, and ØkO is the
internal wave function of a single exciton). Moreover
the energy denominator in (26) is >- so. II behaves as
shown on figure 6, with a range q - Ilao and a
maximum - NIBO. In the opposite limit Nat » 1,
Vk drops sharply near the Fermi level, on a width
-
’JIVF’ When q > ’JIVFI the polarizability is the
same as for a normal gas : it is the sum (voa + vob)
of electron and hole density of states when q kF,
and it drops down to zero when q >> kF. If instead
q d/vF, Bose condensation becomes essential and II vanishes quadratically (see Fig. 6). In between
these two limits, the evolution is smooth : as the density increases, II grows and eventually « flattens »
while broadening its q-range.
Fig. 6.
-A sketch of the static polarizability II (q, 0) as a
function of q for high densities Nalo > 1 (full curve) and for
low densities Nal 0 1 (dashed curve).
3.3 THE DILUTE LIMIT. - We assume Na 3 1
-
yet large enough for molecular biexcitons to be dissociated. The above program can then be carried
out explicitly. The lowest order propagators have the shape (23), with
The self energies E and ± are very small, respectively 0(N) and 0(.,I-N). In leading order one may neglect
them, so that
The resulting polarizability is of order N (as expected).
If we note that ge + Ilh = 9 - so, we may write
it as
(29) represents in fact a simple approximation to the polarizability of a single exciton. Let q6,,(r, - r,)
be the latter’s internal wave function. The correspond- ing charge density operator is
The resulting polarizability should then be
where n is any excited state of the particle-hole Hamil-
tonian Ho, while (0110
=En - Eo. If we took for Ho the full Hamiltonian, including the Coulomb interac- tion, 77 would be the usual Kramers polarizability.
The result (29), involving a single loop of BCS propa- gators, is equivalent to retaining only the kinetic
energy in Ho. The final state n then involves a free
electron-hole pair, with momenta (k + q) and k.
The excitation energy (En - Eo) is just (Okq defined in (29). The corresponding matrix element (pg)o" is
We recover (29). Put in more physical terms, the BCS polarizability (29) takes into account electron-hole correlations in the ground state 10 >, but not in
the excited states In>. The dynamics of electron-hole response is described incorrectly
-but it is not
grossly distorted : if we include Bose condensation,
the RPA is a sensible approximation to 77. (We note
that the result (29) differs from that of Anderson,
Brinkman and Chui [4], who used an incorrect expres- sion for the density operator pq.)
The static polarizability n(q, 0) may be calculated explicitly, yielding the dielectric constant
When qao « 1, we have
Using the expression (I.8) of OkO, we find in reduced
units ao = so
=1
At short wavelength qao > 1, the overlap of Øk and ok+q is negligible, and the summation in (29) is
dominated by k - 0 or k - - q. We find easily
In between 17 goes through a maximum ; its exact shape may be obtained numerically if needed.
We now turn to the ground state energy, which
we want to calculate to order N ’. Within a pertur- bation expansion, terms of that order may arise :
(i) either from corrections 6G to the one particle propagator in the first order « mean field » diagram (Hartree-Fock-Bogoliubov)
(ii) or from higher order skeleton diagrams.
We should remember however that the zeroth order Go has been chosen in such a way that the mean
field energy be minimal : corrections due to bG are
therefore of order bG 1. Since 6G - N, the corres- ponding corrections to the ground state energy are of order N’, negligible for our purpose. We are left
only with higher order diagrams, calculated with the zeroth order propagators Go defined in (23)
and (27) (5).
Let us first stay within RPA. The polarizability is
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