Hamiltonian of a many-electron system with single-electron and electron-pair states in a two-dimensional periodic potential

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by  G. -Q. Hai

Hamiltonian of a many-electron system with single-electron and electron-pair states in
a two-dimensional periodic potential
G.-Q. Hai1 and F. M. Peeters2
1Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo, 13560-970, S˜ao Carlos, SP, Brazil and
2Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, 2020 Antwerpen, Belgium
Based on the metastable electron-pair energy band in a two-dimensional (2D) periodic potential
5 obtainedpreviouslybyHaiandCastelano[J.Phys.: Condens. Matter26,115502(2014)],wepresent
1 inthisworkaHamiltonian ofmanyelectronsconsisting ofsingleelectronsandelectron pairsinthe
0 2D system. The electron-pair states are metastable of energies higher than those of the single-
2 electron states at low electron density. We assume two different scenarios for the single-electron
band. When it is considered as the lowest conduction band of a crystal, we compare the obtained
n
Hamiltonian with the phenomenological model Hamiltonian of a boson-fermion mixture proposed
a
by Friedberg and Lee [Phys. Rev. B 40, 6745 (1989)]. Single-electron-electron-pair and electron-
J
pair-electron-pairinteractiontermsappearinourHamiltonianandtheinteractionpotentialscanbe
0
determined from the electron-electron Coulomb interactions. When we consider the single-electron
2
bandasthehighestvalencebandofacrystal,weshowthatholesinthisvalencebandareimportant

for stabilization of theelectron-pair states in thesystem.
]
n
o PACSnumbers: 71.10.Li,71.10.-w,73.20.At,71.15.-m
c
-
r
The mechanism of electron pairing in unconventional teraction.
p
u superconductingmaterialsstillremainsasoneofthema-
s jorandunresolvedproblemsincondensedmatterphysics. We startwiththesingle-electronstatesina2Dsquare
.
t It is usually believed that the 2D characteristic (such lattice crystal. The crystalperiodic potential is givenby
a
m as a layered crystal structure of the cuprate supercon- Vc(x,y)= V0[cos(qx)+cos(qy)], where q =2π/λ with λ
ductors) and strong electron-electroncorrelationare im- the periodandV the amplitude ofthe crystalpotential.
0
d- portantfor unconventionalsuperconductivity.[1–3] From The Schr¨odinger equation for a single electron is given
n theexperimentalobservationoftheverysmallcoherence by H(r)ψn,k(r) = En,kψn,k(r) with the single-electron
o length in hight Tc superconductors Friedberg and Lee[4] HamiltonianH(r)= 2+Vc(x,y) inthe units ofeffec-
c (referred to as FL below) proposed a phenomenological tiveBohrradiusa a−nd∇effectiveRydbergR ,wherekis
B y
[
theory, the so-called s-channel theory. In this theory of the wavevector in the first Brillouin zone, n is the band
1 the boson-fermion model, they introduced phenomeno- index,En,kandψn,k(r)aretheeigenvalueandeigenfunc-
v logically a local boson field φ representing a pair state tion, respectively.
4 which is localized in coordinate space. Each individual
1 φ quantum is assumed to be unstable with an excita-
Whenwe considertwointeractingelectronsinthis pe-
9
4 tion energy 2ν. It has been shown that this assump- riodicpotential, the HamiltonianisgivenbyH(r1,r2)=
0 tionmakesitpossibleforthe s-channeltheoryto exhibit H(r1)+H(r2)+2/r1 r2 . Thegroundstateofthistwo-
. many superconducting characteristics[4–9]. This model electron system ca|n b−e fo|und for r r due to
1 has been studied both for three-dimensional (3D) and | 1 − 2| → ∞
theelectron-electronrepulsion. Itmeansthattheground
0
two-dimensional (2D) systems. As a matter of fact, the
5 state consistsoftwonon-interactingsingle electronssep-
boson-fermionmodelhas been extensivelystudied inthe
1 arated by an infinitely long distance. However, it was
: area of unconventional superconductivity.[4–13] It can found previouslythat metastable electron-pairstates[15]
v
be traced back before the BCS theory for conventional
may exist in this system. In such an electron-pair state,
i
X superconductivity.[10] two electrons of spin singlet stay in the same unit cell of
r The mechanism of electron pairing and the nature of the crystal lattice in the relative coordinates r=r r
1 2
a −
the bosonic excitations are still key issues for the under- with an average distance between them only about one
standing of unconventional superconductivity.[14] Based third of the lattice period. Because two electrons in the
on previously obtained metastable electron-pair states sameunitcellareofmaximumoverlap,theyarestrongly
in a 2D crystal[15], we present in this work a Hamilto- correlated. Strong electron-electron correlation reduces
nianofa2Delectronsystemconsistingofsingleelectrons greatlythe Coulombrepulsionbetween them. When the
(fermions) and electronpairs (bosons). This new Hamil- amplitude V of the crystal potential is larger than a
0
tonian obtained from electronic structure calculations of certain value, strong correlation between two electrons
a 2D crystalincludes single-electronstates, electron-pair togetherwiththecrystalpotentialmodulationleadstoa
states, as well as Coulomb interactions between single metastable electron-pair state. The electron-pair states
electrons and electron pairs. The interaction potentials inthecenterofmasscoordinatesR=(r +r )/2aregiven
1 2
presentintheHamiltoniancanbeobtainedfromtheelec- by Bloch wavefunctions and form energy bands. The
tronic structure calculations with electron Coulomb in- eigenenergiesofthe electronpairsaregivenbyEpair and
k
2
the wavefunctions can be written as tonian of the system can be written as,
N N N
1 2
H = H (r )+
0 i 2 r r
X XX i j
i=1 i=1 j=i | − |
6
NF 1 NF NF 2
Ψk(R,r)= e√ikA·R X alxly;nmeiGl·RRnm(r)co√sb(mmπθ), = Xi=1H0(ri)+ 2Xi=1Xj6=i |ri−rj|
lxly;nm
NB 2 1 2 2 2
+  H (r )+ 
0 α,ν 2 r r
X X XX α,ν α,µ
α=1 ν=1 ν=1µ=ν | − |
 6 
1 NB NB 2 2 2
+
2 r r
X XXX α,ν β,µ
where G = l qi + l qj (with l ,l = 0, 1, 2,...) is α=1β=αν=1µ=1| − |
l x y x y 6
the reciprocal-lattice vector, Rnm(r) (with±n=±0,1,2,..., NF NB 2 2
and m=0,2,..., n) is taken from the eigenfunction of a + , (1)
r r
XXX i α,ν
hydrogenatomwithascalingfactorofLaguerrefunction i=1α=1ν=1| − |
asgiveninRef.[15], b =2andb =1form 1. Itwas
foundthatthelowest0eigenvalueomfanelectron≥pairisspin where ri denotes single electron i, and rα,ν electron pair
α(withtwoelectronsν=1and2). Thefirsttwotermsin
singletandformsanenergyband. Intermsoftheenergy
the above Hamiltonian are N single electrons and the
per electron, this electron-pair band is found in between F
Coulombinteractionsbetweensingleelectrons. Thenext
the two lowest single-electron bands. The minimum of
two terms are N electron-pairsand the Coulomb inter-
theelectron-pairbandattheΓpointinthefirstBrillouin B
actions betweenthe pairs. The lastterm is the Coulomb
zone is higher than the maximum of the lowest single-
interactions between single electrons and electron pairs.
electron band (at the M point) in the studied square
Due to possible transitions: two single electrons ⇄ an
lattice potential in Ref.[15]. Their difference is defined
electron pair, N and N are not constants. Two single
as an energy gap E . F B
g electronscanformanelectronpairthroughtheCoulomb
interaction between them. In the opposite process, an
electron pair can decouple into two single electrons. A
In the presence of many electrons in the crystal, we
macroscopic number of electron pairs in the system de-
expect that for a certain electron density many-body in-
pendsontheelectronicbandstructureofthecrystal(de-
teractions may stabilize the electron-pair states leading
terminedbythecrystalpotential),thetotalelectronden-
to a macroscopic electron-pair density. Without loss of
sity and temperature, etc.. When there are no electron
generality, we consider two energy bands: a lower band
pairs in the system, the above Hamiltonian reduces to
Es of single-electron states ψ (r) and a higher band
k k that of interacting single electrons with N =NF.
ν0 +Ekpair/2 (given by energy per electron) of electron- Using the eigenfunctions of the single-electron and
pair states. As was shown in Ref.[15], the minima of the electron-pair states, we can obtain the field operators of
two bands in the square lattice are at the Γ point in the single electrons and electron pairs. The single electron
first Brillouin zone and their difference at low-electron (fermion) field operators can be written as,
density is given by ν = E +E , where E is the
0 g BW BW
band width of the single-electron band. ψ˜ (r)= ψ (r)c (2)
σ k k,σ
X
k
In order to obtain the many-electron Hamiltonian, we and
willconsidertwodifferentscenariosforthesingle-electron
band. We consider first the single-electron band as the ψ˜σ†(r)= ψk∗(r)c†k,σ, (3)
X
lowestconductionbandinarealcrystal. Freeconduction k
electrons occupy this band. It is empty in an undoped
insulator at low temperature. When the electron-pair where the operators c†k,σ and ck,σ are creation and an-
nihilation operators, respectively, for a single electron
band is taken into account, some electrons may form
of momentum ~k and spin σ. They obey the fermion
metastable pairs in this band. We assume that there
are N electrons consisting of NF single electrons and anti-commutation relations {ck,σ,c†k′,σ′} = δk,k′δσ,σ′,
NB electron pairs in the system (N = NF +2NB). At {ck,σ,ck′,σ′}=0, and {c†k,σ,c†k′,σ′}=0.
low electron density, all conduction electrons occupy the The electron-pair (boson) field operators are given by
single-electron band, i.e., N = N and N = 0. At
F B
higher densities, electron pairs may appear and many- Ψ˜(R,r)= Ψ (R,r)b (4)
k k
body interaction may stabilize these pairs. The Hamil- X
k
3
and
Ψ˜†(R,r)= Ψ∗k(R,r)b†k, (5) 1.0
X
k 0.8
=1.5
where the operators b†k and bk are creation and anni- q) 0.6 VV00==1125
hmiloamtieonntuomper~akt.orsT,hreeyspoebcteiyvetlhy,efcoormamnuetlaetcitornonreplaatiironosf f(pp 0.4 V0=18
[bk,b†k′]=δk,k′, [bk,bk′]=0, and [b†k,b†k′]=0. 0.2
Consideringthattheelectron-pairnumberN =(N
B
−
N )/2 in the system depends on many-particle interac-
F 0.0
tions and band re-normalization,the Hamiltonian of the 0 5 10 15
system in second quantization with single-electron and q (a-B1)
electron-pair states can be written in two parts,

1.0
H˜ =H˜ +H˜ . (6)
0 1
0.8
The first part H˜ corresponds formally to the five terms
0
in Eq.(1), given by q) 0.6
f(ep
0.4
H˜0 = Eksc†k,σck,σ
X
k,σ 0.2
1
+ 2Ak1X,k2,qXσ,σ′vqc†k1−q,σc†k2+q,σ′ck2,σ′ck1,σ 0.00 5 -1 10 15
+ XhEkpair+2ν0ib†kbk q ( aB)
k
1
+ 2A vpp(q)b†k1 qb†k2+qbk2bk1 0.2
X −
k1,k2,q
1 R)y
+ 2Ak,kX1,q,σvesp(q)b†k1−qc†k+q,σbk1ck,σ, (7) t v(q) ( 0.1
where v = 22π is the electron-electron Coulomb poten-
q q
tial, v (q) = 4v f (q) the pair-pair interaction poten-
pp q pp 0.0
tialwithaformfactorfpp(q), andvesp(q)=2vqfep(q)the 0 2 4 6
single-electron-electron-pair interaction (scattering) po- -1
q (aB)
tentialwithformfactorf (q). Noticethat,inEq.(7)we
ep FIG. 1: The form factors for the electron-pair-electron-pair
havetakenthe bottom of the single-electronbandas the
reference for energy. The summation over q does not in- (pfoptpe(nqt)i)alasn,dasnidnglteh-eeleicnttreorna-cetlieocntropno-tpeanitria(lfevpt((qq))) ifnotrersaicntgiolen-
clude q=0 because it cancelled out with the background electrons-electron-pair transition. λ = 1.5 aB and V0 = 12,
ion-ion interaction and the system is neutral. The form 15, and 18 Ry.
factorsf (q)andf (q)reflectfinitesizeoftheelectron-
ep pp
pair (against point charge) and are given by
fep(q)= a∗lxly;n′m′alxly;nmFnm;n′m′(q), (8)
XlxlyXnmnX′m′
and
f (q)=[f (q)]2, (9) where J (x) is the Bessel function of the first kind.
pp ep m
respectively, with
( 1)m+2m′ ∞ ThesecondpartH˜1 ofthetotalHamiltonianEq.(6)is
Fnm;n′m′(q)= −√bmbm′ Z0 rdrRnm(r)Rn′m′(r) responsible for the transition or transformation between
twosingleelectronsandanelectronpair. Theinteraction
qr qr
×hJm+m′( 2 )+J|m−m′|( 2 )i, (10) potential is just the Coulomb potential between the two
4
electrons. It is given by We consider now another possible scenario for the
single-electronband. In this scenario,the single-electron
2
H˜ = dr dr Ψ˜ (R,r) ψ˜ (r )ψ˜ (r ) band is considered as the highest valence band of an in-
1 Z 1Z 2 † r1 r2 ↑ 1 ↓ 2 sulator. The conduction carriers in this band are holes
| − |
2 with positive charge and dispersion relation Eh. For an
+ Z dr1Z dr2ψ˜†(r2)ψ˜†(r1) r r Ψ˜(R,r) intrinsicinsulator,thisbandisfullyoccupiedbykelectrons
↑ ↓ 1 2
| − | at low temperature and consequently the hole density is
= √1A Xk,qvt(q)(cid:16)b†kck2+q,↑ck2−q,↓+bkc†k2−q,↓c†k2+q,↑(cid:17), zinegroi.niItfiawlleya)sisnumtheetvhaaltentcheerbeaanrdeaNnFd(h)Nholeelsec(ftrroomn pdaoiprs-
B
(11) in the electron-pair band (and N(h) > 2N ), the total
F B
carrier charge number (i.e., the net hole number) in the
where vt(q) is the single-electron-electron-pair interac-
tion potential for transition (transformation) which is system is given by Nh =NF(h)−2NB. Transition of two
given by electronsfromthevalencebandtotheelectron-pairband
creates an electron pair leaving two holes in the valence
vt(q)=lxXly;nm2√π√(b−m1)m2 a∗lxly;nmZ0∞drRnm(r)Jm(qr). bTenaaenkrdign.yg,Ttthhheeetmsocpaennoyaf-rpitohaeritsivcaslhleeonHwcaenmbsiaclthnoednmiaaasntitcihanelltyrheifisnerseFcneicgne.afro2ior.
(12)
can be written as,
The form factors f (q) and f (q) and the potential
pp ep
vt(q) can be obtained from numerical calculation. They H˜(v) =H˜(v)+H˜(v), (14)
aregiveninFig. 1 for the squarelattice with λ=1.5 a 0 1
B
and V0 =12, 15, and 18 Ry. with
The total electron number operator is given by
H˜0(v) = Ekhd†k,σdk,σ
N˜ = dr ψ˜ (r)ψ˜ (r)+2 dR drΨ˜ (R,r)Ψ˜(R,r) Xk,σ
Z σ† σ Z Z †
X 1
σ + 2A vqd†k1 q,σd†k2+q,σ′dk2,σ′dk1,σ
= c†k,σck,σ+2 b†kbk. (13) k1X,k2,qXσ,σ′ −
X X
k,σ k + Ekpair+2Eg b†kbk
It commutateswith the totalHamiltonianEq.(6)and is Xk (cid:16) (cid:17)
conserved. 1
IncomparisonwiththeFLHamiltonian[4],theinterac- + 2A vpp(q)b†k1 qb†k2+qbk2bk1
X −
tion terms due to pair-pair and single-electron-electron- k1,k2,q
1
pEaqisr. i(n6t)e,r(a7c)t,ioannsda(p1p1e)a.rTinheoFurLHmaomdeilltoHnaiamniltgoinveiannboyf + 2A X vhsp(q)b†k1−qd†k+q,σbk1dk,σ, (15)
k,k1,q,σ
the boson-fermion mixture was given by Eqs.(1.7), (1.8)
and (1.9) in Ref. [4] containing three terms only, which and
correspond to the single electrons (or holes), the single
1
beloescotnrosn(si.(eo.,rehloecletsr)onanpdaitrhse)baonsdonths.eTinhteerbaocstoionnqbueatnwteuemn H˜1(v) = √AXk,qvt(q)(cid:16)b†kd†k2+q,↑d†k2−q,↓+bkdk2−q,↓dk2+q,↑(cid:17),
was introduced phenomenologically with spin zero and (16)
with twice the mass and charge of a single electron, and wherethe operatorsd†k,σ anddk,σ arecreationandanni-
itwasfurtherassumedunstablewithanexcitationenergy hilation operators, respectively, for a hole of momentum
2ν. But, the detailed microscopic structure of the boson ~kandspinσ. Theyobeythefermionanti-commutation
qkunaowntnu.mTahnedreiftosroer,itghineoinfttehreapctaiiorninpgomteencthiaanlibsmetwweeerneuthne- relations {dk,σ,d†k′,σ′} = δk,k′δσ,σ′, {dk,σ,dk′,σ′} = 0,
electrons and bosons was described through a coupling and {d†k,σ,d†k′,σ′} = 0. The hole-hole interaction poten-
tial v is the same as that of the electron-electron in-
constantasaparameter. Inthiswork,however,webuild q
teraction. The pair-pair interaction potential is also the
a more complete theory with the Hamiltonian given in
same as that in Eq. (7), but the hole-electron-pair inter-
Eqs. (6), (7), and (11) based on the electronic structure
action (scattering) potential is different from the corre-
calculations of a two-dimensional square lattice. From
spondingsingle-electron-electron-pairpotentialbyasign,
the spin-singlet metastable states of the electron pairs,
vs (q) = vs (q), because the hole has positive charge.
togetherwiththesingle-electronstatesinthesystem,we hp − ep
ThecorrespondingformfactorsarealsogivenbyEqs.(8)
have obtained quantitatively the interaction potentials,
suchaspair-pairinteractionandsingle-electron-electron- and (9). The second part H˜(v) of the total Hamiltonian
1
pairinteraction(bothforscatteringandtransformation), is equivalentto H˜ givenin Eq.(11). It is responsible for
1
presented in the Hamiltonian starting from the electron- creating (destructing) an electron pair in the electron-
electron Coulomb potential. pair bandandtwo holes in the valence bandat the same
5
time. Fromthe definition ofholes, hole creation(annihi- throughmany-particleinteractions. Becausetheelectron
lation) is actually electron annihilation (creation) in the pairsandholescannotbindtogetherdueto stronginter-
valence band. So the creation of an electron pair in the action or screening of many particles, in this case, the
electron-pair band and two holes in the valence band is band-gap renormalizationdue to pair-pair and hole-hole
equivalent to the transition of two single electrons from interactions are essential to stabilize the electron pairs.
the valence band to the electron-pair band, and the in- Band-gap renormalization (BGR) has been extensively
teraction potential vt(q) keeps the same form giving by studied innonlinear optics ofsemiconductors,where one
Eq.(12). deals with an electron-hole plasma (EHP). In an EHP,
The total (net hole) number operator is given by the BGR is given by a sum of electron and hole self-
energies and is a function of the particle density and
N˜h = d†k,σdk,σ−2 b†kbk. (17) temperature.[18–20] It is known that the BGR at finite
X X density can be several times the exciton binding energy
k,σ k
in a 2D system. For a high density EHP, the BGR is
It commutates with the total Hamiltonian Eq. (14) and mainlyinducedbytheCoulombrepulsionamongthepar-
is conserved. ticles (the Coulomb hole effect). Therefore, we believe
that the BGR in our present electron-pair-hole system
shouldbeofsimilarvalue,i.e.,therenormalizationofthe
electron-pairenergywillbeenhancedathighdensityand

canbe severaltimes the biexciton energy. Consequently,
n
ctro theband-gapnormalizationinthepresentsystemcanbe
ele larger than the single particle gap Eg leading to a neg-
er Electron-pair band ative gap. As soon as the bottom of the electron-pair
p
Energy Ehk Ehk baamndactroouscchoepsictheelecFterromni-peanierrgdyenEsiFtyoifsthsteabvaillieznedceibnatnhde,
Single-electron valence band EF system.
In summary, we present a Hamiltonian for a 2D elec-
tronsystemwithsingleelectronsandelectronpairs. The
electron-pairstatesaremetastableatlowelectrondensity
X and they are expected to be stabilized through many-
particle interactions. We have considered two different
scenarios for the single-electron band. In the first case,
the single-electron band is considered as the lowest con-
FIG. 2: A schematic plot of the energy band structure with
duction band. In comparison with the FL Hamiltonian,
electron pairs and holes.
extratermssuchassingle-electron-electron-pairandpair-
pair interactions appear in our Hamiltonian. The inter-
In this scenario, the stabilization of the electron pairs
action potentials in the obtained Hamiltonian are deter-
can be understood as following. At low density, the at-
mined from the electro-electronCoulombinteraction. In
traction between electron pairs and holes are dominant
the second scenario, the single-electron band is consid-
leading to the formation of exciton-type states.[16] The
ered as the highest valence band. We make a connec-
most possible scenario should be an electron pair binds
tion of the electron pairing and its stabilization to hole
two holes into a biexciton (X ). In two dimensions, the
2 density in the system. Our present study has shown ba-
energy of an exciton (for m = m ) is approximately
e h sic insights into possible electron paring mechanism in a
EX = 2R . The energy of a biexciton is given by
b − y crystal.
EX2 = 2EX + ∆EX2, where ∆EX2 is the binding en-
b b b b
ergy of the biexciton in relation to two free excitons. In
a 2D system, this binding energy is about 20% of the
exciton energy EX.[17] Therefore, the energy of an elec-
b Acknowledgments
tron pair can be reduced with about 4.4 R . In other
y
words, the gap energy E can be reduced by 2.2 R . At
g y
highdensity,thesystemtendstobeanelectron-pairand This work was supported by FAPESP and CNPq
hole plasma as soon as the electron pairs can stabilized (Brazil).
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