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>
HAL Id: jpa-00209484
https://hal.archives-ouvertes.fr/jpa-00209484
Submitted on 1 Jan 1982
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J.
Physique 43 (1982)
1083-1098
JUILLET
1982,
1083
Classification
Physics Abstracts
71.35
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
(Reçu
Laue-Langevin,
(*)
BP 156X, 38042 Grenoble Cedex, France
le 17 décembre 1981,
accepté le 4 mars 1982)
Nous généralisons tout d’abord la méthode développée dans l’article précédent en y incluant les degrés
Résumé.
de liberté de spin. Nous classons les états correspondants, et nous discutons brièvement l’effet de l’échange interbande. 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 préliminaires 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.
2014
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.
1. Introduction.
In a preceding paper [1], we
discussed the ground state of an oversimplified electron-hole gas : spinless carriers, direct gap semiconductor, isotropic non degenerate bands. Using
a simple mean field variational ansatz, equivalent to
the BCS wave function in superconductors, we discussed the nature of Bose condensation for bound
electron-hole pairs as a function of density. In the
present paper, we explore the problemfurther, and
-
(*) On leave from Laboratoire de Spectroscopie et
du Corps Solide (associ6 au C.N.R.S. no 232),
5, rue de l’Universit6, 67000 Strasbourg.
d’Optique
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
we
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019820043070108300
=
1084
number), but rather their
we define precisely.
which
state of polarization »,
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 modified 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 corrections 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
(JJ) self consistently : screening
modifies electron-hole pairing, which in turn modifies 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
E(q,
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
plasma state at some critical density n
-
a
a
normal
feature
that will enhance first order transitions.
Altogether, we leave many questions open : our
goal is only to stress the importance of Bose condensation 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 accumulation 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
-
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 normalization 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
(within
(’) 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.
1085
invariance, we see that the spin state depends on 6
independent parameters. We may quote simple examples :
Such
field
simple result
approximation,
a
will break down beyond mean
when we take screening into
account.
us return to the general wave function (1).
neglect interband exchange, we may rotate Se
and Sh independently without affecting the Hamiltonian. 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 :
Let
If
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 factorization of the ground state wave function, which may be
written as
we
À.î and A’ are the eigenvalues of AA’, invariant under
( U and Y are respectively the matrices that
diagonalize À.À. + and A’A). The positive numbers A,
A2 are the significant parameters characterizing
(the operator Aa involving only electrons and holes and
the
spin state : they fix the state of polarization of the
with spin Q). In such a state, TT and 11 pairs condense
exciton (defined independently of any
condensed
separately, with decoupled order parameters
An
unpolarized exciton corresponds to
rotation).
1
(equal weight on the two spin states).
À.1 Å.2
At the other extreme, full polarization corresponds to
A2 0, À.1 1. The energy depends only on A, and
The only difference between singlet and Sz
0
A2, not on the total spin S Se + Sh.
triplet excitons lies in the relative phase of xk1 and
Consider for instance an unpolarized state : the
a kind of « internal gauge symmetry ». In the
the role
xkl
carriers split into two independent groups
singlet, they are in phase, in the triplet they are out of the exclusion principle is minimized. Returning to
of phase.
the original basis, we see that A
U + V is a unitary
Singlet and triplet states are only extreme cases matrix : within a global phase factor, we can write it
for the wave function of condensed excitons
any
as exp[iQ.S], where G is an arbitrary vector and S
combination of them is also possible. It should be are the Pauli spin matrices. Without any loss of
stressed that the total spin S
Se + Sh of condensed generality, we may take the z-axis along 0 : A then
excitons is not a good quantum number, despite takes the diagonal form
rotations
=
=
=
=
=
=
-
-
=
-
=
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 momentum 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 question 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 approximation, up and down spins are thus dynamically decoupled : 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 (’).
-
Depending
(0
as
=
0),
on
or an
Q, the spin state may be a singlet
unpolarized S,- 0 triplet (Q 2 n) :
=
=
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
=
(2)
The parameter rs being defined in terms of the
one spin direction.
Nt for
density
Within a mean field approximation, the
energy in a volume V is simply
ground
state
1086
where e is the energy per particle in the absence of
spin, calculated in I. In that framework, spin polarization 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)), polarization 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 vi1 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 undetermined 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 intraband 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
=
Fig. 1.
-
A sketch of the energy
various
as a
function of density for
spin
unpolarized triplet To (full curve),
polarized triplet T1 (dashed curve), singlet (dotted curve).
8s and ET are the energies of isolated singlet and triplet
states :
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.
spin states : singlet (S), unpolarized triplet (To), fully
polarized triplet (T1) (remember they are only extreme
3. The effect of screening on the ground state. cases). In the dilute limit, the effect of the exclusion 3 .1 RPA FOR A NORMAL
UNCONDENSED PLASMA. principle is small as compared to interband exchange : The standard
for a high density one
approximation
To and T, are close, well separated from the singlet S.
the
random
is
phase approximacomponent
plasma
The reverse holds at high density : the polarization
which
tion
sums all the ring diagrams in a
(RPA),
state is dominant; Sand To are very close, while T,
perturbation expansion. The resulting interaction
is way up. Such a behaviour is sketched in figure 1.
part of the ground state energy is
(3) Such a conclusion may look surprising in view of the
well known Stoner criterion for the appearance of ferromagnetism in an electron gas. The difference comes, from the
Hartree interaction term, which is not zero in the latter
and which is in fact responsible for the magnetic
case
instability (density changes are anyway precluded by electrical neutrality).
-
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
=
1087
particle polarizability, corresponding to a single loop
in the
perturbation series :
It clearly displays virtual excitation of two electronhole 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).
°
=
(Ek = h2 k2/2 m
If
we
,71
+
is the
single particle kinetic energy).
separate H into its real and imaginary parts,
in 2’ (4)
reduces to
Admittedly, the approximations are very bad, especially for small q : we do not claim accuracy, we only
want to stress the underlying physics. The whole
interaction energy (6) thus becomes
-
which
E(q, co)
of the
=
1
we
may rewrite
as
+ V q Il is the dynamic dielectric constant
plasma.
To first order in Vq, Arctg (B2/Bl) reduces to B2 :
the integral in (6) is straightforward, yielding a contribution
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
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
Vq Pq p.,.
Higher order corrections to, AEO may be viewed as
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 expectation 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
a
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 correction. The spin weight is respectively 4 and 2 for the two
diagrams.
-
1088
replaces the denominator (1
empirical form
+
V q lI 1 )
in
(6) by the
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.
-
With such modifications, the RPA is considered
satisfactory at metallic densities, r, - 1.
Such an approach is easily adapted to an electronhole 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 electronhole gas. Indices a )&#x3E; and « b » refer respectively to the
conduction and to the valence band.
-
In the
preceding discussion, we treated holes as
which is perfectly all right. It is
positive particles
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
Elementary polarization loops. 17. and II b exist
in a normal, uncondensed plasma. i7 results from hibridization of the two bands : it appears as a consequence of Bose
Fig. 4.
-
condensation.
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.
-
-
The RPA expression (6) is then replaced by
3. 2 GENERALIZATION TO A BOSE CONDENSED GAS.
Bose condensation of excitons implies an hybridization of the two bands, leading to « anomalous »
-
propagators
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
=
=
and bZa refer respectively to the conduction and
to the valence band). In a strict perturbation approach,
these propagators must be determined se!f-consistently : 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 :
(aZa
1089
If
we
retain
only
the first order
figure 5, equations (14) reduce
diagrams, shown
on
The
eigenvalues of (17) may then be written as
to
They describe two types of quasiparticles, one predominantly electron-like with energy 11:, the other
rather hole-like with energy - ilk
The ground state is the vacuum of these quasiparticles. Its internal structure depends only on the
difference 11: - 11;- = En which controls the eigenvectors 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
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-
should recover the mean field approximation of I. In order to see the equivalence in detail, we
Z. (15)
note that in a neutral system (4) Ea = Lb
then leads to an effective one-electron Hamiltonian
dent) :
(rn* is the reduced mass). If we further note that
we
=
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 perturbative 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 electronhole symmetry, which implies
=
=
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
In
=
=
=
=
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
For convenience
(’)
we use
Distribution in
the
k-space
same
is the
notations
same
as
in I :
for electrons and
holes, since they are produced by pairs of equal
out of the original normal plasma ground state.
momenta
(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
1090
polarizability H has
shown on figure 4 :
one more «
anomalous » term,
In the dilute limit
Nao 1, we know that
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 &#x3E;- 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 &#x3E; ’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 &#x3E;&#x3E; 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.
(N (1
=
-
A question then arises : what should we take for G ?
In ordinary RPA, one takes the free particle propagator : clearly, that would ignore completely the
a
reaction of binding on the dielectric constant
crucial feature ! Ideally, we should maintain the
exact self-consistency equations (14) : for a given set
of graphs, E and ± are obtained by functional derivation. Such an approach was used by Zimmermann [12] :
the resulting self energies are retarded, and some sort
of approximation is needed to eliminate energy integration. 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 intermediate 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 possibilities 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 electronhole binding, which precludes free carrier Debye
screening. In order to clarify orders of magnitude, we
may consider the static polarizability
Fig. 6. A sketch of the static polarizability II (q, 0) as a
function of q for high densities Nalo &#x3E; 1 (full curve) and for
low densities Nal 0 1 (dashed curve).
-
- We assume Na 3 1
for
molecular
biexcitons to be
yet large enough
dissociated. The above program can then be carried
out explicitly. The lowest order propagators have the
shape (23), with
3.3 THE
DILUTE
LIMIT.
-
The self energies E and ± are very small, respectively
In leading order one may neglect
0(N) and
them, so that
0(.,I-N).
The resulting polarizability is of order N (as expected).
If we note that ge + Ilh = 9 - so, we may write
it as
1091
in fact a simple approximation to the
of
a single exciton. Let q6,,(r, - r,)
polarizability
be the latter’s internal wave function. The corresponding charge density operator is
(29) represents
The resulting polarizability should then be
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 perturbation 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.
where n is any excited state of the particle-hole Hamiltonian 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 propagators, is equivalent to retaining only the kinetic
energy in Ho. The final state n then involves afree
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 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 corresponding 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
N : in the general expression (4) we may expand
the log and retain only the second order term
which is tantamount to calculating the second order
ring diagram of figure 2a (properly modified by Bose
condensation). Replacing H by its explicit form (29)
and performing the energy integration, we find
-
-
(29). Put in more physical terms, the BCS
polarizability (29) takes into account electron-hole
correlations in the ground state10 &#x3E;, but not in
the excited statesIn&#x3E;. The dynamics of electron-hole
but it is not
response is described incorrectly
if
Bose
we
include
distorted
:
condensation,
grossly
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 expression for the density operator pq.)
The static polarizability n(q, 0) may be calculated
explicitly, yielding the dielectric constant
We
recover
-
When qao « 1, we have
Using the expression
units ao = so
1
(I.8) of OkO,
recognize the second order perturbation due to
virtual excitation of two quasiparticle pairs with
opposite momenta out of the mean field ground state
(note that (31) involves a summation over the spins a
and a’ of the two polarization loops, which has been
absorbed in the factor N 1). Using the expression (28)
of 11:, the calculation reduces to an integral. We
only performed it in the symmetric case me = nth (’).
We
(5) As in ordinary perturbation theory, lowest order
corrections to the energy involve only changes in the hamiltonian, not in the wave function.
(6) The integration of (31) is rather tedious. We first
calculate the spectral density
we
find in reduced
=
which can be obtained analytically using bipolar coordinates
with basis q, We then carry the resulting integral numerically:
At short wavelength qao &#x3E; 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
A
upper bound is obtained if
rough
we
replace
denominator by the lower quantity 2(1 +
tion
can
37 ;t2
then be done
q2 .
the energy
The integra-
analytically, leading toI E (2,,)
= 7.26 N’ : this is just the result of Zimmermann
[12], obtained with the
same
simplification.
1092
The energy denominator in (31 ) then reduces to
Using
our
reduced units ao = go
=
1,
we
find
expected, the direct screening correction acts to
depress J1. Note that (32) is considerably lower than
the value quoted by Silin [11] and Zimmermann [12].
Unfortunately, (31 ) is not the only contribution - N 2
to the energy. The exchange conjugate diagram of
figure 2b gives for instance a term of the same order :
As
electron and/or valence hole lines (without introducing
any additional anomalous line G). Physically, such
diagrams describe correlations in the final state that
were ignored in RPA. (They will account, for instance,
for the virtual transition of the interacting excitons
towards excited bound states.) Calculating these terms
explicitly is impossible, since it would involve solution
of a four body problem. In principle, final state correlations should lower energy denominators in the
perturbation expansion : they should therefore enhance
the RPA term E(2a). It is difficult however to assess
the importance of the effect. Since the first hydrogenic
excited state (2s) already requires an energy 3/4 go,
higher order corrections should not be dramatic.
For lack of a better argument, we shall ignore them
and we shall assume that the screening correction to
the ground state energy of a dilute system is simply
E (2a)
.
That correction should now be combined with the
term (I.15) found in the unscreened mean field
approach of I. Since the latter only involves parallel
spin exchange, we reduce our former coefficient by a
factor 1/2. The net expansion of the chemical potential
is
((33) follows from the expression (30) of the density
note the factor 1/2 due to spin
matrix element
conservation). A numerical calculation of E(2b) looks
appalling. In a normal system, E (2b) is supposed to
cancel half the direct term E(2a) for large values of q.
Here the matrix elements can change sign so that the
role of E (2b) should be largely reduced. On physical
grounds, we may expect that it will slightly decrease
E(2a), although mathematically it is not obvious.
The most general energy corrections of order N 2
were described formally by Kjeldysh and Kozlov [3].
Once two quasiparticle pairs are created, their four
constituent particles can scatter any number of times
without introducing additional factors N. The corresponding diagrams are shown on figure 7. They are
obtained from either figure 2a or figure 2b by inserting
any number of interactions between the excited
i.e. between the ascending conduction
particles
-
The net compressibility is therefore positive : a
dilute exciton gas is thermodynamically stable (at
least when me
mh)-contrary to previous estimates.
The disagreement with Zimmermann [12] is only
minor, due to his overestimate of E(2a). On the other
hand, we are in clear contradiction with Anderson
et al. [4] and with Silin [11], who claim that the screening
term is much bigger than (I . 25 ). We have no explanation for reference [11]. In reference [4], the screening
correction is calculated as a difference of two terms,
one positive due to reduction by screening of the
exciton binding, the other negative due to « vacuum
polarization » (see section 3.1). Starting from the
first order energy
=
-
Fig. 7. - The most general energy diagram of order N2.
The four extreme vertices may belong to either the conduction or the valence bands (a or b operators). They produce
the factor N 2. The shaded box denotes any numbers of
interactions between the four virtually excited quasiparticles.
these two contributions represent the screening corrections to respectively the cross term and the square
term in (35). These terms cancel to a large extent, and
having a consistent approximation in both is crucial.
In reference [4], different dielectric constants are
used in the two terms (and moreover the polarizability
is calculated with an incorrect density matrix element) : that enhances unduly the final result.
The second order « screening » terms we just
calculated are nothing but the Yan der Waals attraction between excitons : they describe virtual polarization of the two atoms due to Coulomb interactions.
1093
At large distances, the wave functions do not overlap
and only the direct diagram of figure 2a is important.
The exchange conjugate terms of figure 2b come into
play when the distance is ao but then the exclusion
principle is crucial and one can no longer separate
« mean field » and « screening » effects. It is gratifying
that the same approximation, namely the RPA,
known to be exact in the limit of high densities also
yields a sensible physical picture in dilute systems :
there it represents an approximation to the V an der
Waals interaction. As a result, it should provide a
reasonable interpolation scheme in the intermediate
-
-
range.
Our result (34) means that the hard core repulsion
supersedes the Van der Waals attraction in determining the compressibility of Bose condensed atomic
excitons. In our BCS-like description, these excitons
all have the same unpolarized spin state : scattering
of two excitons occurs in all possible spin channels
for the colliding electrons (S,,l + S,,2) and holes
(Shl + Sh2). The effective interaction involves a
weighted average over these spin channels : it may
well be repulsive even though bound molecules exist
in the double singlet state. Clearly, such a result only
makes sense if molecular correlations are negligible,
i.e. if the exciton spacing is much smaller than the
molecular radius aM. When NaM decreases, molecular
correlations build up progressively, while the weight
of the singlet state grows. At a critical density the
atomic exciton condensate disappears, leaving way
to an isotropic singlet biexciton condensate. Such an
evolution is not described by our variational ansatz,
which is equivalent to a zeroth order Bogoliubov
approximation for Bose gases. In order to account
for molecular correlations, we should include in (1)
quartet terms, a*a*bb, which looks untractable.
One may nevertheless understand the transition
qualitatively by treating excitons as point bosons :
this is done elsewhere [14].
3.4 APPROXIMATE TREATMENTS AT INTERMEDIATE
We want to describe screening and the
formation of bound electron-hole pairs self-consistently, while retaining « mean field » propagators
(23), with suitable parameters vk, llk 1. A variational
approach seems best suited. We cannot take both
we would
vk and’ql as variational parameters
then loose all the information on the underlying
dynamics (effective masses, etc...). Moreover, we should
not forget that in a variational method one acts on the
wave function, not on the Hamiltonian operator.
We therefore propose the following procedure :
(i) Consider Uk and vk as variational parameters
DENSITIES.
-
-
(subject
to
(u’
+
v’) = I).
bare propagators) ; at low densities, E is
N, therefore small. Anyhow, only the k-dependence of Ek
would be relevant : a constant term may be absorbed
in /i ; this should be contrasted with ±k which, even
if constant, directly affects the ground state wave
function.
(iii) We then calculate the ground state energy by
for
summing any prescribed set of diagrams
instance the ring diagrams of RPA, using (12) and
-
-
(25) (’).
(iv) The resulting energy Eo is a functional of vk.
We determine vk in such a way that Eo be minimum.
(The chemical potentials Jle and are fixed by requiring given values Ne Nh N).
We should emphasize that the above approach is
not a genuine variational procedure in the sense of
Ritz. We can claim that uk, Vk correspond to a specific
choice of the ground state wave function only in
first order : the mean field analysis of I was indeed
=
=
variational. When we include a selected class of
higher order diagrams, we calculate the expectation
value of an approximate Hamiltonian with an approximate wave function : one cannot be sure that changing
vk acts only on the latter. In that respect, our procedure
is hybrid and somewhat doubtful. Nevertheless, we
believe it is sensible : Uk and vk act primarily on the
wave function (in contrast to ill). Moreover the
procedure is exact at high density (where it reduces
to RPA) and at low density (where screening is
weak) : it may be viewed as an interpolation in between.
variational nature, the procedure is
self-consistent : the polarizability depends on Uk Vk,
which in turn depend on the polarizability. One should
thus achieve an explicit description of the Mott
transition, i.e. the progressive disappearance of the
bound exciton due to screening of the electron-hole
interaction. (In the present case there is. no sharp
transition since we always retain a finite d .)
A similar approach was used by Silin [11], who
also calculated the ground state energy within RPA
while treating Uk and vk as variational parameters.
In calculating the polarizability n, he however relied
on quasiparticle energies q k ± obtained from the
unscreened Hartree Fock self energies E and i. The
feedback of screening on pair formation (via t)
is thus treated inconsistently. Moreover, his variational ansatz is not very realistic, and the interpolation
scheme he uses to treat correlations is somewhat
doubtful for neutral excitons : his results should be
considered with caution (anyhow, they have the
wrong slope when N --+ 0). In contrast, the perturbation approach of Zimmermann [ 1 2] is self-consistent-
Owing
to its
A
(ii) ?I’- k depend on the self energies 2;k and fk
(see (18) and (19)). Given Uk and v’" these two parameters are related by (20). In order to determine r¡t
completely, we set arbitrarily Zk 0. Such a choice
is reasonable at high densities (ordinary RPA involves
=
(7) One may try to account for exchange conjugate
diagrams via the Hubbard trick : while good at high density,
such a procedure would probably overestimate the correction in dilute systems.
1094
it is not variational in nature, and it
is difficult to assess the many approximations needed
in order to carry the calculation to the end.
In principle, the variational program we just
sketched should be simpler and more realistic.
Nevertheless, the numerical work is fairly heavy
(as in I, one may simplify it by using a parametrized
form of vk depending on a small number of variational
quantities). Anyhow, it was well beyond our computing
abilities and we did not attempt it. It may be worth
the effort in the future, since a self-consistent treatment would clarify the interplay of screening and
unfortunately,
binding
-
a
(ii) qD should be
the usual Fermi Thomas result :
where vo, vo are the electron and hole Fermi level
densities of states, g being the Kohn function. We
approximate g by a step function (g 1 if q 2 kF).
Moreover, in order to account for the error by a
factor - 4 mentioned earlier (in the second order
term), we arbitrarily divide q’5 by 4. We thus set
=
(for spin 1/2 particles)
long standing problem.
A more primitive approach of the above type
consists in using the approximation (9) and (10)
instead of genuine RPA : in (12) we replace Arctg x
by x and we approximate 17, (q, m) by III (q, 0). The
ground state energy thus. becomes
Calculations were only performed in the
case me = mb
2 m*. (39) then reduces to
symmetric
=
(iii) We then carry the variational calculation of I.
We choose a parametrized ansatz for uk and vk (in
reduced units ao
1)
Eo
=
given by (26). If we minimize with respect
(36) also provides a self-consistent description, equivalent to a screened mean field approximation. The latter, however is not very good. Accepting
the linearization Arctg x - x as granted (?), one can
carry the energy integration exactly in (12) : the mean
field term in (36) should be screened by E(q, Cokq),
not by the static 8(q, 0)
where Q)kq is the excitation
energy, ( Ilk++ q - ?k ). The error is particularly obvious
in the second order term. Compared to the exact
result (31), the approximation (36) amounts to the
replacement
17(q, 0)
=
is
to Uk and Vkl
-
1
(see
the exact expression (25)). In the dilute limit,
the error is nearly a factor 4 ! Quite generally, (36)
screening corrections, leading to a
ground state energy which is too low.
At the present stage, even the simpler calculation
(36) was beyond our reach. In order to achieve a
qualitative rough estimate, we used the following
overestimates
trick :
(i) We replace I7(q, 0) by the free
bility, setting
carrier
polariza-
approximation, while reasonable in dense
is
completely wrong in the dilute limit :
systems,
there it grossly overestimates screening. Indeed, the
screening correction to dy/dN is so large that it
overcomes the mean field result of I : the net compressibility changes its sign !
Such
an
minimize E - MN with respect to C and 0.
Such a procedure is extremely primitive. It can at
best indicate qualitative trends : the following results
should be considered with extreme caution.
In order to assess our rough estimate, we first
consider a normal electron-hole gas with no pairing
(vk is a step-function). In figure 8, we plot the ground
state energy per particle, EOIN, as a function of the
density parameter rS, defined for a single spin species :
and
we
We compare the above treatment of screening with
the Hartree Fock result of I (no screening at all)
and with the genuine RPA calculation of Brinkman
et al. [10] (including the Hubbard modification). As
expected, the effect of screening is very large. That
effect, however, is overestimated in our primitive
approach (our arbitrary reduction of ql by a factor 4
is not enough). Still, considering the crudeness of
our description, the discrepancy with an honest RPA
calculation is not so large.
We now turn to the Bose condensed states, described
by the ansatz (41). The ground state energy is plotted
in figure 9, with and without screening; the corresponding variational parameters C and Q are shown
in figure 10 : the following comments are in order :
(i) The
reduction in energy due to screening is still
yet somewhat smaller than in a normal
plasma : as expected, exciton binding reduces the
dielectric constant. Taken at face value, figure 9 leads
to spontaneous droplet formation at zero pressure
quite large
-
"
1095
8.
The ground state energy per particle as a function
of the density parameter rp in reduced units ao
1,
co
in the absence of Bose condensation. The full curve is the
Hartree Fock result, the dashed curve is the result of a
genuine RPA calculation carried by Brinkman et al. [10].
The dotted curve is the result of our rough static screening
procedure (36), using the dielectric constant (37) and a
step function v,.
Fig.
-
=
=
Fig. 10. - Comparison of the variational parameters C
and Sl for the unscreened model of I (full curve) and for the
rough estimate of screening in this section (dotted curve).
that question reliably
and also decide
whether that minimum is above or below (-,so)
possible behaviours are sketched on figure 11. Probably, the minimum lies above ( - 80x leading to a
first order phase separation AB, in full agreement
with the original ideas of Mott (note that here the
two phases are insulating because of the excitonic
insulator instability).
can answer
(in an isotropic model). Such a conclusion however,
cannot be trusted. We know that the downward
curvature at large rs is an artefact; quite generally,
overestimate screening, first because of our crude
procedure (see Fig. 8), and also because we use a
free carrier polarizability. The actual curve in figure 9
should be higher. Nevertheless, the existence of
inflection points is most likely : matching the low
density expansion with the RPA curve strongly
suggests the existence of a secondary minimum.
Only a full numerical analysis as described above
we
-
-
Fig. 11. A sketch of what a real self-consistent calculation
might yield. The full curve corresponds to a normal plasma
state. The dashed curves are possible solutions in the Bose
condensed state. In case (a), a first order Mott transition
-
occurs
Fig. 9.
-
The
ground state energy per particle as a function
of r, in reduced units, in the presence of
Bose condensation.
The full curve is the mean field result of I. The dotted curve
is the result of our procedure (36), in which we carry minimization with respect to v,. H is still given by the non selfconsistent form (37).
at finite pressure.
(ii) What is more significant is a comparison of
figures 8 and 9, which displays the effect of electronhole pairing within the same (band !) treatment of
screening. We see that pairing is definitely less efficient
in a screened theory, in agreement with the idea of
a
Mott transition. Such
10 : the parameter
figure
dency
to
a
trend is also apparent in
C, which measures the tenform bound pairs, is largely reduced in the
1096
presence of screening at intermediate densities. Near
the minimum the energy lowering due to electronhole pairing is of order 0.05 go, small but not negligible.
In order to assess the sign of errors, we should remember that screening and pairing oppose each other.
On the average, our crude procedure certainly overestimates screening, which would imply that binding
corrections are underestimated. However, our arbitrary reduction of the Fermi Thomas qt by 1/4,
supposed to account for dynamic corrections does not
make much sense in dealing with weakly bound
electron-hole pairs (the relevant interactions then
involve low frequencies, and static screening is a
good approximation). Out of these two conflicting
errors, which one wins is not clear. Our guess is that
but that needs
the order of magnitude is all right
proof via an honest variational calculation. Note
that our estimate of pairing effects is definitely larger
than the value quoted by Brinkman and Rice [15].
What they calculate is the binding energy w of a
« Cooper » bound state in a frozen Fermi sea, using
as we do a Coulomb interaction reduced by free
carrier static screening. It is known that such an
energy is much smaller than the BCS gap d (the
exponent is twice as large) : our result should be
compared with their co, not with co’/EF’ Nevertheless,
their result is lower than ours by an order of magnitude, probably because they use the bare Fermi
Thomas qD, while we reduce it by 1/4 (exponentials
very quickly !). Anyhow, the result is very sensitive to
the nature of screening and to self consistency : both
estimates are rather crude, and one cannot really
trust them.
-
Altogether, electron-hole pairing, although less
dramatic than in I, remains significant : a joint treatment of screening and pairing seems to be needed.
That is in fact the main message of our crude numerical
analysis, which does not claim any accuracy. A bona
fide self-consistent variational calculation is- indeed
possible, along the lines described before. If the
results are to be trusted, one should not use any
uncontrollable trick as we did here : just plain numerical minimization. The computing work is heavy,
but it looks feasible (the advantage of such a formulation is that physical features may be introduced one
at a time : screening only, binding only, or both). Such
a calculation should answer the following questions
unambiguously:
Is there a first order Mott transition ?
Is the minimum of E/N below (- go) ?
How large are pairing effects at metallic densities ?
-
-
-
Since this approach is (i) variational, (ii) exact in
both limits of small and large N, it seems reliable
enough to settle present arguments.
4. Band structure effects.
Until now, we only
considered a simple model with isotropic, non degenerate bands. In more realistic band structures, our
-
results would be completely modified. Some features
are relatively minor. One may envisage for instance
stable polyexciton complexes, exploiting bonding in
all possible channels. Strictly speaking, they would be
the stable ground state at very small N - however,
their radius is probably large and they rapidly dissociate. The real changes originate in two ways. First of
all, degeneracies and anisotropies lower the energy of
the plasma phabe, which penally goes below the single
exciton level [10] : Ge and Si are the standard examples.
Liquid droplets then form spontaneously at zero
pressure. Moreover, these same features hinder electron-hole pairing at large densities, due to the mismatch
oj }’ermi surfaces. As a result Bose condensation will
only occur at low density N; it will disappear at a
critical Nc beyond which the ground state is a normal
metallic plasma. If N c lies inside the first order phase
separation curve, one has a genuine Mott transition,
between a dilute insulating phase and a conducting
dense liquid. If moreover the vapour pressure is zero
at T
0, Bose condensation is not observable at all
(the B.E. critical temperature, - N’I’, lies inside the
phase separation curve). That puts severe restrictions
on the observability of Bose condensed excitons :
they can exist only if liquid droplets are not stable
- a situation which seems quite rare.
Putting aside the question of phase separation, we
may follow the evolution of the ground state as a
function of N. At low densities, band structure effects
are not drastic : they preclude neither the formation
of bound excitons, nor their Bose condensation.
However, when N increases, they rapidly destroy
pairing : the ground state returns to the normal
RPA result much earlier than in the preceding section
(thereby enhancing first order phase transitions). We
already mentioned in the introduction the effect
of anisotropies. We now add a few comments on the
case of degenerate bands. Such a degeneracy may
arise from an internal structure of the atomic states
(s-bands, p-bands, etc...). In indirect gap materials,
it can also arise as a multivalley structure in either of
the bands.
=
In a dilute gas, electrons and holes will still form
bound states
degeneracies will show up as internal
of
of the resulting bosons. Bose
freedom
degrees
condensation still occurs; it implies coherence of the
internal degrees of freedom of all condensed pairs.
Such coherence is known to lead to interesting physical effects. The standard example is superfluid 3He
(recently, Siggia and Ruckenstein [16] also discussed
properties of Bose condensed atomic hydrogen with
a nuclear spin internal degree of freedom). Similar
effects occur for excitons. For instance, in optically
active direct gap materials, condensed excitons have
aligned electric dipole moments. In indirect gap
materials, intervalley phase coherence may lead to
structural transitions triggered by Bose condensation.
These effects will be discussed elsewhere.
In the opposite dense case, Bose condensation will
-
1097
arise only if the Fermi surfaces match, i.e. if the two
bands have the same degeneracy (the spin structure of
the exciton was one such example). If the degeneracies
are different, the system returns to a normal plasma
state, in such a way as to minimize its kinetics energy.
In such a normal state, there is no symmetry breaking
whatsoever, in contrast to the Bose condensed dilute
gas, which breaks both the gauge symmetry (ordinary
phase locking), and some sort of rotational symmetry
related to the internal structure of the excitons. For
1 = 1 active excitons, for instance, the condensed
state has a definite polarization in a given direction
(whether linear, circular or elliptic). In contrast, in
the normal electron-hole plasma, all angular momentum states are equally populated and there is no
preferred direction. It follows that the transition
between a dilute condensed state and a dense normal
plasma cannot be smooth.
The Kjeldysh Kozlov wave function used
throughout these two papers always pairs one electron
with one hole : the corresponding manifolds must have
either because degeneracies are
the same dimension
identical, or because we retain only part of the states.
In the former case, our formulation applies at arbitrary
density (I .po &#x3E; factorizes as in section 2). In the latter
case, we can only describe polarized states (involving
for instance pairing of an s-electron with a p,,-hole).
The Kjeldysh Kozlov wave function cannot describe
the very simple state in which normal electrons and
holes have different Fermi surfaces.
In the absence of a single wave function that describes properly the two limits, a discussion of the
ground state as a function of density is much more
difficult. We can of course consider separately
-
(i) The polarized condensed states, described within
Kjeldysh Kozlov formulation;
(ii) The normal unpolarized plasma, described within
a standard Hartree Fock (or RPA) language.
In figure 12, we sketch the corresponding E(V)
curves in the unlikely event that droplets are unstable.
The curves necessarily cross somewhere : consequently,
we should observe a first order liquid gas phase
separation, characterized by the double tangent AB
in the figure. The dense «liquid phase should be a
normal plasma, while the dilute « gas » should be
Bose condensed, with a breakdown of internal
a
rotational symmetry.
Such an argument, however, is not conclusive :
by considering only the two extreme cases, we have
really produced the first order transition by hand !
One may well imagine a continuous evolution, in
which a polarized condensate shrinks progressively,
while a normal population of the other internal
states builds up. Unfortunately, the Kjeldysh wave
function cannot describe such an intermediate situation : the problem remains completely open. In our
Fig. 12. A sketch of the (E, V ) curve for electrons and holes
with different degeneracies. The full curve is the normal
plasma with maximal degeneracy (different Fermi surfaces
for electrons and holes). The dashed curve refers to Bose
condensed polarized excitons. The double tangent AB
signals a liquid-gas phase separation between a normal
liquid and a Bose condensed gas.
-
opinion, a first order transition is plausible but we
cannot prove it. What is missing here is a language
in which to describe the transition, in contrast to non
degenerate systems for which the difficulties were
mostly numerical.
-
5. Conclusion.
Our discussion has been mostly
tried to set up a language both
methodological :
in
and
reliable
order to formulate the interplay
simple
of screening and pairing. As in the earlier work of
Silin [11] and Zimmermann [12], we interpolate
between dilute condensed excitons and the dense
plasma phase. All along, we emphasized the underlying
physics but the real numerical work still lies ahead.
It is worth the effort, both in the simple isotropic
model and in more realistic band structures.
A few words of caution are in order. First, one cannot
make real experimental predictions at this stage.
For instance, Bose condensation should give characteristic coherence effects, unusual luminescence
line shapes, etc... Before one looks for them, one
should make sure that it is not hidden by droplet
condensation.
Next, our calculation, based on the Kjeldysh
Kozlov language, is strictly limited to the ground state :
we cannot extend it to finite temperatures. Thus, we
can say nothing on the conjectured existence of two
transitions (a Mott transition and a liquid-gas separation between two conducting phases). Similarly,
thermal ionization of excitons, critical temperatures,
etc... are outside our scope. Recently, a number of
papers appeared that treat approximately the behaviour at finite T [17]. The problem is difficult, and
it is hard to assess the validity of the complicated
machinery that is used. Here we take the opposite
stand ; in the simpler T 0 case, we try to control
a simple formulation and to extend it to more realistic
-
we
-
=
physical systems.
1098
References
[1] COMTE, C., NOZIÈRES, P., J. Physique, hereafter referred
as « I ». Equations of that paper will be denoted
[2]
[3]
as I...
See for instance the review article of RICE, T. M.,
Solid State Phys. 32 (1977) 1 (Academic Press).
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(1968) 521.
[4] ANDERSON, P. W., BRINKMAN, W. F., CHUI, S. T.,
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[5] KOHN, W., Phys. Rev. Lett. 19 (1967) 439.
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643.
There exist many excellent reviews on the subject,
for instance the article of RICE, T. M., Solid State
Phys. 32 (1977) 1 (Academic press). A recent
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NATO International Institute on Electron correlations in solids, Antwerp, July 1981 (to be
published).
[11] SILIN, A. P., Sov. Phys. Solid State 19 (1977) 77.
[12] ZIMMERMANN, R., Phys. Status Solidi 76 (1976) 191.
[13] See for instance COMBESCOT, M., NOZIÈRES, P., J.
Physique 32 (1971) 913.
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