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On the problem of the spontaneous exchange­driven electron interwell re­population in
semiconductor quantum wells
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1999 Semicond. Sci. Technol. 14 852
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Semicond. Sci. Technol. 14 (1999) 852--856. Printed in the UK PII: S0268­1242(99)03819­5
On the problem of the spontaneous
exchange­driven electron interwell
re­population in semiconductor
quantum wells
A V Dmitriev
Department of Low Temperature Physics, Faculty of Physics,
M V Lomonosov Moscow State University, Moscow 119899, Russia
E­mail: dmitriev@lt.phys.msu.su
Received 28 April 1999, accepted for publication 8 June 1999
Abstract. We reconsider the problem of the so­called exchange instability in symmetric
semiconductor quantum wells, that is, an earlier proposed mechanism of a spontaneous
interwell electron re­population due to different carrier density dependence of the Hartree and
exchange energies. We show that the instability presents only in straightforward
Hartree--Fock calculations and disappears as one takes into account interwell electron
correlations.
1. Introduction
The study of phase transitions in many­particle systems
with Coulomb interaction has a long history. One can
mention the spontaneous spin polarization transition in
a low­density free electron gas proposed by Bloch [1],
the Wigner crystallization in a one­component electron
gas [2], the Peierls metal--insulator transition in a crystal
[3], the Kosterlitz--Thouless transition in a two­component
plasmas [4], the transition to an exciton dielectric phase
in a semiconductor having similar electron and hole Fermi
surfaces predicted by Keldysh and Kopaev [5] etc. All
these works treated an ordinary bulk 3D situation. In
the 1980s, when systems with reduced dimension attracted
attention, a new electron transition was suggested that could
take place in a semiconductor quantum well system [6, 7].
Using the Hartree--Fock approximation for the treatment of
electron--electron interaction, the authors of the articles [6, 7]
argued that due to different carrier density dependence of
the Hartree and exchange energies, an interwell electron
redistribution might take place under conditions of low
carrier density, leading to a spontaneous breaking of electro­
neutrality in the system.
The arguments used in [6, 7] were quite simple. Let us
consider a symmetrical double­well system with electrons
populating only the lowest size­quantized levels, at zero
temperature. In the Hartree--Fock approximation, the
ground­state energy of electrons in the wells is equal to
E = K + EHartree - E exchange (1)
where K is the kinetic energy of the electrons. The three
terms in the right­hand part have different n dependence,
n being the 2D electron density in the well. The kinetic
energy of a degenerate 2D electron gas is proportional to
n 2 . The Hartree energy is simply the Coulomb interaction
energy under the approximation of uniform electron charge
distribution, that is, the ordinary electrostatic energy of the
system under consideration. If one uses the jelly model
for the positive background then the Hartree energy equals
zero in the electro­neutral state when electrons are equally
distributed between the wells+. If the neutrality fails due to
carrier redistribution between the wells, then
EHartree # (#n) 2
where #n is a difference between the electron densities in
the wells, because EHartree is equal simply to the energy of
a plane capacitor constituted by two wells.
The last term in equation (1), the exchange energy, is
proportional to n 3/2 for 2D degenerate electron gas [8, 9].
So at a sufficiently low n the negative exchange energy
increases more rapidly with n than other, positive members
of equation (1). This means that under such conditions the
electrons must move to one well, thus ensuring maximum
carrier density there. The corresponding loss in the kinetic
and electrostatic energies is overwhelmed by the gain in the
exchange energy, so the total energy (1) decreases. This
transition is sometimes called the exchange instability.
This simple picture, however, brings about questions.
The most fundamental one is the following. It is clear from
+ One can consider this as a definition of zero energy. It means that EHartree
includes not only the electron--electron interaction energy, but also the energy
of the electron interaction with the background and the electrostatic energy
of the background itself.
0268­1242/99/090852+05$30.00 © 1999 IOP Publishing Ltd

Exchange instability in semiconductor quantum wells
above that the proposed exchange instability is an entirely
electron phenomenon that does not include any change in
the symmetrical positive charge distribution. But electrons
interact via repulsive forces. What is then the source of the
effective attraction between the electrons, which makes them
move from two wells into one, breaking electro­neutrality?
Really, the exchange instability is equivalent to the presence
of an attraction between two negatively charged electron
planes in the wells, but there is no source of such attraction
in the system. The only real attractive force, that between
electrons and ions, acts just in the opposite direction, trying
to maintain the electro­neutrality and to equalize the electron
populations in the wells.
We will show below that the answer to this question is
that the approximations made during the derivation of the
expression (1) in [6, 7] were too crude, and if one treats the
electron--electron interaction more accurately, the exchange
instability disappears. The weak point of equation (1) is that
it completely neglects interlayer electron correlation energy
and treats electrons in two wells as two identical separate
systems, which is incorrect. The interlayer correlations
decrease the energy of the state with equal electron population
in the wells, making this state stable.
Our results support the conclusion of [10] where the
authors also argued against the exchange instability from
the point of view of the Hartree--Fock approximation. Our
approach, however, is more general as we do not use any
approximation in treating the electron--electron interaction.
2. Calculations
Let us consider a quantum well system similar to one used
in [6], that is, a symmetrical double quantum well system
where electrons occupy only the lowest size­quantized levels
in the wells (even when all particles move to a single well),
at zero temperature. We neglect an overlap of electron
wave functions in different wells because interwell tunneling
plays no role in the proposed mechanism of the exchange
instability. Then one can treat the electron degree of freedom
that corresponds to the direction perpendicular to the layers in
a site representation with a site index, i, that can take only two
values, 1 and 2, because the system contains two wells. For
example, a one­electron wave function of a particle localized,
say, in the well number 1, in this representation has the form
# # (#, i) = # # (#)# 1i (2)
where # =#,# is a spin quantum number, # is an in­plane
2D electron coordinate vector, # # is a 2D wave function in
the plane of the well and # ij is the Kronecker delta. The
wave function for an electron in the second well has a similar
form.
The electron Hamiltonian of the system in this
representation can be written as
“
H =
N
X k=1
“
p 2
k
2m
+ U = “
K + U (3)
where N is the number of electrons in the system, “
p k is the
2D in­plane momentum of the kth electron, “
K is the kinetic
energy, U is the electron--electron interaction energy
U =
1
2
N
X k=1
N
X n=1
{u 11 (# k - # n )# i k i n
+ u 12 (# k - # n )[1 -# i k i n
]}
(4)
where i k = 1, 2 is the site index of the kth electron, # k is its
2D coordinate vector and u 11 and u 12 are intra­ and interwell
two­particle electron--electron interaction energies. In the
simplest case of infinitely thin wells
u 11 (#) =
e 2
# 0 |#|
u 12 (#) =
e 2
# 0 p # 2 + d 2
(5)
where d is an interwell distance (the barrier width) and # 0
is the dielectric constant of a medium surrounding the wells.
If the well width is finite, the simple expressions (5) are no
longer valid, but in any case
u 11 (#) > u 12 (#) (6)
and this is the only their property which is important for us.
We assume that members with k = n responsible for
electron self­energy are omitted from the sum in the right­
hand part of equation (4) and from similar sums below.
The electron interaction energy with the positive
background of the symmetrical double­well system is
symmetric in site index and hence it does not change on
carrier redistribution between the wells, so it is omitted in
equation (3) as well as other invariable terms such as the
electrostatic energy of the background itself. As we are
going only to compare energies of different electron states
in the system and not to calculate the absolute values of these
energies, we omit such constant terms from the Hamiltonian.
In other words, the one­electron energy in equation (3) is
measured from the level of size quantization. Taking the
Hamiltonian in the form (3) we neglect the boundary effects
at the edges of the layers, which is not important for bulk
effects such as the exchange instability.
Now our goal is to prove that for any electron state with
all electrons in one well one can build another state with
electrons equally populating both wells, which is lower in
energy. We will call the first state the single well state, and
the second one the symmetrical state. Then we will be able to
conclude that no single­well state can ever be the ground state
of the system in question and hence the proposed exchange
instability cannot take place in reality because it would lead
just to the formation of a single­well ground state.
The single­well state wave function that corresponds to
a state with all electrons in, say, well number 1, must have
the form
# sw = #(# 1 . . . #N )# 1i 1 . . . # 1i n
(7)
where # is a normalized 2D (in­plane) N ­electron wave
function that satisfies the antisymmetry condition at an
interchange of any two particles. # depends also on spin
variables of all particles but they are not explicitly shown
here because the Hamiltonian is spin independent and the
spins are not directly involved in the calculations.
It is easy to show why any N ­electron single­well
state wave function in our system has this form. Really,
one can expand the wave function in terms of the Slater
853

A V Dmitriev
determinants which describe different states of a non­
interacting electron system with all N particles in the first
well. Consequently, all determinants must include the one­
electron wave functions from the first well, every one of
which is given by equation (2). So each term in every Slater
determinant that enters the N ­electron single­well wave
function contains # 1i 1 # 1i 2 . . . # 1i N
as a common multiplier,
and thus one comes to the expression (7) for the wave function
as a whole.
Let us consider another N ­electron wave function
s =
1
2 N/2 #(# 1 . . . #N )(# 1i 1 + # 2i 1 ) . . . (# 1i N
+ # 2i N ) (8)
with the same # as in equation (7). The trial function (8)
describes a symmetrical state with equal electron populations
in both wells. Let us now compare mean energies in the states
sw and s .
It is clear that
# sw | “
K| sw # = # s | “
K| s # (9)
because the 2D kinetic energy operator “
K acts only on #. In
contrast, the Coulomb interaction energies in these two states
differ. Using equation (4) one obtains, on the one hand,
# sw |U | sw # =
2
X i 1 =1
. . .
2
X N =1
Z d# 1 . . .
. . . d#N U |#(# 1 . . . #N )| 2 # 1i 1 . . . # 1i N
=
1
2
N
X
k,n=1
Z d# 1 . . . d#N u 11 (# k - # n )|#(# 1 . . . #N )| 2 .
(10)
We assume here and later on that the summation over spin
variables is included in the integration over #s.
On the other hand,
# s | “
U | s # =
1
2
N
X
k,n=1
1
2 N
2
X
i 1 ...i N =1
Z d# 1 . . .
. . . d#N |#(# 1 . . . #N )| 2 (# 1i 1
+ # 2i 1 ) 2 . . .
. . . (# 1i N
+ # 2i N
) 2 [u 11 # i k n
+ u 12 (1 -# i k i n
)]
=
1
8
N
X
k,n=1
2
X
i k ,i n =1
Z d# 1 . . . d#N |#(# 1 . . . #N )| 2
â[u 11 # i k i n (# 1i k
+ # 2i k )(# 1i n
+ # 2i n )
+u 12 (1 -# k i n )(# 1i k
+ # 2i k )(# 1i n
+ # 2i n )]
= # sw | “
U | sw # -
1
4
N
X
k,n=1
Z d# 1 . . . d#N |#(# 1 . . . #N )| 2
â[u 11 (# k - # n ) - u 12 (# k - # n )]. (11)
Since for given # k and # n the intrawell interaction
u 11 is always greater than the interwell one u 12 (see
equations (5)--(6)), the integrating function in the second
term of the final expression is positive and hence the
integral is also positive. This means that the mean
electron--electron interaction energy #U# is less when
electrons are symmetrically distributed between the wells
than when they are all concentrated in a single well. As the
kinetic energies in both states coincide (see equation (9)), we
come to the conclusion that the energy of the single­well state
is always greater than that of the corresponding symmetrical
state with the same in­plane wave function.
If one takes a more general trial wave function
= #(# 1 . . . #N )[ # ## 1i 1
+ p ## 2i 1
] . . .
. . . [ # ## 1i N
+ p ## 2i N
]
# + # = 1 (12)
then it is easy to find that
# |U | # =
1
4
N
X
k,n=1
Z d# 1 . . . d#N |#| 2
â[(# 2 + # 2 )u 11 (# k - # n ) + 2##u 12 (# k - # n )]. (13)
The interaction energy (13) has a minimum at # = # = 1/2,
that is, just in the symmetrical state. The kinetic energy
corresponding to the wave function (12) does not depend on
# and #, so the full energy also has a minimum in this state.
The existence of the minimum is a direct consequence of the
fact that u 12 < u 11 , which holds for any non­zero interwell
distance.
3. Discussion
The results of the preceding section show that for any given
state of our system with all electrons in one well there is
another state with the symmetrical electron distribution over
wells and a lower energy. Hence the single­well state cannot
be the ground state of our system, which means that the
exchange instability cannot take place in it.
So a question arises of what is wrong with the arguments
of [6, 7] (see section 1) that lead to existence of the instability.
Our opinion is that the source of the error is in the neglect of
the interlayer correlation energy that decreases the full energy
of the symmetrical state as compared with its straightforward
Hartree--Fock estimate used in [6]. The energy of the
symmetrical state of the system has been calculated there
simply as twice the energy of a single well with N/2
electrons. This method of calculation is acceptable if the
interwell distance, d, is much greater than r n , the mean
interelectron distance in the well (r n # 1/ # n). In this
case the interlayer correlations are negligible because the
electric field produced by the electrons in one well becomes
almost uniform at the position of the other well. However, an
elementary estimate of different terms in equation (1) shows
that the exchange instability could take place only when the
electron density n is so low that r n # d or r n > d or, which
is the same, when two wells are so close to each other that
the interwell barrier width is of the order of or less than the
intercarrier distance in the wells. But at the interwell distance
d # r n the field of the electrons in one well is still non­
uniform in the other well, so there is no reason to neglect
the interwell correlations. If one still forgets about them and
describes electrons in both wells by identical wave functions,
then maxima and minima of the charge densities in the wells
will be in identical positions in the planes of two wells. If we
described this quantum mechanical picture in classical terms,
we would say that the electrons in different wells are situated
`just in front of each other' across the barrier.
To further clarify things, let us consider this situation in
a system with thin barrier d # r n and in the limit of low
854

Exchange instability in semiconductor quantum wells
electron density. It is well known that at a sufficiently low
density a one­component electron gas experiences the Wigner
transition to a crystalline phase [2, 11] where the particles
form a periodical lattice with one electron in each site. If
we now consider for example the 2D carrier concentration
in the wells to be so low that the Wigner crystallization
takes place in each of them, then within the simple approach
described above both 2D electron crystals in two layers would
be situated in a position with their sites just in front of each
other at a distance much less than the mean inter­electron
distance in the plane (d # r n ). So it becomes clear that when
one intends to treat electrons in different wells as identical and
independent systems, actually one unintentionally introduces
rigid interlayer correlations and does it in a way that is
unfavourable from the point of view of the electron--electron
interaction energy. In this example one could significantly
reduce this energy in two manners: first, putting all electrons
in one well; the mean interelectron distance then increases
because r n # d; and second, keeping the electrons in
two wells but changing the electron arrangement so that the
electrons in one well are situated in front of empty places in
the other.
The first way obviously leads to the single­well state, and
the result of the second method can be approximated by the
symmetrical wave function built as was shown above. Really,
if we could look at the electrons in the system along the
growth direction, we would notice that the electron structures
in the plane are similar in the symmetrical and single­well
states because both structures must simply coincide when
the barrier width tends to zero. As our symmetrical wave
function is just based on the electron in­plane arrangement
in the single­well state, it accounts for the main features of
the actual electron ordering in the symmetrical state as long
as the barrier is thin, that is, just in the area of our interest.
These two states where the electron--electron correla­
tions are properly taken into account are considerably closer
in energy to the ground state of our system than the state we
started from, that with the electrons in two wells treated as
independent. Of these two, the electro­neutrality is broken
in the single­well state so there is an electrostatic energy loss
in it as compared to the electro­neutral symmetrical state that
corresponds therefore to the minimum energy of all three
states.
If the electron density in the wells is now increased above
the threshold of the Wigner crystallization, there will be no
2D crystalline long range order any more. However, short
range correlations in the in­plane electron distribution will
survive, and so the above picture of the interwell correlations
still holds although it may not be so evident.
Up to now we have spoken only about the interaction
energy. Let us consider the electron kinetic energy in all
three states. Evidently, there is a kinetic energy increase
in the second and third states as compared to the first one
because the Fermi energy in the N ­particle gas is greater
than twice the Fermi energy in the N/2 one. However, this
effect is not important because at low electron density the
main factor is the interaction energy, otherwise no Wigner
crystallization could ever take place. Really, the kinetic
energy of a particle in the 2D electron gas is proportional
to n whereas its Coulomb energy to # n.
We remind the reader that during all the discussion above
we bore in mind the case of low carrier density and thin
barrier, which is important for the study of the exchange
instability. For thick barriers, the role of the interwell
correlations is negligible.
From this point of view the reason for the difference
between the results obtained in [6, 7] and those of the present
paper is clear. One has only to adapt the general picture
described above to the approximate Hartree--Fock scheme
used in [6, 7].
The use [6] of two identical N/2­electron Slater
determinants in two wells as an N ­electron wave function
allows us to partially take into account only the intralayer
electron correlations+ and the corresponding intralayer
exchange energy. However, the interlayer correlation energy,
another negative contribution to the full energy of the system,
is completely lost in this state.
Let us now turn to the state with all N electrons situated
within a single layer. Here all correlations are the intrawell
ones and so all them are taken into account (approximately, of
course) even within the Hartree--Fock method. As a result,
there is a definite gain in the electron--electron interaction
energy as compared to the preceding state. There is also
some energy loss due to breaking of the electro­neutrality
in the system but it is proportional to the barrier thickness,
d, and hence is small for thin barriers. In contrast, the
correlation energy gain is d independent. So evidently the
single­well state is more advantageous in the small d limit
than the previous one. This is just the result obtained in [6, 7].
Finally, our trial symmetrical wave function (8) based on
the single­well N ­electron in­plane wave function combines
advantages of both preceding variants. On the one hand, it
accounts for both intra­ and interwell correlations in a good
approximation because the actual electron ordering in the
two­well system with a thin barrier is quite similar to the
ordering in the N ­electron single­well state, as we tried to
show above. Actually, the symmetrical function accounts for
the correlations as well as the N ­electron single­well function
because both are based on one in­plane wave function. On
the other hand, in the symmetrical state there is no loss in
the electrostatic energy because the system keeps electro­
neutrality. This function better approximates the true ground
state wave function of our system than the other two functions
considered above, and it corresponds to the lowest energy
value among all three states.
So one can conclude that the arguments leading to
the exchange instability are based on the use of a poor
approximation for the wave function of the electro­neutral
state, which does not account for the interlayer electron
correlations. The use of the better approximation for the
function immediately removes the instability.
+ The approximate Hartree--Fock method does not take into account any
correlations caused by the Coulomb electron--electron interaction. However,
the method includes the quantum correlation effect known as the exchange
hole formation (see, for example, [12]), which follows from the Pauli
principle and hence exists even in the ideal gas of fermions, decreasing the
probability of finding two particles close to each other. In a gas of interacting
particles, these quantum correlations manifest themselves in the exchange
energy, which is hence a kind of correlation energy corresponding to the
quantum correlations.
855

A V Dmitriev
4. Conclusion
We considered here the old­standing problem of the so­called
exchange instability in symmetric quantum well systems
[6, 7]. We showed that if one accurately calculates the
electron--electron interaction energy with proper account of
the interwell correlations, there is no exchange instability.
Its appearance in the simple Hartree--Fock approach is
connected with the overestimation of the symmetrical state
energy due to complete neglect of the interwell correlations.
We did not use any approximation in our treatment of the
electron--electron interaction in the system.
Acknowledgments
The author is grateful to Professor P Thomas for hospitality
in Phillips­Universit˜ at Marburg, FRG, during the initial stage
of this work, and to Dr A G Mironov for useful discussions,
support and encouragement. The work was supported in part
by the Russian Foundation for Basic Research.
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