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On the transport properties of a “normal” metal in the resonating-valence-bond model
Volume 137, number 6
PHYSICS LETTERS A
22 May 1989
ON THE TRANSPORT PROPERTIES OF A “NORMAL” METAL IN THE RESONATING-VALENCE-BOND MODEL S.N. ARTEMENKO Institute of Radioengineering and Electronics oftheAcademy ofSciences, Marx av. 18, 103907 Moscow, USSR Received 20 February 1989; accepted for publication 13 March 1989 Communicated by V.M. Agranovich
The linear response ofa metal in the “normal” RVB state to an electric field, IR-absorption by free carriers, and current—voltage characteristics of tunnel junctions are studied using Keldysh’s diagram technique. The temperature dependence of the in-plane conductivity obtained differs from the experimentally observed dependence. The form of the tunneling I—V curves and IR-absorption reflects the spectrum of elementary excitations.
1. Introduction Many different mechanisms of high T~superconductivity have been proposed soon after it was discovered. Nevertheless, there is no generally accepted theory, which can account for the different unusual magnetic and transport properties of high T~materials both in normal and in superconducting states. One of the most interesting mechanisms of this kind was proposed by Anderson [1]. This mechanism is based on the resonatingvalence-bond (RVB) model (see also refs. [2—8]),according to which the low-energy excitations in the strongly correlated electron system consist of neutral fermions (spinons) and charged bosons (holons). Since the RVB state is strongly different from the ordinary metallic state which can be described by Landau Fermi-liquid theory, it is interesting to study the properties ofthe RVB metal and compare them with the properties of existing materials, especially with the properties of high T~superconductors, even if the high T~superconductivity is not generated by the RVB state. Thus, it may be useful to study the properties of the material in the RVB state assuming that this state is realized. Investigations of this kind (see refs. [5,11—13]) proposed the explanation of some experimental data for high T~superconductors. In this article the in-plane dc resistivity and tunneling in the “normal” RYB state are studied further, and the absorption of the electromagnetic radiation by free carriers at frequencies w>> 1 /‘r (t is the momentum relaxation time) is calculated.
2. Formulation In this section we consider the formulation of the RVB model, which is based on the results of refs. [4,5,10]. We shall use the mean-field approximation for the Hubbard Hamiltonian. According to ref. [4] electron field operators are expressed by field operators of holon e and of spinon S~,with spin a at site i. These operators satisfy the completeness condition e~e, +ZaS~Sj,.= 1 (we neglect the possibility of the filling of the upper Hubbard band assuming the limit of a large on-site Coulomb potential). Therefore, to calculate the excitation spectrum it is necessaryto minimize the Hamiltonian, expressed in terms of holon and spinon field operators, provided the completenesscondition is satisfied. As a result a nonzero chemical potential of spinons ~z,appears. In the momentum representation the Hamiltonian has the form 294
0375-960 11891$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 137, number 6
PHYSICS LETTERS A
22 May 1989
fl=2t ~ ~ p,k,q,a T
V
pa (
\(C1+C2±
L~ y~qJk pa p,k,q,a
(‘
k+qfi
(‘
~C’±~
kfr-’p±qa
pa
(1
~
ka
k+q/3
p+qfl/
,
where t is a transfer integral, J is the superexchange interaction, y (q) = cos q~a+ cos q~a,i~h= jiis the chemical potential of holons, and j~is the electron chemical potential. Using a Hartree—Fock factorization and diagonalizing eq. (1) by a Bogoliubov transformation with u = ~[1— (~ +~~)/E~] and v~= 1 —u, we find the holon and spinon spectra, —
2+Cu,+~)2] R=!c~cosp~a+K~cosp~a, E=[A A=4~cosp~a+A~cosp~a, ~~=2oty(j~)+R, ö=/2, E~= —~ —4,~t/J,
J ~ /2,
Kx,y
1/2
4x,yJ
The existence of different components ofthe order parameter A and K reflects SU (2) invariance [9]. At zero hole concentration (ö= 0) the states with different A and K are equivalent, but at ô~0 the degeneracy is broken and the symmetry of the order parameter in the ground state depends on ö. We shall consider the case pc~= = K, 4, = Ae A~ = 4. The spinon spectrum contains in general a gap Eg proportional to ~ ‘~‘,
Eg~UssJ(l+c)/(l+c+21~/42)
where
c~ = 2öt +
E5=,.JE~+A2(l
,c, c= cos ~‘. The spinon spectrum can be expressed as +c+2~~/A2)~’2+42(l—c)i2
with ~=
~~(cospxa+cos~ya_ 42(l+21~~2/42)),
~=
(cosap~—cosap~).
From these expressions it follows that Eg = 0 only if c= 1, and since Egs’~ (i.e. Eg is small) the minimum of Es is situated near Ip~ I I I it/2a. In the general case of I Cl ~ 1 the minimum is achieved near these points too, but Eg ~ 0. In the special case of s-wave pairing c = 1, which was considered by Anderson and coworkers [2,4—61,the minimum of E5 is achieved in the vicinity of the lines p~= ±p,,± it/a. The parameters which describe the excitation spectrum can be found from the self-consistency conditions and from equations expressing the electron density N 0 (or hole density SN=N— N0, where N is the number of sites), —
6N=~N,,
~N=~~~°(l—2n,),
(2)
where ~j,,and N~are the distribution functions of spinons and holons, respectively. If these equations are satisfied, then the completeness condition for spinon and holon operators is fulfilled automatically. In the general case C 1 one can obtain the estimation A K J, I~5 ÔJ. In the case c = 1 (s-wave pairing) it is easy to obtain explicit expressions for the parameters at 6N/N<< t/J and T<< ic A=J,
ö=~N/N, ,c=
2
m —4
ôt,
j.t~—
ÔJ
log[A/max(T,p~)]
1u~+8K~m_E~=Tlog(1_e_T~T), T~=8iu&t/J. 295
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22 May 1989
3. dc conductivity Now we shall study the linear response of holons in the nonsuperconducting state to an electric field parallel to the conducting layers. This problem was studied in refs. [5,12,13]. Conclusions were made that the scattering of holons by spinons [5] or by phonons [121 can explain the experimental temperature dependence of the resistivity. To calculate the current we use a kinetic equation for holons, which is obtained from the Hamiltonian (1) by means of Keldysh’s method [141. The kinetic equation has the standard form oN
ON
ON
(3)
An expression for the current equation can be obtained from the well-known relation between the current and electron Green function. The latter may be written as a convolution of holon and spinon Green functions. As a result we getj= —JpN~dp. First, we consider the scattering of holons by spinons. In this case the contribution to the collision integral ~ in eq. (3) originates from the first term of the Hamiltonian (1). For I,~,we obtain [I~(r,
t;p, )G(r, t;p, ~)—.~(r,
t;p,
E)G~(r,t;p, ~)]
~—,
(4)
where G ±are holon Green functions in Keldysh’s technique, and the mass operators ~ are expressed in terms of Green functions according to fig. 1. Using the explicit expressions for the Green functions of holons and spinons we obtain the collision integral which has an evident physical meaning,
ist=J ~ _(l—flp+q)flk±q(l+Nk)Np]ô(E~~—E~—E;+q+E~+q)+...}.
(5)
In eq. (5) three terms are not written out. These terms can be obtained from the expression inside the curly E~±q by 1—n, v2 and _Es. brackets by the substitution of flp+q, Up2+q, E+q and/or nk+q, ~ Similar to the case of ordinary electron collisions the effectivemomentum relaxation due to holon scattering is efficient if the umklapp-processes are present. Umklapp-scattering is possible if there are spinons near the Brillouin zone boundary, i.e. the conductivity may be determined by holon—spinon scattering in the case c= 1 only. Therefore we shall study scattering by spinons for this case only. To calculate the scattering time we linearize the collision integral with respect to N~.At T>> TB (but T<< J), when N~<< 1, we obtain
!
dkdq t2y2(q){u+qu~+q(1—n,+q)nk±qö(E~ _E~_E;+q+E~±q)+...}.
(6)
The main difference between this expression and the corresponding results of refs. [5] and [131 is the dependence of the matrix element of the holon—spinon interaction on the momentum q. In ref. [5] this dependence was not taken into account, and in ref. [131 in the place of y2 (q) in eq. (6) there is a matrix element depending on the momentum difference in the initial and final states of the holon~.The evaluation of eq.
I”
296
Fig. 1. Mass operator for holon—spinon scattenng. Solid and wavy lines correspond to the holon and spinon Green functions respectively.
Volume 137, number 6
PHYSICS LETFERS A
22 May 1989
K
P1 p
p Fig. 2. Mass operator for scattering by impurities.
(6) gives a more rapid variation of2tK l/tand andcharacteristic p with temperature the dependence p— T. If mh3v,~ <> T, effective then p—T512. results contradict with the experimental data for high superconductors. It should be noted, that the minimum of the matrix element at the same momentum where there is a minimum of spinon energy may be a consequence of the tight-binding approximation with the interaction between the nearest neighbours, and a deviation from this approximation may give the observed dependence. Let us consider impurity scattering. In ref. [12] scattering ofbosons by phonons and by charged impurities was studied, the presence of spinons being neglected. We shall start with a different approach, according to which the impurities interact with electrons, and not with holons, i.e. we add to the Hamiltonian a term ~p,qCp~+qCpUq,where c,, is the electron field operator, and Uq is the impurity potential. Then we substitute electron operators for holon and spinon operators. As a result we obtain that the interaction of holons with impurities depends on the spinon density. The mass operator for impurity scattering is shown in fig. 2. Calculating the mass operator we obtain the collision integral ~
—(1
t?q)flk(l
+I 1)N,]o(E~_E,h1 _E+E~)+...},
2 and ES for 1
(7) —
where ellipsis stands for the terms which Linearizing can be obtained by substitution n, u v2 and the —E5, and again n, is the impurity concentration. eq. (7) we obtain theofscattering time at temperatures T> Eg, TB, when N~<< 1, flJIUI[uufl(lfl)o(Eh_Li1_ES+~5)+1
.
(8)
If we consider the short-range interaction of electrons with impurities, like in ordinary metals, then I UI2 has no temperature dependence. In the case C= 1, when the spinon density-of-states may be considered as independent of energy, two independent integrations over spinon energies Eq and Ek in eq. (8) yield 1 Iv-- T2, p~—T2. For C 1, when the spinon density-of-states increases with energy, we obtain p— T4. Thus, again there is no agreement with experimental data on high T, superconductors. These results may be applied to the case of scattering by phonons at temperatures higher than the Debye temperature, if one substitutes n, U2 in eq. (8) for a factor which takes into account the density of phonons and the matrix element of electron—phonon interaction. Since the phonon density increases with temperature, the corresponding 1 /t increases with temperature more rapidly than in the case of impurity scattering, and differs from the experimental dependence p’-’ T. 4. High-frequency response Now we consider the absorption of electromagnetic radiation by free charge carriers. If charge carriers are ordinary electrons, then this absorption is absent, because it is impossible to satisfy both energy and momentum ~ When this work was finished, ref. [16] appeared in which p( T) was calculated. In ref. [16] the matrix element has aform similar to eq. (6), but it was replaced by a constant. As a result a different form ofp ( T) was obtained.
297
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22 May 1989
&) ~
Fig. 3. Diagram for the high-frequency conductivity.
conservation laws. In the RVB model the absorption is possible due to the presence of spinons and holons. To calculate a linear response we replace in the Hamiltonian the electron momentum p by p (e/C )A, where A is the vector potential. This is equivalent to the addition of (e/C)A to the holon momenta in the first term in eq. (1). Then we calculate the correction to the electron Green function in Keldysh’s representation that is linear with respect to A. At frequencies w>> 1/i the conductivity can be calculated according to the diagram in fig. 3, 2N 2 f dp dp e 2(2it)6 0 e 1 dk 1 2 2 + rn J N 1U~U~+kt (p—pi +k) V,y(p—p1 +k) —
x ~
—n,~ +N,~)— (l—np)(l+Z\~)np+klY,l_k + iw[E+k+E~_k—E—E~—a—i0] ( 2, E5 for 1 n, v2, —E5 are dropped again. The where theofterms thatwhich can becontains obtained by substitution of n, u to the energy conservation, describes the abreal part eq. (9) a ö-function corresponding sorption of a photon due to the transition of both holon and spinon. We shall estimate the high-frequency conductivity a in the case of frequencies smaller than the energy bandwidths. First we consider the state with c 1. Let Ti~Eg, TB. Then at w>>wC=mhv~the conductivity is frequency independent, c=a~—~ (e2N 0/m)(h/t); 3 and t—.~1 eV gives a,-— 102 ~ cm1 which is close to the experimental value estimation for N0— 1021 cm [15]. In the low-frequency limit the absorption starts from the frequency w = 2E, and near the edge it has the form a-~[~JEg(a—2Eg)/wc]ac, and at Eg<>Eg, TB, then the threshold at low frequency is absent and a-— ~,/‘~f7~ a~,provided T<< w,; and a-..a,, provided T>> w,. At C= 1 the spinon density-of-state differs from the case C= 1, thus the dependence of a(w, T) changes. For example, at w >> w,, increases with a linearly. ...
‘
—
5. Tunneling current The tunneling current within the frame of the RVB model was studied in refs. [5,11]. The results of this section are close to the results of ref. [11],but there are some differences. Besides, we consider different relations between temperature and the parameters of the excitation spectrum. We use the method of the tunneling Hamiltonian. The tunneling current I at a voltage V is given by the formula I_i$dpoG~~:;t~t) where G + is the electron Green function for one of the electrodes. Starting from the equations for the Green functions introduced by Keldysh [14], and taking into account terms proportional to the square of the tunneling matrix elements TM, with the help of the factorization of the electron Green function, we obtain expressions for the current, which again have a simple physical meaning. For a tunnel junction of the RVB metal with ordinary metal we have
298
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PHYSICS LETTERS A
22 May 1989
i~~f ~ +v[(l—np)(l+Np)(l—nq)—n~pnq]ô(V+E~+E~+E~)},
(10)
where the first term, for example, describes the contribution of the process in which in the RVB electrode spinon with energy E5 annihilates and a holon with energy Eh is created, and in the normal metal an electron with energy EN vanishes. For tunneling between two RVB metals we obtain I~j IlqI2dPdqdIPidlqi [uuq2[n,(l+Np)Nq(l—nq)
(lnp)Npt(l+Nqi)nq]~(VE~+EphiE5qEqhi)+...}.
(11) (11)
Similar to eq. (5) it is necessary to add to eq. (11) terms in which u2, n, E5 are substituted for v2, 1—n, —E5. Let us discuss the results for the current at voltages V small in comparison with the bandwidths. First, we consider the state with C= 1. For the contact RVB-metal—normal-metal at low temperatures T<< Eg, TB, when holons are degenerate and one can neglect thermally excited spinons, we have I~O(IV~Eg)[signV+O(~V)(V~Eg)/2TB].
(12)
If the voltage at the normal metal electrode is negative, then the current—voltage characteristic is linear. If this voltage is positive and the electrons flow to the normal metal, then the I— V curve becomes quadratic. This is a consequence of the Bose statistics of holons and may be easily understood, because in the first case annihilation of holons takes place, and in the second case we have creation of holons, the probabilities of these processes being proportional to N~and 1 + N~respectively. At T~i~ Eg, TB, when the holon gas is not degenerate and one can neglect the gap in the spinon spectrum, I—~2sinh~(O( V+EB)(V~EB+T)e~12T+O(~ V—EB) (J/+EB)2+2T2)
(13)
This expression has a characteristic feature at a voltage V equal to the difference between the minimum energy of holons and their chemical potential. In the case of contact of two RVB metals, in one of the electrodes creation ofholons with probability 1 + N 0 2 sign V. For example, at T>>Eg, TB, takes place for both signs of voltage V. Thus, at large V, i—V I~~(V2+2IVIT)sinh(V/2T)e~1/2T. (14) If C ~ 1 the density-of-states of spinons increases with energy, therefore, at large V we obtain the dependence I—V2 and [—V3 instead of[—Vand [-‘V2.
6. Conclusion We have studied some transport properties of a metal in the “normal” RVB state. According to the results obtained in the preceding sections, the temperature dependence of the in-plane resistivity p—-T, which was observed in high T~materials, cannot be explained by the RVB model, at least, within the limits of the tightbinding approximation with nearest-neighbour interaction, provided the spinons and holons are considered as weakly interacting particles. Both the shape of tunneling I—V curves and the frequency dependence of optical absorption in the RVB state and their dependence on temperature reflects the character of the spectra of spi299
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nons and holons. Large magnitude of the optical absorption and the voltage dependence of the conductivity of tunnel junctions formed by high 7’, materials may be interpreted by means of the RVB model.
References [I]P.W. Anderson, Science 235 (1987) 1196. [2] G. Bascaran, Z. Zou and P.W. Anderson, Solid State Commun. 63 (1987) 973. [3] S. Kivelson, D. Rokshar and Z. Sethna, Phys. Rev. B 35 (1987) 8865. [4] Z. Zou and P.W. Anderson, Phys. Rev. B 37 (1988) 627. [5] P.W. Anderson and Z. Zou, Phys. Rev. Lett. 60 (1988) 132. [6] J.M. Wheatley, T.C. Hsu and P.W. Anderson, Phys. Rev. B 37 (1988)5897. [7]1. Dzyaloshinskii, A. Polyakov and P. Wiegmann, Phys. Lett. A 127 (1988) 112. [8] P.B. Wiegmann, Phys. Rev. Lett. 60 (1988) 821. [9] F.C. Zhang, C. Gros, T.M. Rice and H. Shiba, Physica C 153—155 (1988) 1251. [10] H. Fukuyama, Y. Hasegawa and V. Suzumura, Physica C 153—155(1988) 1630. [11] K. Flensberg, P. Hedegard and M. Brix, Phys. Rev. B 38 (1988) 841. [12] D.Y. Xing, W.Y. Lai, W.P. Su and R.S. Ting, Solid State Commun. 65 (1988) 1319. [131 C. Kallin and A.J. Berlinsky, Phys. Rev. Lett. 60 (1988) 2556. [141 L.V. Keldysh, Zh. Eksp. Teor. Fiz. 47 (1964) 1515. [IS) S. Tajima, S. Uchida, M. Ishii, M. Takagi, S. Tanaka, V. Kawabe, H. Hasegawa, T. Aita and T. Ishiba, Mod. Phys. Lett. B 1 (1988) 353. [16]A. Oguri and S. Maekawa, Physica C 156 (1988) 679.
300
PHYSICS LETTERS A
22 May 1989
ON THE TRANSPORT PROPERTIES OF A “NORMAL” METAL IN THE RESONATING-VALENCE-BOND MODEL S.N. ARTEMENKO Institute of Radioengineering and Electronics oftheAcademy ofSciences, Marx av. 18, 103907 Moscow, USSR Received 20 February 1989; accepted for publication 13 March 1989 Communicated by V.M. Agranovich
The linear response ofa metal in the “normal” RVB state to an electric field, IR-absorption by free carriers, and current—voltage characteristics of tunnel junctions are studied using Keldysh’s diagram technique. The temperature dependence of the in-plane conductivity obtained differs from the experimentally observed dependence. The form of the tunneling I—V curves and IR-absorption reflects the spectrum of elementary excitations.
1. Introduction Many different mechanisms of high T~superconductivity have been proposed soon after it was discovered. Nevertheless, there is no generally accepted theory, which can account for the different unusual magnetic and transport properties of high T~materials both in normal and in superconducting states. One of the most interesting mechanisms of this kind was proposed by Anderson [1]. This mechanism is based on the resonatingvalence-bond (RVB) model (see also refs. [2—8]),according to which the low-energy excitations in the strongly correlated electron system consist of neutral fermions (spinons) and charged bosons (holons). Since the RVB state is strongly different from the ordinary metallic state which can be described by Landau Fermi-liquid theory, it is interesting to study the properties ofthe RVB metal and compare them with the properties of existing materials, especially with the properties of high T~superconductors, even if the high T~superconductivity is not generated by the RVB state. Thus, it may be useful to study the properties of the material in the RVB state assuming that this state is realized. Investigations of this kind (see refs. [5,11—13]) proposed the explanation of some experimental data for high T~superconductors. In this article the in-plane dc resistivity and tunneling in the “normal” RYB state are studied further, and the absorption of the electromagnetic radiation by free carriers at frequencies w>> 1 /‘r (t is the momentum relaxation time) is calculated.
2. Formulation In this section we consider the formulation of the RVB model, which is based on the results of refs. [4,5,10]. We shall use the mean-field approximation for the Hubbard Hamiltonian. According to ref. [4] electron field operators are expressed by field operators of holon e and of spinon S~,with spin a at site i. These operators satisfy the completeness condition e~e, +ZaS~Sj,.= 1 (we neglect the possibility of the filling of the upper Hubbard band assuming the limit of a large on-site Coulomb potential). Therefore, to calculate the excitation spectrum it is necessaryto minimize the Hamiltonian, expressed in terms of holon and spinon field operators, provided the completenesscondition is satisfied. As a result a nonzero chemical potential of spinons ~z,appears. In the momentum representation the Hamiltonian has the form 294
0375-960 11891$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 137, number 6
PHYSICS LETTERS A
22 May 1989
fl=2t ~ ~ p,k,q,a T
V
pa (
\(C1+C2±
L~ y~qJk pa p,k,q,a
(‘
k+qfi
(‘
~C’±~
kfr-’p±qa
pa
(1
~
ka
k+q/3
p+qfl/
,
where t is a transfer integral, J is the superexchange interaction, y (q) = cos q~a+ cos q~a,i~h= jiis the chemical potential of holons, and j~is the electron chemical potential. Using a Hartree—Fock factorization and diagonalizing eq. (1) by a Bogoliubov transformation with u = ~[1— (~ +~~)/E~] and v~= 1 —u, we find the holon and spinon spectra, —
2+Cu,+~)2] R=!c~cosp~a+K~cosp~a, E=[A A=4~cosp~a+A~cosp~a, ~~=2oty(j~)+R, ö=
J ~
Kx,y
1/2
4x,yJ
The existence of different components ofthe order parameter A and K reflects SU (2) invariance [9]. At zero hole concentration (ö= 0) the states with different A and K are equivalent, but at ô~0 the degeneracy is broken and the symmetry of the order parameter in the ground state depends on ö. We shall consider the case pc~= = K, 4, = Ae A~ = 4. The spinon spectrum contains in general a gap Eg proportional to ~ ‘~‘,
Eg~UssJ(l+c)/(l+c+21~/42)
where
c~ = 2öt +
E5=,.JE~+A2(l
,c, c= cos ~‘. The spinon spectrum can be expressed as +c+2~~/A2)~’2+42(l—c)i2
with ~=
~~(cospxa+cos~ya_ 42(l+21~~2/42)),
~=
(cosap~—cosap~).
From these expressions it follows that Eg = 0 only if c= 1, and since Egs’~ (i.e. Eg is small) the minimum of Es is situated near Ip~ I I I it/2a. In the general case of I Cl ~ 1 the minimum is achieved near these points too, but Eg ~ 0. In the special case of s-wave pairing c = 1, which was considered by Anderson and coworkers [2,4—61,the minimum of E5 is achieved in the vicinity of the lines p~= ±p,,± it/a. The parameters which describe the excitation spectrum can be found from the self-consistency conditions and from equations expressing the electron density N 0 (or hole density SN=N— N0, where N is the number of sites), —
6N=~N,,
~N=~~~°(l—2n,),
(2)
where ~j,,and N~are the distribution functions of spinons and holons, respectively. If these equations are satisfied, then the completeness condition for spinon and holon operators is fulfilled automatically. In the general case C 1 one can obtain the estimation A K J, I~5 ÔJ. In the case c = 1 (s-wave pairing) it is easy to obtain explicit expressions for the parameters at 6N/N<< t/J and T<< ic A=J,
ö=~N/N, ,c=
2
m —4
ôt,
j.t~—
ÔJ
log[A/max(T,p~)]
1u~+8K~m_E~=Tlog(1_e_T~T), T~=8iu&t/J. 295
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3. dc conductivity Now we shall study the linear response of holons in the nonsuperconducting state to an electric field parallel to the conducting layers. This problem was studied in refs. [5,12,13]. Conclusions were made that the scattering of holons by spinons [5] or by phonons [121 can explain the experimental temperature dependence of the resistivity. To calculate the current we use a kinetic equation for holons, which is obtained from the Hamiltonian (1) by means of Keldysh’s method [141. The kinetic equation has the standard form oN
ON
ON
(3)
An expression for the current equation can be obtained from the well-known relation between the current and electron Green function. The latter may be written as a convolution of holon and spinon Green functions. As a result we getj= —JpN~dp. First, we consider the scattering of holons by spinons. In this case the contribution to the collision integral ~ in eq. (3) originates from the first term of the Hamiltonian (1). For I,~,we obtain [I~(r,
t;p, )G(r, t;p, ~)—.~(r,
t;p,
E)G~(r,t;p, ~)]
~—,
(4)
where G ±are holon Green functions in Keldysh’s technique, and the mass operators ~ are expressed in terms of Green functions according to fig. 1. Using the explicit expressions for the Green functions of holons and spinons we obtain the collision integral which has an evident physical meaning,
ist=J ~ _(l—flp+q)flk±q(l+Nk)Np]ô(E~~—E~—E;+q+E~+q)+...}.
(5)
In eq. (5) three terms are not written out. These terms can be obtained from the expression inside the curly E~±q by 1—n, v2 and _Es. brackets by the substitution of flp+q, Up2+q, E+q and/or nk+q, ~ Similar to the case of ordinary electron collisions the effectivemomentum relaxation due to holon scattering is efficient if the umklapp-processes are present. Umklapp-scattering is possible if there are spinons near the Brillouin zone boundary, i.e. the conductivity may be determined by holon—spinon scattering in the case c= 1 only. Therefore we shall study scattering by spinons for this case only. To calculate the scattering time we linearize the collision integral with respect to N~.At T>> TB (but T<< J), when N~<< 1, we obtain
!
dkdq t2y2(q){u+qu~+q(1—n,+q)nk±qö(E~ _E~_E;+q+E~±q)+...}.
(6)
The main difference between this expression and the corresponding results of refs. [5] and [131 is the dependence of the matrix element of the holon—spinon interaction on the momentum q. In ref. [5] this dependence was not taken into account, and in ref. [131 in the place of y2 (q) in eq. (6) there is a matrix element depending on the momentum difference in the initial and final states of the holon~.The evaluation of eq.
I”
296
Fig. 1. Mass operator for holon—spinon scattenng. Solid and wavy lines correspond to the holon and spinon Green functions respectively.
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K
P1 p
p Fig. 2. Mass operator for scattering by impurities.
(6) gives a more rapid variation of2tK l/tand andcharacteristic p with temperature the dependence p— T. If mh3v,~ <
—(1
t?q)flk(l
+I 1)N,]o(E~_E,h1 _E+E~)+...},
2 and ES for 1
(7) —
where ellipsis stands for the terms which Linearizing can be obtained by substitution n, u v2 and the —E5, and again n, is the impurity concentration. eq. (7) we obtain theofscattering time at temperatures T> Eg, TB, when N~<< 1, flJIUI[uufl(lfl)o(Eh_Li1_ES+~5)+1
.
(8)
If we consider the short-range interaction of electrons with impurities, like in ordinary metals, then I UI2 has no temperature dependence. In the case C= 1, when the spinon density-of-states may be considered as independent of energy, two independent integrations over spinon energies Eq and Ek in eq. (8) yield 1 Iv-- T2, p~—T2. For C 1, when the spinon density-of-states increases with energy, we obtain p— T4. Thus, again there is no agreement with experimental data on high T, superconductors. These results may be applied to the case of scattering by phonons at temperatures higher than the Debye temperature, if one substitutes n, U2 in eq. (8) for a factor which takes into account the density of phonons and the matrix element of electron—phonon interaction. Since the phonon density increases with temperature, the corresponding 1 /t increases with temperature more rapidly than in the case of impurity scattering, and differs from the experimental dependence p’-’ T. 4. High-frequency response Now we consider the absorption of electromagnetic radiation by free charge carriers. If charge carriers are ordinary electrons, then this absorption is absent, because it is impossible to satisfy both energy and momentum ~ When this work was finished, ref. [16] appeared in which p( T) was calculated. In ref. [16] the matrix element has aform similar to eq. (6), but it was replaced by a constant. As a result a different form ofp ( T) was obtained.
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&) ~
Fig. 3. Diagram for the high-frequency conductivity.
conservation laws. In the RVB model the absorption is possible due to the presence of spinons and holons. To calculate a linear response we replace in the Hamiltonian the electron momentum p by p (e/C )A, where A is the vector potential. This is equivalent to the addition of (e/C)A to the holon momenta in the first term in eq. (1). Then we calculate the correction to the electron Green function in Keldysh’s representation that is linear with respect to A. At frequencies w>> 1/i the conductivity can be calculated according to the diagram in fig. 3, 2N 2 f dp dp e 2(2it)6 0 e 1 dk 1 2 2 + rn J N 1U~U~+kt (p—pi +k) V,y(p—p1 +k) —
x ~
—n,~ +N,~)— (l—np)(l+Z\~)np+klY,l_k + iw[E+k+E~_k—E—E~—a—i0] ( 2, E5 for 1 n, v2, —E5 are dropped again. The where theofterms thatwhich can becontains obtained by substitution of n, u to the energy conservation, describes the abreal part eq. (9) a ö-function corresponding sorption of a photon due to the transition of both holon and spinon. We shall estimate the high-frequency conductivity a in the case of frequencies smaller than the energy bandwidths. First we consider the state with c 1. Let Ti~Eg, TB. Then at w>>wC=mhv~the conductivity is frequency independent, c=a~—~ (e2N 0/m)(h/t); 3 and t—.~1 eV gives a,-— 102 ~ cm1 which is close to the experimental value estimation for N0— 1021 cm [15]. In the low-frequency limit the absorption starts from the frequency w = 2E, and near the edge it has the form a-~[~JEg(a—2Eg)/wc]ac, and at Eg<
‘
—
5. Tunneling current The tunneling current within the frame of the RVB model was studied in refs. [5,11]. The results of this section are close to the results of ref. [11],but there are some differences. Besides, we consider different relations between temperature and the parameters of the excitation spectrum. We use the method of the tunneling Hamiltonian. The tunneling current I at a voltage V is given by the formula I_i$dpoG~~:;t~t) where G + is the electron Green function for one of the electrodes. Starting from the equations for the Green functions introduced by Keldysh [14], and taking into account terms proportional to the square of the tunneling matrix elements TM, with the help of the factorization of the electron Green function, we obtain expressions for the current, which again have a simple physical meaning. For a tunnel junction of the RVB metal with ordinary metal we have
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i~~f ~ +v[(l—np)(l+Np)(l—nq)—n~pnq]ô(V+E~+E~+E~)},
(10)
where the first term, for example, describes the contribution of the process in which in the RVB electrode spinon with energy E5 annihilates and a holon with energy Eh is created, and in the normal metal an electron with energy EN vanishes. For tunneling between two RVB metals we obtain I~j IlqI2dPdqdIPidlqi [uuq2[n,(l+Np)Nq(l—nq)
(lnp)Npt(l+Nqi)nq]~(VE~+EphiE5qEqhi)+...}.
(11) (11)
Similar to eq. (5) it is necessary to add to eq. (11) terms in which u2, n, E5 are substituted for v2, 1—n, —E5. Let us discuss the results for the current at voltages V small in comparison with the bandwidths. First, we consider the state with C= 1. For the contact RVB-metal—normal-metal at low temperatures T<< Eg, TB, when holons are degenerate and one can neglect thermally excited spinons, we have I~O(IV~Eg)[signV+O(~V)(V~Eg)/2TB].
(12)
If the voltage at the normal metal electrode is negative, then the current—voltage characteristic is linear. If this voltage is positive and the electrons flow to the normal metal, then the I— V curve becomes quadratic. This is a consequence of the Bose statistics of holons and may be easily understood, because in the first case annihilation of holons takes place, and in the second case we have creation of holons, the probabilities of these processes being proportional to N~and 1 + N~respectively. At T~i~ Eg, TB, when the holon gas is not degenerate and one can neglect the gap in the spinon spectrum, I—~2sinh~(O( V+EB)(V~EB+T)e~12T+O(~ V—EB) (J/+EB)2+2T2)
(13)
This expression has a characteristic feature at a voltage V equal to the difference between the minimum energy of holons and their chemical potential. In the case of contact of two RVB metals, in one of the electrodes creation ofholons with probability 1 + N 0 2 sign V. For example, at T>>Eg, TB, takes place for both signs of voltage V. Thus, at large V, i—V I~~(V2+2IVIT)sinh(V/2T)e~1/2T. (14) If C ~ 1 the density-of-states of spinons increases with energy, therefore, at large V we obtain the dependence I—V2 and [—V3 instead of[—Vand [-‘V2.
6. Conclusion We have studied some transport properties of a metal in the “normal” RVB state. According to the results obtained in the preceding sections, the temperature dependence of the in-plane resistivity p—-T, which was observed in high T~materials, cannot be explained by the RVB model, at least, within the limits of the tightbinding approximation with nearest-neighbour interaction, provided the spinons and holons are considered as weakly interacting particles. Both the shape of tunneling I—V curves and the frequency dependence of optical absorption in the RVB state and their dependence on temperature reflects the character of the spectra of spi299
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nons and holons. Large magnitude of the optical absorption and the voltage dependence of the conductivity of tunnel junctions formed by high 7’, materials may be interpreted by means of the RVB model.
References [I]P.W. Anderson, Science 235 (1987) 1196. [2] G. Bascaran, Z. Zou and P.W. Anderson, Solid State Commun. 63 (1987) 973. [3] S. Kivelson, D. Rokshar and Z. Sethna, Phys. Rev. B 35 (1987) 8865. [4] Z. Zou and P.W. Anderson, Phys. Rev. B 37 (1988) 627. [5] P.W. Anderson and Z. Zou, Phys. Rev. Lett. 60 (1988) 132. [6] J.M. Wheatley, T.C. Hsu and P.W. Anderson, Phys. Rev. B 37 (1988)5897. [7]1. Dzyaloshinskii, A. Polyakov and P. Wiegmann, Phys. Lett. A 127 (1988) 112. [8] P.B. Wiegmann, Phys. Rev. Lett. 60 (1988) 821. [9] F.C. Zhang, C. Gros, T.M. Rice and H. Shiba, Physica C 153—155 (1988) 1251. [10] H. Fukuyama, Y. Hasegawa and V. Suzumura, Physica C 153—155(1988) 1630. [11] K. Flensberg, P. Hedegard and M. Brix, Phys. Rev. B 38 (1988) 841. [12] D.Y. Xing, W.Y. Lai, W.P. Su and R.S. Ting, Solid State Commun. 65 (1988) 1319. [131 C. Kallin and A.J. Berlinsky, Phys. Rev. Lett. 60 (1988) 2556. [141 L.V. Keldysh, Zh. Eksp. Teor. Fiz. 47 (1964) 1515. [IS) S. Tajima, S. Uchida, M. Ishii, M. Takagi, S. Tanaka, V. Kawabe, H. Hasegawa, T. Aita and T. Ishiba, Mod. Phys. Lett. B 1 (1988) 353. [16]A. Oguri and S. Maekawa, Physica C 156 (1988) 679.
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