- Email: [email protected]
Possible electron-exciton (E-E) interaction and high Tc superconductivity in bulk semimetals
Solid State Communications,Vo1.63,No.7, pp.645-648, 1987. Printed in Great Britain.
0038-1098/87 $3.00 + .OO Pergamon Journals Ltd.
Possible Electron-Exciton (E-E)Interaction and High Tc.superconductivity in Bulk Semimetals Yin Daole, Hsn Hushan (Departmentof Physics,Peking University,Beijing) (China Center of AdvancedScience and Technology) (received1 May 1987 by W.Y. man.1 Abstract In relationto the high PC recentlyobservedin Y-Ba-Cu-Csystems,the post sibilityof formationof Cooper pairs by virtual excitonexchangein a single couplixg matrix elementsand possible phase is analysed. Electron-Exciton optimum conditionof superconducting Tc:and its upper limit are evaluated. It shows that this kind of superconductivity may displaysome differentfeatures comparedwith usual BCS superconducting phase. PACS mumbers: 74.20 Z 71.55+2
The mechanism of superconductivity with Tc much higher than 3OK observedin Y-Ba-Cu-C systems"12*3) is still an open question for discussion. Superconductivity of Ba (Pb,_xBix)03 system with Tc 13k were explaninedas electronphnnnn (E-P , i.e., usual BCS mechanism(415)and bipolaron(61 . However, the Tc of Y-Ba-Cu-0 seems much too high comparedwith the popular predicted limitationof E-P superconductivity about Tc'30-40k(7'8). E-E mechanismhas been proposed for some imaginarylow-dimensional compositesystems(gllO*ll).Nevertheless,some recent study shows that the high 'PCof Y-Ba-Cu-0 is likely singlephasesuperconductivity.Thus, a modelling estimationof B-E couplingwith its possible contributionto superconductivity in bulk semimetalshas to be carried out. In this letter we will begin with the considerationof the conditionfor the possible coexistenceof excitons and metallic conductivity in a single phase matter. A modelling electronicstructureof this type is shown in Fig 1. Similar multiminimumconductionband can be found frequentlyin realisticnonmetals, however. now we need a specificposition of Fermi level relative to the minimums of conduction band so to realize a peculiar semimetal. The occupancyof k space by the small Fermi seas (Fig 2 solid line sphere) with w of the order nne tenth for usual metals and the region for excitons near r are shown in Fig 2 (dottedline sphere). Momentum conservationof electron exciton interactionin such a system holds evidently.
r
or A
L
Fig 1 direct transitionexcitationenergy band model in semimetal
Fig 2
direct transitionexcitationand Fermi surfacemodel in semimetal
Both exciton and phonon are boson exoitations and can play role likewise in a general BCS form Hamiltonian
This is correspondingto direct exciton transiThere are other ways of E-E interaction Ition. 'via indirect excitons,e.g., Fig 3. 645
HIGH Tc SUPERCONDUCTIVITY
646
Fig 3
’
IN BULK SEMIMETALS
61/“
Vol. 63, No. 7
is the perturbing potential due to
indirect transition excitation energy
where
band model in semimetal
the jth mode of bosons. In the case of electron-phonon
interaction
and one finds
Where Cip- and Cz6 anihilation
denote the creation and
of electrons with 2 and spin r in
conduction
band. and Vi;? denotes the pairing
attractive
interaction
is the polarization vector; end Ua
where
is the crystal potential, so we have the parameter A
in (2) equ. as
(9)
(2) via any boson intermediate Superconducting
(phonon or exiton 1.
Tc of such a system is
(10)
(3) Considering
the strong coupling correction
and Coultirepulsive
psuedopotential /'we
Alternatively,
for the electron-exciton
mechanism
get
(11)
with the binding energy of exciton
where
f._&is a characteristic
intermediate
energy of the (12)
bosons, and the parameter
which is usually much smaller than the energy Eg (specially for the case of Wannier gap exciton) and it can be neglected in the first (4)
approximation.
We denote
Vol. 63, No. 7
HIGH Tc SUPERCONDUCTIVITY
647
IN BULK SEMIMETALS
(13) and find
(14)
where we note that since <-c/and T-c'ere small,one may approximate
There are three main distinctions between At first, since the excitons e-e anaA e-p' are of electronic nature, A e-e is independent h
to the atomic mass M and one expects no isotope effect in E-E superconductivity.
2> in (9) eque, for Ae+ in (14) equ. the %h in the we have only first power of Wex 'v Eg denominator, so the priority of low energy excitations in contribution to )Cee
may be less
remarkable than in the E-P case. McMillan-Hopfiedly
(19)
Next. unlike
Finally, in the
type parameter
?e_e of (14)
the dielectric constant'14).
withk,
In comparison with the electron-phonon matrix element ISI
in (10) equ., we see the term in
square brackets of (18) equ. as
E&
.vu a. up
is of the same order
(1) s(r)*
So
<2*)
can be
ronghlg evaluated as
equ-, the square average of matrix elements(Zz> originates from the perturbing potential with no certain Pl (coso)
dUex
symmetry, so the
angular momentum hybridization
in electronic
wave functions will be less influential in E-B coupling strength (12). To make a further evaluation, the matrix element
(20) Thus, we can expect
Z;;/ can be expressed in the form:
(21') In viev of the formulae (3) (11) and (14). and assuming
&*kO.*
we have
(16)
where the foot notes 1 and 2 distinguish electrons 1 and 2, c and v denote and valence band respectively.
(22)
the
the conduction
Me replaced the
for a given
ye_,
the optimum E-B Tc appears
while
coulomb interaction in usual exciton problem by a screened coulomb potential vith screening length 1 due to the semimetallic nature of material so that
is Bohr radius. Follow a derivation a0 similar to that in ref. 13, we get
where
Thus, fcr a moderate have
i?,,?-0.05
U.u.ve may
648
HIGH Tc SUPERCONDUCTIVITYIN BULK SEMIMETALS
Vol. 63, No. 7
The role of dex in E-E superconductivity is similar to Debye frequency dd in BC3 mechanism. However the former is much more changeful and susceptibledue to its eletronicnature.
(24)
Although this predictionmay be somewhat enccuraging, one should not overlookthe difficulties to realize the condition (22) equ., Because the limitationof p e_e (21) equ. here we need a very narrow gap with pg- 10 -‘ev Incoordinationbetveen fg and [e_e will cause drastic decreasein Tc as could be seen from Fig 4. l
Another very critical factor in E-E superconductivityis the Fermi level ff . Since we are dealing with a very narrow gapt a shift of ff in some lo-'ev may radicallychange the picture in Fig 1.2 and 3. Thus, if it were so, ve will expect a superconductorwith high Tc but very sensitiveto dopping. ln multi-phase systems, owing to the varied inhomogeneityeffects, the optimum condition of E-E superconductivity might be realized in some local region, say, in the vicinity of specific boundaries,then some percolationcharacter of this kind of superconductivity could be observed. Though the estimationof B-E superconduvtivity gives a much higher possible fc than that has been done for B-P mechanism(798)9 we should keep in mind that the latters were based, explicitlyor implicitly.on the usual cation Muffin-Tinapproximation. A very interestingquestionwould be whether an anion perturbingpotentialcontributionto E-P superconductivity in some specific semimetals is possible? A study along this line is going on and will be published elsewhere.
we.1 T)e-e Fig 4
superconductingTc vi.% exciton energy c3ex
Reference
1)
2. Zhao, et. al..,to appear in KeXue TongBao 2) C-W. Chu, et. al., to appear in Phys.Rev. Lett.
8) C.M. Varma, Supercond.in d- and f- band metals" (Proc. of the IV - th conf., Karlsruhe, 1982) p 603
3) Q. Wen, W. zhang, et. al., to appear in. KeXue TongBao (in Chinese)
9) W.A. Little, Phys. Rev. m A (1964)pI416 IO) V.L. Ginsburg,Phys. Lett. Q (1964) ~101
4) B. Batlcgg etal. "Supemonductivity in d-and f- band metals" (Proc. of the TV-th Confer., Karlsnubep 1982) p 401.
Il)D. Allender, J. Bray, J.Bardeen,Phys. Rev. B7 (1973) ~I020
6) A. Alexandrov,J. &nninger, Phys. Rev. B 3 (1981) ~1164
12) about angular hybridizationin E-P mechanism see ref.7 and J. J. Hopfield, Phys. Rev. 186 (1969)~443; Yin Daole, Xhang Liyuan, Dai Yuandong, Acta. Phgs. Sin. a (1979) ~841 I31 J.:Singh,Solid State Physics, Vol B (1984)
7) W. L. McMillan, Phys. Rev., 167
p295 I41 H. Haken, J. Phys. Chem. Solids 8 (1959)~166
5) K. Kitasawa et al. Proc. of Joint JapanChain Seminar II on Superconductivity, Sendai, 1986, ~117
(1968)~331
0038-1098/87 $3.00 + .OO Pergamon Journals Ltd.
Possible Electron-Exciton (E-E)Interaction and High Tc.superconductivity in Bulk Semimetals Yin Daole, Hsn Hushan (Departmentof Physics,Peking University,Beijing) (China Center of AdvancedScience and Technology) (received1 May 1987 by W.Y. man.1 Abstract In relationto the high PC recentlyobservedin Y-Ba-Cu-Csystems,the post sibilityof formationof Cooper pairs by virtual excitonexchangein a single couplixg matrix elementsand possible phase is analysed. Electron-Exciton optimum conditionof superconducting Tc:and its upper limit are evaluated. It shows that this kind of superconductivity may displaysome differentfeatures comparedwith usual BCS superconducting phase. PACS mumbers: 74.20 Z 71.55+2
The mechanism of superconductivity with Tc much higher than 3OK observedin Y-Ba-Cu-C systems"12*3) is still an open question for discussion. Superconductivity of Ba (Pb,_xBix)03 system with Tc 13k were explaninedas electronphnnnn (E-P , i.e., usual BCS mechanism(415)and bipolaron(61 . However, the Tc of Y-Ba-Cu-0 seems much too high comparedwith the popular predicted limitationof E-P superconductivity about Tc'30-40k(7'8). E-E mechanismhas been proposed for some imaginarylow-dimensional compositesystems(gllO*ll).Nevertheless,some recent study shows that the high 'PCof Y-Ba-Cu-0 is likely singlephasesuperconductivity.Thus, a modelling estimationof B-E couplingwith its possible contributionto superconductivity in bulk semimetalshas to be carried out. In this letter we will begin with the considerationof the conditionfor the possible coexistenceof excitons and metallic conductivity in a single phase matter. A modelling electronicstructureof this type is shown in Fig 1. Similar multiminimumconductionband can be found frequentlyin realisticnonmetals, however. now we need a specificposition of Fermi level relative to the minimums of conduction band so to realize a peculiar semimetal. The occupancyof k space by the small Fermi seas (Fig 2 solid line sphere) with w of the order nne tenth for usual metals and the region for excitons near r are shown in Fig 2 (dottedline sphere). Momentum conservationof electron exciton interactionin such a system holds evidently.
r
or A
L
Fig 1 direct transitionexcitationenergy band model in semimetal
Fig 2
direct transitionexcitationand Fermi surfacemodel in semimetal
Both exciton and phonon are boson exoitations and can play role likewise in a general BCS form Hamiltonian
This is correspondingto direct exciton transiThere are other ways of E-E interaction Ition. 'via indirect excitons,e.g., Fig 3. 645
HIGH Tc SUPERCONDUCTIVITY
646
Fig 3
’
IN BULK SEMIMETALS
61/“
Vol. 63, No. 7
is the perturbing potential due to
indirect transition excitation energy
where
band model in semimetal
the jth mode of bosons. In the case of electron-phonon
interaction
and one finds
Where Cip- and Cz6 anihilation
denote the creation and
of electrons with 2 and spin r in
conduction
band. and Vi;? denotes the pairing
attractive
interaction
is the polarization vector; end Ua
where
is the crystal potential, so we have the parameter A
in (2) equ. as
(9)
(2) via any boson intermediate Superconducting
(phonon or exiton 1.
Tc of such a system is
(10)
(3) Considering
the strong coupling correction
and Coultirepulsive
psuedopotential /'we
Alternatively,
for the electron-exciton
mechanism
get
(11)
with the binding energy of exciton
where
f._&is a characteristic
intermediate
energy of the (12)
bosons, and the parameter
which is usually much smaller than the energy Eg (specially for the case of Wannier gap exciton) and it can be neglected in the first (4)
approximation.
We denote
Vol. 63, No. 7
HIGH Tc SUPERCONDUCTIVITY
647
IN BULK SEMIMETALS
(13) and find
(14)
where we note that since <-c/and T-c'ere small,one may approximate
There are three main distinctions between At first, since the excitons e-e anaA e-p' are of electronic nature, A e-e is independent h
to the atomic mass M and one expects no isotope effect in E-E superconductivity.
2> in (9) eque, for Ae+ in (14) equ. the %h in the we have only first power of Wex 'v Eg denominator, so the priority of low energy excitations in contribution to )Cee
may be less
remarkable than in the E-P case. McMillan-Hopfiedly
(19)
Next. unlike
Finally, in the
type parameter
?e_e of (14)
the dielectric constant'14).
withk,
In comparison with the electron-phonon matrix element ISI
in (10) equ., we see the term in
square brackets of (18) equ. as
E&
.vu a. up
is of the same order
(1) s(r)*
So
<2*)
can be
ronghlg evaluated as
equ-, the square average of matrix elements(Zz> originates from the perturbing potential with no certain Pl (coso)
dUex
symmetry, so the
angular momentum hybridization
in electronic
wave functions will be less influential in E-B coupling strength (12). To make a further evaluation, the matrix element
(20) Thus, we can expect
Z;;/ can be expressed in the form:
(21') In viev of the formulae (3) (11) and (14). and assuming
&*kO.*
we have
(16)
where the foot notes 1 and 2 distinguish electrons 1 and 2, c and v denote and valence band respectively.
(22)
the
the conduction
Me replaced the
for a given
ye_,
the optimum E-B Tc appears
while
coulomb interaction in usual exciton problem by a screened coulomb potential vith screening length 1 due to the semimetallic nature of material so that
is Bohr radius. Follow a derivation a0 similar to that in ref. 13, we get
where
Thus, fcr a moderate have
i?,,?-0.05
U.u.ve may
648
HIGH Tc SUPERCONDUCTIVITYIN BULK SEMIMETALS
Vol. 63, No. 7
The role of dex in E-E superconductivity is similar to Debye frequency dd in BC3 mechanism. However the former is much more changeful and susceptibledue to its eletronicnature.
(24)
Although this predictionmay be somewhat enccuraging, one should not overlookthe difficulties to realize the condition (22) equ., Because the limitationof p e_e (21) equ. here we need a very narrow gap with pg- 10 -‘ev Incoordinationbetveen fg and [e_e will cause drastic decreasein Tc as could be seen from Fig 4. l
Another very critical factor in E-E superconductivityis the Fermi level ff . Since we are dealing with a very narrow gapt a shift of ff in some lo-'ev may radicallychange the picture in Fig 1.2 and 3. Thus, if it were so, ve will expect a superconductorwith high Tc but very sensitiveto dopping. ln multi-phase systems, owing to the varied inhomogeneityeffects, the optimum condition of E-E superconductivity might be realized in some local region, say, in the vicinity of specific boundaries,then some percolationcharacter of this kind of superconductivity could be observed. Though the estimationof B-E superconduvtivity gives a much higher possible fc than that has been done for B-P mechanism(798)9 we should keep in mind that the latters were based, explicitlyor implicitly.on the usual cation Muffin-Tinapproximation. A very interestingquestionwould be whether an anion perturbingpotentialcontributionto E-P superconductivity in some specific semimetals is possible? A study along this line is going on and will be published elsewhere.
we.1 T)e-e Fig 4
superconductingTc vi.% exciton energy c3ex
Reference
1)
2. Zhao, et. al..,to appear in KeXue TongBao 2) C-W. Chu, et. al., to appear in Phys.Rev. Lett.
8) C.M. Varma, Supercond.in d- and f- band metals" (Proc. of the IV - th conf., Karlsruhe, 1982) p 603
3) Q. Wen, W. zhang, et. al., to appear in. KeXue TongBao (in Chinese)
9) W.A. Little, Phys. Rev. m A (1964)pI416 IO) V.L. Ginsburg,Phys. Lett. Q (1964) ~101
4) B. Batlcgg etal. "Supemonductivity in d-and f- band metals" (Proc. of the TV-th Confer., Karlsnubep 1982) p 401.
Il)D. Allender, J. Bray, J.Bardeen,Phys. Rev. B7 (1973) ~I020
6) A. Alexandrov,J. &nninger, Phys. Rev. B 3 (1981) ~1164
12) about angular hybridizationin E-P mechanism see ref.7 and J. J. Hopfield, Phys. Rev. 186 (1969)~443; Yin Daole, Xhang Liyuan, Dai Yuandong, Acta. Phgs. Sin. a (1979) ~841 I31 J.:Singh,Solid State Physics, Vol B (1984)
7) W. L. McMillan, Phys. Rev., 167
p295 I41 H. Haken, J. Phys. Chem. Solids 8 (1959)~166
5) K. Kitasawa et al. Proc. of Joint JapanChain Seminar II on Superconductivity, Sendai, 1986, ~117
(1968)~331