- Email: [email protected]
Superconductivity of triplet bipolarons
~
Solid State Communications, Vol.57,No.8, pp.553-557, 1986. Printed in Great Britain.
SUPERCONDUCTIVITY
Institute
of Physics,
Department
0038-1098/86 $3.00 + .00 Pergamon Press Ltd.
OF TRIPLET BIPOLARONS
Yan-Min Li Chinese Academy of Sciences,
of Physics,
Li-Yuan Zhang Peking University,
Received 3 November
BeiJing,
BeiJlng,
China
China
1985 by W. Y. Kuan
Superconductivity of triplet blpolarons has been investigated in a phase analogous to the superfluid A-phase of 3He by use of the broken-symmetry Hartree approximation. The transition temperature T c , the chemical potential and the order-parameOers have been obtained, and the specific heat and the thermodynamic critical field have also been calculated. It has been found that there exist two energy gaps in the superconducting state. In the normal phase there also exists a gap, whose magnitude is consistent with the recent experimental evidence of the heavy-fermlon superconductor tHFS) , UPt% . Temperature dependence of the critical field of anothe@ HFS UBel% , Which is not yet understood, can also be qualitatively~accounted for by this model. These show the possibility that triplet bipolarons might be formed in the HFSs.
I. I n t r o d u c t i o n
s o - c a l l e d heavy-fermion superconductors (HFSs) recently discovered,
The local electron pairs, i.e. bipolaro~s, have attracted much attention and many investigations have been made on the singlet bipolarons for the possibility of having hightemperature superconductivity and its various interesting properties.
20
~d
such as CeC~2Si219, UBe13
UPt321, hay? been ~ g a r d e d as a klnd
of triplet superconauctors . Kulik 25 has suggested the possibility of triplet local electron pairs. In this communication we will first consider the possibility of triplet bipolarons as the ground state, and derive the effective pair-Ramiltonian of triplet bipolarons (Section YI). In Section IIl, the Hartree approximation via the Bogoliubov variational principle 24 will be used to discuss superconductivity of triplet bipolarons. The transition t e m p e r a t u r e , the chemical p o t e n t i a l and
Anderson I first proposed that the singlet bipolaron on the same site could be formed as a result of the strong local lattice deformation which produces centers of negative effective correlation energies (the -U centers). Many electrical, magnetic and optical properties of amorphous materials can be interpreted by this kind of on-site bipolarons 2-4. Some other possible mechanisms of forming the g U centers have been suggested and discussed9-~-. Theoretical investigations of superconductivity of the on-site bipolarons have also been made I0-11 • Rice and Sneddon 12 ascribed BaPbl-x BixO 3 as an example of this kind of superconductors.
the order~parameters are o b t a i n e d . I n S e c t i o n IV. the s p e c i f i c heat and the thermodynamic critical field will be calculated. The concluding remarks is given in Section V.
II. EFFECTIVE HAMILTONIAN We first consider the ease of intersite bipolarons. The essentials for the formation of them is the electron-phonon coupling. When the bipolarons are formed, two atoms occupied by one electron~pair lie closer than the other atoms, wlth a canonical transformation which removes the electron-phonon coupling, one can derive an e~ective Hamiltonian
Lakkis et al. 15 extended the idea to the intersite slnglet electron-palring, and used it as the ground state of Ti407. The similar ground state of vanadit~m bronzes was proposed later by Chakraverty et al.14. Alexander and Ranninger 15 discussed the excitation and superconductivity of this kind of bipolarons. LiTi204 16-17 is considered as a superconductor induced by them.
H=PIo+
We have pointed 18 that in the case of the interslte pairing, the triplet biolarons are almost as stable as the singlet blpolarons, and will have great effects on the thermodynamic properties of bipolaronlc systems. Therefore it is necessary to investigate superconductivity of triplet bipolarons. The
~,÷ w
(v,
,
+
+
+
+
W
±
•
n...
+
where m labels the pair of atoms and the • 553
SUPERCONDUCTIVITY OF TRIPLET BIPOLARONS
554
+
t ,2 specifies the atoms in the pair. Cm~cr
,
Ca@or and mm~Cm+oco4~m~; are, respectively, the creation, destruction and number operators associated to the electronic state localized at the ~th atom with mr-spin in the mth pair. H o describes the intra-pair motion of electrons; H I and W are, respectively, the inter-pair hopping and the sum of the inter-pair Coulomb interaction and the inter-pair exchange interaction of electrons. We will treat H o as the unperturbated Hamiltonlan, H I and W as the perturbations. The elgenvalues of He can easily be obtained 18. The lowest energies of singlet and triplet blpolarons when unperterbated, respectively,
Vol. 57, No. 8
I
..
~is the dissociation energy of triplet bipolarons when unperturbated. If the lattice is ordered, 6 m is independent of m. When Jmm' is negative, ferromagnetism can happen; while Jmm' is possltive, antiferromafnetlsm may appear. In this communication we only consider superconductivity of triplet blpolarons in the ordered lattice. Coexistence of magnetism with superconductivity as well as the effect of the disorder will be discussed elsewhere.
are
Es = (V+v1+J1~2_112
[(U_vI_J I)2 + 16t12
Et = v1-J1
]I12 (2.2a) (2.2b)
Normaly only when v I ~O, may bipolarons be stable 18. In the ordinary blpolaronlc materis/s, tl is small, so U-v1-J1>>~t1~. From E t < F~s, the condition of the triplet blpolarons as the ground state is approximately given by J1 > 2tl/(U+I.Vl I )" (2.3) since electrons are very localized, the exchange interaction is not much affected by the eleotron-phonon interaction. In some cases, (2.3) could be satisfied. When the perturbations exist, the stability of the bipolarons generally decreases. But in some materials, expecially in the quasi-lowdimensions/ ones, stable bipolarons can still be formed 18. In this paper, we will assume that triplet bipolarons are deeply bound. When the minimum dissociation energy of triplet bipolaronsAl>>T (kB = 1 ) , the density of free electrons is considerably small, and will be neglected. We Can then project the origins/ Hamiltonian (2.1) into the eubspace of triplet bipolarons to obtain the effective pair-Hamiltonian. For convenience to discuss and without losing physical meaning, we will only consider the phase having two kinds of pairs ~}, ~ , analogous to the superfluid A-phase of ~He 25. Introducing the triplet blpolaron operators
Am = Cml.f- Cm24" Bml = Cm14, Cm24, ,
(2.4)
and using t h e c a n o n i c a l t r a n s f o r m a t i o n 15 o r t h e p r o j e c t o r a p p r o a c h 26, one can o b t a i n t h e effective triplet pair-Hamiltonian
H = ~ E m nm - mm' tmm' (Am Am' + Bm
For the on-slte bipolorons, the Hamitonian (2.1) can still describe their motion as long as ~ 1,2 is understood as the two degenerate orbltals28f each atom. Recently, Marel and Sawatzky " considered the possibility of triplet pairs as the ground state. When the onsite triplet bipolarons are deeply bound, the same type palr-Hamitonlan as (2.5) can be obtained. Thus, our discussion can be applicable for both intersite and on-slte triplet blpolarons. In the expression of vmm' of (2.6), the term in the bracket should be small compared to the first term in the superconductors, otherwise the systems normallYlbecome the charge-density-wave insulators O. Thus we have yam' --~Itm~ m' ~'~ 2 / ~ . For on-site blpolarons, one can easily obtained from (2.6) that tmm, ~ I/2 Vmm'. This conclusion will be used later.
Ill. SUPERCONDUCTIVITY In this section, we will discuss superconductivity of triplet bipolarons according to pair-Hamiltonian in (2.5). For the sake of convenience, only the nearest neighour interaction of pairs are considered. So the subscripts of parameters tam , 9 Yam, and Jam, are discarded from now on. The superconducting order-parameters are defined as h
C~m
A
=
'
~m =
~Bm>
,
(3.1)
which determine the transition temperature, at which 0Lm, ~m = O. The broken-symmetry Hartree approximation is introduced via the Bogoliubo¥ variational principle for the thermodynamic potential 24
Bm' ) + (5.2)
+ ~I~
Cvmm, ~= ~=, + Jm=' s= s,),
C2.5)
where nm"
Am =
£m = ~ 6"o +
tram, =~'z'
+ Bm ~m' 8m " Am
Am - Bm Bm,(2"6)
v,-J~-X='~-Itm=,='=']2/A
t,,i=,~=#~<~,~,<< , I A
,
where N e is the number of pairs, ~! =I/T, is the trial Hamiltonian, <... "2@ the average , with the density matrix ~o = exp[-~'(Ho-/~el~/Tr~z~-~ -~Ne) ,)~the chemical potential. When the system is homogeneous, i.e. ~ m and ~m is independent of the subscript m, the brokensymmetry Hamiltonian H O is chosen as
A,-~. A,-gzRm-~B m ~nm), (3.~1
SUPERCONDUCTIVITY OF TRIPLET BIPOLARONS
Vol. 57, No. 8
where ~ I , 62 and ~ are variational parameters to minimize the thermodynamic p@tential h o • Since at each site there are only three possible states ~ 0 > , |0> and ~ 0 > , the opereters ~, ~ and ~ can be denoted as
/ooo~ A-UOOl
/coo I ~* /ool /
~O00t,
/loo/ ~* |ooo/
~O00l,
~001/
06 05
where = ~
, E ± = ( A± R , , [ ~ 2 + 4 ( i~,l 2--)/,
.
R)I2,
,k.
-~Ne)]+
where n = N ~ N is the average density per site, N the number of pair sites, z the number of the nearest neighbours, n, ~ a n d ~ are, respectively, given by
e ~''~/~15h({~11) T_( P_~ ' + '7~IR" n =--,n,o,=~ i[, ,'~/~ , I'
(5.7~)
c h ( ~ RI2)]
t+2e ~'vz c~('~',~/z')
--4
34 2
Of
Then the trial thermodynamic poten-
+2e
/
(3.~)
@'(~
i-5
0.2
(3.4)
+ 1~,=12- ))1/2,-
~nTr exp [-
-~
o '-'o.3
tial per site is readily derived as
i~Ao/N = -I/(N ~')
6
0,4
Substitute (3.4) into (3.3), one can easily obtain the elgen-energies of ~m +// n m
~0 = h
555
0
I
I
I
I
I
I
I
t
I
0.1
Q2
0.3
0.4
0.5 n
0.6
0.7
0.8
0.9
Fig. 1. Transition temperature T c (the solid surve) and the ratio of the gap A I and T c (the dashed curve). The unit of T o is iS. From (3o11,b), we can obtain the asymptotic behavieurs of the chemical potential for n <41 as vzn - (I-2n) tz-tz exp [-(1-n)tz/T~, T<
i
(3.13)
/(=
'
vzn
n/2 (tz)2(Tc-T)/Tc~
(I-3n/2) tz + Tc
'
- T <4 To .
The asymptotic behavlours of the order-parameters are A minimization of the trial thermodynamic potential El S yields the optimum values
~
= tzo~
~z 7~
= tz~ =vzn
I coI ItS (To-T)/TcZ] '/2, Xc-T <<7~ .(S'14) (3.8)
If we substitute (3.8) into (3.7a-c), we can get the couplig equations for the chemical potential and the order parsmeters. These equations should be solved self-consistently. If (3.8) is substituted into (3.6), the free energy can be obtained, by which we can discuss the thermodynamic properties in the superconcucting state. This will be given in the next section. From (3.7b) and (3.7c), we have
= ~,
(3.9)
which is the trivial solution without polarization. When T > To, let ~ = ~ = O,we have
~= V Z n -
T In[2
(1-n)/nJ, T ~ T c.
(4.10)
(I-2n) tz + R exp ( - 1 ~ ' / 2 )
l[2sh(~'R/2~(3.11a)
R ~[2(1-2n) t z -.~.] t h (~'R/2)
(3.11b)
Let Or, ~ - * 0 in (3.11a,b), we obtain the superconducting transition temperature T c = tz (I-3n/2) / ~n[2(1-n)/n] ,
0,5
-0.5
0.4
n = 0,2
(3.12)
which agrees with the result by Eulik 23. The dependence of T c with n is shown in Fig. 1 (the solid curve). T o is independent of v~which results from the Hartree approximation, since the Hartree approximation neglects the correlations between electrons. But the result so obtained is resonable when n << I.
-0.6
~ 0.2, " -
-0,7
_o2_ o.~ r- . . . . . . ~ ,
~,
~_
o.,
oJ
-0.8
%%
o
when T < To, (3.7) can be rewritten as k=
The t e m p e r a t u r e d e p e n d e n c e s o f t h e c h e m i c a l potential ( t h e s o l i d c u r v e s ) and t h e o r d e r p a r a m e t e r s ( t h e d a s h e d c u r v e s ) a r e shown i n Fig. 2.
I
T~
-0.9
T
-I
Fig. 2. Temperature dependences of the orderparameter ~0~ (the dashed curves) and the chemical potential ,41(the solid curves ). Note//is actually ( ~ - v z n ) /
(tz). In the conventional BCS superconductors, as we know, there are a lot of normal electrons in the superconducting state, and only a part of electrons form Cooper pairs. The superconducting gap, which appears only when the Cooper pairs are formed, is defined as the average energy per electron needed to destroy a Cooper pair. In the bipolaronic superconductors, however, "pairs" exist before the
Vol. 57, No. 8
SUPERCONDUCTIVITY OF TRIPLET BIPOLARONS
556
occurring of the superconducting state. This to the different definition of the superconducting gap from BCS one. We will discuss it in the next section.
b
will lead
IV. THERMODYNAMIC PROPPRTIES From (3.6) and (3.8), we obtain energy
the free
FJm=oolIN.Jr~(1-n)~+ t , ( I ~ l 2 + ;~ 12) - T Z n + 2 axp (BgJ2) oh (@'R/2); + vzn212.(4.1) The specific heat in the superconducting state is then o b t a i n e d a s
C m - T~2
re
- - 2Ntz
~T 2
al~(~
(4.25
o
~
B~ ^
using
(3.14)
~ .= 2 e ~'(~-v )
(4.7) (4.~5
e.p[-(1-n) tzlT] }
activation type energies, ~ 1 : t z , ~ = ( 1 - n )
tz,
which result from the three eigen-snergies in the superconducting state, as shown in Fig. 3. AccordinF to ~ . 5 ) we have
E+ (0)~ (l-n)
t,,~=
A 1 = R(o)=
tz
where R'= [ (I-2 e ~?~-v)) + 8 ~ ~']/ j I~2 and ~ is the chemical potential. For n << I we have < n i n~ j ~ ~ n 2 exp (-v/T). (4.8) Using (4.6) and (4.8), we get Cn(T) ~
I/2
Nn 2 (v/T) 2 exp
(-v/T),
(4.9)
(4.4)
<
from which we can see that in the normal state there also exists a gap, 43 = v. According to the last discussions in Section ll. v.~-~2t, and noting z = 2 in the one-disensional case, we obtain A 3 ~ A I . Consequently we have
E.
l
A,
l
Eo
A2
A31Tc~_AIITc~Z.8
Fig. 3. Eigen-energies in the superconducting state. Thus there exist two gaps in the superconducting stats. This contrasts with the case of singlet bipolarons, where there is o~ly one gap, whose magnitude is equal to A I '. appearence of the superconducting gaps shows that the superconducting condensation is different from the Bose one without any gaps. When T -*T c , we have =
2 N)C((O)I2(tz/Tc 52 .
(4.5)
The temperature dependence of C (T) is shown in Fig. 4 (the heavy soIid curves) In the frame of the Hartree approximation, the specific heat in the normal phase is equal to zero owing to neglecting the correlation. ~owever, we will obtain the non-zero specific heat when the correlation is included. If we do not consider the effect of the hopping term of pairs, the specific heat is On (T) me I S T
(4.10)
It is interesting to compare this result with the recent experimental evidence by Buyers et al. 28, who have measured the spin-fluctuation spectrum of the HFS UPt 3 with neutron scatterinF just above the superconducting transition temperature. They found that the activated intensity of the spin fluctuations is consistent with the existence of a gap]ike structure in the normal phase, and the gap.~-~ 3To. This is very well consistent with our theoretical result. For the thermodynamic critical field, Using the well-known formula
He 2 /8~=
I/V (Fo(nl-Fo (s))
,
(4.11)
we obtain the asymptotic behaviours of H c ~Hc (0)
(]-TI(2To)) ~
~<
n ~ [(Tc-T)ITc] ~
Tc-TZ4T c (4.12)
= v12 .Y-a.<~i ~j> I a T . ( 4 . 6 ) sJ
I n t h e one-demensional case, the c o r r e l a t o r < n i n i > can be e x a c t l y c a l c u i a t e d . I t can
that
.
The density dependence of ~I/TC is shown in Fig. I (the dashed surve). ~ i / T c is not sensitive to n. In the normal density regime, the ratio is about 3.0. Yt can be expected that this conclusion is also valid in the higher dimensional case.
Hc = ( H J 0 ) I
be sho~n
~- Z c ~'(~-v~
(-tzlT) + (I-2n).
one can see from (4.3) that there are two
O(T c )
F.
and (4.2), for n <<1 we have
C = " (tzlT)2{exp
(n
r"
Fig.4. Temperature dependences of the specific heat C (the heavy solid curve) and the critical field H c (the thin curve). The dashed curve is the specific heat in the normal state.
where He(O) = (8~ntzN/V) ~.
The temperature
dependence of H e is shown in Fig. 4. When T-*Tc, - ~Hc/ ~T--~oo; and when temperature is
557
SUPERCONDUCTIVITY OF TRIPLET BIPOLARONS
Vol. 57, No. 8
somewhat lower than To, Hc increases linearly on cooling. We would like t o qualitatively compare these results with the experimental evidences of the upper critical field of another HFS UBs1329, which cannot be accounted
how, s u p e r c o n d u c t i v i t y o f b i p o l a r o n s i s much different from t h e c o n v e n t i o n a l one. It can be e x p e c t e d t h a t many a n o m a l o u s b e h a v i o u r s o f e x o t i c s u p e r c o n d u c t o r s c a n be a c c o u n t e d f o r by t h i s m o d e l .
f o r by c u r r e n t t h e o r i e s o f e i t h e r c o n v e n t i o n a l or p-wave s u p e r c o n d u c t i v i t y . It is found that
We have adopted the Hartree approximation, which neglects the correlation between electrons. Therefore only when n <~I, is the result o b t a i n e d r e s o n a b l e . Triplet blpolarons can show magnetism,which may lead t o coexslstence with superconductivity. Recently Batlogg et al.30 reported measurements of the ultrasound attenuation of (U, Th) Bet3. They found a A-shaped attenuation peak below To, which may correspond t~,a antiferromagnetic trmnsion. Buyers et s-l.~v
HO~ (T) of UBe13 has a considerably large initial slope and a linear dependence with T below about 0.9 T c. This behaviour is qualitatively consistent with our result. V. CONCLUSIONS
I n t h i s c o m m u n i c a t i o n we h a v e d i s c u s s e d superconductivity of triplet bipolarons in a
also found that the spin-fluctuations Just a b o v e Tc a r e a n t i - f e r r o m a g n e t i c . Coexistence o f m a g n e t i s m and s u p e r c o n d u c t i v i t y o f t r i p l e %
hase ~ He.
similar to the superfluid A-phase of It has been found that there exist two g a p s in t h e superconducting s t a t e . In the
normal phase there
also
exists
bipolarons will be discussed in a separate publication.
a g a p , whose
magnitude is consistent with the experimental result of UPt328. The thermodynamic critical field H e has a infinite slope at To, and has a linear dependence with T at low temperature. This hehaviour can qualitatively accounted for the unusual shape of the upper critical field of UBe13. Consequently, it is necessary to investlgate theoretically the possibility that triplet blpolarons may be formed in HFSs. Any-
ACKNOWLEDGMENT---One of authors (YML) would llke to express his sincere thanks ts professor Y, Y. Li~ Institute of physics and Professor K. A. Chao, department of Physics, ~,iversity of LinkSplng, Sweden, for their having interested him i n t h e b i p o l a r o n p r o b l e m and many s t i m u l a ting
discussions.
REFERENCES IP.W.Anderson, Phys. Rev. Lett. ~ , 9 5 3
(1975)
2p.A. Street and N.F.Mott, Phys. Rev. Lett. 35, 1293 (1975). 3p.w. Anderson,J.Phys.
(Paris)37,C4-339(1976).
4D.Adler and E.J.Yoffa, Phys. Rev.Lett. 36, 1197 (1976). 5p.Pincus, Solid State Commun. 11, 51 (1972). 6 p.M.Ckaikin,A.F.Garlto and A.J.Heeger, Phys. Rev. B 5, 4966 (1972).
17K.W.Ng, R.N. Shelton,E.L.Wolf,
Phys.Lett. A 110,
423 (1985). 18y. M. Li and K.A.Chao,Acta Phys. Sin.(Chlnese) 33,273 (1984) (English trans: Chin.Phys. 4 , 804 (1984); and to be published. 19F. Steglish, J.Arts,C.D.Bredle,W. Lieke,D. Meschede,W. Franz and H.S Chafer, Phys. Rev. Lett. 43,1892 (1979). 20H. R. Ott,H. Rudigier, Z. Fisk and J.L. Smlth, Phys. Rev. Lett. 50,1593 (1985).
7G.Beni,P. Pincus and J. Kansmori, Phys. Rev. B 10, 1896 (1974).
21G. R. Steward,Z.Fisk,J.O.Willis and J.L, Smlth, Phys. Rev. Lett. 52,679 (1984).
8G.V. Ionova,E.F.Makarov and S.P. Ionov, Phys. Status Solidi (b) 8_!,671 (1977).
22p.W.Anderson,Phys. Rev.B 30,1549 (1984); C.M. Varma,Bull.Am. Phys. Soc.29,404 (1984). 23I. O.Eulik, Physics 126B,280 (1984).
9K. Weiser, Phys. Rev.B 25,1408 (1982). los. Robaszkiewicz, R. Micnas and K.A.Chao, Phys. Rev. b 23, 1447(1981); ibid,2_~4,1579(1981). 11 I.O.Kulik and A.G.Pedan, Sov. Phys. JETP 52,
742 (1980). 12T.M. Rice and L. Sneddon, Phys. Rev. Lett. 47, 689 (1981). 13S. Lakkis,C. Schlenker, B.E. Chakravsrty,R. Buder, M. Marizlo, Phys. Rev. B 14,1429 (1976). 14B.K.Chakraverty,M.J.Sienko and J.Bonnerot, Phys. Rev.B 17,3781 (1978). 5A.Alexandrov and J. Ranninger, Phys. Roy. B 233, 1796 (1981); ibid,24,1164 (1981). 16D. C.Jonston,J.Low. Temp. Phys. 25,145 (1976); D.C.Jonston,R.W. McOallum,C.A.Luengo and M.B. Maple, ibid, 25,177 (1976).
24S.V.Tyablikov, Methods in the Quantum Theory of Magnetism (Plenum,New York, 1967).
25A.J.Leggett,Rev.Mod.Phys.47,331
(1975).
26V.J.Rmery, Phys. Rev.B1_~4, 2989 (1976). 27D.Van Der Marel and O.A.Sawatzky, Solid State Commun. 55, 937(1985). 28W.J.L. Buysrs,J.E.Ejems and J.D. Garett,Phys. Rev. Lett. 55, 1223(1985). 29M.B.Maple,J.W.Chen,S.E.Lambert,Z.Fisk,J.L. Smith,H.R. Ott,J.S.Brooks and M.J.Naughton, Phys. Rev. Lett,54, 477 (1985). 30B.Batlegg,D.Bishop,B.Golding,C.M.Varma, Z. Fisk, J.L. Smith,and H.R. Ott, Phys.Rev. Lett. 55,1319 (1985).
Solid State Communications, Vol.57,No.8, pp.553-557, 1986. Printed in Great Britain.
SUPERCONDUCTIVITY
Institute
of Physics,
Department
0038-1098/86 $3.00 + .00 Pergamon Press Ltd.
OF TRIPLET BIPOLARONS
Yan-Min Li Chinese Academy of Sciences,
of Physics,
Li-Yuan Zhang Peking University,
Received 3 November
BeiJing,
BeiJlng,
China
China
1985 by W. Y. Kuan
Superconductivity of triplet blpolarons has been investigated in a phase analogous to the superfluid A-phase of 3He by use of the broken-symmetry Hartree approximation. The transition temperature T c , the chemical potential and the order-parameOers have been obtained, and the specific heat and the thermodynamic critical field have also been calculated. It has been found that there exist two energy gaps in the superconducting state. In the normal phase there also exists a gap, whose magnitude is consistent with the recent experimental evidence of the heavy-fermlon superconductor tHFS) , UPt% . Temperature dependence of the critical field of anothe@ HFS UBel% , Which is not yet understood, can also be qualitatively~accounted for by this model. These show the possibility that triplet bipolarons might be formed in the HFSs.
I. I n t r o d u c t i o n
s o - c a l l e d heavy-fermion superconductors (HFSs) recently discovered,
The local electron pairs, i.e. bipolaro~s, have attracted much attention and many investigations have been made on the singlet bipolarons for the possibility of having hightemperature superconductivity and its various interesting properties.
20
~d
such as CeC~2Si219, UBe13
UPt321, hay? been ~ g a r d e d as a klnd
of triplet superconauctors . Kulik 25 has suggested the possibility of triplet local electron pairs. In this communication we will first consider the possibility of triplet bipolarons as the ground state, and derive the effective pair-Ramiltonian of triplet bipolarons (Section YI). In Section IIl, the Hartree approximation via the Bogoliubov variational principle 24 will be used to discuss superconductivity of triplet bipolarons. The transition t e m p e r a t u r e , the chemical p o t e n t i a l and
Anderson I first proposed that the singlet bipolaron on the same site could be formed as a result of the strong local lattice deformation which produces centers of negative effective correlation energies (the -U centers). Many electrical, magnetic and optical properties of amorphous materials can be interpreted by this kind of on-site bipolarons 2-4. Some other possible mechanisms of forming the g U centers have been suggested and discussed9-~-. Theoretical investigations of superconductivity of the on-site bipolarons have also been made I0-11 • Rice and Sneddon 12 ascribed BaPbl-x BixO 3 as an example of this kind of superconductors.
the order~parameters are o b t a i n e d . I n S e c t i o n IV. the s p e c i f i c heat and the thermodynamic critical field will be calculated. The concluding remarks is given in Section V.
II. EFFECTIVE HAMILTONIAN We first consider the ease of intersite bipolarons. The essentials for the formation of them is the electron-phonon coupling. When the bipolarons are formed, two atoms occupied by one electron~pair lie closer than the other atoms, wlth a canonical transformation which removes the electron-phonon coupling, one can derive an e~ective Hamiltonian
Lakkis et al. 15 extended the idea to the intersite slnglet electron-palring, and used it as the ground state of Ti407. The similar ground state of vanadit~m bronzes was proposed later by Chakraverty et al.14. Alexander and Ranninger 15 discussed the excitation and superconductivity of this kind of bipolarons. LiTi204 16-17 is considered as a superconductor induced by them.
H=PIo+
We have pointed 18 that in the case of the interslte pairing, the triplet biolarons are almost as stable as the singlet blpolarons, and will have great effects on the thermodynamic properties of bipolaronlc systems. Therefore it is necessary to investigate superconductivity of triplet bipolarons. The
~,÷ w
(v,
,
+
+
+
+
W
±
•
n...
+
where m labels the pair of atoms and the • 553
SUPERCONDUCTIVITY OF TRIPLET BIPOLARONS
554
+
t ,2 specifies the atoms in the pair. Cm~cr
,
Ca@or and mm~Cm+oco4~m~; are, respectively, the creation, destruction and number operators associated to the electronic state localized at the ~th atom with mr-spin in the mth pair. H o describes the intra-pair motion of electrons; H I and W are, respectively, the inter-pair hopping and the sum of the inter-pair Coulomb interaction and the inter-pair exchange interaction of electrons. We will treat H o as the unperturbated Hamiltonlan, H I and W as the perturbations. The elgenvalues of He can easily be obtained 18. The lowest energies of singlet and triplet blpolarons when unperterbated, respectively,
Vol. 57, No. 8
I
..
~is the dissociation energy of triplet bipolarons when unperturbated. If the lattice is ordered, 6 m is independent of m. When Jmm' is negative, ferromagnetism can happen; while Jmm' is possltive, antiferromafnetlsm may appear. In this communication we only consider superconductivity of triplet blpolarons in the ordered lattice. Coexistence of magnetism with superconductivity as well as the effect of the disorder will be discussed elsewhere.
are
Es = (V+v1+J1~2_112
[(U_vI_J I)2 + 16t12
Et = v1-J1
]I12 (2.2a) (2.2b)
Normaly only when v I ~O, may bipolarons be stable 18. In the ordinary blpolaronlc materis/s, tl is small, so U-v1-J1>>~t1~. From E t < F~s, the condition of the triplet blpolarons as the ground state is approximately given by J1 > 2tl/(U+I.Vl I )" (2.3) since electrons are very localized, the exchange interaction is not much affected by the eleotron-phonon interaction. In some cases, (2.3) could be satisfied. When the perturbations exist, the stability of the bipolarons generally decreases. But in some materials, expecially in the quasi-lowdimensions/ ones, stable bipolarons can still be formed 18. In this paper, we will assume that triplet bipolarons are deeply bound. When the minimum dissociation energy of triplet bipolaronsAl>>T (kB = 1 ) , the density of free electrons is considerably small, and will be neglected. We Can then project the origins/ Hamiltonian (2.1) into the eubspace of triplet bipolarons to obtain the effective pair-Hamiltonian. For convenience to discuss and without losing physical meaning, we will only consider the phase having two kinds of pairs ~}, ~ , analogous to the superfluid A-phase of ~He 25. Introducing the triplet blpolaron operators
Am = Cml.f- Cm24" Bml = Cm14, Cm24, ,
(2.4)
and using t h e c a n o n i c a l t r a n s f o r m a t i o n 15 o r t h e p r o j e c t o r a p p r o a c h 26, one can o b t a i n t h e effective triplet pair-Hamiltonian
H = ~ E m nm - mm' tmm' (Am Am' + Bm
For the on-slte bipolorons, the Hamitonian (2.1) can still describe their motion as long as ~ 1,2 is understood as the two degenerate orbltals28f each atom. Recently, Marel and Sawatzky " considered the possibility of triplet pairs as the ground state. When the onsite triplet bipolarons are deeply bound, the same type palr-Hamitonlan as (2.5) can be obtained. Thus, our discussion can be applicable for both intersite and on-slte triplet blpolarons. In the expression of vmm' of (2.6), the term in the bracket should be small compared to the first term in the superconductors, otherwise the systems normallYlbecome the charge-density-wave insulators O. Thus we have yam' --~Itm~ m' ~'~ 2 / ~ . For on-site blpolarons, one can easily obtained from (2.6) that tmm, ~ I/2 Vmm'. This conclusion will be used later.
Ill. SUPERCONDUCTIVITY In this section, we will discuss superconductivity of triplet bipolarons according to pair-Hamiltonian in (2.5). For the sake of convenience, only the nearest neighour interaction of pairs are considered. So the subscripts of parameters tam , 9 Yam, and Jam, are discarded from now on. The superconducting order-parameters are defined as h
C~m
A
=
'
~m =
~Bm>
,
(3.1)
which determine the transition temperature, at which 0Lm, ~m = O. The broken-symmetry Hartree approximation is introduced via the Bogoliubo¥ variational principle for the thermodynamic potential 24
Bm' ) + (5.2)
+ ~I~
Cvmm, ~= ~=, + Jm=' s= s,),
C2.5)
where nm"
Am =
£m = ~ 6"o +
tram, =~'z'
+ Bm ~m' 8m " Am
Am - Bm Bm,(2"6)
v,-J~-X='~-Itm=,='=']2/A
t,,i=,~=#~<~,~,<< , I A
,
where N e is the number of pairs, ~! =I/T, is the trial Hamiltonian, <... "2@ the average , with the density matrix ~o = exp[-~'(Ho-/~el~/Tr~z~-~ -~Ne) ,)~the chemical potential. When the system is homogeneous, i.e. ~ m and ~m is independent of the subscript m, the brokensymmetry Hamiltonian H O is chosen as
A,-~. A,-gzRm-~B m ~nm), (3.~1
SUPERCONDUCTIVITY OF TRIPLET BIPOLARONS
Vol. 57, No. 8
where ~ I , 62 and ~ are variational parameters to minimize the thermodynamic p@tential h o • Since at each site there are only three possible states ~ 0 > , |0> and ~ 0 > , the opereters ~, ~ and ~ can be denoted as
/ooo~ A-UOOl
/coo I ~* /ool /
~O00t,
/loo/ ~* |ooo/
~O00l,
~001/
06 05
where = ~
, E ± = ( A± R , , [ ~ 2 + 4 ( i~,l 2--)/,
.
R)I2,
,k.
-~Ne)]+
where n = N ~ N is the average density per site, N the number of pair sites, z the number of the nearest neighbours, n, ~ a n d ~ are, respectively, given by
e ~''~/~15h({~11) T_( P_~ ' + '7~IR" n =--,n,o,=~ i[, ,'~/~ , I'
(5.7~)
c h ( ~ RI2)]
t+2e ~'vz c~('~',~/z')
--4
34 2
Of
Then the trial thermodynamic poten-
+2e
/
(3.~)
@'(~
i-5
0.2
(3.4)
+ 1~,=12- ))1/2,-
~nTr exp [-
-~
o '-'o.3
tial per site is readily derived as
i~Ao/N = -I/(N ~')
6
0,4
Substitute (3.4) into (3.3), one can easily obtain the elgen-energies of ~m +// n m
~0 = h
555
0
I
I
I
I
I
I
I
t
I
0.1
Q2
0.3
0.4
0.5 n
0.6
0.7
0.8
0.9
Fig. 1. Transition temperature T c (the solid surve) and the ratio of the gap A I and T c (the dashed curve). The unit of T o is iS. From (3o11,b), we can obtain the asymptotic behavieurs of the chemical potential for n <41 as vzn - (I-2n) tz-tz exp [-(1-n)tz/T~, T<
i
(3.13)
/(=
'
vzn
n/2 (tz)2(Tc-T)/Tc~
(I-3n/2) tz + Tc
'
- T <4 To .
The asymptotic behavlours of the order-parameters are A minimization of the trial thermodynamic potential El S yields the optimum values
~
= tzo~
~z 7~
= tz~ =vzn
I coI ItS (To-T)/TcZ] '/2, Xc-T <<7~ .(S'14) (3.8)
If we substitute (3.8) into (3.7a-c), we can get the couplig equations for the chemical potential and the order parsmeters. These equations should be solved self-consistently. If (3.8) is substituted into (3.6), the free energy can be obtained, by which we can discuss the thermodynamic properties in the superconcucting state. This will be given in the next section. From (3.7b) and (3.7c), we have
= ~,
(3.9)
which is the trivial solution without polarization. When T > To, let ~ = ~ = O,we have
~= V Z n -
T In[2
(1-n)/nJ, T ~ T c.
(4.10)
(I-2n) tz + R exp ( - 1 ~ ' / 2 )
l[2sh(~'R/2~(3.11a)
R ~[2(1-2n) t z -.~.] t h (~'R/2)
(3.11b)
Let Or, ~ - * 0 in (3.11a,b), we obtain the superconducting transition temperature T c = tz (I-3n/2) / ~n[2(1-n)/n] ,
0,5
-0.5
0.4
n = 0,2
(3.12)
which agrees with the result by Eulik 23. The dependence of T c with n is shown in Fig. 1 (the solid curve). T o is independent of v~which results from the Hartree approximation, since the Hartree approximation neglects the correlations between electrons. But the result so obtained is resonable when n << I.
-0.6
~ 0.2, " -
-0,7
_o2_ o.~ r- . . . . . . ~ ,
~,
~_
o.,
oJ
-0.8
%%
o
when T < To, (3.7) can be rewritten as k=
The t e m p e r a t u r e d e p e n d e n c e s o f t h e c h e m i c a l potential ( t h e s o l i d c u r v e s ) and t h e o r d e r p a r a m e t e r s ( t h e d a s h e d c u r v e s ) a r e shown i n Fig. 2.
I
T~
-0.9
T
-I
Fig. 2. Temperature dependences of the orderparameter ~0~ (the dashed curves) and the chemical potential ,41(the solid curves ). Note//is actually ( ~ - v z n ) /
(tz). In the conventional BCS superconductors, as we know, there are a lot of normal electrons in the superconducting state, and only a part of electrons form Cooper pairs. The superconducting gap, which appears only when the Cooper pairs are formed, is defined as the average energy per electron needed to destroy a Cooper pair. In the bipolaronic superconductors, however, "pairs" exist before the
Vol. 57, No. 8
SUPERCONDUCTIVITY OF TRIPLET BIPOLARONS
556
occurring of the superconducting state. This to the different definition of the superconducting gap from BCS one. We will discuss it in the next section.
b
will lead
IV. THERMODYNAMIC PROPPRTIES From (3.6) and (3.8), we obtain energy
the free
FJm=oolIN.Jr~(1-n)~+ t , ( I ~ l 2 + ;~ 12) - T Z n + 2 axp (BgJ2) oh (@'R/2); + vzn212.(4.1) The specific heat in the superconducting state is then o b t a i n e d a s
C m - T~2
re
- - 2Ntz
~T 2
al~(~
(4.25
o
~
B~ ^
using
(3.14)
~ .= 2 e ~'(~-v )
(4.7) (4.~5
e.p[-(1-n) tzlT] }
activation type energies, ~ 1 : t z , ~ = ( 1 - n )
tz,
which result from the three eigen-snergies in the superconducting state, as shown in Fig. 3. AccordinF to ~ . 5 ) we have
E+ (0)~ (l-n)
t,,~=
A 1 = R(o)=
tz
where R'= [ (I-2 e ~?~-v)) + 8 ~ ~']/ j I~2 and ~ is the chemical potential. For n << I we have < n i n~ j ~ ~ n 2 exp (-v/T). (4.8) Using (4.6) and (4.8), we get Cn(T) ~
I/2
Nn 2 (v/T) 2 exp
(-v/T),
(4.9)
(4.4)
<
from which we can see that in the normal state there also exists a gap, 43 = v. According to the last discussions in Section ll. v.~-~2t, and noting z = 2 in the one-disensional case, we obtain A 3 ~ A I . Consequently we have
E.
l
A,
l
Eo
A2
A31Tc~_AIITc~Z.8
Fig. 3. Eigen-energies in the superconducting state. Thus there exist two gaps in the superconducting stats. This contrasts with the case of singlet bipolarons, where there is o~ly one gap, whose magnitude is equal to A I '. appearence of the superconducting gaps shows that the superconducting condensation is different from the Bose one without any gaps. When T -*T c , we have =
2 N)C((O)I2(tz/Tc 52 .
(4.5)
The temperature dependence of C (T) is shown in Fig. 4 (the heavy soIid curves) In the frame of the Hartree approximation, the specific heat in the normal phase is equal to zero owing to neglecting the correlation. ~owever, we will obtain the non-zero specific heat when the correlation is included. If we do not consider the effect of the hopping term of pairs, the specific heat is On (T) me
(4.10)
It is interesting to compare this result with the recent experimental evidence by Buyers et al. 28, who have measured the spin-fluctuation spectrum of the HFS UPt 3 with neutron scatterinF just above the superconducting transition temperature. They found that the activated intensity of the spin fluctuations is consistent with the existence of a gap]ike structure in the normal phase, and the gap.~-~ 3To. This is very well consistent with our theoretical result. For the thermodynamic critical field, Using the well-known formula
He 2 /8~=
I/V (Fo(nl-Fo (s))
,
(4.11)
we obtain the asymptotic behaviours of H c ~Hc (0)
(]-TI(2To)) ~
~<
n ~ [(Tc-T)ITc] ~
Tc-TZ4T c (4.12)
= v12 .Y-a.<~i ~j> I a T . ( 4 . 6 ) sJ
I n t h e one-demensional case, the c o r r e l a t o r < n i n i > can be e x a c t l y c a l c u i a t e d . I t can
that
.
The density dependence of ~I/TC is shown in Fig. I (the dashed surve). ~ i / T c is not sensitive to n. In the normal density regime, the ratio is about 3.0. Yt can be expected that this conclusion is also valid in the higher dimensional case.
Hc = ( H J 0 ) I
be sho~n
~- Z c ~'(~-v~
(-tzlT) + (I-2n).
one can see from (4.3) that there are two
O(T c )
F.
and (4.2), for n <<1 we have
C = " (tzlT)2{exp
(n
r"
Fig.4. Temperature dependences of the specific heat C (the heavy solid curve) and the critical field H c (the thin curve). The dashed curve is the specific heat in the normal state.
where He(O) = (8~ntzN/V) ~.
The temperature
dependence of H e is shown in Fig. 4. When T-*Tc, - ~Hc/ ~T--~oo; and when temperature is
557
SUPERCONDUCTIVITY OF TRIPLET BIPOLARONS
Vol. 57, No. 8
somewhat lower than To, Hc increases linearly on cooling. We would like t o qualitatively compare these results with the experimental evidences of the upper critical field of another HFS UBs1329, which cannot be accounted
how, s u p e r c o n d u c t i v i t y o f b i p o l a r o n s i s much different from t h e c o n v e n t i o n a l one. It can be e x p e c t e d t h a t many a n o m a l o u s b e h a v i o u r s o f e x o t i c s u p e r c o n d u c t o r s c a n be a c c o u n t e d f o r by t h i s m o d e l .
f o r by c u r r e n t t h e o r i e s o f e i t h e r c o n v e n t i o n a l or p-wave s u p e r c o n d u c t i v i t y . It is found that
We have adopted the Hartree approximation, which neglects the correlation between electrons. Therefore only when n <~I, is the result o b t a i n e d r e s o n a b l e . Triplet blpolarons can show magnetism,which may lead t o coexslstence with superconductivity. Recently Batlogg et al.30 reported measurements of the ultrasound attenuation of (U, Th) Bet3. They found a A-shaped attenuation peak below To, which may correspond t~,a antiferromagnetic trmnsion. Buyers et s-l.~v
HO~ (T) of UBe13 has a considerably large initial slope and a linear dependence with T below about 0.9 T c. This behaviour is qualitatively consistent with our result. V. CONCLUSIONS
I n t h i s c o m m u n i c a t i o n we h a v e d i s c u s s e d superconductivity of triplet bipolarons in a
also found that the spin-fluctuations Just a b o v e Tc a r e a n t i - f e r r o m a g n e t i c . Coexistence o f m a g n e t i s m and s u p e r c o n d u c t i v i t y o f t r i p l e %
hase ~ He.
similar to the superfluid A-phase of It has been found that there exist two g a p s in t h e superconducting s t a t e . In the
normal phase there
also
exists
bipolarons will be discussed in a separate publication.
a g a p , whose
magnitude is consistent with the experimental result of UPt328. The thermodynamic critical field H e has a infinite slope at To, and has a linear dependence with T at low temperature. This hehaviour can qualitatively accounted for the unusual shape of the upper critical field of UBe13. Consequently, it is necessary to investlgate theoretically the possibility that triplet blpolarons may be formed in HFSs. Any-
ACKNOWLEDGMENT---One of authors (YML) would llke to express his sincere thanks ts professor Y, Y. Li~ Institute of physics and Professor K. A. Chao, department of Physics, ~,iversity of LinkSplng, Sweden, for their having interested him i n t h e b i p o l a r o n p r o b l e m and many s t i m u l a ting
discussions.
REFERENCES IP.W.Anderson, Phys. Rev. Lett. ~ , 9 5 3
(1975)
2p.A. Street and N.F.Mott, Phys. Rev. Lett. 35, 1293 (1975). 3p.w. Anderson,J.Phys.
(Paris)37,C4-339(1976).
4D.Adler and E.J.Yoffa, Phys. Rev.Lett. 36, 1197 (1976). 5p.Pincus, Solid State Commun. 11, 51 (1972). 6 p.M.Ckaikin,A.F.Garlto and A.J.Heeger, Phys. Rev. B 5, 4966 (1972).
17K.W.Ng, R.N. Shelton,E.L.Wolf,
Phys.Lett. A 110,
423 (1985). 18y. M. Li and K.A.Chao,Acta Phys. Sin.(Chlnese) 33,273 (1984) (English trans: Chin.Phys. 4 , 804 (1984); and to be published. 19F. Steglish, J.Arts,C.D.Bredle,W. Lieke,D. Meschede,W. Franz and H.S Chafer, Phys. Rev. Lett. 43,1892 (1979). 20H. R. Ott,H. Rudigier, Z. Fisk and J.L. Smlth, Phys. Rev. Lett. 50,1593 (1985).
7G.Beni,P. Pincus and J. Kansmori, Phys. Rev. B 10, 1896 (1974).
21G. R. Steward,Z.Fisk,J.O.Willis and J.L, Smlth, Phys. Rev. Lett. 52,679 (1984).
8G.V. Ionova,E.F.Makarov and S.P. Ionov, Phys. Status Solidi (b) 8_!,671 (1977).
22p.W.Anderson,Phys. Rev.B 30,1549 (1984); C.M. Varma,Bull.Am. Phys. Soc.29,404 (1984). 23I. O.Eulik, Physics 126B,280 (1984).
9K. Weiser, Phys. Rev.B 25,1408 (1982). los. Robaszkiewicz, R. Micnas and K.A.Chao, Phys. Rev. b 23, 1447(1981); ibid,2_~4,1579(1981). 11 I.O.Kulik and A.G.Pedan, Sov. Phys. JETP 52,
742 (1980). 12T.M. Rice and L. Sneddon, Phys. Rev. Lett. 47, 689 (1981). 13S. Lakkis,C. Schlenker, B.E. Chakravsrty,R. Buder, M. Marizlo, Phys. Rev. B 14,1429 (1976). 14B.K.Chakraverty,M.J.Sienko and J.Bonnerot, Phys. Rev.B 17,3781 (1978). 5A.Alexandrov and J. Ranninger, Phys. Roy. B 233, 1796 (1981); ibid,24,1164 (1981). 16D. C.Jonston,J.Low. Temp. Phys. 25,145 (1976); D.C.Jonston,R.W. McOallum,C.A.Luengo and M.B. Maple, ibid, 25,177 (1976).
24S.V.Tyablikov, Methods in the Quantum Theory of Magnetism (Plenum,New York, 1967).
25A.J.Leggett,Rev.Mod.Phys.47,331
(1975).
26V.J.Rmery, Phys. Rev.B1_~4, 2989 (1976). 27D.Van Der Marel and O.A.Sawatzky, Solid State Commun. 55, 937(1985). 28W.J.L. Buysrs,J.E.Ejems and J.D. Garett,Phys. Rev. Lett. 55, 1223(1985). 29M.B.Maple,J.W.Chen,S.E.Lambert,Z.Fisk,J.L. Smith,H.R. Ott,J.S.Brooks and M.J.Naughton, Phys. Rev. Lett,54, 477 (1985). 30B.Batlegg,D.Bishop,B.Golding,C.M.Varma, Z. Fisk, J.L. Smith,and H.R. Ott, Phys.Rev. Lett. 55,1319 (1985).