Why holes are not like electrons. III. How holes in the normal state turn into electrons in the superconducting state

0.43 MB·English

by  J. E. Hirsch

Why holes are not like electrons. III. How holes in the normal state turn into
electrons in the superconducting state
J. E. Hirsch
Department of Physics, University of California, San Diego
La Jolla, CA 92093-0319
(Dated: January 26, 2009)
In recent work, we discussed the difference between electrons and holes in energy band in solids
9 from a many-particle point of view, originating in the electron-electron interaction[1], and from a
0 singleparticlepoint ofview,originating intheelectron-ion interaction[2]. Weproposed thatsuper-
0 conductivityinsolids only occurswhen theFermilevelis close tothetop of aband (holecarriers),
2 that it originates in ‘undressing’ of carriers from both the electron-electron and the electron-ion
interaction, and that as a consequenceholes in thenormal state behavelikeelectrons in thesuper-
n
conducting state[3]. However, the connection between both undressing effects was left unclear, as
a
J was left unclear how the transformation from hole behavior to electron behavior occurs. Here we
clarifythesequestionsbyshowingthatthesameelectron-electroninteractionphysicsthatpromotes
6
pairing of hole carriers and undressing of carriers from the electron-electron interaction leads to
2
undressingofcarriersfromtheelectron-ioninteractionandtransformsthebehaviorofcarriersfrom

hole-liketoelectron-like. Furthermorethisphenomenonisconnectedwiththeexpulsionofnegative
]
n charge that we predict to occur in superconductors. These unexpected connections support the
o validityofourtheoreticalframework, thetheoryofholesuperconductivity,toexplainsuperconduc-
c tivity in solids.
-
r
p PACSnumbers:
u
s
. I. INTRODUCTION an electronicenergyband is almost full,inparticulargiv-
t
a ing rise to superconductivity.
m
Weproposethatthenormalmetallicandthesupercon-
When Felix Bloch formulated his epoch-making the-
- ducting state are in a sense mirror images of each other,
d ory of electrons in metals[4], he had an “uneasy feeling
wherethe‘mirror’switchesthesignoftheelectriccharge,
n that the model of independent electrons might repre-
or the bottom and top of the band, or the electron-ion
o
sent a rather poor approximation and would turn out
c andthe electron-electroninteractions. The ‘mirror’isby
in some respects to be entirely inadequate”[5] (italic
[ no means perfect because of the vastly different masses
ours). Bloch realized full well the arbitrariness involved
associatedwiththenegativeelectronandthepositiveion,
2 in his privileging the electron-ion interaction over the
hencetherearesignificantdifferencesbetweenthenormal
v
direct electron-electron Coulomb interaction, given that
2 e2 =14.4eVAisjust aslargeaninteractionbetweentwo metallic and the superconducting state, and the ‘mirror
1 switching’doesnotoccurrightatthehalffilledbandbut
electrons as between an electron and a monovalent ion
6 insteadonlywhenabandisalmostfull. Nevertheless,an
3 at the same distance[6]. In the statement quoted above, essentialcommonalitybetweenbothstatesemergesfrom
. with“insomerespects”Blochhadundoubtedlythe phe-
1 consideringthe electron-ionandthe electron-electronin-
nomenon of superconductivity in mind[6], which was
0 teraction on an equal footing, leading to the conclusion
9 widely believedatthe time to be causedbythe electron- that normal metallic behavior ensues when the electron-
0 electron Coulomb interaction[7]. However, Bloch’s the- ioninteractiondominates,andsuperconductingbehavior
: ory’s myopic point of view became even more myopic
v when the electron-electroninteraction dominates.
with the conventional theory of superconductivity[8],
i Inthenormalmetal,therearefewelectronsintheband
X thatadscribesthephenomenontotheinteractionofelec-
with a lot of room to move around, the positive ions are
r trons with ions displacedfromtheir equilibrium position
a rigid and the electrons will act as nearly independent of
(electron-phonon interaction)[9], bringing the disregard
eachotherandadjusttheirwavefunctionsindividuallyto
for the role of the electron-electron Coulomb interaction
optimize the electron-ion interaction, with the electron-
to a new high.
electron interaction being non-optimized. In the super-
Instead, the theory of hole superconductivity[10] pro- conductor instead, the almost-full band is crowded with
poses that while electron-electron Coulomb interactions a lot of electrons that become ’rigid’ because of their
can indeed be neglected when a band is almost empty, mutual strong interaction, with their wavefunction ad-
they become increasingly dominant as the filling of a justed to optimize the electron-electron interaction and
band increases. Unlike the prevalent point of view with the electron-ion interaction being non-optimized.
nowadays, that electron-electron interactions are most This means in particular that states that are electron-
important near half-filling of a band[11], we propose ion-interaction-energy costly will be occupied in the su-
that electron-electroninteraction physics dominates and perconductingstate. Aconsequenceofthispointofview
qualitativelychangesthenormalmetallicbehaviorwhen is also that the theory predicts that superconductivity
2
is particularly favored when the electron-ion interaction
atom
strength Ze2 is weak, i.e. when the ionic charge Z is

small[12].
Inthefollowing,weanalyzethataspectoftheCoulomb s i n gle-electron ! U'+2!
interaction that we believe to be essential to understand e n e r g y le v e l s U
superconductivity, at the level of a single atom, a di-
atomic molecule and a solid. We show that the essen-
tial physics manifests itself in both real and momentum
real space
space in a remarkably parallel fashion in going from the
charge distribution
atomtothesolid. Itinvolvesexpansionoftheelectronic
wavefunction to achieve lowering of electron-electron in-
teraction and of quantum kinetic energy, at the expense
of electron-ion interaction energy.
FIG. 1: The two-electron atom. When the two electrons
occupy the unexpanded orbital, they pay a high price in
II. THE ATOM Coulomb energy, U. Instead, they prefer to reside in an ex-
panded orbital, paying each the price ǫ in single-particle en-
ergy,yet achieving a lower total energy U′+2ǫ.
Consider the wavefunction of an electron in the low-
est energy state (1s) of a hydrogen-like atom of nuclear
charge Z 5 5
E (Z¯)= Z¯ =E (Z) y (6c)
e−e e−e
4 − 4
Z3
ϕ (r)=( )1/2e−Zr (1)
Z π with Ee−e(Z) = U. Therefore, expansion of the orbital
costs electron-ion energy
with r measured in units of the Bohr radius a0. For two
electrons in the atom in a singlet state, the spatial wave ∆E =4Zy (7a)
e−i
function is not
but gives a lowering of both kinetic energy
ψ(r1,r2)=ϕZ(r1)ϕZ(r2) (2)
∆E = 4Zy+2y2 (7b)
kin
−
because the strong Coulomb interaction between two
electrons in this state and of Coulomb energy
e2 5
U =Z d3rd3r′|ϕZ(r1)|2 ~r1 ~r2 |ϕZ(r2)|2 (3) ∆Ee−e =−4y. (7c)
| − |
The single-particle energy of each electron in the ex-
makes this state energetically too costly. The true wave
panded orbital is the sum of its kinetic and electron-ion
function in the two-electron atom contains both radial
energy, and is larger than the single-particle energy in
and angular correlations between the electrons, and is
the unexpanded orbital by
well described by the Hylleraas wave function[13]. How-
ever, the main effect is captured by the Hartree approx-
1
imation that allows for expansion of the single-particle ǫ= (∆Ekin+∆Ee−i)=y2 (8)
2
orbital:
because the costinelectron-ionenergyis largerthanthe
ψH(r1,r2)=ϕZ¯(r1)ϕZ¯(r2) (4a) kinetic-energylowering. However,thisismorethanoffset
by the reduction in Coulomb repulsion in the expanded
with
orbital:
Z¯ =Z y (4b) 5
− U′ E (Z¯)=U y (9)
e−e
≡ − 4
The energy of the electrons in that state is (expressedin
Rydbergs=13.6eV) so that the total energy
E(Z¯)=Ekin(Z¯)+Ee−i(Z¯)+Ee−e(Z¯) (5) U′+2ǫ=U +2y2 5y (10)
− 4
with
is lowerthan the energy in the non-expandeddoubly oc-
E (Z¯)=2Z¯2 =E (Z) 4Zy+2y2 (6a) cupied orbital provided y <5/8.
kin kin
− We represent the situation schematically in Fig. 1,
with two “orbitals” per atom. The two electrons
E (Z¯)= 4ZZ¯ =E (Z)+4Zy (6b) “choose”tooccupytheorbitalwithhighersingle-particle
e−i e−i
−
3
energy because the lower Coulomb repulsion more than
offsets the cost in single-particle energy. The space- diatomic molecule

chargedistributionbecomesmoreextendedinrealspace:

negative charge is “expelled” outward when the second an ti-

electron is added to the orbital, leading to the lowering bo nding

of both electron-electron Coulomb energy (Eq. 7c)) and

kinetic energy (Eq. (7b)).

The optimal value of y that minimizes the energy e-ion e-e
bo nding dominates dominates
Eq. (10) is y = 5/16, so the orbital Eq. (1) expands

to ϕZ¯(r) with Z¯ = Z − 156, which costs electron-ion (a) (b) (c)
Coulombenergy ∆E = 5Z but gives a loweringof ki-
e−i 4
neticenergy∆E = 5Z+ 25 andofelectron-electron FIG.2: (a)Schematicdepictionofthewavefunctionforbond-
kin −4 128 ing and antibonding orbitals in a diatomic molecule. The
Coulomb energy ∆E = 25. The single-particle en-
e−e −64 bondingorbitalhashigherchargedensitybetweentheionsto
ergy of each electron in the expanded orbital is larger get maximum benefit from theelectron-ion interaction and a
than the single-particle energy in the unexpanded or- smoothwavefunctiontogivelowerkineticenergy. (b)and(c)
bital by ǫ = 25 which is more than offset by the re- shows the bonding (lower) and antibonding (upper) single-
256
duction in Coulomb repulsion in the expanded orbital particle energy levels, with different occupations of two elec-
U′ =U(Z 5/16)=U(Z) 25, so that the total energy trons of opposite spin.
− −64
isU′+2ǫ=U 25 andtheenergyloweringachievedby
−128
expanding the orbital is 25/128Ry=2.7eV.
where, in terms of the Coulomb integrals
The physics of the two-electron atom just described
is a remarkable microcosm of the physics of electrons
e2
in a nearly filled electronic energy band: orbital expan- (ij kl) d3rd3r′ϕ∗(r)ϕ∗(r′) ϕ(r′)ϕ (r) (13)
sionandpromotiontohighersingle-particleenergylevels | ≡Z i j ~r ~r′ l k
| − |
driven by electron-electron repulsion and kinetic energy
U = (iiii), V = (ij ij), J = (ij ji) = (iijj) and
loweringwillplayakeyrole,aswediscussinthefollowing | | | |
∆t = (iiij). All these matrix elements are positive. In
sections. |
particular, the hybrid Coulomb matrix element
∆t=(iiij) (14)
III. THE DIATOMIC MOLECULE
|
lowers the Coulomb repulsion for two electrons in anti-
Within a linear combination of atomic orbitals ap-
bonding states[15], as seen from Eq. (12b). For suffi-
proach, the bonding and antibonding orbitals of a di-
ciently large ∆t, the ‘inverted occupation’ shown in Fig.
atomic molecule are given by
2(c) with the two electrons in the antibonding orbital
ϕ1(r) ϕ2(r) would have lower total energy (kinetic + electron-ion +
ψb,a(r)= (2(1 ±S12))1/2 (11) electron-electron) than the usual one where both elec-
± trons are in the bonding orbital, Fig. 2(b).
with the + ( ) sign corresponding to bonding (b) (anti- Note also that for one electron in the bonding state
−
bonding (a)), assuming s-orbitals for definiteness. S12 = and one in the antibonding state, the direct Coulomb
(ϕ1,ϕ2) is the overlap matrix element. The bonding repulsion is
orbital has larger amplitude and correspondingly larger
charge density in the region between the atoms, while U +V 2J
U = − (15)
the antibonding orbital changes sign and has vanishing ba 2(1 S2 )
12
chargedensityatapointbetweentheatoms,asdepicted −
in Fig. 2(a). As emphasized in II, this is a real physi- which indicates that the Coulomb matrix element J
cal difference between bonding and antibonding orbitals favors ferromagnetism, particularly near the half-filled
thatcannotbeeliminatedbyacanonicaltransformation. band[14].
Putting two electrons in the bonding orbitalresults in Letus nowconsider the diatomic molecule asa micro-
an electron-electron repulsion energy cosm for an electronic energy band, as depicted in Fig.
3. If the electron-ion energy dominates, the occupation
e2
U = d3rd3r′ ψ (r)2 ψ (r′)2 will be as shown on the left side. In particular, for three
bb Z | b | ~r r~′ | b | electrons in the molecule (analogous to a ‘nearly filled
| − |
U +V +2J +4∆t band’) two electrons will go into the bonding state and
= (12a)
2(1+S12))2 one electron into the antibonding state. Instead, if the
electron-electron interaction dominates, the occupation
and for two electrons in the antibonding orbital
will be as depicted on the right side of Fig. 3, where
U +V +2J 4∆t twoelectronsoccupythe antibondingstate andone elec-
U = − (12b)
aa 2(1 S12))2 tron the bonding state. Equivalently, the single hole in
−
4

givenby Eq. (17): itisincreasinglypositiveasthe inter-
atomic distance R decreases and the ionic charge Z de-
creases. The ‘correlated hopping’ parameter ∆t changes
the hopping amplitude depending on the electronic oc-
cupation of the sites involved in the hopping process:
the hopping amplitude for an electron is t0, t0 ∆t and
−
t0 2∆t depending on whether there are 0, 1 or 2 other
−
electrons at the sites involved in the hopping process.
For one electron in the diatomic molecule, the eigen-
states of the Hamiltonian Eq. (16) are the bonding and
electron-ion energy band electron-electron
dominated dominated antibonding states c†bσ|0>,c†aσ|0>, with |0> the empty
molecule and
FIG.3: Thediatomic moleculeasamicrocosm for anenergy c† +c†
c† = 1σ 2σ (18a)
band. When theelectron-electron interaction dominates, the bσ √2
occupation of single-particle energy levels changes, as shown
ontherightsideofthefigure: themiddlediagram represents
a ferromagnet and the upperdiagram a superconductor. c† c†
c† = 1σ− 2σ (18b)
aσ √2
the filled band resides in the bonding rather than in the
and energies ǫb = t0, ǫa = +t0. For three electrons in
antibonding state. −
the molecule, the eigenstates are
Inatightbindingdescription,theHamiltonianforelec-
trons in a diatomic molecule with one orbital per atom ˜b> =c† c† c† 0> (19a)
is given by[16] | σ aσ b↑ b↓|
H = [t0 ∆t(n1,−σ+n2,−σ)]+U ni↑ni↓ a˜> =c† c† c† 0> (19b)
−Xσ − Xi | σ a↑ a↓ bσ|
+V n1σn2σ+J( (c†1σc2σ+h.c.))2 (16) with energies
Xσ Xσ
ǫ˜b = (t0 2∆t)+U +2V (20a)
− −
assuming higher order interactions (involving 6 or more
fermion operators) can be neglected. t0 is the hopping
amplitude for a single electron in the molecule. The cre- ǫ˜a =+(t0−2∆t)+U +2V (20b)
ation and annihilation operators in Eq. (16) refer to or-
respectively. Theorderingofthesestatesdependsonthe
thogonal orbitals φ1(r),φ2(r) obtained from linear com- magnitude of the single hole hopping amplitude
binations of the atomic orbitals ϕ1(r),ϕ2(r)[16]. The
Coulomb matrix elements are given by the expression
th t0 2∆t (21)
Eq. (13) with the orthogonal orbitals. In particular, the ≡ −
hybrid matrix element ∆t is given in terms of Eq. (13) For t > 0, ǫ˜ < ˜ǫ and the state ˜b > , correspond-
h b a σ
with atomic orbitals, ∆ta.o., by ing to two electrons in the bonding or|bital and one elec-
tronintheantibondingorbitalhaslowerenergythanthe
S12
∆t=∆t (U +V) (17) state a˜> ,sothelowestenergystatecorrespondstothe
a.o.− 2 | σ
three-electron state on the left side of Fig. 3. Instead,
to lowest order in the nearest neighbor overlap S12. To for th < 0 the ordering is reversed and two electrons
the extentthatthe Mullikenapproximationforthe over- will occupy the antibonding orbital, as depicted on the
lapchargedistributionholds[17],theexpressionEq. (17) right side of Fig. 3. The latter situation will occur if
exactly vanishes, and in practice it is found that devia- ∆t>t0/2.
tionsfromtheMullikenapproximationareverysmall[16]. The parameter ∆t represents a ‘bond charge
For that reason, the ”correlated hopping” interaction repulsion’[20]betweenelectrons: itpusheselectronsaway
term ∆t is usually ignored in formulating tight binding fromthebond(regionbetweentheions)towardsthesites
Hamiltonians to describe interacting electrons in solids. (ions)bysuppressingtheoccupationofthebondingstate
However,in considering the interactions between elec- and increasing the occupation of the antibonding state.
tronsasthenumberofelectronsinthediatomicmolecule Themagnitudeofthehoppingparametert0,whichisde-
(ortheband)increasesitisessentialtotakeintoaccount termined by the strength of the electron-ion attraction,
the expansion of the atomic orbital that occurs for the decreases as the ionic charge Z decreases. Instead, as
doubly-occupiedatom,discussedinSect. II.Whendoing already mentioned, the magnitude of ∆t increases as
so, we showed in Refs. [12, 18, 19] that the correlated the ionic charge decreases because of the increased or-
hopping term ∆t has a value very different from that bital expansion under double occupancy. So when the
5
electron-electroninteractiondominatesovertheelectron- sect. II, the atomic overlapmatrix element is given by
ion attraction in the diatomic molecule, t changes sign
h
and this leads to the inverted occupation shown on the (ZZ¯)3/2
right side of Fig. 3. S =(ϕZ,ϕZ¯)= (Z+Z¯)3 (22)
2
Furthermore, as the electrons doubly-occupy the an-
tibonding orbital, they have larger amplitude for their withZ¯ =Z 5/16. S isadecreasingfunctionoftheionic
−
wavefunction at the atomic site, rather than in the in- charge Z, and as Z 5/16 it becomes arbitrarilysmall.
→
teratomic bond, as depicted by the upper picture in Fig. Thehoppingamplitudeforanelectronwhentherearem
2(a). Larger on-site occupation implies a larger expan- other electrons in the two sites involved in the hopping
sionofthe atomicwavefunction,asdepictedinthe lower process is tm = Smt0, and has the same sign as t0. If
right panel of Fig. 1, and hence negative charge is ex- we define ∆t as the difference in hopping amplitude for
pelled outward, away from the interatomic region. This a single hole and when there is a second hole in one of
in turn also causes a relative lowering of kinetic energy, the sites, it is given by
as discussed in Sect. II for the single atom.
In summary, our analysis of the diatomic molecule ∆t=t0S(1 S) (23)
−
shows strong parallels with the discussion in Sect. II for
Thisphysicswillpromotepairingofholecarriers[21],but
the single atom: when the electron-electron interaction
will not change the sign of the hopping amplitude for
dominates over the electron-ion interaction, the single-
holes, since for a single hole we have
electronhigher energylevel(antibonding state) becomes
more occupied and the single-electronlower energy level th =t0S2 (24)
(bonding state) becomes less occupied. In other words,
holestendtoincreasinglyoccupythelower-energysingle- of the same sign as t0.
electron energy levels. At the same time, the negative However,thereareadditionalcontributionstothevari-
charge spatial distribution increases outward, which im- ation of hopping amplitude with occupation, that arise
plies a higher positive charge distribution in the region frommatrix elements ofthe Coulombinteractioninvolv-
inside the structure. That is, negative charge is “ex- ing electrons on neighboring sites. In the absence of or-
pelled” from the interior region of the molecule towards bital expansion these contributions essentially cancel, as
the exterior. The interactionmatrix element responsible discussed in Sect. III, however in combination with or-
forthiseffectinthediatomicmolecule(andinthesolidas bital expansion they have a fundamental effect, and Eq.
wewillsee))is∆t,andtheessentialunderlyingphysicsis (24) ceases to be correct. We discussed in Ref.[12] a cal-
orbital expansion driven by electron-electron interaction culationofthehoppingamplitude forasingleholeinthe
(since, as mentioned, in the absence of orbitalexpansion diatomic molecule obtained by approximating the bond-
the parameter∆tis foundtobe zero)andkinetic energy ingandantibondingstatesofthethree-electrondiatomic
lowering(sincetheatomicorbitalexpansiongiveskinetic molecule by the wavefunction
energy lowering as seen in Sect. I).
ψ˜b,a(r1,r2,r3)=
ϕ¯1(r1)ϕ¯1(r2)ϕ2(r3) ϕ1(r1)ϕ¯2(r2)ϕ¯2(r3)
± (25)
IV. FURTHER ANALYSIS OF ∆t (2(1 S˜12))1/2
±
where ϕ¯ (r ) and ϕ (r ) are the expanded and unex-
In this section we discuss in more detail the physical i j i j
panded orbitals for the j th electron in atom i = 1,2
origin of the occupation inversion that we argue can oc-
−
and the overlap is
curinthediatomicmolecule,becausethesameprinciples
will apply to the solid state. In Refs. [12, 18, 19] we cal- S˜12 =(ϕ¯1,ϕ1)2(ϕ¯1,ϕ2). (26)
culated from first principles the hopping amplitude for a
singleelectron,t0,andforasinglehole,th,inadiatomic where (ϕ¯1,ϕ1) = S ((Eq. (22)). The energies of these
molecule, taking into account the orbital relaxation ef- states are
fect. These hopping amplitudes were obtained from the
difference in energy of an electron or a hole in the bond- (¯1¯12H3 ¯1¯12) (¯1¯12H3 1¯2¯2)
ǫ˜ = | | ± | | (27)
ing and antibonding states. As can be seen for example b,a 1 S˜12
±
in Fig. 7(c) of Ref. [19], the hole hopping amplitude
th (denoted by t2 in ref.[19]) goes to zero and in fact wherethelabels1,¯1areshorthandfortheorbitalsϕ1,ϕ¯1,
etc. The three-electron Hamiltonian is given by
changes signforsufficientlysmallinteratomicdistanceR
and small ionic charge Z. In this section we discuss in
more detail how the change in sign of t comes about. H3 =H3sp+H3ee (28a)
h
In paper I of this series[1] we have emphasized the with the single-particleandelectron-electronparts given
changeinhoppingamplitudewithoccupationthatarises by
fromtheoverlapmatrixelementoftheexpandedandun-
expanded atomic orbital. For the 1s orbital discussed in H3sp =h1+h2+h3 (28b)
6
H3ee =h12+h23+h13 (28c)
where h is the single-particle Hamiltonian for the i-th
i
electron and h the interaction potential between the
ij
i-th and j-th electron, given by (in atomic units)
2Z 2Z
h= 2 (29a)
−∇ − r1 − r2
2
h12 = (29b)
r12
with r1,r2 the electronic coordinate relative to atom 1,
2, and r12 the distance between electrons. The differ-
encebetweentheenergiesEq. (27)givetheholehopping
amplitude t
h
ǫ˜ ˜ǫ FIG. 4: Dependence of the sign of the single hole hopping
th = a− b. (30) amplitude in the diatomic molecule th on the ionic charge
2 Z and the interatomic distance R for the 1s orbital. In the
regionabovethecurvethesingleholehoppingamplitudehas
In the regime where the overlap S is small (small ionic
charge Z) we can neglect the overlap Eq. (26), being the same sign as the single electron hopping amplitude t0
proportional to S2. The difference in energies between (t0>0),correspondingtotheorderingofenergylevelsshown
on the left side of Fig. 3. In the region below the curve the
the bonding and antibonding states is then determined
sign of the single hole hopping amplitude is opposite to that
by the second matrix element in Eq. (27), given by
of the single electron hopping amplitude and the occupation
of the levels for three electrons in the diatomic molecule is
(¯1¯12H3 1¯2¯2)=(¯1¯12H3sp 1¯2¯2)+(¯1¯12H3ee 1¯2¯2) (31) inverted as shown on the right side of Fig. 3.
| | | | | |
The first term in Eq. (31) (single-particle term) is
frompositivetonegative,andthestate ofinvertedoccu-
(¯1¯12|H3sp|1¯2¯2)=(¯1,h,¯2)S2+2(¯1,h,¯1)SS¯1¯2 (32) pation becomes the low-energy state.
In Ref.[12] we only explored the parameter regime
and is negative, so it lowers the energy of the bonding
stateandraisestheenergyoftheantibondingstate. The where th has the same sign as t0, i.e. where there is no
occupation inversion. For sufficiently small Z and small
firstterminEq. (32)correspondsessentiallytoEq. (24),
interatomic distance R, t is found to change sign, as
the secondtermwouldbe absentif the orbitalsatneigh- h
shown in Fig. 4. (For the calculation in Fig. 4 the over-
boring sites were constructed to be orthogonal to each
lap Eq. (26) was included hence there are other terms
other.
in t beyond those given in Eq. (34), see Ref.[12]). The
The second term in Eq. (31) (electron-electron inter- h
region t < 0 corresponds to the electron-electron dom-
action term) is h
inated regime shown on the right side of Fig. 3, where
(¯1¯12H3ee 1¯2¯2)=2(¯1,¯1,h12,1,¯2)S+(¯1,2,h12,1,¯2)S¯1¯2 occupation inversion occurs. For further details on the
| | (33) form of the matrix elements entering in the calculation
and is positive, so it raises the energy of the bonding leading to Fig. 4 the reader is referred to the Appendix
state and lowers the energy of the antibonding state. It of Ref.[12].
is clear then that the state of inverted occupation (an-
tibonding state) has lower electron-electron interaction
V. THE SOLID
energy than the ordinary state (bonding state). The dif-
ference in energy between the states is the hole hopping
amplitude We consider the Hamiltonian for electrons in a band
in a tight binding representation given by[15]
th = (¯1,h,¯2)S2 2(¯1,h,¯1)SS¯1¯2
− 2−(¯1,¯1,h12,1,−¯2)S−(¯1,2,h12,1,¯2)S¯1¯2 (34) H = −<Xij>,σ(t0−∆t(ni,−σ+nj,−σ))(c†iσcjσ +h.c.)
where the first two terms, involving the electron-ion in-
+U n n +V n n (35)
teraction, are positive (including their sign), and the i↑ i↓ i j
Xi <Xij>
lasttwotermsinvolvingthe electron-electroninteraction
are negative (including their sign) . When the electron- Performing a particle hole transformation
electron interaction dominates over the electron-ion in-
teraction(forsmallionicchargeZ)thesignoft switches h† =c (36)
h iσ iσ
7
electron-ion dominated
Hole superconductivity

antibonding state
bonding cool
state normal metal superconductor
FIG. 6: When the system goes superconducting, the anti-
FIG. 5: The electronic states at the Fermi energy look very bonding states near the top of the band become fully occu-
differentwhentheFermilevelisnearthebottomandnearthe pied with electrons and the bonding states at the bottom of
topofabandinanormalmetal. Nearthebottomoftheband the band become empty. In other words, the holes “Bose-
the wavelength is large and the state is smooth, resembling condense” to the bottom of the band. The promotion of
thefree-electronplanewavefunction,nearlyunaffectedbythe electrons to the higher electron-ion energy states is driven
electron-ion potential. Instead, near the top of the band the principally bythe electron-electron interaction.
wavelength approaches a single lattice spacing and the state
is strongly modified by theelectron-ion interaction.
electron-ion interaction. In addition, as discussed in I,
as the hole propagates it causes a large deformation in
the Hamiltonian becomes
the atomic cloudwhere itlands (Fig. 1), andasa conse-
H = + (t +∆t(n +n ))(h† h +h.c.) quence it is also highly dressed by the electron-electron
h i,−σ j,−σ iσ jσ interaction.
<Xij>,σ
Asthesystemgoessuperconductingthecarriersatthe
+U ni↑ni↓+V ninj (37) Fermi energy undress from both the electron-electron[1]
Xi <Xij> and the electron-ion interaction[2] in order to conduct
better. Their wavelength expands, and they no longer
where the number operators are now n =h† h and
iσ iσ iσ ‘see’thediscreteelectron-ionpotential[2]. Figure 6 illus-
trates how this occurs.
th =t0 2∆t. (38)
− Indeed, we propose that in the transitionto supercon-
We have shown that this Hamiltonian leads to pairing ductivity,theelectronicoccupationofthebandisshifted
of holes and to superconductivity in a BCS formalismin upward,tooccupyallthesingle-particleenergylevelsup
the regime of low hole concentration[21]. Furthermore tothetopoftheband,leavingthelowestenergylevelsin
we have discussedthat the transitionto superconductiv- the band empty. In other words, the holes that were oc-
ity drivenby ∆t is associatedwith “undressing”[22]: the cupyingtheuppermostelectroniclevels“Bose-condense”
effective massof the carriersdecreasesandthe quasipar- andnowoccupythelowestelectron-ionenergylevels. By
ticle weight increases, and a transfer of spectral weight sodoing,theelectronicstatesthatarenowatthebound-
fromhightolowfrequenciestakesplacebothinthesingle ary between occupied and empty states (dashed line on
particle spectral function (detectable in photoemission the right side of Fig. 6) become “smooth” states, of the
experiments)andintheopticalabsorptionspectrum[23]. ’bonding type’, with large amplitude in the region be-
In addition, in paper II of this series we have argued tween the ions[24].
that superconductivity is also associated with ‘undress- How can such complete reorganization take place?
ing’ofcarriersfromtheelectron-ioninteraction,andthat Just like in the diatomic molecule, it will occur for
itinvolvesawavelengthexpansion: thewavelengthofcar- low hole concentration if the hopping amplitude t(n)
riersattheFermienergygrowsfromamicroscopiclength (n=number of electrons per site) changes sign:
(k−1 ˚A−1 interatomic distance) to a much larger
F ∼ ∼
wavelength. Here we explain how this occurs. t(n)=t0 n∆t<0 (39)
−
Figure5showsschematicallytheformofthewavefunc-
tionofthe carriersatthe Fermienergywhenthe bandis so that in particular the single hole hopping amplitude
almost empty and when it is almost full. We are assum- th = t0 2∆t < 0. Then, in the Hamiltonian Eq. (37)
−
ing that the hopping amplitude t0 is positive, as occurs the lowestenergystate for a hole occurs for k =0 rather
for s orbitals, which implies that the minimum in the than for k = π. This, however, is not just a ‘canonical
band occurs for k = 0. When the Fermi level is close to transformation’: the states near the bottom and the top
the top of the band (hole carriers) the state is “bumpy” of the band are qualitatively different.
rather than smooth. As discussed in II, it is not a state As function of temperature, the hopping amplitude
well-suited to conduct electricity: it is strongly coupled t(n) can change sign if the number of electrons n in-
tothelatticeandis(nearly)Bragg-scatteredbyit,trans- creasesasthe temperature islowered. Indeedweshowed
ferring the momentum it acquires from an external field in Ref. [25] that several anomalous properties of high
to the ionic lattice in the process,andacquiringmomen- T cuprates are explained by the assumption that the
c
tum in opposite direction. It is highly “dressed” by the number of holes in the Cu O planes decreases as the
−
8
el ectron-electron dominated energy band = superconductor atom d i a t o m i c m o lecule solid

FIG. 7: In the superconducting state, the holes occupy
smoothbondingstates,thatwerepreviouslyoccupiedbyelec- FIG.8: Single-electronenergylevels(upperpartofthefigure)
tronsinthenormalstate. Theholes‘ride’ontopoftherigid and real space charge distribution (lower part of the figure).
negatively charged ions without disturbing their charge dis- Aselectronsmoveuptheenergylevels,negativechargemoves
tribution as they do in the normal state[24]. The electron outward in the atom, the diatomic molecule and the solid in
energy band depicted on theright side includes theelectron- the superconducting state. In the solid in the normal state,
electron interaction energy n∆t and therefore it is inverted. thechargedistributionisspatially homogeneous. Thecharge
redistribution isdrivenbyelectron-electron repulsion and ki-
neticenergylowering,andislargerforsmallerionicchargeZ
in all three cases.
temperature is lowered, and that such a charge transfer
process between the planes and off-plane charge reser- VI. SUPERCONDUCTIVITY FROM HOLE
voir atoms is aided by the Coulomb matrix element ∆t. UNDRESSING
In other systems the change in n with temperature may
be associated with redistribution of occupation between
AsdiscussedinIIandRef. [3],thereisplentyofexper-
differentbandswithoutchargetransferbetweendifferent imentalevidencethatdressedholecarriersinthenormal
regions in real space.
statebecomeundressedfromtheelectron-ioninteraction
andbehavelike undressedelectron carriersinthe super-
InFigure7werepresentthesituationinanalternative
conducting state. How this occurs becomes clear from
way, where the ’energy band’ now includes the electron-
the considerations in this paper.
electron interaction energy n∆t. Because the sign of the
Let us review the experimental evidence for undress-
hoppingamplitude haschangedforthealmost-fullband,
ing of carriers from the electron-ion interaction in the
the lowest energy electronic states occur at k = π and
superconducting state:
areoccupiedbyelectrons,andthehighestoccuratk =0
(1) Rotating superconductor: A superconducting body
andareunoccupied. Becausethebandisalmostfull,the
rotatingwithangularvelocityω~ developsauniformmag-
carriers at the Fermi energy now have a smooth wave-
netic field in the interior[29], given by
function, as depicted on the left side of Fig. 7, just like
the smooth wavefunction of carriers at the Fermi energy
2m c
in the normal state of a metal when the band is almost B~ = e ~ω (40)
− e
empty. Theholesride‘ontop’ofthenegativeions,with-
out disturbing them and without noticing the discrete- with m the free electron mass. This has been measured
e
ness of the potential. This corresponds to the ‘mirror in conventional, heavy fermion, and high T cuprate
c
image’ of the lower left panel of Fig. 5. superconductors[30]. The magnetic field always points
parallel to the angular velocity, as given by Eq. (40),
This reorganization in energy level occupation has a never antiparallel, indicating that the superfluid carriers
counterpartin the realspace chargedistribution. Figure behave as negatively charged carriers[31].
8 shows the situation schematically for the atom, the di- (2) Gyromagnetic effect: If a magnetic field B~ is sud-
atomic molecule and the solid. As higher single-electron
denlyappliedtoasuperconductoratrest,theentirebody
energy levels become occupied, electronic charge moves
is found to start rotating with angular momentum
outwardinthethreecases. Thetheoryofholesupercon-
ductivity predicts that negative charge is expelled from m c
L~ = e VB~ (41)
the interior of the metal towards the surface as it un- body 2πe
dergoes the transition to the superconducting state[26],
and that as a consequence an excess negative charge with me the free electron mass, and V the volume[32].
density ρ exists within a London penetration depth of This angular momentum is always antiparallel to the
−
the surface of superconductors, with ρ = H /4π or applied magnetic field for the same reason as in (1)
− c
ρ− = Hc1/4π for type I and type II super−conductors above[31].
respect−ively[27]. (3) Bernoulli potential: Wherethere isa spatialvaria-
tionofthesuperfluidvelocityanelectricfieldisexpected
The superconductor looks like a ‘giant atom’[28] with to develop, the resulting potential is termed Bernoulli
expanded orbitals. potential[33]. Experimental measurements[34] are con-
9
sistent with an electric field given by
1 1
E~ = ~ m v2 (42)
e∇2 e s
T > T 0 < T < T T = 0
c c
where e and m are the free electron charge and mass,
e
andv isthesuperfluidvelocity. Inparticularthesignof
s
theBernoullipotentialmeasuredcorrespondstonegative
charge carriers.
(4) Hall effect: The Hall coefficient is found to be es-
sentiallyalwayspositiveinthenormalstate,correspond-
ing to hole carriers[35], and to change its sign from pos- FIG. 9: Occupied states in the band (upper part of the fig-
itive to negative at temperatures slightly below T , in- ure) and in the Brillouin zone (lower part of the figure) as
c
dicating that carriers change from hole-like to electron- thetemperatureislowered. TheshadedareasoftheBrillouin
like[36]. zoneareoccupiedbyelectrons. Whenthetemperatureislow-
(5) Wavelength expansion: We have proposed an ex- ered below Tc a ‘hole pocket’ is pierced at k ∼ 0 that grows
as the temperature is further lowered, while the hole pock-
planation for the Meissner effect in superconductors in-
etsneartheedges of theBrillouin zonebecome progressively
volving a wavelength expansion of the carriers at the smaller and eventually disappear at T =0. At temperatures
Fermi energy[2, 37] from the microscopic length kF−1 0 < T < Tc the empty states near k ∼ π correspond to the
to the mesoscopic length 2λL (λL =London penetra- normal fluid hole carriers and the empty states near k ∼ 0
tion depth), corresponding to the wavevector change correspond to thesuperfluid hole carriers.
kF q0 =1/(2λL).
→
These experimental observations and theoretical con-
siderations indicate that the carriers of electric current bottom of the band is opposite to that of carriers at the
in the normal state, dressed hole carriers, morph into top of the band. For that reason, these superfluid hole
undressed electron-like carriers in the superconducting carriersbehave like electrons, andin particularexhibit a
state. How does this happen? In II and Ref.[3], we sug- negative Hall coefficient, as well as the other signatures
gested that the antibonding electrons at the top of the of electron transport reviewed at the beginning of this
Fermi distribution in the normal state condense to the section.
bottom of the band in the transition to superconductiv- Figure9alsosuggestsanewinterpretationofthe‘two-
ity. However,thiswouldconflictwiththePauliprinciple, fluid model’[38] of superconductivity. The normalquasi-
because the states at the bottom of the band are occu- particles are the holes near the top of the band, i.e.
pied! k π, while the superfluid are the holes at the bottom
Furthermore,itisimportanttopointoutthatitwould of∼the band, with k 0. The conventional BCS theory
be incorrectto assume that all the electronsin the band is argued to be consi∼stent with the two-fluid model[39],
become ‘undressed’ from the electron-ion interaction in however it does not provide a clear separation of both
thesuperconductingstate. Becausethesuperfluidweight components as Fig. 9 does.
ns that enters into the London penetration depth λL Note also that the continuous process by which the
holesmovefromthetoptothebottomofthebandasthe
1 4πn e2
= s (43) superfluidcondensatedevelopsrepresentsthemomentum
λ2L mec2 spacecounterpartoftheorbitexpansiondiscussedincon-
nection with the Spin Meissner effect[37].
most definitely corresponds to the hole concentration
rather than to the electron concentration in the band
(which is much larger). It is essentially the same charge
VII. TWO-ORBITAL MODEL
carrier density that carries the current in the normal
state.
So the holes in the normal state have to behave like In previous work we have introduced an electronic
electrons in the superconducting state. The solution of model with two orbitalsper site to describe the essential
this puzzle is what was depicted in Fig. 6. In Fig. 9 we physics of electron-hole asymmetry (electronic dynamic
showthe expectedevolutionwith temperatureof the oc- Hubbardmodel)[40]. Thespacingbetweensingle-particle
cupationsinthebandstructureandintheBrillouinzone. energy levels is ǫ, and the interactions are such that two
Asthemetaliscooledintothesuperconductingstate,the electrons will occupy predominantly the higher single-
holes condense to the bottom of the band, which corre- particle level because the Coulomb repulsion there (U′)
sponds to long-wavelength states near the center of the ismuchsmallerthanbothinthelowerlevelandbetween
Brilloin zone, which are smooth bonding states as de- one electron in each level. This corresponds to the sit-
picted on the left side of Fig. 7. So it may be said that uation depicted schematically on the right side of Fig.
thesuperfluidcarriersarestillholes,notelectrons. How- 1.
ever, the sign of the effective mass for carriers at the
10

parameters for the lowest eigenvalue of H1 to be smaller
than the lowest eigenvalue of H2 is
ǫU′
√zt> +ǫ2 (45)
r 2
Here, t is the bare electron hopping amplitude for an
empty band. For a cubic lattice it is given by
~2 3.81eV
t= = (46)
2mea2 a(˚A)2
(a) (b) Therefore,theconditionEq. (45)canbesatisfiedforrea-
sonable parameters, e.g. a = 2˚A, t = 0.95eV, ǫ = 1eV,
FIG. 10: Propagation of a pair in the strong coupling limit. U′ = 5eV. In that case, the hole propagates without
The fact that theholes undresscompletely and propagate as disturbing the background, as shown schematically in
freeelectronsimpliesthereisnorelaxationofthechargecloud the left diagram of Fig. 7. The propagation shown in
asdepictedin(b),ratherthepropagationisasdepictedin(a), Fig. 10(b) corresponds to ‘partial’ undressing from the
with theholes occupyingthelower atomic energy level. electron-electron interaction, while that shown in Fig.
10(a) corresponds to full undressing
Considerapairofholespropagatingwithinthismodel
in the limit where the size of the pair is a single lattice VIII. DISCUSSION
spacing. This corresponds to the strong coupling limit
where the single hole hopping amplitude goes to zero as
In this paper we have discussed the point of view that
discussed in Ref.[41]. In Fig. 10 we show two possi-
the competition between electron-ion interaction domi-
ble ways the pair propagation can occur. As in Ref.[40]
nance versus electron-electron interaction dominance in
(Eq. (4)), we assume for simplicity that the bare hop-
solids leads to normal metallic or superconducting be-
ping amplitude is the same between all nearest neigh-
havior depending on whether the former or the latter
bor orbitals, t. If the propagation is as shown in Fig.
wins. Electron-electron interaction will dominate when
10(b), the electrons in the singly occupied site relax to
an electronic energy band is almost full, i.e. when the
the lower orbital, and the hopping amplitude is reduced
carriers in the normal state are hole-like. Correspond-
by the overlap matrix element S (Eq. 27(a) of Ref.[40]).
ingly, superconductors in nature are found to have posi-
Instead,if the propagationis as showninFig. 10(a),the
tiveHallcoefficientinthenormalstate[35]. Furthermore,
hopping amplitude is not reduced. This corresponds to
the electron-electron interaction strength e2 will domi-
the case where the holes occupy the lowest orbital, as
nate overthe electron-ioninteractionstrengthZe2 when
depicted on the right side of Fig. 6.
the ionic charge Z is small. Correspondingly, high T
The wavefunction for a pair of holes is a linear combi- c
superconductivity is found in materials with highly neg-
nation of the states
atively charged substructures (planes) containing nega-
|Ψp >= √1N |↑↓>i (44a) t(FiveeionAss,)l−ikepltahnees(Cinut−heOa2r)s=enpildaenseasnindtthheecBu−prpaltaens,esthine
Xi −
MgB2.
Here we restricted the discussion of the atom and the
2 diatomic molecule to the simplest case of a 1s orbital.
Ψ >= [ > > > > ] (44b)
| ex rzN |↑ i |↓ j −|↓ i |↑ j However exactly the same physics should take place in
<Xij> bands originating from other atomic orbitals. For exam-
where i,j are nearest neighbor sites and z is the number ple we showed in Ref.[19] that the quantities of interest
of nearest neighbors to a site. Its energy is the lowest for 2p orbitals in a diatomic molecule behave very simi-
eigenvalue of the matrix larlytothoseinthe 1sorbital. With porbitals,theroles
of k 0 and k π switch, howeverthe essential physics
∼ ∼
0 2t√z determinedbywhetherastateisnearthebottomornear
H1 =(cid:12) 2t√z − U′ (cid:12) the top of the band remains the same.
(cid:12)− − (cid:12)
(cid:12) (cid:12) When one talks about electronic energy bands one is
or the matrix (cid:12) (cid:12)
implicitlyprivilegingtheelectron-ioninteractionoverthe
0 2tS√z electron-electron interaction. The states at the bottom
H2 =(cid:12) 2tS√z −U′ 2ǫ(cid:12) of a band have low electron-ion energy, and those at the
(cid:12) − − − (cid:12) top of a band have high electron-ion energy. However,
(cid:12) (cid:12)
for the case of Fig.(cid:12) 10(a) and 10(b) re(cid:12)spectively. In conversely,aswe havearguedin this paper,electrons re-
the limit where S is very small, the condition on the siding in states near the bottom of the band have high