Metastable electron-pair states in a two-dimensional crystal

Abstract

We study possible quantum states of two correlated electrons in a two-dimensional periodic potential and find a metastable energy band of electron pairs between the two lowest single-electron bands. These metastable states result from interplay of the electron-electron Coulomb interaction and the strength of the crystal potential. The paired electrons are bound in the same unit cell in relative coordinates with an average distance between them of approximately one third of the crystal period. Furthermore, we discuss how such electron pairs can possibly be stabilized in a many-electron system.

Full text

arXiv:1306.3849v1 [cond-mat.str-el] 17 Jun 2013
Metastable electron-pair states in a two-dimensional crystal
G.-Q. Hai1∗and L. K. Castelano2
1Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo, 13560-970, S˜ao Carlos, SP, Brazil and
2Departamento de F´ısica, Universidade Federal de S˜ao Carlos, 13565-905, S˜ao Carlos, SP, Brazil
We study possible quantum states of two correlated electrons in a two-dimensional crystal and
find a metastable energy band of the electron pair between the two lowest single-electron bands.
These metastable states result from the interplay of electron correlations and the strength of the
crystal potential. The two paired electrons are bound in the same unit cell in relative coordinates
with an average distance between them about one third of the crystal period. Furthermore, we show
how such electron-pairs can be stabilized in a many-electron system.
From the experimental observations of very small co-
herence lengths in hight Tcsuperconductors Friedberg
and Lee1proposed the s-channel theory. In this theory
of the boson-fermion model, they introduced phenomeno-
logically a local boson field φrepresenting a pair state
which is localized in coordinate space. Each individual φ
quantum is assumed to be unstable with an excitation en-
ergy 2ν. They showed that this assumption makes it pos-
sible for the s-channel theory to exhibit many supercon-
ductivity characteristics. At temperature T < Tcthere
is always a macroscopic distribution of zero-momentum
bosons coexisting with a Fermi distribution of electrons
(or holes). For instance, when the Fermi energy ǫFis
equal to νat zero temperature, the decay φ→2ecannot
take place. Therefore the bosons are present. Even for
ǫF< ν there is still a macroscopic number of (virtual)
zero-momentum bosons in the form of a static coherent
field amplitude whose source is the fermion pairs.1
As a matter of fact, the boson-fermion model has been
extensively studied in unconventional superconductivity.
The theory proposed in Refs. 2,3 consists of a mixture of
bipolarons (bosons) and itinerant electrons (fermions).
The bipolarons can spontaneously decay into itinerant
electrons and vice versa. However, recent experimen-
tal study on disentangling the electronic and phononic
contributions to the bosonic excitations in high Tcsu-
perconductor does not support the polaronic formation
of electron pairs.4It demonstrates though that bosonic
excitations of electronic origin are responsible for the for-
mation of the superconducting state in the cuprates.
The mechanism of electron paring and the nature of
the bosonic excitations are still key issues for understand-
ing unconventional superconductivity. In this work, we
study the quantum states of two correlated electrons in
a two-dimensional (2D) crystal. We find a metastable
energy band of electron pairs in this system. In contrast
to the single-electron bands, which are stable at any elec-
tron densities and can be easily detected experimentally,
these electron-pair states are metastable in the absence
of other electrons in the crystal. Considering many-body
effects, however, the electron pairs could be stabilized
∗E-mail: hai@ifsc.usp.br
due to many-particle interactions. From our results, the
unstable bosonic φquantum introduced in the s-channel
theory1can be understood as the metastable electron
pair obtained in the present work. The electron-pair-
hole interaction is responsible for the renormalization of
this unstable individual φquantum.
We consider a 2D crystal with a potential V(x, y) =
V0[cos(qx) + cos(qy)], where q= 2π/λ,λis the period,
and V0is the amplitude of the crystal potential. The
single-electron states are well known in this potential.
The Schr¨odinger equation for a single electron is given by
H0ψk+Gl(r) = El,kψk+Gl(r), with H0=−∇2+V(x, y),
where kis the wavevector in the first Brillouin zone, l
is the band index, and Gl=lxqi+lyqj(with lx, ly=
0,±1,±2, ...) is the reciprocal-lattice vector; El,kand
ψk+Gl(r) are the eigenvalue and eigenfunction, respec-
tively. Here the length and energy are measured in units
of the effective Bohr radius aBand effective Rydberg
Ry= ¯h2/2mea2
B, respectively. When we consider two
electrons in this periodic potential, their Hamiltonian
is given by H=H0(r1) + H0(r2) + 2/|r1−r2|. Be-
fore solving the corresponding Schr¨odinger equation, we
should bear in mind that, due to electron-electron re-
pulsion, the ground state of this system can be found
for |r1−r2| → ∞. In other words, the ground state
of two electrons in this system corresponds to two non-
interacting single particles separated by an infinitely long
distance. However, in a real system the electron den-
sity is not zero and, consequently, the distance between
two electrons is finite. Therefore, we intend to solve the
relevant equation for two electrons as a function of the
distance between them. The idea is to try first to find
any possible metastable states of electron pairs due to
the competition between the electron-electron interaction
and the 2D crystal lattice potential. Then we will study
the stability of the electron pairs.
We now introduce the center of mass and relative coor-
dinates, R=1
2(r1+r2) = (X, Y ) and r=r1−r2= (x, y),
respectively. The two-electron Hamiltonian becomes
H=−1
2∇2
R−2∇2
r+2
r
+ 2V0[cos(qX) cos(qx/2) + cos(qY ) cos(qy/2)] .(1)
This Hamiltonian is periodic in Xand Ywith period
λ. We can choose a Bloch wavefunction in the center-

2
of-mass coordinates. As to the function in the relative
coordinates r, we have to consider the symmetry of the
electron-electron Coulomb potential and the periodic po-
tential representing a 2D square lattice. We use the fol-
lowing basis for our trial wavefunction,
ψlx,ly;n,m(R,r) = 1
√Aei(k+Gl)·RRn,m(r)φm(θ),(2)
with
Rn,m(r) = βcn,m (2βξnr)me−β ξnrL2m
n−m(2βξnr),(3)
and
φm(θ) = 1
√bmπcos(mθ),(4)
where n= 0,1,2,···,m= 0,1,2,···, n,ξn= 2/(2n+ 1),
cn,m =2ξ3
n(n−m)!/(n+m)!1/2,b0= 2, bm= 1 for
m≥1, and L2m
n−m(x) is the generalized Laguerre poly-
nomial. The function Rn,m (r) is taken from the wave-
function of a 2D hydrogen atom5with a modification
introduced by a dimensionless scaling parameter β. It
satisfies the following orthogonality relation6,
Z∞
0
rdrRn′,m(r)Rn,m(r) = δn,n′.(5)
The two-electron wavefunction can be written as
Ψk(R,r) = X
lx,lyX
n,m
alx,ly;n,m(k)ψlx,ly;n,m (R,r).(6)
Considering the antisymmetry of the electron wavefunc-
tions with spin states, we find that the two-electron wave-
functions of the singlet and triplet states are given by the
above expression with the sum over even mand odd m
only, respectively. Here we are especially interested in
the singlet states.
Using Eqs. (1), (2), and (6) we obtain the matrix
eigenvalue equation
X
l′
x,l′
yX
n′,m′hlx, ly;n, m|H|l′
x, l′
y;n′, m′ial′
x,l′
y;n′,m′(k)
=Epair
kalx,ly;n,m(k).(7)
We solve Eq. (7) as a function of βlooking for local min-
ima in the lowest eigenvalue. The parameter βplays an
important role when solving this equation because it de-
termines the average distance between the two electrons
hri. For fixed βwe can obtain a full set of eigenvalues and
eigenfunctions. Consequently, the average distance hri
between the two electrons can be calculated. The eigen-
value of Eq. (7) has to converge upon increasing the size
of the matrix up to a maximum n=nmax and maximum
lx=ly=lmax. We find that, for fixed period λ, the low-
est eigenvalue of the two coupled electrons in the 2D peri-
odic potential develops a local minimum when V0is larger
than a certain value. Figure 1(a) shows the two lowest
0.0 0.5 1.0 1.5 2.0 2.5 3.0
-15
-10
-5
0
(a)
Energy (R
y
)
= 1.5 a
B
V
0
=12 R
y
V
0
=15 R
y
V
0
=18 R
y
0.6 0.8 1.0 1.2 1.4 1.6
0.5
1.0
1.5
2.0
2.5
(b)
Average distance <r> (a
B
)
01234
0.0
0.5
1.0
1.5
(c)
=0.5 <r>=1.3397 a
B
=0.9084 <r>=0.5474 a
B
=1.1885 <r>=0.4427 a
B
=1.5 <r>=0.9266 a
B
Radial probability density (a
-1
B
)
r (a
B
)
1 2 3 4 5
1
10
(d)
pair
V
0
(R
y
)
(a
B
)
X
M
FIG. 1: (a) The two lowest eigenvalues at k=0 as a function
of βfor λ= 1.5 aBand V0= 12,15, and 18 Ry. The solid
(dotted) curves indicate the lowest (second lowest) eigenvalue.
(b) The average distance hribetween the two electrons as a
function of βfor k=0, λ= 1.5 aBand V0= 15 Ry. (c) The
electron radial probability densities in relative coordinates for
different βgiven in the inset (the corresponding hriare shown
in (b) by the dots). (d) A diagram in the (V0,λ) plane showing
where a metastable state of an electron pair exists. Γ, Xand
Mindicate different points in the first Brillouin zone.
eigenvalues of spin singlet states at k=0 (the Γ point) as
a function of βfor λ= 1.5 aBand V0= 12,15, and 18
Ry. These curves converge at nmax = 8 and lmax = 4
within an error of 10−3. We have checked the calcula-

3
tions with nmax = 18 and lmax = 6 and the obtained re-
sults do not basically change. For small βthe eigenvalues
are almost zero because the two electrons are not bound.
Upon increasing βa local minimum is found in the lowest
eigenvalue. For large β, beyond a local maximum eigen-
value, the eigenvalue decreases as expected because the
ground state of the system is the single-particle state for
r→ ∞. We have checked the calculation for very large
β. In this case the lowest eigenvalue approaches an en-
ergy value twice the eigenvalue of a single-electron plus
the Coulomb repulsion 2/hri. We now consider the case
of λ= 1.5 aBand V0= 15 Ryfor a more detailed anal-
ysis. In Fig. 1(b), we plot hrias a function of βat k=0
around the local minimum eigenvalue (1.70 > β > 0.48).
The minimum of the lowest eigenvalue E=-8.35716 Ryis
found at β=β0= 0.9084 with hri0=0.5474 aB. A min-
imum average distance is found at β= 1.1885 with value
hrimin = 0.4427 aB. Figure 1(c) shows the corresponding
electron radial probability density in relative coordinates
for k=0 and β= 0.5,0.9084,1.1885, and 1.5.
These results show that there exists a metastable state
of an electron pair with energy equal to the minimum
eigenvalue found at β0. The two electrons are localized
in the same unit cell in relative coordinates and the av-
erage distance between them hri0is about one third of
the period λ. For λ= 1.5 aBwe find hri0=0.5474 aB.
Starting from β0and decreasing β, the radial probability
density becomes broader and the corresponding eigen-
value increases. On the contrary, with increasing βfrom
β0,hrifirst decreases until a minimum value hrimin is
reached, where we obtain the narrowest radial probabil-
ity density and the Coulomb repulsion between the two
electrons is greatly enhanced. The eigenvalue of the elec-
tron pair increases as well. As βincreases further, one of
the electrons is pushed into the nearest-neighbour unit
cells as it can be deduced from Fig. 1(c). In this case a
local maximum in the lowest eigenvalue appears where
the two lowest eigenvalues approach each other.
We understand this metastable state as a manifesta-
tion in the periodic potential of the electron-pair states
5
10
Single-electron band
X
-4.18
-4.16
-4.14
Single-electron band
Electron-pair band
Energy per electron (R
y
)
-10
-9
FIG. 2: Dispersion relations of the electron-pair and single-
electron states for λ= 1.5 aBand V0= 15 Ry.
existing in some individual atoms or ions,7,8 , such as the
negative hydrogen ion H−. It is the result of strong
electron-electron correlations and local confinement in
each unit cell of the 2D crystal potential. The metastable
state appears when the potential amplitude V0is larger
than a certain value for fixed period λ. In Fig. 1(d) we
show in the (V0, λ) plane where this metastable electron
pair can appear. We find that at the M points (kx=±q/2,
ky=±q/2) this local minimum appears at smaller V0(λ)
than at the Γ point for a fixed λ(V0). It means that a
pair of short wavelengths is easier to form than that of
long wavelengths in the crystal. For smaller V0(or λ),
this metastable state cannot survive when the Coulomb
repulsion overcomes the electron-electron correlation and
local electron confinement from the 2D potential.
0 5 10 15 20 25 30
-30
-20
-10
0
(a)
Energy per electron (R
y
)
0 5 10 15 20 25 30
2
4
6
(b)
E
g
(R
y
)
V
0
(R
y
)
=1
=1.5
=2
=3
=4
=5
FIG. 3: Dependence of (a) Epair
Γ/2 (solid curves) and Esingle
M
(dashed curves) and (b) the energy gap Egon V0for different
λ. The horizontal dotted line in (b) indicates the minimum
band gap renormalization |∆|= 2.2 Ryfor a low-density sys-
tem.
Fig. 2 shows the dispersion relation of this metastable
pair, for λ= 1.5 aBand V0= 15 Ry, together with
the two lowest single-electron bands. In order to better
compare with the single-electron energy, the energy of
the electron-pair is given by its value divided by 2, i.e.,
the energy per electron. The electron-pair band remains
above the lowest single-electron band. Their difference
will be shown in the next figure.
In Fig. 3(a) we plot the energy per electron of the
metastable pair at the Γ point (Epair
Γ/2) as a function
of V0for different λ. We also show the maximum energy

4
Esingle
Mat the Mpoint of the lowest single-electron band.
We define the difference between these two energies as
an energy gap, Eg=Epair
Γ/2−Esingle
M, and plot it in
Fig. 3(b).
So far we have obtained the single-particle states in
the system as shown in Fig. 2, i.e., the single-electron
(fermion) and single electron-pair (boson) states. In the
following, we will consider the case in which many elec-
trons are presented in the system. Now the 2D periodic
potential should be understood as an effective potential
of a 2D square lattice crystal. We further assume the
lowest single-electron band as the valence band with an
electron filling factor νe. For νe= 1 there are two elec-
trons per unit cell on average in real space. It is reason-
able to assume that the system is neutral in this case.
For νe<1 we can understand that holes are presented
in the valence band with a filling factor νh= 1 −νe.
When there are electrons in the electron-pair band, the
electron pairs in the metastable states will attract holes.
Notice that our electron pair is formed by two strongly
correlated electrons and, therefore, it does not recombine
directly with holes. The Hamiltonian of a many-particle
system consisting of electron pairs in the Γ valley of the
electron-pair band and holes in the Mvalleys of the va-
lence band can be written as
e
H=X
k
[Epair
k+ 2Eg]ˆnp,k+X
k,σ
Ehole
k,σ ˆnh,k,σ
+1
2AX
q
vp−p(q)[ρ†
q,pρq,p −Np]
+1
2AX
q
vh−h(q)[ρ†
q,hρq,−Nh] + Hp−h,(8)
where ˆnp,k(ˆnh,k,σ) is the particle number operator
for the electron-pair (hole), σis the spin, vp−p(q)
(vh−h(q)) is the pair-pair (hole-hole) interaction poten-
tial, and Np(Nh) is the electron-pair (hole) number;
ρq,p =Pk2a†
kak+q(ρ†
q,p =Pk2a†
k+qak) and ρq,h =
Pk,σ c†
k,σck+q,σ (ρ†
q,h =Pk,σ c†
k+q,σck,σ) are the density
operators, the operators a†and a(c†and c) are creation
and annihilation operators for the electron-pairs (holes),
respectively, and Hp−his the pair-hole interaction. In
this many-body boson-fermion system, the energies of
an electron pair and a hole are given by Epair
k+ Σpand
Ehole
k,σ + Σh, respectively. The many-body corrections are
included in the self-energies Σpand Σh. Many-particle
interactions renormalize the total energy of the system
and also reduce the energy gap from Egto Eg− |∆|.
For low densities the self-energies are small. The in-
teractions between electron pairs and holes are dominant
and lead to the formation of exciton-type states.9The
most probable scenario should be an electron pair bound
to two holes to form a biexciton (X2). Another important
detail is that our system is of a multi-valley valence band.
This can introduce changes in the nature of the biexciton
states and permit polyexcitons Xnwith n > 2.9In two
dimensions, the energy of an exciton is approximately
EX
b=−4(µ/me)Ry, where µ=memh/(me+mh) is the
reduced mass. For me=mhwe have EX
b=−2Ry. The
energy of a biexciton is EX2
b= 2EX
b+∆EX2
b, where ∆EX2
b
is the binding energy of the biexciton relative to two free
excitons. In a 2D system this binding energy has a value
about 20% of the exciton energy EX
b.10,11 Therefore, we
can estimate the reduction of the energy gap by a quan-
tity ∆ = EX2
b/2 due to biexciton effects. For me=mh
we have ∆ ≃2.2EX
b/2 = −2.2Ry. In this way the elec-
tron pair can be stabilized for Eg<|∆|= 2.2Ryas
indicated in Fig. 3(b). This should be the minimum re-
duction of the electron-pair energy. Since the two paired
electrons here are bound together, this should enhance
the electron-pair-hole attractions. Possible polyexcitons
in a multi-valley band system can also further reduce the
electron-pair energy for low densities.
For high densities the system is an electron-pair and
hole plasma. The electron pairs and holes cannot bind
because of strong screening of many-particle nature. In
this case band-gap renormalization (BGR) due to pair-
pair and hole-hole interactions are essential in stabilizing
the electron pairs. BGR has been extensively studied in
nonlinear optics of semiconductors, where one deals with
an electron-hole plasma (EHP). In an EHP the BGR
is given by the sum of electron and hole self-energies
and is a function of the interparticle distance rsand
temperature.12,13 At low densities electron-hole interac-
tion leads to formation of excitons. With increasing par-
ticle density the BGR increases and can be several times
larger that the exciton binding energy. In a 2D EHP at
zero temperature, for instance, the BGR is about 8Ryat
rs= 1.12. For a high density EHP the BGR is mainly
induced by the Coulomb repulsion among the particles
(the Coulomb hole effect). Therefore, we believe that
the BGR in our present electron-pair-hole system should
be of similar behaviour, i.e., the BGR will be enhanced
at high densities and can be a few times larger than the
biexciton energy.
On the other hand, 2D charged boson fluids have been
studied over the last few decades14 with artificially in-
troduced bosons considered as point charges. From the
electron-pair states obtained in this work, we can cal-
culate the pair-pair and pair-hole interaction potentials.
Consequently, one can study the ground-state properties
of the present narrow-gap and multi-valley boson-fermion
system with electron pairs and holes. A progress is that
now the bosons (i.e., the electron pairs) are obtained from
the crystal band structure and they are not point charges.
In conclusion, we have obtained an electron-pair en-
ergy band in a two-dimensional crystal. The electron-pair
states are metastable in the absence of other electrons in
the system. The two correlated electrons are bound in
the same unit cell in relative coordinates with an average
separation about 1/3 of the period λof the crystal po-
tential. Furthermore, we have discussed how the electron
pairs can be stabilized in a many-particle boson-fermion
system with electron pairs and holes. The present calcu-

5
lations can be carried out for a three-dimensional system.
Acknowledgments
This work was supported by FAPESP and CNPq
(Brazil). GQH thanks A. Bruno-Alfonso for checking the
numerical calculation and discussion and P. Vasilopoulos
for critical reading of the manuscript.
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