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Near-infrared absorption of YBa2Cu3O7−δ
PHYSICA
Physica C 215 ( 1993) 359-370 North-Holland
Near-infrared absorption of YBa2Cu307_a Evidence for Bose-Einstein
condensation
of small bipolarons
A.S. A l e x a n d r o v , A . M . B r a t k o v s k y , N.F. M o t t a n d E . K . H . Salje Interdisciplinary Research Centre in Superconductivity, University of Cambridge, Madingley Road, Cambridge, CB3 0HE, UK Received 24 February 1993 Revised manuscript received 17 May 1993
A significant change with temperature in intensity of the optical absorption spectrum of YBa2CuaO7_a in the range 2000 to 10000 cm-1 at the superconducting transition is shown to be a manifestation of Bose-Einstein condensation of small bipolarons. The sum rule for bipolaronic infrared absorption is derived and applied for the explanation of this unusual effect on the basis of the polaron theory of high-Tosuperconductivity. The temperature dependence of the normal state absorption is controlled by the thermal excitation of singlet inter-site bipolarons into triplet states. As determined from optical absorption experiments singlettriplet exchange energy has the same value as the spin gap discovered by Rossat Mignod et al. by neutron scattering.
1. Introduction
The physics of small polarons and bipolarons has become of particular interest recently in the context of theories of high temperature superconductors. Some microscopic models [ 1,2 ] suggest that charged bosons, formed by strong electron-phonon and electron-electron exchange interactions (latticeand spin bipolarons) might be responsible for the puzzling thermodynamic and kinetic properties of high-Tc oxides. Such characteristic features of high-To copper based superconductors as the curious absence of coherent effects and of the Korringa law in the nuclear spin relaxation rate, the "spin" pseudogap, unexpected enhancement below T¢ of low-frequency conductivity, linear in T resistivity are indicative of charged bosons (small lattice and spin bipolarons) [3]. One should add here the heat capacity [4,5], which is reminiscent of He-4 and unusual temperature dependences of lower and upper critical field as well as of the sound attenuation and velocity which however can be expected for charged bosons [4,6]. Unusual features of the oxygen isotope effect and of the softening of optical phonons at energies above the (pseudo) gap [7 ] clearly show that phonons arc involved, and high-T~ oxides are in the transition region from a polaronic Fermi-liquid to a bipolaronic
charged Bose-liquid, as discussed by one o f us [ 2,8 ] (for other manifestations o f the Bose-liquid behavior o f superconducting oxides see also ref. [9 ] and recent review papers [ 10-12 ] ). In this paper we shall argue that the recent observation [ 13 ] o f the significant influence o f the superconducting phase transition on the near infrared absorption with the characteristic frequency z,___0.7 eV is a clear manifestation o f Bose-Einstein condensation in YBa2Cu307_ 6- Using a simple but rather general model we show how the superfluid phase transition o f a Bose-gas on a lattice can influence the high-frequency optical absorption. With the sum-rule for the optical conductivity of a coupled electronphonon system we demonstrate that the effect is a consequence o f the coherent tunneling and Bosecondensation o f small bipolarons. Earlier evidence for small (bi)polarons in copper based oxides comes from the photo-induced infrared absorption, measured by Heeger's and by Taliany's groups [ 14,15 ] and from the observation o f Sugai [ 16 ] o f both infrared and Raman-active vibration mode. There is an indication o f the fine structure o f the incoherent part o f the angle resolved photoemission spectra ( A R P E S ) [17], which can be explained as a superposition o f multiphonon and elec-
0921-4534/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
360
A.S. Alexandrov et al. / Bose-Einstein condensation of small bipolarons
tronic excitations in accordance with the polaron theory of ARPES [ 18 ]. Several attempts were made to eliminate bipolaronic superconductivity [ 19 ] as mechanism for highTo, however. The widespread argument against the electron-phonon coupling in high-T¢ materials, based on the absence of the oxygen isotope effect [20 ] seems to be irrelevant in view of more recent measurements [ 8 ]. A small temperature slope of resistivity can be interpreted as caused by a weak scattering of carriers by phonons [ 21 ] without telling much about the band structure renormalization due to the strong coupling with high-frequency optical and molecular vibrations. The fact that these phonons are strongly coupled with electrons is confirmed by tunneling [ 22 ], Raman [ 16 ] and photo-induced infrared [ 15,14 ] measurements. The carriers are heavy due to this coupling ( m * ~ - 10 m) as is confirmed by the optical measurement [23 ], and this mass enhancement could be responsible for a low value of 2 [21 ] estimated from d p / d T . Thus the above mentioned objections may be irrelevant. The more serious problem for the bipolaronic mechanism of high-To is the possibility of the coherent tunneling of (bi)polarons with a reasonable value of the effective mass. Chakraverty et al. [ 24 ], Emin [25] and Anderson [26] assert that their bandwidth should not exceed 10-4-10 -5 eV, and then, in accordance with the theory [ 19 ] the maximal T¢ attainable with small bipolarons should be a few K or less. However, this is not the case for the intermediate value of the coupling constant 2 ~- 1 if the high-frequency phonons ~o_~0.1 eV are involved in the polaron formation. With high-frequency phonons and with a relatively narrow bare band (such that the adiabatic ratio D / o J < 10, D is the bare band half-width) one can expect (bi)polaronic bandwidth of the order of a few hundreds K, and the same estimate for T¢. It is worth mentioning that a similar estimation for the polaronic bandwidth ( 1 0 - 2 - 1 0 - 4 of the electronic bandwidth) has been obtained by Holstein in his original paper [ 27 ]. The condition for the small polaron formation, known in the literature, is the smallness of the transfer integral J compared with the polaronic binding energy Ep [27]. Because the optical phonon frequencies in metal oxides are usually high (0.05-0.1 eV) the polaronic binding energy is large ( E v = 0.2-0.4 eV) and
thus the allowed value for the bare electronic bandwidth compatible with the small polaron formation is high enough being of the order of 0.5-1.0 eV. In this case Holstein's estimate of the small-polaron bandwidth and also some variational and MonteCarlo simulations give the value of a few hundred K (>0.01 eV), which is at least two orders of magnitude larger than those of Chakraverty, Emin and Anderson. Of course, a short-range Coulomb repulsion (intersite) should be suppressed, approximately below 1 eV to ensure mobile bipolaron formation, and that is quite feasible because the high-frequency or shortrange dielectric constant in metal oxides is usually large (~> 5). We now going to compare our arguments with experiment. To our knowledge there is only one clear observation of the coherent small polaron tunneling in a reasonable narrow band (or the order of 100 K). In the case of WO3 Gehling and Salje [ 28] found at temperatures below 130 K a crossover from a regime of small polaron hopping conductivity with constant activation energy to a regime with zero activation energy, thus expecting "polaronic band conduction" in accordance with the canonical theory of the small polaron transport developed by Tjablikov [29], Yamashita and Kurosawa [ 30 ], Holstein [ 27 ], and in a more general form by Firsov and co-workers [ 31 ]. As we show in this paper that the experiment by Dewing and Salje [l 3] confirms the possibility of the coherent (bi)polaronic tunneling in the structural CuOE-plane. The observed infrared absorption is closely related to the conductivity in the CuOEplane and are much less related to the CuO-chains. The experimental evidence stems from the observation [32,33] that doping of YBaCuO with Zn in the planes does shift the spectral function proportional to the degree of doping whereas there is no effect from doping the chains with Fe. In section 2 we discuss a simple model of bosonic pairs on a lattice to show with general symmetry arguments how statistics leads to the pronounced influence of Bose-Einstein condensation on the optical conductivity. We derive the conductivity sum rule for bipolarons in section 3 and discuss the integrated optical absorption of on-site and of inter-site bipolarons in section 4 and section 5, respectively. In section 5 we compare our theoretical results with ex-
A.S. Alexandrov et al. ~Bose-Einstein condensation of small bipolarons periment. The singlet-triplet exchange energy, found from this comparison, has the same value as the spin gap, observed with neutron scattering [ 34 ]. In conclusion some alternative explanations, including the BCS and Hirsh's "hole superconductivity" [ 35 ] are discussed.
2. Qualitative consideraton with the "toy" model Let us discuss a simple but rather general model which shows a hundred percent decrease of the optical absorption while temperature changes from Tc to zero: an ideal gas of molecules, composed of two identical panicles (electrons) with charge e on a lattice. To stabilize their bound state one should assume that the pair attraction operates, compensating the Coulomb repulsion. All molecules are in the ground internal state with the binding energy being much higher than temperature. The periodic crystal field is assumed to be large compared with the binding energy, so all molecular states, including continuum, are built from the single-band single-electron Bloch wave functions. Because of the translation symmetry the molecular wave function is a two-particle Bloch function described with the total quasi-momentum K: 7/i:(r~, rE ) = exp (iKR) U ~ ( R , r ) ,
( 1)
where R = (rt + r 2 ) / 2 , r = r l - r 2 , and Ux(R, r) is periodic with the lattice translations. At low temperature only states with small K (near F point), are relevant, so one can expand Ux in a series of K:
U~(R, r) ~- U~(R, r) + K.D':(R, r)
(2)
with D = limx~oVxUx. A linear interaction of a molecule with light is described by the Hamiltonian: Hint-
-icE my
(v~ + v 2 ) ,
(3)
361
where D~:= ~ f dR dr( U:o(R, r)*VR'Di(R, r) - U~o(R,
r)VR.DY(R, r)*). As usual one can neglect a field inhomogeneity (photon momentum) and put h = c = 1. From eq. (4) the rate of the optical absorption is proportional to the kinetic energy of a molecule, and an immobile molecule, composed from two identical particles cannot absorb, which is a trivial consequence of a parity and spin conservation. The dipole matrix element for the F point is equal to zero because all singlet states of a molecule with K-- 0 are even and all triplets are odd under the inversion transformation, R, r - - , - R , - r , for a lattice with the inversion symmetry. This is similar to the dipole-forbidden singleelectron transitions in semiconductors. The matrix element, eq. (4), increases with increasing K. It is now the essence of our argument that the occupancy at K S 0 is a thermodynamic function which "sees" the Bose condensation and, thus, allows us to measure the condensation directly via optical absorbance [ 36 ]. The Fermi-golden rule yields the optical conductivity: 4xe 2
a(v)= ~mvEk
~ ID~:IZc~(E:--ei-- v) ,
(5)
where t~o, is the energy spectrum of a single molecule, including its continuous part and K: E k = ~ -~-mmn ( K )
(6)
is the kinetic energy of all molecules, which obey the Bose-Einstein distribution with the chemical potential #, n ( K ) = ( e x p ( ( K 2 / 4 r n - l z ) / T ) - 1 )-1. Integrating eq. (7) over frequency we obtain the conductivity sum rule [37 ] for our system:
I ( T ) = f du a(v) = rce2a2Ej,, o
(7)
where
with E being an electric field. I f the frequency is high enough the molecule is excited or dissociated by light, absorbing one photon to the final s t a t e f The matrix element of the transition is given by:
(J] Hint [i) - - ieE" K D~f, ?nip
(4)
4
a:= ~mm ~: IV'Jl2/(t:-ti) "
(8)
a is a temperature independent characteristic length of the order of the molecular radius (the cubic lattice is assumed). Thus the optical absorption of our "toy" model
A.S. Alexandrovet al. ~Bose-Einsteincondensationof smallbipolarons
362
(holes). Starting from a single band Frohlich Hamiltonian:
Toy model 3.0
I4= E E(k)cLck,~
2.5
k,$
+ (2N) -~/2 ~
2.0
+Hph
-~1.5
E 0.5
0.0 0 .0
0.5
1.0 T/Tc
1.5
2.0
Fig. I. Temperaturedependenceof the integrated optical absorption of free Ix)sons on a lattice, composed from two identical particles. shows a drastic temperature dependence, being proportional to T s/2 below the Bose-Einstein condensation temperature To= 3.3nE/3/2m and linear well above T~, fig. 1, (n is the molecular concentration):
I(T¢) -
F(5/2)((5/2)
exp(x-/t)-I
'
(9)
where t = T~ T~ is the reduced temperature and/z = 0 for t< 1. In the normal state ( t > 1 ) the following equation holds for #:
0
+ Vc .
(11)
Alexandrov and Ranninger [ 19 ] derived the bipolaronic Hamiltonian, describing the repulsion and the tunneling of small bipolarons, which are hard-core bosons on a lattice:
1.0
i
7(q)~o(q)ci+¢,sck,s(d¢+dtq)
k,q,s
xl/2 dx
exp (x-- #) -- 1
=t-3/2F(3/2)((3/2) .
Hb= ~
m,m'
(-t=m,b~bm,+v (2) , m,m'
× n=(nm,- 1) +4Vr~,,.,nmnm,),
(12)
Hph=Y~q to(q)d~dq is the phonon energy, n==b~bm is the bipolaronic density operator, b== c.,,tc=,+. The interaction vm,=, does not depend
where
on the kinetic energy and includes the direct (density-density) Coulomb (Vc) repulsion and the attraction via phonons between two small polarons on different sites m, tram' and ,,(2) , ~ m , m ' are the bipolaron transfer integral and the bipolaron repulsion due to the virtual polaron exchange, correspondingly, both of the second order in the electronic kinetic energy E(k) [11]: oo
tm,~, = 2 i
J dz( #m,m'(Z)~=,m' ( 0 ) )
pheXp(--izJl:)
0
(10)
,
(13)
oo
.Um,m' (2) = 2 i | dz(#,,,,,,, (z)6,,,,,,,,(0))phexp(-iAz) , The temperature derivative of the integrated optical conductivity (or absorption), dI(T)/dT is proportional to the heat capacity of an ideal Bose-gas, tabulated in textbooks. The condensed fraction of molecules, composed of two identical particles, does not absorb light, if bosons are condensed at F point of a lattice with the inversion symmetry.
o
(14)
where 6m,,,,, (z) = T,,,,,,,,einpb~ 1
× exp(--r--- ~ \x/2N × e - iH~h~,
3. Conductivity sum rule for on-site small bipolarons Now we know at least one realistic example of charged bosons, composed from two electrons
• " 7(q)d~[e-~'m-e-l*n"l-H.c.)
( 15 )
( )vh is the averaging with the phonon density matrix and T,,,,,~,=(1/N)~kE(k)exp(ik(m-rn')) being a bare electronic transfer integral between sites m and m'. The transformation of the Frohlich Hamiltonian,
363
A.S. Alexandrov et al. /Bose-Einstein condensation of small bipolarons
eq. (11 ), into bipolaronic one, eq. (12), is performed with two canonical transformation, exp $1 and exp $2: $1 = (2N) -1/2 ~ q,m,$
y(q)c~,scm,s(d~e-lq=-n.c.) , (16)
(flO=,=,(O)c~,~c=,,~lp) (&)~,. =
m,=',$ Z
E~-Ep
(17) '
where Ezp and I f ) , IP) are the energy levels and the eigenstates of the original Frohlich Hamiltonian, eq. ( 11 ) without the kinetic energy. The first canonical transformation eliminates the Frohlich interaction and the second one eliminates the one-electron hopping. The bipolaronic Hamiltonian, eq. (12) describes low-energy excitations of a strongly-coupled (,~> 1 ) many electron-phonon system at low enough temperature T<<,4=2Ev- V¢, where Ep is the familiar polaronic shift of the electronic band: 1
Ep= ~--N~ y(q)2o)(q) ,
(18)
with re(q) being the phonon frequency and N t h e total number of sites (cells). Because the Frohlich interaction is diagonal in the site-representation it does not change the conductivity sum rule, derived by Maldague [ 38 ] for a single band Hubbard model (for more details see Appendix A): I(T)=-~(-
T.,,,,,c*,,,,~c,,,,,),
~
the nearest neighbor hopping) and n the bipolaron atomic density. The conductivity sum rule, eq. (20), includes optical absorption due to the bipolaron dissociation with and without phonon emission and absorption. It includes also the low-frequency Drude conductivity of small bipolarons tunneling in a bipolaronic narrow band (the half-bandwidth w = zt=,,~,<<,4) due to their scattering by phonons, impurities and by each other without their dissociation. If one is interested only in the optical path /opt ( v ~ ,4 or higher) one should subtract the bipolaronic Drude contribution ID~de from the total absorption intensity: I ( T) =Iopt( T ) + IDrude( T ) .
(21)
To derive the integrated bipolaronic Drude conductivity, IDr~d~ one can apply the sum rule to the bipolaronic Hamiltonian, eq. (12), keeping in mind that bipolarons have charge 2e (Appendix A):
IDruae(T) - 4n~2a2 /\m,=, E t=,=, b~,b~,,). 2
(22)
Subtracting eq. (22) from eq. (20) one obtains:
/opt = Ige2a2(v(2)( n-
( nmnm+,) )
(19)
In,m',$
where a is a lattice constant. To calculate the kinetic energy of the strongly coupled electron-phonon system up to the second order in the transfer integral with the accuracy ~ (Tm,=,/Ep)2= 1/22 one can apply the polaronic $1 and bipolaronic $2 canonical transformations to eq. (19) with the following result: I(T)=
~
(2v'2) ( n- ( n,,,n,,,+,,) ) +2(Y~
(20)
with ~,,~2)~.,,~2)_ -~.,.., ..., z the coordination number (for simplicity we consider cubic or quadratic lattice and
4. Temperature dependence of the integrated optical absorption of on-site small bipolarons
Let us consider a low-density regime. It was demonstrated in ref. [39] that the ground state of our system is a homogeneous Bose-liquid, in which the tendency to the charge order is suppressed by quantum fluctuations if n < no. The critical value of the concentration nc above which the charge density wave develops turns out to be independent of the repulsion if the latter is strong v+ v (2) >> w. For a cubic lattice no---0.08. If the repulsion between bipolarons has a moderate value v+ v (2) < 3w the homogeneous Bose-liquid is stable versus the charge order practically in the whole density region.
A.S. Alexandrovet al. / Bose-Einstein condensation of small bipolarons
364
In a homogeneous repulsive Bose-liquid the short range pair correlation function g ( r ) = ( n (0) n ( r ) ) / n 2 is small; for example, in liquid He4 g(r) < 0.2 for r < 2.5 A [40]. The temperature dependent part of this correlator is even smaller (Appendix B). Thus the contribution to the conductivity sum rule, eq. (22) of the term quadratic in the bipolaron density is practically temperature independent and negligible for the homogeneous strongly repulsive Bose-liquid. In the dilute limit one can also neglect the corrections to the free particle energy spectrum, which are small while the gas parameter is small. These simplifications yield: /opt(T)
=
~e2a2((v(2)--w)n(T)
+f
2w
0
y-lexp(~/T)--I
)
'
where Nb(E) is the density of states in a narrow bipolaronic band, and 2w
n(T)=
"
0
Nb(E) de
y-lexp(~/T)-I
(25)
is the bipolaron atomic density determined by the chemical potential/t, y = exp (/~/T). The first term in eq. (24) describes the incoherent absorption of light accompanied by emission and absorption of many phonons. The frequency dependence of this absorption was discussed by Bryksin and co-workers [41 ]. The absorption m a x i m u m lies at a frequency 4Ep-V~. Comparing the m a x i m u m position with experiment [ 13 ] one obtains a realistic value for the polaronic level shift of the order of a few hundred meV. Bryksin et al. [ 41 ] ignored the tunneling motion of bipolarons, so their results are not valid in the temperature range compared with T~ as far as the temperature dependence of the absorption intensity is concerned. But the gross spectral features on a frequency scale of the order of Ep are, of course, temperature independent. The second term in the integrated absorption, eq. (24) is due to the coherent tunneling contribution and identical to that discussed above within the "toy" model. This term is responsible for the influence of the superconducting transition on the optical absorption.
In the limit of very high characteristic phonon frequencies to >> A, phonons can not be emitted or absorbed in the relevant frequency range. In this limit v(2)=w, as was shown in ref. [42] and the bipolaronic Hamiltonian can be mapped on the local-pair Hamiltonian [ 39 ]. If v (2) = w the incoherent contribution to/opt turns out to be zero, and the integrated optical absorption for a nonretarded interaction is just identical to that of the "toy model", eq. (7), at least in the low-temperature region when the band energy dispersion is practically parabolic. In this case Iovt (0) = 0 (fig. 1 ), which clearly contradicts the experimental observation, fig. 5. From this we have to conclude that a naive local-pair picture, derived from the negative " U " Hubbard Hamiltonian cannot explain the temperature dependence of the near infrared absorption in Y B a E C U 3 0 7 _ 6. On the other hand if the phonon frequencies are comparable or less than the bipolaron binding energy, v (2) does not contain the polaronic narrowing factor and is much larger than w [42 ]. In this more realistic case the absorption at T = 0 is finite and the coherent contribution is not very big, less than l 0 percent of the total intensity. To describe the temperature dependence of the infrared absorption in a wide temperature range in the normal state, compared with A one has to take into account the thermal dissociation of a bipolaron on two small polarons. Restricting ourselves to a quasi two-dimensional lattice with a constant density of polaronic and bipolaronic states within the polaronic and bipolaronic bands Np (E) = 1/2Wp, Nb (E) = I / 2 W respectively, fig. 2 (a), one obtains:
2w/T T2 !
/opt(T) 2 n ( T ) + ~c ~--~ /opt(O) n~ Tc
xdx
y_lexp(x)_ 1 (26)
with
n(T)
determined
by
eq.
(25)
and
~ = TE / w ( v (2)- W)ne. The parameter Oc determines the relative contribution of the coherent motion to the kinetic energy of the system and because v (2) > w> TJne [ 11 ] this parameter is usually small, of the order of 0.1 or less. Here ne is the electron (hole) atomic concentration, which is assumed to be temperature independent. Taking into account the upper polaronic band, fig. 2(a), with the half band-width wp=wm**/m* one
A.S. Alexandrovet al. ~Bose-Einstein condensationof small bipolarons
glect the polaronic contribution to the integrated optical absorption, eq. (26), compared with the bipolaronic one (see also ref. [41]). The polaronic absorption is shifted to the low-frequency (Drude) region. In the superconducting phase T < T¢ the chemical potential is zero, so y = 1. The temperature dependence of the integrated intensity:
Polarons
+ Polarons
-l-
2wp
A//2
~T
t Triplet bipolarons
zx/2
-+
t Bipolarons
t
d
2w
//-///////////$//
Singlet bipolarons
2w
_ _ _ ///,///////////-~t//
a
m*T - ~
[ 1 +w/y e x p ( - A / 2 T ) ) ln~l + x/~ exp( - d / 2 T - 2 w m * * / m * T ) } 2w/T
y-lexp(x) - 1 '
13
t/ ~\\
1.2
l? ~. x
I.I
1: /... /....
1,0
".......... .
.
.
.
.
.
~
.
0
.
........
4/
0.9
""'
r i
~ 2
~ 3
4
T/Te
Fig. 3. Temperaturedependenceof the integratedoptical absorption of on-sitebipolaronsfor three differentvaluesof the binding energy: A/T¢=0.1 (solid line), 1.0 (dotted line), 2.0 (dashed line), and tic=0.1, no=0.9, w/T¢= 1.9 and m*/m**=0.3. obtains the following equation for the chemical potential in the normal state ( T> T¢): in(1 - y e x p ( - - 2 w / T ) 1-y ] + m~,ln(l
-
/opt (T) = 1 Iop,(O)
b
Fig. 2. On-site (a) and inter-site (b) bipolaronicenergybands.
"~.
365
wne T"
1 + x/% e x p ( - ` 4 / 2 T ) +x/Y exp( - . 4 / 2 T - 2 w m * * / m * T ) } (27)
The ratio of the polaronic m* and bipolaronic m** effective masses for the lattice on-site bipolarons should be small [ 19 ]. This fact enables one to ne-
calculated for three different values of the binding energy A/Tc using eqs. (27) and (28) is shown in fig. 3. Because the relative density of states in the polaronic band is small ( ~ m*/m**) the temperature dependence of the on-site bipolaron concentration is weak with the exception of the unphysical region of very low binding energy, fig. 3 (the experimental value for YBa2Cu307 is .4/Tc = 7-8 [ 111 ). It is worth mentioning that for quasi-two dimensional bosons T~= new/L and three dimensional corrections to the chemical potential are small as m , / nem±, where L is a large logarithm of the ratio of the inter-plane mass m i to the in-plane one m, [3 ].
5. Integrated optical absorption for inter-site bipolarons and comparison with experimental results
The Hubbard repulsion on copper is estimated to be of the order of a few eV and on oxygen it is of the order of 1 eV. Thus the formation of on-site bipolarons seems to be unrealistic because the twice polaronic level shift is less than 1 eV. However the formation of inter-site small bipolarons on copper and (or) on oxygen, fig. 4 is quite feasible as in Ti407, MxV205 [43], WO3_x [44-47] andNbO2.5_x [48], where they were experimentally detected with high reliability. The bipolaronic Bose-liquid stabilized by oxygen dynamic distortion was proposed as the
A.S. Alexandrovet aL / Bose-Einstein condensationof small bipolarons
366
0
0 0 O:O
a ~ x ~
0 O 0
O
0 0 0 0 © 0
Iopt(T___~)= 81. (T) + 81¢( T ) ,
(29)
top,(0)
where the noncoherent contribution, responsible mainly for the temperature dependence of the normal state absorption is
3AT, { 1 - y e x p ( - 2 w / T - J / T ) ' ~ ~I. (T) = 1 - ~ in ~. 1- - ~ e ~
]
(30) and the coherent one: 2w/T
0
(y Fig. 4. Inter-sitesinglet (a) and triplet (b) bipolaronson oxygen sites (circles) surroundedby copper (crosses). The internal optical transition ofa singlet is shownby a dashed line. ground state of high-T¢ oxides by one of us [2]. It was shown [ 3 ] that triplet inter-site bipolarons describe the temperature dependence of the Korringa ratio and the singlet-triplet exchange energy might be responsible for the pseudogap in the neutron spectra [ 34 ]. It was proposed also [ 13 ] that triplet bipolarons are important in the temperature dependence of the normal state infrared absorptiont. In this section we derive the temperature dependence of the integrated optical absorption for intersite bipolarons, assuming that the temperature is small for their dissociation, but is comparable with the singlet-triplet exchange energy J, fig. 2(b), estimated from NMR and neutron spectra to be of the order of a few hundred K [ 3 ]. In general the absorption spectra and the integrated optical conductivities in a finite frequency window for singlet and triplet bipolarons are different. This can be due to an additional absorption with the internal transition like the bonding-antibonding transition of a singlet which is absent from a triplet bipolaronic state (see fig. 4). While A >> T the total number of bipolarons is practically temperature independent. Using the bipolaronic Hamiltonian, eq. (12), with an additional spin quantum number I for bipolaronic singlet (l= 0) and triplet (l= 0, +_ 1 ) states bm,z one obtains instead of eq. (28):
×
l
3
-~ e x p ( x ) - 1 + y - i e x p ( x + J / T ) -
) 1 " (31)
y is determined now from the thermal equilibrium of singlet and triplet bipolarons if T > Tc [ 3 ] ln(1-lYe_yW/r)+3 l n ( 1 - y e ( - Z ~ - J ) / z ~ _ new
-
1 - y e -J/r ] -
T ' (32)
and y = 1 if T< To. The temperature independent constant A is proportional to the difference of the internal absorption of a singlet and of a triplet bipolaron, fig. 4, and should be determined from the comparison with experiment. The triplet bandwidth is assumed to be the same as for the singlet, fig. 2 (b). Equation (32) is a two-dimensional approximation and it gives an exponentially small but nonzero chemical potential just at T¢. This leads to a small artificial jump of the optical absorption at T¢. However, 3d-corrections to p (proportional to ( T - T o ) 2 as described in ref. [3]) eliminate this small jump. Different from the on-site bipolarons the concentration of singlet inter-site ones is strongly temperature dependent on the scale T ~ J because the density of triplet states is high, ~ 3. This fact allows us to describe the pronounced decrease of the integrated absorption intensity in the normal state with the reasonable value of the singlet-triplet exchange energy J/T~ ~-4.4, corresponding to the characteristic value of the spin pseudogap in the neutron scattering [34], Eo,p/Tc~-3.5-5 (depending on the doping), fig. 5.
A.S. Alexandrov et aL / Bose-Einstein condensation of small bipolarons
367
,2
S
" q ~
o 9000 9o~u
0.8
0.6
8000
7000 6000 Wavenumbers
5ooo (cm -I)
4ooo
3000
2000
o
I
I
1
2
3
4
5
T/To Fig. 5. Integrated optical absorption of inter-site bipolarons (solid line) and experiment [ 13], no=0.9, J/Tc=4.4, 6c=0.014, A= 1.6, w~T¢= 1.9. YBa2Cu30~ (triangles) and YBaCuO/Fe (circles). Insert: the spectral dependence of the near-infrared absorption.
A remarkable drop of the slope of the optical absorption below T¢, fig. 5, is due to the temperature dependent coherent contribution to the sum rule. This contribution, eq. (31) vanishes at T = 0 because of Bose-Einstein condensation of singlet bipolarons as in the "toy" model, section 2. A small discrepancy with experiment below T~ might be due to the pair correlations (see Appendix B), because the best fit is achieved with ne-~ 1.
6. Discussion and conclusions
With the conductivity sum rule for small bipolarons we have described the unusual temperature dependence of the near-infrared absorption, measured experimentally [ 13 ] in YBa2Cu307, and YBaCuO/ Fe, fig. 5. Of course, polaronic description of the optical properties of high-T¢ oxides is not a unique possibility. As was argued by Hirsch [ 35 ] within the BCS approach the kinetic energy is essentially unchanged below To. This is because the occupation numbers of electrons (holes) is very similar at T = T¢ and at T = 0: at T = T¢ it is a step function broadened in a range 2T¢ and at T = 0 it is a step function broadened in a range of the BCS-gap A. Both function are extremely
close for 2A/T¢=4 as is usually the case. Applying the conductivity sum rule one can reach the conclusion that it should be a decrease in the slope of the intraband absorption at T¢ with the temperature lowering in a BCS superconductor. However the real trouble with the BCS approach is that for frequencies much higher than the superconducting energy gap, no change in optical absorption is predicted [49 ]. Any change in the optical absorption due to pairing seems to be practically invisible. The BCS approach gives also a Drude-like spectral dependence of the intra-band absorption This is in contradiction with the experimental observations which show a relative change of the absorption of more than 10% and a spectral shape which is drastically different from the Drude one, fig. 5. On the other side if one assumes that the frequency window 2000-10000 cm -~ corresponds to the inter-band transitions, then again the BCS-model is in a qualitative disagreement with the experiment, predicting the small increase in the inter-band absorption with increasing temperature in the normal state instead of the strong ( ~ 2 0 % ) decrease, observed experimentally. (To reach this conclusion one should take into account that the total intensity, integrated from 0 to oo is temperature independent. ) As alternative to the BCS model Hirsch has considered models of superconductivity where pairing
368
A.S. Alexandrov et al. /Bose-Einstein condensation o f small bipolarons
originates in gain of kinetic rather than potential energy of carders due to strong correlations in a Hubbard sub-bands (the "hole" superconductivity) [ 35,50 ]. In such systems, a change in the frequencydependent conductivity occurs at frequencies much higher than the scale set by the superconducting energy gap. The models predict a decrease in the high frequency inter-band optical absorption upon the superconducting transition. On the contrary the intraband absorption should increase below Tc in a "hole" superconductor. In our view it is difficult to reconcile the "hole" superconductivity [35 ] with the optical experimental data now at hand. First of all the relevant frequency region u~0.5-0.7 eV lies entirely in the charge-transfer gap, estimated to be higher than 1.5 eV, so the near infrared spectrum, fig. 5 (insert), corresponds to intra-band transitions within the oxygen band hybridised with the copper one. If this is the case then the model with the kinematic pairing is in a qualitative disagreement with the experiment, which shows a decrease of intensity just below Tc instead of an increase, predicted by the model for the intra-band absorption. If, nevertheless, the experiment, fig. 5, measures inter-band absorption, then it is difficult to understand with this model the decrease of the absorption in the normal state with increasing temperature because the model [35 ] is a mean-field BCS-like approximation, which for the normal state predicts an increase of the inter-band optical absorption with temperature. On the other hand it was demonstrated by Mihailovic et al. [23 ] that the infrared absorption of different copper-based oxides results from small (bi) polaronic states within the charge-transfer gap (see also ref. [ 51 ] ). We have shown in the present paper that its temperature dependence below Tc [ 13 ] is a manifestation of the coherent bipolaron tunneling within a narrow band and of Bose-Einstein condensation. The simple "toy" model is proposes and the conductivity sum rule for bipolaronic systems is derived which illustrate the strong influence of BoseEinstein condensation of charged bosons on their high-frequency optical properties. In the present analysis, we have not considered the effect of spin-wave "shakeoff" resulting from the coupling of carders to spin excitations as a source of the near infrared absorption. This coupling leads to
the spin bipolaron formation as proposed by one of us [ 1 ]. It was argued [23 ] that any contribution to or(z,) from magnon (quasi) shakeoff should be significantly weaker than the direct contribution to a(v) arising from the electron-photon coupling.
Acknowledgements We thank our colleagues at Cambridge, in particular A. Campbell, J. Cooper, W. Liang, J. Loram, J. Waldram, and J. Wheatley for very valuable discussions. One of us (A.S.A.) is grateful to Ph. Nozieres for his calling my attention to Bose-Einstein condensation of paraexcitons [36 ] and to H. Capellmann, J. Ranninger and A. Gogolin for their enlightening comments on the magnetic contribution to the optical absorption, on the photo-induced infrared conductivity and on our "toy" model.
Appendix A To derive the sum-rule, eq. (19), one can rewrite the Frohlich Hamiltonian, eq. (11) in a site representation: H= E m,m
T-,-'c~,sc-',s+(2N) -1/2 ' ,s
× ~ y(q)og(q)c~,:m,s(dcei¢"+dt-qe -iq') m,q,S
+ H p h -~- Vc .
(33)
The real part of the conductivity is
a~(u)=n ~ I12g(u-Ef+E°) Ef-Eo '
(34)
where 10) and I f ) are the ground and excited states of H, respectively. The current operator is: (m ' - m ) T = , , , c ,*, : , , , , ,
J=ie ~ m,m
(35)
' ,s
and satisfies the equation J=ie[H,X] ,
(36)
where X is the sum of the position operators of the electrons: X = ~, m c ~ , : . . . .
(37)
A.S. Alexandrov et al. / Bose-Einstein condensation of small bipolarons
Integrating over frequency eq. (34) and using the identity eq. (36) one obtains: oo
ien O'x~(Z,) d~,= -~- <0l
[X, Jx] 10>,
(38)
0
which yields eq. (19) after the direct substitution of eq. (35) and eq. (37). To derive the sum rule for the bipolaronic Drude conductivity, eq. (22), one can apply the Peierls substitution for the electron transfer integral with a vector potential A:
T,.,,.,=~"T.,,,n,eieA('-'').
(39)
This substitution yields for the bipolaronic transfer integral in the bipolaronic Hamiltonian, eq. (12) tm,,n, ~ tm,.,e 2ieA(m - m')
(40)
and for the bipolaronic current operator:
8Hb
J--- &4 =2ie ~
(m-m')tm,,.,b~bm,.
(41)
The rest is identical to the derivation of the electronic sum rule, eqs. ( 3 4 ) - ( 3 8 ) if one replace c~,~ and e on bm and 2e, respectively.
Appendix B In an ideal Bose-gas the presence of a particle at some point in space increases the probability that another particle is near that point, i.e. the particles "attract" one another. The short-range pair-correlation function turns out to be twice of the classical value, g ( 0 ) = 2 for all temperatures above Tc [52]. In the condensed state (below To) this correlator drops to its classical value ( = 1 ) at T = 0. To analyze the value and the temperature dependence of the pair-correlation function in the Boseliquid one can apply an empirical relation [ 53 ]
g(r, T)=l+(1-no(T)/n)2[g*(r, T ) - I ] ,
(42)
where no is the condensate density and g* (r, T) is the extrapolation of the normal-state function to below T~. This relation is found to be in excellent agreement with the X-ray data for He4 [ 40 ]. For a short distance g* (0, T) turns to be small in a contrast with an ideal Bose-gas; for He4 it is below 0.2
369
(see fig. 7 of ref. [40] ). In this case eq. (42) yields a negative temperature derivative of g(r, T) in the superfluid state. Thus the temperature dependence of the pair-correlation function is determined by the repulsion. For the strong repulsion this correlator gives an additional suppression of the optical absorption in the superconducting state, [eq. (23) ], to that from the temperature dependence of the coherent contribution (the last term in eq. (23)). However this additional suppression is expected to be small because the relative number of particles in a Bose-condensate is small, less than 10% of the total number as in He4.
References [ 1 ] N.E Mott, Adv. Phys. 39 (1990) 55. [2] A.S. Alexandrov, Pis'ma Zh. Eksp. Teor. Fiz. (Prilozh.) 46 (1987) 128 (JETP Lett Suppl. 46 (1987) 107 ]; Phys. Rev. B 38 (1988) 925. [3] A.S. Alexandrov, J. Low Temp. Phys. 87 (1992) 721. [4] A.S. Alexandrov and J. Ranninger, Solid State Commun. 81 (1992) 403. [5] N.F. Mon, Physica C 196 (1992) 369. [6] A.S. Alexandrov and J. Ranninger, Physica C 159 (1989) 367. [ 7 ] H.S. Obhi and E.K.H. Salje, J. Phys. C 4 (1992) 195; Physica C 141 (1990) 547. [ 8] A.S. Alexandrov, Phys. Rev. B 46 (1992) 14932. [9] Y.J. Uemura, A. Keren, L.P. Le, G.M. Luke, B.J. Sternlieb, W.D. Wu, J.H. Brewer, R.L. Whetten, S.M. Huang, S. Lin, R.V. Kaner, F. Diedrich, S. Donovan, G. Gruner and K. Hoiczer, Nature 352 ( 1991 ) 605. [10] N.F. Mort, Physiea C 205 (1993) 191. [ 11 ] A.S. Alexandrov and A.B. Krebs, Usp. Fiz. Nauk 162 (May 1992) 1. [Soy. Phys. Usp. 35 (May 1992) 345]. [ 12] A.S. Alexandrov and N.F. Mott, submitted to Supercond. Sci. Technol. ( 1993 ). [ 13] H.L. Dewing and E.K.H. Salje, Supercond. Sei. Technol. 5
(1992) 50. [ 14] Y.H. Kim, C.M.Foster, A.J. Heeger, S. Cos and G.Stucky, Phys. Rev. B 38 (1988) 6478. [ 15] C.Taliani, A.J. Pal, G. Ruani, R. Zamboni, X.Wei and Z.V. Vardini, Electronic Properties of HTcSC and Related Compounds, Springer Series of Solid State Science (Springer-Verlag, Berlin) 99 (1990) 280. [ 16] S. Sugai, Physica C 185-189 ( 1991 ) 76. [ 17] R. Liu, B.W. Veal, A.P. Paulikas, J.W. Downey, P.J. Kostic, S. Fleshier, U. Welp, C.G. Olson, X. Wu, A.J. Arko and J.J. Joyce, Phys. Rev. B 46 (1992) 11056; C.G. Olson et al., Phys. Rev. B 42 (1990) 381. [ 18 ] A.S. Alexandrov and J. Ranninger, Phys. Rev. B 45 (1992) 7730; Physica C 198 (1992) 360.
370
A.S. Alexandrov et al. / Bose-Einstein condensation o f small bipolarons
[ 19 ] A.S. Alexandrov and J. Ranninger, Phys. Rev. B 23 ( 1981 ) 1796;24 (1981) 1164. [20] P.W. Anderson and E. Abrahams, Nature 327 (1987) 363. [21]M. Gurvitch and A.T. Fiory, Phys. Rev. Lett. 59 (1987) 1337. [22] S.I. Vedeneev and V.A. Stepanov, Physica C 162-164 (1989) 1131. [23] D. Mihailovic, C.M. Foster, K. Voss and A.J. Heeger, Phys. Rev. B 42 (1990) 7989. [24] B.K. Chakraverty, D. Feinberg, Z. Hang and M. Avignon, Solid State Commun. 64 (1987) 1147. [25 ] D. Emin, Phys. Rev. Lett. 62 (1989) 1544; D. Emin and S. Hillery, Phys. Rev. B 39 (1989) 6575. [26 ] P.W. Anderson, Theories of Fullerene Tc's which will not work, Preprint, Princeton University ( 1991 ). [27] T. Holstein, Ann. Phys. (NY) 8 (1959) 325, 343. [ 28 ] R. Gehlig and E. Salje, Philos. Mag. 47 ( 1983 ) 229. [29] S.V. Tjablikov, Zh. Eksp. Teor. Phys. 23 (1952) 381. [30 ] J. Yamashita and T. Kurosawa, J. Phys. Chem. Solids 5 (1958) 34. [ 31 ] I.G. Lang and Yu.A. Firsov, Zh. Eksp. Teor. Fiz. 43 ( 1962 ) 1843 [Sov. Phys. JETP 16, (1963) 1301]; Polarons, ed. Yu. Firsov (Nauka, Moscow, 1975 ). [32] H.L. Dewing and E.K.H. Salje, J. Solid State Chem. 100 (1992) 363. [33] H.L. Dewing, E.K.H. Salje, K. Scott and A. Mackenzie, J. Phys. Cond. Matter 4 (1992) L109. [ 34 ] J. Rossat-Mignod, L.P. Regnault, P. Bourges, C. Vettier, P. Buffet and J.Y. Henry, Physica Scripta 45 (1992) 74. [35 ] J.E. Hirsch, Physica C 201 (1992) 347. [36]This is similar to the influence of Bose-Einstein condensation of paraexcitons in Cu20 on luminescence as discussed by A. Mysyrowicz, D.W. Snoke and J.P. Wolfe, Phys. Status Solidi B 159 (1990) 387.
[ 37 ] R. Kubo, J. Phys. Soc. Jpn. 12 ( 1957 ) 570. [38] P.F. Maldague, Phys. Rev. B 16 (1977) 2437. [ 39 ] A.S. Alexandrov, J. Ranninger and S. Robaszkiewicz, Phys. Rev. B 33 (1986) 4526. [ 40 ] F.H. Wirth and R.B. Hallock, Phys. Rev. B 35 ( 1987 ) 89. [41 ] V.V. Bryksin, V.S. Voloshin and A.V. Raitsev, Fiz. Tverd. Tela (Leningrad) 25 ( 1983 ) 1427 [Sov. Phys. Solid. State 25 (1983) 820]; V.V. Bryksin and A.V. Gol'tsev, Fiz. Tverd. Tela (Leningrad) 30 (1988) 1476 [SOv. Phys. Solid. State 30 (1988) 851. [42]A.S. Alexandrov and V.V. Kabanov, Fiz. Tverd. Tela (Leningrad) 28 (1986) 1129. [ 43 ] S. Lakkis, C. Schlenker, B.K. Chakraverty, R. Buder and M. Marezio, Phys. Rev. B 14 (1976) 1429; V.K. Chakraverty, M.T. Sienko and J. Bonnerot, Phys. Rev. B 17 (1978) 3781. [44] O.F. Schirmer and E. Salje, Solid State Commun. 33 (1980) 333. [ 45 ] E. Salje and G. Hoppmann, Philos. Mag. B 43 ( 1981 ) 145. [ 46 ] O.F. Schirmer and E. Salje, J. Phys. C 13 ( 1981 ) L 1067. [ 47 ] E. Salje and B. Guttler, Philos. Mag. B 50 (1984) 607. [48] C. Ruscher, E. Salje and A. Hussain, J. Phys. C 21 (1988) 3734. [ 49 ] D.C. Mattis and J. Bardeen, Phys. Rev. 111 ( 1958 ) 412. [50] J.E. Hirsch, Physica C 158 (1989) 326; J.E. Hirsch and F. Marsiglio, Phys. Rev. B 39 (1989) 11515. [51 ] G.A. Thomas, D.H. Rapkine, S.L. Cooper, S-W. Cheong, A.S. Cooper, L.F. Schneemeyer and J.V. Waszczak, Phys. Rev. B 45 (1992) 2474. [ 52 ] L.D. Landau and E.M. Lifshitz, Statistical Physics 5, Part 1 (Pergamon, Oxford, 1980). [ 53 ] G.J. Hyland, G. Rowlands and F.W. Cummings, Phys. Lett. A31 (1970) 465.
Physica C 215 ( 1993) 359-370 North-Holland
Near-infrared absorption of YBa2Cu307_a Evidence for Bose-Einstein
condensation
of small bipolarons
A.S. A l e x a n d r o v , A . M . B r a t k o v s k y , N.F. M o t t a n d E . K . H . Salje Interdisciplinary Research Centre in Superconductivity, University of Cambridge, Madingley Road, Cambridge, CB3 0HE, UK Received 24 February 1993 Revised manuscript received 17 May 1993
A significant change with temperature in intensity of the optical absorption spectrum of YBa2CuaO7_a in the range 2000 to 10000 cm-1 at the superconducting transition is shown to be a manifestation of Bose-Einstein condensation of small bipolarons. The sum rule for bipolaronic infrared absorption is derived and applied for the explanation of this unusual effect on the basis of the polaron theory of high-Tosuperconductivity. The temperature dependence of the normal state absorption is controlled by the thermal excitation of singlet inter-site bipolarons into triplet states. As determined from optical absorption experiments singlettriplet exchange energy has the same value as the spin gap discovered by Rossat Mignod et al. by neutron scattering.
1. Introduction
The physics of small polarons and bipolarons has become of particular interest recently in the context of theories of high temperature superconductors. Some microscopic models [ 1,2 ] suggest that charged bosons, formed by strong electron-phonon and electron-electron exchange interactions (latticeand spin bipolarons) might be responsible for the puzzling thermodynamic and kinetic properties of high-Tc oxides. Such characteristic features of high-To copper based superconductors as the curious absence of coherent effects and of the Korringa law in the nuclear spin relaxation rate, the "spin" pseudogap, unexpected enhancement below T¢ of low-frequency conductivity, linear in T resistivity are indicative of charged bosons (small lattice and spin bipolarons) [3]. One should add here the heat capacity [4,5], which is reminiscent of He-4 and unusual temperature dependences of lower and upper critical field as well as of the sound attenuation and velocity which however can be expected for charged bosons [4,6]. Unusual features of the oxygen isotope effect and of the softening of optical phonons at energies above the (pseudo) gap [7 ] clearly show that phonons arc involved, and high-T~ oxides are in the transition region from a polaronic Fermi-liquid to a bipolaronic
charged Bose-liquid, as discussed by one o f us [ 2,8 ] (for other manifestations o f the Bose-liquid behavior o f superconducting oxides see also ref. [9 ] and recent review papers [ 10-12 ] ). In this paper we shall argue that the recent observation [ 13 ] o f the significant influence o f the superconducting phase transition on the near infrared absorption with the characteristic frequency z,___0.7 eV is a clear manifestation o f Bose-Einstein condensation in YBa2Cu307_ 6- Using a simple but rather general model we show how the superfluid phase transition o f a Bose-gas on a lattice can influence the high-frequency optical absorption. With the sum-rule for the optical conductivity of a coupled electronphonon system we demonstrate that the effect is a consequence o f the coherent tunneling and Bosecondensation o f small bipolarons. Earlier evidence for small (bi)polarons in copper based oxides comes from the photo-induced infrared absorption, measured by Heeger's and by Taliany's groups [ 14,15 ] and from the observation o f Sugai [ 16 ] o f both infrared and Raman-active vibration mode. There is an indication o f the fine structure o f the incoherent part o f the angle resolved photoemission spectra ( A R P E S ) [17], which can be explained as a superposition o f multiphonon and elec-
0921-4534/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
360
A.S. Alexandrov et al. / Bose-Einstein condensation of small bipolarons
tronic excitations in accordance with the polaron theory of ARPES [ 18 ]. Several attempts were made to eliminate bipolaronic superconductivity [ 19 ] as mechanism for highTo, however. The widespread argument against the electron-phonon coupling in high-T¢ materials, based on the absence of the oxygen isotope effect [20 ] seems to be irrelevant in view of more recent measurements [ 8 ]. A small temperature slope of resistivity can be interpreted as caused by a weak scattering of carriers by phonons [ 21 ] without telling much about the band structure renormalization due to the strong coupling with high-frequency optical and molecular vibrations. The fact that these phonons are strongly coupled with electrons is confirmed by tunneling [ 22 ], Raman [ 16 ] and photo-induced infrared [ 15,14 ] measurements. The carriers are heavy due to this coupling ( m * ~ - 10 m) as is confirmed by the optical measurement [23 ], and this mass enhancement could be responsible for a low value of 2 [21 ] estimated from d p / d T . Thus the above mentioned objections may be irrelevant. The more serious problem for the bipolaronic mechanism of high-To is the possibility of the coherent tunneling of (bi)polarons with a reasonable value of the effective mass. Chakraverty et al. [ 24 ], Emin [25] and Anderson [26] assert that their bandwidth should not exceed 10-4-10 -5 eV, and then, in accordance with the theory [ 19 ] the maximal T¢ attainable with small bipolarons should be a few K or less. However, this is not the case for the intermediate value of the coupling constant 2 ~- 1 if the high-frequency phonons ~o_~0.1 eV are involved in the polaron formation. With high-frequency phonons and with a relatively narrow bare band (such that the adiabatic ratio D / o J < 10, D is the bare band half-width) one can expect (bi)polaronic bandwidth of the order of a few hundreds K, and the same estimate for T¢. It is worth mentioning that a similar estimation for the polaronic bandwidth ( 1 0 - 2 - 1 0 - 4 of the electronic bandwidth) has been obtained by Holstein in his original paper [ 27 ]. The condition for the small polaron formation, known in the literature, is the smallness of the transfer integral J compared with the polaronic binding energy Ep [27]. Because the optical phonon frequencies in metal oxides are usually high (0.05-0.1 eV) the polaronic binding energy is large ( E v = 0.2-0.4 eV) and
thus the allowed value for the bare electronic bandwidth compatible with the small polaron formation is high enough being of the order of 0.5-1.0 eV. In this case Holstein's estimate of the small-polaron bandwidth and also some variational and MonteCarlo simulations give the value of a few hundred K (>0.01 eV), which is at least two orders of magnitude larger than those of Chakraverty, Emin and Anderson. Of course, a short-range Coulomb repulsion (intersite) should be suppressed, approximately below 1 eV to ensure mobile bipolaron formation, and that is quite feasible because the high-frequency or shortrange dielectric constant in metal oxides is usually large (~> 5). We now going to compare our arguments with experiment. To our knowledge there is only one clear observation of the coherent small polaron tunneling in a reasonable narrow band (or the order of 100 K). In the case of WO3 Gehling and Salje [ 28] found at temperatures below 130 K a crossover from a regime of small polaron hopping conductivity with constant activation energy to a regime with zero activation energy, thus expecting "polaronic band conduction" in accordance with the canonical theory of the small polaron transport developed by Tjablikov [29], Yamashita and Kurosawa [ 30 ], Holstein [ 27 ], and in a more general form by Firsov and co-workers [ 31 ]. As we show in this paper that the experiment by Dewing and Salje [l 3] confirms the possibility of the coherent (bi)polaronic tunneling in the structural CuOE-plane. The observed infrared absorption is closely related to the conductivity in the CuOEplane and are much less related to the CuO-chains. The experimental evidence stems from the observation [32,33] that doping of YBaCuO with Zn in the planes does shift the spectral function proportional to the degree of doping whereas there is no effect from doping the chains with Fe. In section 2 we discuss a simple model of bosonic pairs on a lattice to show with general symmetry arguments how statistics leads to the pronounced influence of Bose-Einstein condensation on the optical conductivity. We derive the conductivity sum rule for bipolarons in section 3 and discuss the integrated optical absorption of on-site and of inter-site bipolarons in section 4 and section 5, respectively. In section 5 we compare our theoretical results with ex-
A.S. Alexandrov et al. ~Bose-Einstein condensation of small bipolarons periment. The singlet-triplet exchange energy, found from this comparison, has the same value as the spin gap, observed with neutron scattering [ 34 ]. In conclusion some alternative explanations, including the BCS and Hirsh's "hole superconductivity" [ 35 ] are discussed.
2. Qualitative consideraton with the "toy" model Let us discuss a simple but rather general model which shows a hundred percent decrease of the optical absorption while temperature changes from Tc to zero: an ideal gas of molecules, composed of two identical panicles (electrons) with charge e on a lattice. To stabilize their bound state one should assume that the pair attraction operates, compensating the Coulomb repulsion. All molecules are in the ground internal state with the binding energy being much higher than temperature. The periodic crystal field is assumed to be large compared with the binding energy, so all molecular states, including continuum, are built from the single-band single-electron Bloch wave functions. Because of the translation symmetry the molecular wave function is a two-particle Bloch function described with the total quasi-momentum K: 7/i:(r~, rE ) = exp (iKR) U ~ ( R , r ) ,
( 1)
where R = (rt + r 2 ) / 2 , r = r l - r 2 , and Ux(R, r) is periodic with the lattice translations. At low temperature only states with small K (near F point), are relevant, so one can expand Ux in a series of K:
U~(R, r) ~- U~(R, r) + K.D':(R, r)
(2)
with D = limx~oVxUx. A linear interaction of a molecule with light is described by the Hamiltonian: Hint-
-icE my
(v~ + v 2 ) ,
(3)
361
where D~:= ~ f dR dr( U:o(R, r)*VR'Di(R, r) - U~o(R,
r)VR.DY(R, r)*). As usual one can neglect a field inhomogeneity (photon momentum) and put h = c = 1. From eq. (4) the rate of the optical absorption is proportional to the kinetic energy of a molecule, and an immobile molecule, composed from two identical particles cannot absorb, which is a trivial consequence of a parity and spin conservation. The dipole matrix element for the F point is equal to zero because all singlet states of a molecule with K-- 0 are even and all triplets are odd under the inversion transformation, R, r - - , - R , - r , for a lattice with the inversion symmetry. This is similar to the dipole-forbidden singleelectron transitions in semiconductors. The matrix element, eq. (4), increases with increasing K. It is now the essence of our argument that the occupancy at K S 0 is a thermodynamic function which "sees" the Bose condensation and, thus, allows us to measure the condensation directly via optical absorbance [ 36 ]. The Fermi-golden rule yields the optical conductivity: 4xe 2
a(v)= ~mvEk
~ ID~:IZc~(E:--ei-- v) ,
(5)
where t~o, is the energy spectrum of a single molecule, including its continuous part and K: E k = ~ -~-mmn ( K )
(6)
is the kinetic energy of all molecules, which obey the Bose-Einstein distribution with the chemical potential #, n ( K ) = ( e x p ( ( K 2 / 4 r n - l z ) / T ) - 1 )-1. Integrating eq. (7) over frequency we obtain the conductivity sum rule [37 ] for our system:
I ( T ) = f du a(v) = rce2a2Ej,, o
(7)
where
with E being an electric field. I f the frequency is high enough the molecule is excited or dissociated by light, absorbing one photon to the final s t a t e f The matrix element of the transition is given by:
(J] Hint [i) - - ieE" K D~f, ?nip
(4)
4
a:= ~mm ~: IV'Jl2/(t:-ti) "
(8)
a is a temperature independent characteristic length of the order of the molecular radius (the cubic lattice is assumed). Thus the optical absorption of our "toy" model
A.S. Alexandrovet al. ~Bose-Einsteincondensationof smallbipolarons
362
(holes). Starting from a single band Frohlich Hamiltonian:
Toy model 3.0
I4= E E(k)cLck,~
2.5
k,$
+ (2N) -~/2 ~
2.0
+Hph
-~1.5
E 0.5
0.0 0 .0
0.5
1.0 T/Tc
1.5
2.0
Fig. I. Temperaturedependenceof the integrated optical absorption of free Ix)sons on a lattice, composed from two identical particles. shows a drastic temperature dependence, being proportional to T s/2 below the Bose-Einstein condensation temperature To= 3.3nE/3/2m and linear well above T~, fig. 1, (n is the molecular concentration):
I(T¢) -
F(5/2)((5/2)
exp(x-/t)-I
'
(9)
where t = T~ T~ is the reduced temperature and/z = 0 for t< 1. In the normal state ( t > 1 ) the following equation holds for #:
0
+ Vc .
(11)
Alexandrov and Ranninger [ 19 ] derived the bipolaronic Hamiltonian, describing the repulsion and the tunneling of small bipolarons, which are hard-core bosons on a lattice:
1.0
i
7(q)~o(q)ci+¢,sck,s(d¢+dtq)
k,q,s
xl/2 dx
exp (x-- #) -- 1
=t-3/2F(3/2)((3/2) .
Hb= ~
m,m'
(-t=m,b~bm,+v (2) , m,m'
× n=(nm,- 1) +4Vr~,,.,nmnm,),
(12)
Hph=Y~q to(q)d~dq is the phonon energy, n==b~bm is the bipolaronic density operator, b== c.,,tc=,+. The interaction vm,=, does not depend
where
on the kinetic energy and includes the direct (density-density) Coulomb (Vc) repulsion and the attraction via phonons between two small polarons on different sites m, tram' and ,,(2) , ~ m , m ' are the bipolaron transfer integral and the bipolaron repulsion due to the virtual polaron exchange, correspondingly, both of the second order in the electronic kinetic energy E(k) [11]: oo
tm,~, = 2 i
J dz( #m,m'(Z)~=,m' ( 0 ) )
pheXp(--izJl:)
0
(10)
,
(13)
oo
.Um,m' (2) = 2 i | dz(#,,,,,,, (z)6,,,,,,,,(0))phexp(-iAz) , The temperature derivative of the integrated optical conductivity (or absorption), dI(T)/dT is proportional to the heat capacity of an ideal Bose-gas, tabulated in textbooks. The condensed fraction of molecules, composed of two identical particles, does not absorb light, if bosons are condensed at F point of a lattice with the inversion symmetry.
o
(14)
where 6m,,,,, (z) = T,,,,,,,,einpb~ 1
× exp(--r--- ~ \x/2N × e - iH~h~,
3. Conductivity sum rule for on-site small bipolarons Now we know at least one realistic example of charged bosons, composed from two electrons
• " 7(q)d~[e-~'m-e-l*n"l-H.c.)
( 15 )
( )vh is the averaging with the phonon density matrix and T,,,,,~,=(1/N)~kE(k)exp(ik(m-rn')) being a bare electronic transfer integral between sites m and m'. The transformation of the Frohlich Hamiltonian,
363
A.S. Alexandrov et al. /Bose-Einstein condensation of small bipolarons
eq. (11 ), into bipolaronic one, eq. (12), is performed with two canonical transformation, exp $1 and exp $2: $1 = (2N) -1/2 ~ q,m,$
y(q)c~,scm,s(d~e-lq=-n.c.) , (16)
(flO=,=,(O)c~,~c=,,~lp) (&)~,. =
m,=',$ Z
E~-Ep
(17) '
where Ezp and I f ) , IP) are the energy levels and the eigenstates of the original Frohlich Hamiltonian, eq. ( 11 ) without the kinetic energy. The first canonical transformation eliminates the Frohlich interaction and the second one eliminates the one-electron hopping. The bipolaronic Hamiltonian, eq. (12) describes low-energy excitations of a strongly-coupled (,~> 1 ) many electron-phonon system at low enough temperature T<<,4=2Ev- V¢, where Ep is the familiar polaronic shift of the electronic band: 1
Ep= ~--N~ y(q)2o)(q) ,
(18)
with re(q) being the phonon frequency and N t h e total number of sites (cells). Because the Frohlich interaction is diagonal in the site-representation it does not change the conductivity sum rule, derived by Maldague [ 38 ] for a single band Hubbard model (for more details see Appendix A): I(T)=-~(-
T.,,,,,c*,,,,~c,,,,,),
~
the nearest neighbor hopping) and n the bipolaron atomic density. The conductivity sum rule, eq. (20), includes optical absorption due to the bipolaron dissociation with and without phonon emission and absorption. It includes also the low-frequency Drude conductivity of small bipolarons tunneling in a bipolaronic narrow band (the half-bandwidth w = zt=,,~,<<,4) due to their scattering by phonons, impurities and by each other without their dissociation. If one is interested only in the optical path /opt ( v ~ ,4 or higher) one should subtract the bipolaronic Drude contribution ID~de from the total absorption intensity: I ( T) =Iopt( T ) + IDrude( T ) .
(21)
To derive the integrated bipolaronic Drude conductivity, IDr~d~ one can apply the sum rule to the bipolaronic Hamiltonian, eq. (12), keeping in mind that bipolarons have charge 2e (Appendix A):
IDruae(T) - 4n~2a2 /\m,=, E t=,=, b~,b~,,). 2
(22)
Subtracting eq. (22) from eq. (20) one obtains:
/opt = Ige2a2(v(2)( n-
( nmnm+,) )
(19)
In,m',$
where a is a lattice constant. To calculate the kinetic energy of the strongly coupled electron-phonon system up to the second order in the transfer integral with the accuracy ~ (Tm,=,/Ep)2= 1/22 one can apply the polaronic $1 and bipolaronic $2 canonical transformations to eq. (19) with the following result: I(T)=
~
(2v'2) ( n- ( n,,,n,,,+,,) ) +2(Y~
(20)
with ~,,~2)~.,,~2)_ -~.,.., ..., z the coordination number (for simplicity we consider cubic or quadratic lattice and
4. Temperature dependence of the integrated optical absorption of on-site small bipolarons
Let us consider a low-density regime. It was demonstrated in ref. [39] that the ground state of our system is a homogeneous Bose-liquid, in which the tendency to the charge order is suppressed by quantum fluctuations if n < no. The critical value of the concentration nc above which the charge density wave develops turns out to be independent of the repulsion if the latter is strong v+ v (2) >> w. For a cubic lattice no---0.08. If the repulsion between bipolarons has a moderate value v+ v (2) < 3w the homogeneous Bose-liquid is stable versus the charge order practically in the whole density region.
A.S. Alexandrovet al. / Bose-Einstein condensation of small bipolarons
364
In a homogeneous repulsive Bose-liquid the short range pair correlation function g ( r ) = ( n (0) n ( r ) ) / n 2 is small; for example, in liquid He4 g(r) < 0.2 for r < 2.5 A [40]. The temperature dependent part of this correlator is even smaller (Appendix B). Thus the contribution to the conductivity sum rule, eq. (22) of the term quadratic in the bipolaron density is practically temperature independent and negligible for the homogeneous strongly repulsive Bose-liquid. In the dilute limit one can also neglect the corrections to the free particle energy spectrum, which are small while the gas parameter is small. These simplifications yield: /opt(T)
=
~e2a2((v(2)--w)n(T)
+f
2w
0
y-lexp(~/T)--I
)
'
where Nb(E) is the density of states in a narrow bipolaronic band, and 2w
n(T)=
"
0
Nb(E) de
y-lexp(~/T)-I
(25)
is the bipolaron atomic density determined by the chemical potential/t, y = exp (/~/T). The first term in eq. (24) describes the incoherent absorption of light accompanied by emission and absorption of many phonons. The frequency dependence of this absorption was discussed by Bryksin and co-workers [41 ]. The absorption m a x i m u m lies at a frequency 4Ep-V~. Comparing the m a x i m u m position with experiment [ 13 ] one obtains a realistic value for the polaronic level shift of the order of a few hundred meV. Bryksin et al. [ 41 ] ignored the tunneling motion of bipolarons, so their results are not valid in the temperature range compared with T~ as far as the temperature dependence of the absorption intensity is concerned. But the gross spectral features on a frequency scale of the order of Ep are, of course, temperature independent. The second term in the integrated absorption, eq. (24) is due to the coherent tunneling contribution and identical to that discussed above within the "toy" model. This term is responsible for the influence of the superconducting transition on the optical absorption.
In the limit of very high characteristic phonon frequencies to >> A, phonons can not be emitted or absorbed in the relevant frequency range. In this limit v(2)=w, as was shown in ref. [42] and the bipolaronic Hamiltonian can be mapped on the local-pair Hamiltonian [ 39 ]. If v (2) = w the incoherent contribution to/opt turns out to be zero, and the integrated optical absorption for a nonretarded interaction is just identical to that of the "toy model", eq. (7), at least in the low-temperature region when the band energy dispersion is practically parabolic. In this case Iovt (0) = 0 (fig. 1 ), which clearly contradicts the experimental observation, fig. 5. From this we have to conclude that a naive local-pair picture, derived from the negative " U " Hubbard Hamiltonian cannot explain the temperature dependence of the near infrared absorption in Y B a E C U 3 0 7 _ 6. On the other hand if the phonon frequencies are comparable or less than the bipolaron binding energy, v (2) does not contain the polaronic narrowing factor and is much larger than w [42 ]. In this more realistic case the absorption at T = 0 is finite and the coherent contribution is not very big, less than l 0 percent of the total intensity. To describe the temperature dependence of the infrared absorption in a wide temperature range in the normal state, compared with A one has to take into account the thermal dissociation of a bipolaron on two small polarons. Restricting ourselves to a quasi two-dimensional lattice with a constant density of polaronic and bipolaronic states within the polaronic and bipolaronic bands Np (E) = 1/2Wp, Nb (E) = I / 2 W respectively, fig. 2 (a), one obtains:
2w/T T2 !
/opt(T) 2 n ( T ) + ~c ~--~ /opt(O) n~ Tc
xdx
y_lexp(x)_ 1 (26)
with
n(T)
determined
by
eq.
(25)
and
~ = TE / w ( v (2)- W)ne. The parameter Oc determines the relative contribution of the coherent motion to the kinetic energy of the system and because v (2) > w> TJne [ 11 ] this parameter is usually small, of the order of 0.1 or less. Here ne is the electron (hole) atomic concentration, which is assumed to be temperature independent. Taking into account the upper polaronic band, fig. 2(a), with the half band-width wp=wm**/m* one
A.S. Alexandrovet al. ~Bose-Einstein condensationof small bipolarons
glect the polaronic contribution to the integrated optical absorption, eq. (26), compared with the bipolaronic one (see also ref. [41]). The polaronic absorption is shifted to the low-frequency (Drude) region. In the superconducting phase T < T¢ the chemical potential is zero, so y = 1. The temperature dependence of the integrated intensity:
Polarons
+ Polarons
-l-
2wp
A//2
~T
t Triplet bipolarons
zx/2
-+
t Bipolarons
t
d
2w
//-///////////$//
Singlet bipolarons
2w
_ _ _ ///,///////////-~t//
a
m*T - ~
[ 1 +w/y e x p ( - A / 2 T ) ) ln~l + x/~ exp( - d / 2 T - 2 w m * * / m * T ) } 2w/T
y-lexp(x) - 1 '
13
t/ ~\\
1.2
l? ~. x
I.I
1: /... /....
1,0
".......... .
.
.
.
.
.
~
.
0
.
........
4/
0.9
""'
r i
~ 2
~ 3
4
T/Te
Fig. 3. Temperaturedependenceof the integratedoptical absorption of on-sitebipolaronsfor three differentvaluesof the binding energy: A/T¢=0.1 (solid line), 1.0 (dotted line), 2.0 (dashed line), and tic=0.1, no=0.9, w/T¢= 1.9 and m*/m**=0.3. obtains the following equation for the chemical potential in the normal state ( T> T¢): in(1 - y e x p ( - - 2 w / T ) 1-y ] + m~,ln(l
-
/opt (T) = 1 Iop,(O)
b
Fig. 2. On-site (a) and inter-site (b) bipolaronicenergybands.
"~.
365
wne T"
1 + x/% e x p ( - ` 4 / 2 T ) +x/Y exp( - . 4 / 2 T - 2 w m * * / m * T ) } (27)
The ratio of the polaronic m* and bipolaronic m** effective masses for the lattice on-site bipolarons should be small [ 19 ]. This fact enables one to ne-
calculated for three different values of the binding energy A/Tc using eqs. (27) and (28) is shown in fig. 3. Because the relative density of states in the polaronic band is small ( ~ m*/m**) the temperature dependence of the on-site bipolaron concentration is weak with the exception of the unphysical region of very low binding energy, fig. 3 (the experimental value for YBa2Cu307 is .4/Tc = 7-8 [ 111 ). It is worth mentioning that for quasi-two dimensional bosons T~= new/L and three dimensional corrections to the chemical potential are small as m , / nem±, where L is a large logarithm of the ratio of the inter-plane mass m i to the in-plane one m, [3 ].
5. Integrated optical absorption for inter-site bipolarons and comparison with experimental results
The Hubbard repulsion on copper is estimated to be of the order of a few eV and on oxygen it is of the order of 1 eV. Thus the formation of on-site bipolarons seems to be unrealistic because the twice polaronic level shift is less than 1 eV. However the formation of inter-site small bipolarons on copper and (or) on oxygen, fig. 4 is quite feasible as in Ti407, MxV205 [43], WO3_x [44-47] andNbO2.5_x [48], where they were experimentally detected with high reliability. The bipolaronic Bose-liquid stabilized by oxygen dynamic distortion was proposed as the
A.S. Alexandrovet aL / Bose-Einstein condensationof small bipolarons
366
0
0 0 O:O
a ~ x ~
0 O 0
O
0 0 0 0 © 0
Iopt(T___~)= 81. (T) + 81¢( T ) ,
(29)
top,(0)
where the noncoherent contribution, responsible mainly for the temperature dependence of the normal state absorption is
3AT, { 1 - y e x p ( - 2 w / T - J / T ) ' ~ ~I. (T) = 1 - ~ in ~. 1- - ~ e ~
]
(30) and the coherent one: 2w/T
0
(y Fig. 4. Inter-sitesinglet (a) and triplet (b) bipolaronson oxygen sites (circles) surroundedby copper (crosses). The internal optical transition ofa singlet is shownby a dashed line. ground state of high-T¢ oxides by one of us [2]. It was shown [ 3 ] that triplet inter-site bipolarons describe the temperature dependence of the Korringa ratio and the singlet-triplet exchange energy might be responsible for the pseudogap in the neutron spectra [ 34 ]. It was proposed also [ 13 ] that triplet bipolarons are important in the temperature dependence of the normal state infrared absorptiont. In this section we derive the temperature dependence of the integrated optical absorption for intersite bipolarons, assuming that the temperature is small for their dissociation, but is comparable with the singlet-triplet exchange energy J, fig. 2(b), estimated from NMR and neutron spectra to be of the order of a few hundred K [ 3 ]. In general the absorption spectra and the integrated optical conductivities in a finite frequency window for singlet and triplet bipolarons are different. This can be due to an additional absorption with the internal transition like the bonding-antibonding transition of a singlet which is absent from a triplet bipolaronic state (see fig. 4). While A >> T the total number of bipolarons is practically temperature independent. Using the bipolaronic Hamiltonian, eq. (12), with an additional spin quantum number I for bipolaronic singlet (l= 0) and triplet (l= 0, +_ 1 ) states bm,z one obtains instead of eq. (28):
×
l
3
-~ e x p ( x ) - 1 + y - i e x p ( x + J / T ) -
) 1 " (31)
y is determined now from the thermal equilibrium of singlet and triplet bipolarons if T > Tc [ 3 ] ln(1-lYe_yW/r)+3 l n ( 1 - y e ( - Z ~ - J ) / z ~ _ new
-
1 - y e -J/r ] -
T ' (32)
and y = 1 if T< To. The temperature independent constant A is proportional to the difference of the internal absorption of a singlet and of a triplet bipolaron, fig. 4, and should be determined from the comparison with experiment. The triplet bandwidth is assumed to be the same as for the singlet, fig. 2 (b). Equation (32) is a two-dimensional approximation and it gives an exponentially small but nonzero chemical potential just at T¢. This leads to a small artificial jump of the optical absorption at T¢. However, 3d-corrections to p (proportional to ( T - T o ) 2 as described in ref. [3]) eliminate this small jump. Different from the on-site bipolarons the concentration of singlet inter-site ones is strongly temperature dependent on the scale T ~ J because the density of triplet states is high, ~ 3. This fact allows us to describe the pronounced decrease of the integrated absorption intensity in the normal state with the reasonable value of the singlet-triplet exchange energy J/T~ ~-4.4, corresponding to the characteristic value of the spin pseudogap in the neutron scattering [34], Eo,p/Tc~-3.5-5 (depending on the doping), fig. 5.
A.S. Alexandrov et aL / Bose-Einstein condensation of small bipolarons
367
,2
S
" q ~
o 9000 9o~u
0.8
0.6
8000
7000 6000 Wavenumbers
5ooo (cm -I)
4ooo
3000
2000
o
I
I
1
2
3
4
5
T/To Fig. 5. Integrated optical absorption of inter-site bipolarons (solid line) and experiment [ 13], no=0.9, J/Tc=4.4, 6c=0.014, A= 1.6, w~T¢= 1.9. YBa2Cu30~ (triangles) and YBaCuO/Fe (circles). Insert: the spectral dependence of the near-infrared absorption.
A remarkable drop of the slope of the optical absorption below T¢, fig. 5, is due to the temperature dependent coherent contribution to the sum rule. This contribution, eq. (31) vanishes at T = 0 because of Bose-Einstein condensation of singlet bipolarons as in the "toy" model, section 2. A small discrepancy with experiment below T~ might be due to the pair correlations (see Appendix B), because the best fit is achieved with ne-~ 1.
6. Discussion and conclusions
With the conductivity sum rule for small bipolarons we have described the unusual temperature dependence of the near-infrared absorption, measured experimentally [ 13 ] in YBa2Cu307, and YBaCuO/ Fe, fig. 5. Of course, polaronic description of the optical properties of high-T¢ oxides is not a unique possibility. As was argued by Hirsch [ 35 ] within the BCS approach the kinetic energy is essentially unchanged below To. This is because the occupation numbers of electrons (holes) is very similar at T = T¢ and at T = 0: at T = T¢ it is a step function broadened in a range 2T¢ and at T = 0 it is a step function broadened in a range of the BCS-gap A. Both function are extremely
close for 2A/T¢=4 as is usually the case. Applying the conductivity sum rule one can reach the conclusion that it should be a decrease in the slope of the intraband absorption at T¢ with the temperature lowering in a BCS superconductor. However the real trouble with the BCS approach is that for frequencies much higher than the superconducting energy gap, no change in optical absorption is predicted [49 ]. Any change in the optical absorption due to pairing seems to be practically invisible. The BCS approach gives also a Drude-like spectral dependence of the intra-band absorption This is in contradiction with the experimental observations which show a relative change of the absorption of more than 10% and a spectral shape which is drastically different from the Drude one, fig. 5. On the other side if one assumes that the frequency window 2000-10000 cm -~ corresponds to the inter-band transitions, then again the BCS-model is in a qualitative disagreement with the experiment, predicting the small increase in the inter-band absorption with increasing temperature in the normal state instead of the strong ( ~ 2 0 % ) decrease, observed experimentally. (To reach this conclusion one should take into account that the total intensity, integrated from 0 to oo is temperature independent. ) As alternative to the BCS model Hirsch has considered models of superconductivity where pairing
368
A.S. Alexandrov et al. /Bose-Einstein condensation o f small bipolarons
originates in gain of kinetic rather than potential energy of carders due to strong correlations in a Hubbard sub-bands (the "hole" superconductivity) [ 35,50 ]. In such systems, a change in the frequencydependent conductivity occurs at frequencies much higher than the scale set by the superconducting energy gap. The models predict a decrease in the high frequency inter-band optical absorption upon the superconducting transition. On the contrary the intraband absorption should increase below Tc in a "hole" superconductor. In our view it is difficult to reconcile the "hole" superconductivity [35 ] with the optical experimental data now at hand. First of all the relevant frequency region u~0.5-0.7 eV lies entirely in the charge-transfer gap, estimated to be higher than 1.5 eV, so the near infrared spectrum, fig. 5 (insert), corresponds to intra-band transitions within the oxygen band hybridised with the copper one. If this is the case then the model with the kinematic pairing is in a qualitative disagreement with the experiment, which shows a decrease of intensity just below Tc instead of an increase, predicted by the model for the intra-band absorption. If, nevertheless, the experiment, fig. 5, measures inter-band absorption, then it is difficult to understand with this model the decrease of the absorption in the normal state with increasing temperature because the model [35 ] is a mean-field BCS-like approximation, which for the normal state predicts an increase of the inter-band optical absorption with temperature. On the other hand it was demonstrated by Mihailovic et al. [23 ] that the infrared absorption of different copper-based oxides results from small (bi) polaronic states within the charge-transfer gap (see also ref. [ 51 ] ). We have shown in the present paper that its temperature dependence below Tc [ 13 ] is a manifestation of the coherent bipolaron tunneling within a narrow band and of Bose-Einstein condensation. The simple "toy" model is proposes and the conductivity sum rule for bipolaronic systems is derived which illustrate the strong influence of BoseEinstein condensation of charged bosons on their high-frequency optical properties. In the present analysis, we have not considered the effect of spin-wave "shakeoff" resulting from the coupling of carders to spin excitations as a source of the near infrared absorption. This coupling leads to
the spin bipolaron formation as proposed by one of us [ 1 ]. It was argued [23 ] that any contribution to or(z,) from magnon (quasi) shakeoff should be significantly weaker than the direct contribution to a(v) arising from the electron-photon coupling.
Acknowledgements We thank our colleagues at Cambridge, in particular A. Campbell, J. Cooper, W. Liang, J. Loram, J. Waldram, and J. Wheatley for very valuable discussions. One of us (A.S.A.) is grateful to Ph. Nozieres for his calling my attention to Bose-Einstein condensation of paraexcitons [36 ] and to H. Capellmann, J. Ranninger and A. Gogolin for their enlightening comments on the magnetic contribution to the optical absorption, on the photo-induced infrared conductivity and on our "toy" model.
Appendix A To derive the sum-rule, eq. (19), one can rewrite the Frohlich Hamiltonian, eq. (11) in a site representation: H= E m,m
T-,-'c~,sc-',s+(2N) -1/2 ' ,s
× ~ y(q)og(q)c~,:m,s(dcei¢"+dt-qe -iq') m,q,S
+ H p h -~- Vc .
(33)
The real part of the conductivity is
a~(u)=n ~ I
(34)
where 10) and I f ) are the ground and excited states of H, respectively. The current operator is: (m ' - m ) T = , , , c ,*, : , , , , ,
J=ie ~ m,m
(35)
' ,s
and satisfies the equation J=ie[H,X] ,
(36)
where X is the sum of the position operators of the electrons: X = ~, m c ~ , : . . . .
(37)
A.S. Alexandrov et al. / Bose-Einstein condensation of small bipolarons
Integrating over frequency eq. (34) and using the identity eq. (36) one obtains: oo
ien O'x~(Z,) d~,= -~- <0l
[X, Jx] 10>,
(38)
0
which yields eq. (19) after the direct substitution of eq. (35) and eq. (37). To derive the sum rule for the bipolaronic Drude conductivity, eq. (22), one can apply the Peierls substitution for the electron transfer integral with a vector potential A:
T,.,,.,=~"T.,,,n,eieA('-'').
(39)
This substitution yields for the bipolaronic transfer integral in the bipolaronic Hamiltonian, eq. (12) tm,,n, ~ tm,.,e 2ieA(m - m')
(40)
and for the bipolaronic current operator:
8Hb
J--- &4 =2ie ~
(m-m')tm,,.,b~bm,.
(41)
The rest is identical to the derivation of the electronic sum rule, eqs. ( 3 4 ) - ( 3 8 ) if one replace c~,~ and e on bm and 2e, respectively.
Appendix B In an ideal Bose-gas the presence of a particle at some point in space increases the probability that another particle is near that point, i.e. the particles "attract" one another. The short-range pair-correlation function turns out to be twice of the classical value, g ( 0 ) = 2 for all temperatures above Tc [52]. In the condensed state (below To) this correlator drops to its classical value ( = 1 ) at T = 0. To analyze the value and the temperature dependence of the pair-correlation function in the Boseliquid one can apply an empirical relation [ 53 ]
g(r, T)=l+(1-no(T)/n)2[g*(r, T ) - I ] ,
(42)
where no is the condensate density and g* (r, T) is the extrapolation of the normal-state function to below T~. This relation is found to be in excellent agreement with the X-ray data for He4 [ 40 ]. For a short distance g* (0, T) turns to be small in a contrast with an ideal Bose-gas; for He4 it is below 0.2
369
(see fig. 7 of ref. [40] ). In this case eq. (42) yields a negative temperature derivative of g(r, T) in the superfluid state. Thus the temperature dependence of the pair-correlation function is determined by the repulsion. For the strong repulsion this correlator gives an additional suppression of the optical absorption in the superconducting state, [eq. (23) ], to that from the temperature dependence of the coherent contribution (the last term in eq. (23)). However this additional suppression is expected to be small because the relative number of particles in a Bose-condensate is small, less than 10% of the total number as in He4.
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