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Quasiparticle lifetime behaviour in a simplified self-consistent T-matrix treatment of the attractive Hubbard model in 2D
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Physica B 241 243 (1998) 832 834
A unifying theory of the magnetic and transport properties of Laz_xSrxCuO4 E. Lai, R.J. G o o d i n g * Department qf Physics, Queen's Universio,, Kingston, Ont., Canada K7L 3N6
Abstract
We present a theory of impurity-dominated transport for La 2 xSrxCuO,~ when x<0.05, and use this theory to reanalyze previously published data for x = 0.002 (Chen et al., Phys. Rev. Lett. 63 (1989) 2307; Phys. Rev. B 43 (1991) 392; 51 (1995) 3671) and x = 0.04 (Keimer et al., Phys. Rev. B 46 (1992) 14 034). The starting point of this theory is identical to one used previously in successfully explaining the magnetic spin texture for this same compound. Our chiral impurity theory provides the best fit to the experimental data for both doping levels..F 1998 Published by Elsevier Science B.V. All rights reserved. Keywords." La-based high T c compounds; Transport theory
In this paper we present a theory that describes the conductivity of La2 xSrxCuO4 (hereafter referred to as LSCO) for 0.0 ~ x <0.05 at low temperatures. Our model displays the importance of both impurity effects and strong correlations. It takes into account the coupling of the hole motion to the magnetic background produced by strong correlations, and has proven successful in quantitatively describing the magnetic "spin texture" of this system [1]. In our model, we assume that at low doping levels and low temperatures, the hole is localized in a single CuO2 plane near a Sr impurity. To describe hole motion between localized impurity states, we first determine a continuum approximation for the *Corresponding author. Tel.: + 1-613 545 2696; fax.: + 1-613 545 6463: e-mail: gooding(w, physics.queensu.ca
hole wave function ~(r) by examining the twodimensional Schr6dinger equation in the effective mass approximation, viz. h2
---v
2m*
e2
2 4,
c l ~
4, = E 4',
tl)
where m* is the effective mass of the hole, ~:± is the out-of-plane dielectric constant, d, ~ ! . 8 5 i is the distance between the Sr impurity and the CuO2 plane, and V 2 is the Laplacian operator for two dimensions. The potential energy term follows from the relative separation distance between the Sr impurity centre and the hole. For LSCO, the appropriate numbers are m*/mo ~ 1-2 (we will be using 1.5), and e± ~ 31 + 2 [2-4]. The next step is to determine the ground state wave function 4'(r) for Eq. (1). For r >> d j_, the
0921-4526/98/$19.00 c 1998 Published by Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 7 ) 0 0 7 3 1 -X
E. Lai, R.J. Gooding / Physica B 241-243 (1998) 832-834
ground state wave function is that of a circularly symmetric s-wave state 0 ( r ) ~ e x p ( - r/a). However, we have not yet included the strong coupling of the hole motion to the magnetic background produced by the Cu spins in the LSCO system. At low doping levels and low temperatures, it is now well established (see Refs. [5-7]) that the ground state corresponds to holes circulating either clockwise or counterclockwise around the Sr impurity. Thus, one possible way to treat the coupling of the hole motion to the magnetic background is to realize that the presence of strong correlations changes the ground state from that of a circularly symmetric, s-wave impurity ground state, as is almost always inferred in doped semiconductors [8], to that of a doubly degenerate state with chiral quantum number ~s~-- _+ i [6]. We refer to this as a chiral impurity 9round state. This model has been exploited in quantitatively explaining a variety of experiments concerning the magnetic properties of LSCO [9,10,1]. For example, neutron scattering experiment by Keimer et al. has measured the correlation length as a function of doping; our impurity model tracks this data accurately [1]. Encouraged by these successes, we use the same model to describe this compound's transport properties. Mimicking the circulating character of this state, we determine the continuum approximation for the wave functions of the form exp( _+ iqS). By mapping this problem onto the solutions of Kummer's functions [12], one can show that asymptotically far from the Sr impurity, these chiral impurity ground states have the same form as the p-wave ( / = 1, m / = _+ 1) solution of the hydrogen atom: 0 ~ re r/, e_+i,/,,
(2)
where a = ~:,he/2m * e e ~ 5.48,~. We assume that transport proceeds by hopping between such impurity states, and thus we only include the effects of strong correlations by their influence on the specification of the symmetry of the impurity ground states. To derive the hopping conductivity for LSCO, we use the Miller and Abrahams random resistor network model [8], and modify it accordingly for the 2-D p-wave states in Eq. (2). We find that at high temperatures, where there are sufficient number of high energy phonons to assist hopping be-
833
tween neighbouring sites, nearest neighbour impurity band conduction (IBC) takes place, and the conductivity is given by
where ec is the average activation energy needed for a hole to hop to its neighbouring site. This result is independent of dimensionality and the symmetry of the impurity states. As the temperature is lowered, hopping between neighbouring sites may be forbidden due to the absence of the requisite high energy phonons. In this case, variable range hopping (VRH) conduction takes place. For the 2-D p-wave states given in Eq. (2), the (chiral) hopping conductivity is given by
,4, where To is a characteristic temperature. Finally, for temperatures so low such that the energy difference between the final and initial site is comparable to the Coulomb correlation energy between carriers, Coulomb gap (CG) conduction takes place. For 2-D p-wave states the (chiral) Coulomb-gap conductivity is then given by
where TEs is another characteristic temperature. These types of hopping conduction (as a function of decreasing temperature), with impurity band conduction followed by variable range hopping conduction and then by Coulomb gap conduction, have been observed in many materials [13-15]. We believe that this same sequence, now modified by the chiral impurity ground states, is also observed in LSCO. Upon analyzing the conductivity data from a x = 0.002 sample of LSCO prepared by Chen et al. [4], we find that IBC provides the best fit I for
1To determine the "best fit" for this and the x = 0.04 crystal, we have compared the data to different theories of conduction (activated, VRH, CG, power law, and logarithmic forms of the conductivity), and have maximized the goodness of fit parameter.
834
E. Lai. R.J. Goodin~ / Physica B 241 243 (1998) 832 834
50 K < T <295 K; this same result was given in Ref. [4]. For this sample for 4 K < T < 50 K, our 2-D p-wave VRH conductivity formula gives the best fit; in particular, this theory provides an improvement in the ;{~ by a factor of two compared with the 3-D, s-wave state theory proposed in Ref. [4]. There is no data below 4 K to test the validity of the Coulomb gap model. For a x = 0.04 sample studied by Keimer et al. [11], we find that IBC is observed for 20 K < T <70 K. We note that in Ref. [11] it was proposed that this temperature region corresponded to weak localization; however, 1BC gives a •2 roughly twenty times lower than for the weak localization logarithmic form proposed in Ref. [11], and thus we feel that IBC is a much more appropriate description of the conduction in this temperature range. For 1 K < T <20 K, our 2-D p-wave VRH conductivity formula again provides the best fit. Finally, for T < 1 K, our 2-D p-wave CG conductivity formula gives the best fit. Our chiral impurity theory does provide an improvement in the fits to the experimental data for either VRH (x = 0.002 and 0.04) or for CG conduction, but admittedly these improvements are only a factor of two or three. However, in all three of these cases, the best fit to the data is found to correspond to our 2-D, p-wave theory. We believe that this repeated agreement between theory and experiment lends strong credibility to our theory. It is to be emphasized that the physics that led to our theory, the chiral impurity ground state, and the successful description of the spin texture of LSCO at low temperatures [1], are the same. Dis-
order and strong correlations dominate the low temperature, low doping regime of LSCO. The treatment of other transport properties of LSCO, such as the magnetoresistance, and the extrapolation to doping levels above the superconducting threshold (Xsc ~ 0.05) will be discussed in a future publication. We thank Chih-Yung Chen and Bernhard Keimer for sending us their original transport data, and Marc Kastner for early discussions of this problem. This work was supported by the NSERC of Canada.
References [1] R.J. Gooding, N.M. Salem, R.J. Birgeneau, F.C. Chou, Phys. Rev. B 55 [1997) 6360. [2] C.Y. Chen et al., Phys. Rev. Lett. 63 (1989) 2307. [3] C.Y. Chen et al., Phys. Rev. B 43 (1991) 392. [4] C.Y. Chen et al., Phys. Rev. B 51 [1995) 3671. [51 K.J. von Szczepanski et al., Europhys. Lett. 8 (1989) 797. [6] R.J. Gooding, Phys. Rev. Lett. 66 (1991) 2266. [7] K.M. Rabe, R. Bhatt, J. Appl, Phys. 69 (1991) 4508. [8] Vor a review, see B.I. Shklovskii, A.L. Efros, Electronic Properties of Doped Semiconductors, Springer, New York, 1984. [9] R.J. Gooding, A. Mailhot, Phys. Rev. 48 (1993) 6132. [10] R.J. Gooding, N.M. Salem, A. Mailhot, Phys. Rev. B 49 [1994) 6067. [11] B. Keimer et al., Phys. Rev. B 46 (1992) 14 034. [12] For a review, see M. Abramowitz, I.A. Stegun, Handbook of Mathematical functions, Dover, New York, 1970. [13] N.F. Mort, Phil. Mag. 26 (1972)1015. [14] A.H. Clark~ Phys. Rev. 154 [1967) 750. [15] G. Sadasiv, Phys. Rev. 128 (1962) 1131.
Physica B 241 243 (1998) 832 834
A unifying theory of the magnetic and transport properties of Laz_xSrxCuO4 E. Lai, R.J. G o o d i n g * Department qf Physics, Queen's Universio,, Kingston, Ont., Canada K7L 3N6
Abstract
We present a theory of impurity-dominated transport for La 2 xSrxCuO,~ when x<0.05, and use this theory to reanalyze previously published data for x = 0.002 (Chen et al., Phys. Rev. Lett. 63 (1989) 2307; Phys. Rev. B 43 (1991) 392; 51 (1995) 3671) and x = 0.04 (Keimer et al., Phys. Rev. B 46 (1992) 14 034). The starting point of this theory is identical to one used previously in successfully explaining the magnetic spin texture for this same compound. Our chiral impurity theory provides the best fit to the experimental data for both doping levels..F 1998 Published by Elsevier Science B.V. All rights reserved. Keywords." La-based high T c compounds; Transport theory
In this paper we present a theory that describes the conductivity of La2 xSrxCuO4 (hereafter referred to as LSCO) for 0.0 ~ x <0.05 at low temperatures. Our model displays the importance of both impurity effects and strong correlations. It takes into account the coupling of the hole motion to the magnetic background produced by strong correlations, and has proven successful in quantitatively describing the magnetic "spin texture" of this system [1]. In our model, we assume that at low doping levels and low temperatures, the hole is localized in a single CuO2 plane near a Sr impurity. To describe hole motion between localized impurity states, we first determine a continuum approximation for the *Corresponding author. Tel.: + 1-613 545 2696; fax.: + 1-613 545 6463: e-mail: gooding(w, physics.queensu.ca
hole wave function ~(r) by examining the twodimensional Schr6dinger equation in the effective mass approximation, viz. h2
---v
2m*
e2
2 4,
c l ~
4, = E 4',
tl)
where m* is the effective mass of the hole, ~:± is the out-of-plane dielectric constant, d, ~ ! . 8 5 i is the distance between the Sr impurity and the CuO2 plane, and V 2 is the Laplacian operator for two dimensions. The potential energy term follows from the relative separation distance between the Sr impurity centre and the hole. For LSCO, the appropriate numbers are m*/mo ~ 1-2 (we will be using 1.5), and e± ~ 31 + 2 [2-4]. The next step is to determine the ground state wave function 4'(r) for Eq. (1). For r >> d j_, the
0921-4526/98/$19.00 c 1998 Published by Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 7 ) 0 0 7 3 1 -X
E. Lai, R.J. Gooding / Physica B 241-243 (1998) 832-834
ground state wave function is that of a circularly symmetric s-wave state 0 ( r ) ~ e x p ( - r/a). However, we have not yet included the strong coupling of the hole motion to the magnetic background produced by the Cu spins in the LSCO system. At low doping levels and low temperatures, it is now well established (see Refs. [5-7]) that the ground state corresponds to holes circulating either clockwise or counterclockwise around the Sr impurity. Thus, one possible way to treat the coupling of the hole motion to the magnetic background is to realize that the presence of strong correlations changes the ground state from that of a circularly symmetric, s-wave impurity ground state, as is almost always inferred in doped semiconductors [8], to that of a doubly degenerate state with chiral quantum number ~s~-- _+ i [6]. We refer to this as a chiral impurity 9round state. This model has been exploited in quantitatively explaining a variety of experiments concerning the magnetic properties of LSCO [9,10,1]. For example, neutron scattering experiment by Keimer et al. has measured the correlation length as a function of doping; our impurity model tracks this data accurately [1]. Encouraged by these successes, we use the same model to describe this compound's transport properties. Mimicking the circulating character of this state, we determine the continuum approximation for the wave functions of the form exp( _+ iqS). By mapping this problem onto the solutions of Kummer's functions [12], one can show that asymptotically far from the Sr impurity, these chiral impurity ground states have the same form as the p-wave ( / = 1, m / = _+ 1) solution of the hydrogen atom: 0 ~ re r/, e_+i,/,,
(2)
where a = ~:,he/2m * e e ~ 5.48,~. We assume that transport proceeds by hopping between such impurity states, and thus we only include the effects of strong correlations by their influence on the specification of the symmetry of the impurity ground states. To derive the hopping conductivity for LSCO, we use the Miller and Abrahams random resistor network model [8], and modify it accordingly for the 2-D p-wave states in Eq. (2). We find that at high temperatures, where there are sufficient number of high energy phonons to assist hopping be-
833
tween neighbouring sites, nearest neighbour impurity band conduction (IBC) takes place, and the conductivity is given by
where ec is the average activation energy needed for a hole to hop to its neighbouring site. This result is independent of dimensionality and the symmetry of the impurity states. As the temperature is lowered, hopping between neighbouring sites may be forbidden due to the absence of the requisite high energy phonons. In this case, variable range hopping (VRH) conduction takes place. For the 2-D p-wave states given in Eq. (2), the (chiral) hopping conductivity is given by
,4, where To is a characteristic temperature. Finally, for temperatures so low such that the energy difference between the final and initial site is comparable to the Coulomb correlation energy between carriers, Coulomb gap (CG) conduction takes place. For 2-D p-wave states the (chiral) Coulomb-gap conductivity is then given by
where TEs is another characteristic temperature. These types of hopping conduction (as a function of decreasing temperature), with impurity band conduction followed by variable range hopping conduction and then by Coulomb gap conduction, have been observed in many materials [13-15]. We believe that this same sequence, now modified by the chiral impurity ground states, is also observed in LSCO. Upon analyzing the conductivity data from a x = 0.002 sample of LSCO prepared by Chen et al. [4], we find that IBC provides the best fit I for
1To determine the "best fit" for this and the x = 0.04 crystal, we have compared the data to different theories of conduction (activated, VRH, CG, power law, and logarithmic forms of the conductivity), and have maximized the goodness of fit parameter.
834
E. Lai. R.J. Goodin~ / Physica B 241 243 (1998) 832 834
50 K < T <295 K; this same result was given in Ref. [4]. For this sample for 4 K < T < 50 K, our 2-D p-wave VRH conductivity formula gives the best fit; in particular, this theory provides an improvement in the ;{~ by a factor of two compared with the 3-D, s-wave state theory proposed in Ref. [4]. There is no data below 4 K to test the validity of the Coulomb gap model. For a x = 0.04 sample studied by Keimer et al. [11], we find that IBC is observed for 20 K < T <70 K. We note that in Ref. [11] it was proposed that this temperature region corresponded to weak localization; however, 1BC gives a •2 roughly twenty times lower than for the weak localization logarithmic form proposed in Ref. [11], and thus we feel that IBC is a much more appropriate description of the conduction in this temperature range. For 1 K < T <20 K, our 2-D p-wave VRH conductivity formula again provides the best fit. Finally, for T < 1 K, our 2-D p-wave CG conductivity formula gives the best fit. Our chiral impurity theory does provide an improvement in the fits to the experimental data for either VRH (x = 0.002 and 0.04) or for CG conduction, but admittedly these improvements are only a factor of two or three. However, in all three of these cases, the best fit to the data is found to correspond to our 2-D, p-wave theory. We believe that this repeated agreement between theory and experiment lends strong credibility to our theory. It is to be emphasized that the physics that led to our theory, the chiral impurity ground state, and the successful description of the spin texture of LSCO at low temperatures [1], are the same. Dis-
order and strong correlations dominate the low temperature, low doping regime of LSCO. The treatment of other transport properties of LSCO, such as the magnetoresistance, and the extrapolation to doping levels above the superconducting threshold (Xsc ~ 0.05) will be discussed in a future publication. We thank Chih-Yung Chen and Bernhard Keimer for sending us their original transport data, and Marc Kastner for early discussions of this problem. This work was supported by the NSERC of Canada.
References [1] R.J. Gooding, N.M. Salem, R.J. Birgeneau, F.C. Chou, Phys. Rev. B 55 [1997) 6360. [2] C.Y. Chen et al., Phys. Rev. Lett. 63 (1989) 2307. [3] C.Y. Chen et al., Phys. Rev. B 43 (1991) 392. [4] C.Y. Chen et al., Phys. Rev. B 51 [1995) 3671. [51 K.J. von Szczepanski et al., Europhys. Lett. 8 (1989) 797. [6] R.J. Gooding, Phys. Rev. Lett. 66 (1991) 2266. [7] K.M. Rabe, R. Bhatt, J. Appl, Phys. 69 (1991) 4508. [8] Vor a review, see B.I. Shklovskii, A.L. Efros, Electronic Properties of Doped Semiconductors, Springer, New York, 1984. [9] R.J. Gooding, A. Mailhot, Phys. Rev. 48 (1993) 6132. [10] R.J. Gooding, N.M. Salem, A. Mailhot, Phys. Rev. B 49 [1994) 6067. [11] B. Keimer et al., Phys. Rev. B 46 (1992) 14 034. [12] For a review, see M. Abramowitz, I.A. Stegun, Handbook of Mathematical functions, Dover, New York, 1970. [13] N.F. Mort, Phil. Mag. 26 (1972)1015. [14] A.H. Clark~ Phys. Rev. 154 [1967) 750. [15] G. Sadasiv, Phys. Rev. 128 (1962) 1131.