Lorentzian form for the imaginary part of the dynamic spin susceptibility: comparison with NQR and neutron scattering data in copper oxide superconductors Текст научной статьи по специальности «Физика»

ISSN 2G72-59B1

Volume ll, No. 2, pages 29-32, 2009

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Vadim Atsarkin (Institute of Radio Engineering and Electronics, Moscow) Detlef Brinkmann (University of Zurich, Zurich) Yurij Bunkov (CNRS, Grenoble) John Drumheller (Montana State University, Bozeman) Mikhail Eremin (KSU, Kazan) Yoshio Kitaoka (Osaka University,

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Boris Malkin (KSU, Kazan) Haruhiko Suzuki (Kanazawa University, Kanazava) Murat Tagirov (KSU, Kazan)

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In Kazan State University the Electron Paramagnetic Resonance (EPR) was discovered by Zavoisky E.K. in 1944.

Lorentzian form for the imaginary part of the dynamic spin susceptibility: comparison with NQR and Neutron Scattering data in copper oxide superconductors

We present some new results based on the relaxation function theory for a doped two-dimensional Heisenberg antiferromagnetic system with damping of paramagnon-like excitations. The Lorentzian form for the imaginary part of the dynamic spin susceptibility gives a reasonable agreement with neutron scattering and plane copper nuclear spin-lattice relaxation rate 63(1/7’i) data in right up to optimally doped La2-xSrxCuO4.

PACS: 74.72.-h, 74.25.Ha, 75.40.Gb

Keywords: cuprate superconductors, dynamic spin susceptibility

1. Introduction

Plane copper oxide high-temperature superconductors (high- Tc) are the doped S = 1/2 twodimensional Heisenberg antiferromagnetic (2DHAF) systems. In the carrier free regime, the elementary excitations are spin waves [1-3], magnons in the quasiparticle language. Observations by neutron scattering (NS) of the o / T scaling for the averaged over the Brillouin zone the imaginary

part of the dynamic spin susceptibility, %"(o,T) = \ %"(q,o,T)d2q & %’(o,T ^ 0) f (o / T), in the underdoped high- Tc compounds [2] above Tc is referred to a nearby quantum phase transition [1]. Nuclear Magnetic/Quadrupole Resonance (NMR/NQR) studies [4] revealed the extension of the universal behavior of %" (o, T) down to the MHz frequency range. In this paper we present some new results based on the relaxation function theory with damping of the paramagnon-like excitations [5-7] in connection with plane copper nuclear spin-lattice relaxation rate as obtained by NQR and imaginary part of the dynamic spin susceptibility %" (k,o) as obtained by NS experiments.

2. Basic relations

We employ the t - J Hamiltonian [8] known as the minimal model for high- Tc cuprates:

written in terms of the Hubbard operators X?° that create an electron with spin a at site i and S, are spin-1/2 operators. Here, the hopping integral tj = t between the nearest neighbors (NN) describes the

motion of electrons causing a change in their spins and J = 0.12 eV is the NN AF coupling constant. The spin and density operators are defined as follows:

I.A. Larionov

Kazan State University, Kremlevskaya, 18, Kazan 420008, Russia * E-mail: [email protected] (Received November 10, 2009; accepted December 14, 2009)

(1)

with the standard normalization X,00 + X++ + X. = 1.

The static spin susceptibility as derived within the t - J model [9] is given by,

X(k) =

4 | c,

Jg-(g+ +Yk)

(3)

Table 1. The calculated in the T0 limit antiferromagnetic spin-spin correlation function between the nearest neighbours cb the parameter g_, and the spin stiffness constant pS .

and has the same structure as in the isotropic spin-wave theory [10] at all doping levels. The NN AF spin-spin correlation function is given by

c1 = (1/4)S*S*+p^ , the index p runs over NN, and Yk = (1 / 2)(coskx + cosky). The parameter g+ is related to AF correlation length % via the

expression % = 1 / (2^1 g+ -1) & (Jy/g _ / kBT)exp(2nS /kBT), where pS is spin stiffness. The values of the parameters of the theory [9]: cx, g_, and pS are given in Table 1.

The relaxation shape function is given by [11]

Doping c1 g_ 2nps / J %0

S ii o -0.1152 4.1448 0.38 -

S=0.04 -0.1055 3.913 0.3 1/(2 S)

S=0.15 -0.0617 2.947 0.13 1/S

F (k,o) = -

TA1k A2k /n

[oTk(o2 -Afk -A2k)]2 +(o2 -A2k)2’

where Tk = 2/(nA2k ) , and Aj2k and A^k are related to the frequency moments

0) = j onF(k,o)do,

(4)

as Alk = \®k), A 2k = (( oO) / \®k}) -\®k), the expression for the second moment is given by

(5)

(o2) = i( [Sk, S-k ])/Xk = -( 8 Jc - 4tffT ](1 - Yy )/Xk, (6)

where T = p^k)kfkh , p = (1 + S) / 2, and fkh = [exp(-Ek + u) / kBT +1]-1 is the Fermi function of

holes, where the number of extra holes, S, due to doping, per one plane Cu 2+, can be identified with

where

the Sr content x in La2-x Sr x CuO 4. The excitation spectrum of holes is given by, Ek = the hoppings, t, are affected by electronic and AF spin-spin correlations c1, resulting in effective values [5,8], for which we set tf =SJ / 0.2, in order to match the insulator-metal transition. The

chemical potential u is related to S by S = p^kfkh . Note that F(k,o) is real, even in both k and

and normalized to unity j doF(k,o) = 1. The detailed expression for is given in [5].

We take the Lorentzian form for the imaginary part of the dynamic spin susceptibility,

Xkork , Xkork

o

Xl (k,o)=-

(7)

[o-o™ ]2 +rk [o + o^ ]2 +rk

for k around the AF wave vector (n,n). The spin-wavelike dispersion, renormalized by interactions, is given by the relaxation function [11], given by Eq. (4),

(• to

®kW = 2 j oF(k,o)do, (8)

where the integration over o in Eq. (8) has been performed analytically and exactly [7].

Figure 1. The averaged over the Brillouin zone the imaginary part of dynamic spin susceptibility

xL (m) = 1 xL (q, m )d2q versus mlT . Symbols: NS data for La196Sr004CuO4 at various m values from Ref. [13], the lines show the calculated x'L (®).

The damping of paramagnon-like excitations rk is given by rk = ^(mk) - (®iT)2 .

The plane copper nuclear spin-lattice relaxation rate is given by

2kT

3 (11T) =-

I

|k| >V$ef

63F(k)2Xl(k,o0) ,

(9)

W

E

where m0 = 2n x 34 MHz (■« T, J) is the measuring NQR frequency. The hyperfine formfactor for plane foCu sites is given by, 63,F(k)2 =(Aab +4yji)', where Aab = 1.7-10"7 eV and B = (1 + 4£) • 3.8-10-7 eV are the Cu on-site and transferred hyperfine couplings, respectively [12]. The effective correlation length ^

is given by, = %0-1 + [5,13]. Thus from now on

we replace % by B,eg and %0 values are presented in the Table 1.

The spin diffusive contribution (from small wave vectors H < 11 ) can be calculated from general

physical grounds, namely, the linear response theory, hydrodynamics, and fluctuation-dissipation theorem [5-7,11,14],

3(11T1) Dlf =

3 F (0)2 kBT x(k = 0)

A.

where

nhD

A = [11 (4n)]ln[1 + D21 (Of)]

(10)

and

D = lim[ng2 F (q, 0)] 1 is the spin diffusion constant.

q——0

3. Results

Figure 1 shows the averaged over the Brillouin zone and normalized imaginary part of dynamic spin

T(K)

Figure 2. Temperature and doping dependence of the plane copper nuclear spin-lattice relaxation rate 63(1/T1) = 2W. Experimental data for La2-xSrxCuO4 from Ref. [4]. Solid lines show the results of the calculations with Lorentzian form of the susceptibility and taking into account the damping of the paramagnon-like excitations using Eq. (7). The dashed lines show the results of the calculations without damping of the paramagnon-like excitations, after Refs. [5,6], i.e., using Eq. (11).

susceptibility х"(т,T) versus т/ T . It suggests т/ T scaling for underdoped high- Tc layered

cuprates with a deviations at small т in qualitative agreement with NS data [1,13].

Figure 2 shows the calculated with Eqs. (7) and (9) plane copper nuclear spin-lattice relaxation rate 63(1/T1) (solid lines) without any adjustable parameters. The dashed lines show the calculated 63(1/T1) without damping of the paramagnon-like excitations [5], where F (k,rn) is related to the

imaginary part of the dynamic spin susceptibility x"(k,rn) as [5,11],

X" (k,rn) = rnXkF (k,rn) . (11)

It is worth to mention that the temperature dependence of 63(1/T1) in both theories is governed by the temperature dependence of the correlation length and by the factor kBT in agreement with [12]. At low T, where « const, the plane copper 63(1/T1) <x T, as it should. At high T, the correlation length

shows weak doping dependence and 63(1/T1) of doped samples behaves similarly to that of La2CuO4.

4. Summary

In summary, we developed further a relaxation function theory [5-7] for dynamic spin properties and approved the Lorentzian form for the imaginary part of the dynamic spin susceptibility for layered copper high- Tc in the normal state. The т / T scaling and spin-lattice relaxation at plane copper sites may be explained within the damped spin-wave-like theory, possessing a reasonable agreement with the observations by means of neutron scattering and magnetic resonance in high- Tc copper oxides.

Acknowledgments

This work was supported by RFBR Grant No.09-02-00777-a.

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