Spin dynamics in lightly doped ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$: Relaxation function within the $t\text{−}J$ model

The relaxation function theory of doped two-dimensional $S=1∕2$ Heisenberg antiferromagnetic (AF) systems in the paramagnetic state is presented taking into account the hole subsystem as well as both the electron and AF correlations. The expression for fourth frequency moment of relaxation shape function is derived within the $t\text{−}J$ model. The presentation obeys rotational symmetry of the spin correlation functions and is valid for all wave vectors through the Brillouin zone. The spin diffusion contribution to relaxation rates is evaluated and is shown to play a significant role in carrier free and doped antiferromagnet in agreement with exact diagonalization calculations. At low temperatures the main contribution to the nuclear spin-lattice relaxation rate, ${}^{63}(1∕T_1)$, of plane ${}^{63}\mathrm{Cu}$ arises from the AF fluctuations, and ${}^{17}(1∕T_1)$, of plane ${}^{17}\mathrm{O}$, has the contributions from the wave vectors in the vicinity of $(\pi ,\pi )$ and small $q\sim 0$. It is shown that the theory is able to explain the main features of experimental data on temperature and doping dependence of ${}^{63}(1∕T_1)$ in the paramagnetic state of both carrier free ${\text{La}}_2{\text{CuO}}_4$ and doped ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$ compounds.

Figure 1

The inverse correlation length ${\xi }_{\text{eff}}$ versus temperature fitted (solid lines) to the experimental data as obtained from neutron scattering experiments. For carrier free ${\text{La}}_2{\text{CuO}}_4$: filled circles from Ref. 4 (the fitted data), asterisks from Ref. 44 and open circles from Ref. 26. for doped ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$: squares for $x=0.02$, down triangles for $x=0.03$, and up triangles for $x=0.04$ from Ref. 26.

Figure 2

The calculated doping dependence of the spin diffusion constant $D$ in the $T\to 0$ limit with the fixed decoupling parameter $\zeta =1$ (squares) and with $\zeta$ as obtained from the best fit to NQR data (circles). The solid line is the result of Bonca and Jaklic (Ref. 50) for doping dependence of $D$ in the high temperature limit. The asterisk marks the value of the spin diffusion constant $D$ in the limit of high temperatures $T\to \infty$.

Figure 3

Log-scale mesh of the calculated dynamic structure factor $S(\mathbf{k},{\omega }_c)$ for carrier free antiferromagnet at $T=500\mathrm{K}$ with simple decoupling $\zeta =1$. The cross on the vertical axis marks the maximum of $JS(\mathbf{k},{\omega }_c)$ at $\mathbf{k}={\mathbf{q}}_0$.

Figure 4

Log-scale mesh of the calculated dynamic structure factor $S(\mathbf{k},{\omega }_c)$ for carrier free antiferromagnet at $T=1000\mathrm{K}$ with simple decoupling $\zeta =1$. The cross on the vertical axis marks the maximum of $JS(\mathbf{k},{\omega }_c)$ at $\mathbf{k}={\mathbf{q}}_0$.

Figure 5

Semilog-scale mesh of the calculated dynamic structure factor $S(\mathbf{k},{\omega }_c)$ for carrier free antiferromagnet at $T=500\mathrm{K}$ with the decoupling parameter $\zeta =1.8$. The region of small $\mathbf{q}$ values $(\mathbf{q}⪡{\mathbf{q}}_0)$ for which ${\log }_{10}\left[JS(\mathbf{k},\omega )\right]<-3$ is not shown. The cross on the vertical axis marks the maximum of $JS(\mathbf{k},{\omega }_c)$ at $\mathbf{k}={\mathbf{q}}_0$.

Figure 6

Semilog-scale mesh of the calculated dynamic structure factor $S(\mathbf{k},{\omega }_c)$ for carrier free antiferromagnet at $T=1000\mathrm{K}$ with the decoupling parameter $\zeta =1.8$. The region of small $\mathbf{q}$ values $(\mathbf{q}⪡{\mathbf{q}}_0)$ for which ${\log }_{10}\left[JS(\mathbf{k},\omega )\right]<-3$ is not shown. The cross on the vertical axis marks the maximum of $JS(\mathbf{k},{\omega }_c)$ at $\mathbf{k}={\mathbf{q}}_0$.

Figure 7

Log-scale mesh of the calculated dynamic structure factor $S(\mathbf{k},{\omega }_c)$ for $x=0.04$ hole content at $T=500\mathrm{K}$ with simple decoupling $\zeta =1$. The cross on the vertical axis marks the maximum of $JS(\mathbf{k},{\omega }_c)$ at $\mathbf{k}={\mathbf{q}}_0$.

Figure 8

Log-scale mesh of the calculated dynamic structure factor $S(\mathbf{k},{\omega }_c)$ for $x=0.04$ hole content at $T=1000\mathrm{K}$ with simple decoupling $\zeta =1$. The cross on the vertical axis marks the maximum of $JS(\mathbf{k},{\omega }_c)$ at $\mathbf{k}={\mathbf{q}}_0$.

Figure 9

Log-scale mesh of the calculated dynamic structure factor $S(\mathbf{k},{\omega }_0)$ for $x=0.035$ hole content at $T∕J=0.1(T\simeq 140\mathrm{K})$ with simple decoupling $\zeta =1$. The cross on the vertical axis marks the maximum of $JS(\mathbf{k},{\omega }_0)$ at $\mathbf{k}={\mathbf{q}}_0$.

Figure 10

Semilog-scale mesh of the calculated dynamic structure factor $S(\mathbf{k},{\omega }_0)$ for $x=0.035$ hole content at $T∕J=0.2(T\simeq 280\mathrm{K})$. The region of small $\mathbf{q}$ values $(\mathbf{q}⪡{\mathbf{q}}_0)$ for which ${\log }_{10}\left(JS\right(\mathbf{k},\omega \left)\right)<-3$ is not shown. The cross on the vertical axis marks the maximum of $JS(\mathbf{k},{\omega }_0)$ at $\mathbf{k}={\mathbf{q}}_0$.

Figure 11

Log-scale plot of the quantity ${}^{\alpha }F{\left(\mathbf{k}\right)}^{2}S(\mathbf{k},\omega )∕J$ versus $k_x$ $(k_y=0)$. The solid lines from down until up: ${}^{17}F{\left(\mathbf{k}\right)}^{2}S(\mathbf{k},{\omega }_0)∕J$ is given for $x=0.035$ at $T∕J=0.1$ $(T\simeq 140\mathrm{K})$ and $T∕J=0.2(T\simeq 280\mathrm{K})$ with the decoupling parameter $\zeta =1.3$, ${}^{63}F{\left(\mathbf{k}\right)}^{2}S(\mathbf{k},{\omega }_c)∕J$ is given for carrier free AF at $(x=0)$ at $T=500\mathrm{K}$ and $T=1000\mathrm{K}$ with the decoupling parameter $\zeta =1.8$. The dotted lines are ${}^{\alpha }F{\left({\mathbf{q}}_0\right)}^{2}S({\mathbf{q}}_0,\omega ){k_x}^2∕\left(0.5J{\mathbf{q}}_0^2\right)$ for $k_x<{\mathbf{q}}_0$ and ${}^{\alpha }F{\left({\mathbf{q}}_0\right)}^{2}S({\mathbf{q}}_0,\omega ){\mathbf{q}}_0^2∕\left(0.5J{k_x}^2\right)$ for $k_x>{\mathbf{q}}_0$.

Figure 12

Semilog-scale plot of the quantity ${}^{\alpha }F{\left(\mathbf{k}\right)}^{2}S(\mathbf{k},\omega )∕J$ versus $\mathbf{k}$ along the diagonal of the Brillouin zone $(k_x=k_y)$. ${}^{17}F{\left(\mathbf{k}\right)}^{2}S(\mathbf{k},{\omega }_0)∕J$ is given for $x=0.035$ at $T∕J=0.1(T\simeq 140\mathrm{K}$, lower curve) and $T∕J=0.2(T\simeq 280\mathrm{K}$, upper curve) with the decoupling parameter $\zeta =1.3$. ${}^{63}F{\left(\mathbf{k}\right)}^{2}S(\mathbf{k},{\omega }_c)∕J$ is given for carrier free AF $(x=0)$ at $T=500\mathrm{K}$ (lower curve at small $\mathbf{k}$) and $T=1000\mathrm{K}$ (upper curve at small $\mathbf{k}$) with the decoupling parameter $\zeta =1.8$. The dotted lines are ${}^{\alpha }F{\left({\mathbf{q}}_0\right)}^{2}S({\mathbf{q}}_0,\omega ){\mathbf{q}}_0^2∕\left(0.5J{\mathbf{k}}^2\right)$ for $\mathbf{k}>q_0$. The lower and upper crosses mark the values of ${}^{63}F{\left({\mathbf{q}}_0\right)}^{2}S({\mathbf{q}}_0,{\omega }_c)∕J$ at $T=500\mathrm{K}$ and $T=1000\mathrm{K}$, respectively.

Figure 13

The calculated temperature and doping dependence of the plane oxygen nuclear spin-lattice relaxation rate ${}^{17}(1∕T_1)$ (lines) and the experimental data for ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$ as measured by NMR with $x=0.025$ (triangles) and $x=0.035$ (squares) from Ref. 57. The experimental points have been rearranged with $J=1393\mathrm{K}$. The results of calculations with $\omega =2\pi \times 52\text{MHz}\left(9\mathrm{T}\right)$ are given for $x=0.035(\zeta =1.3)$ by the solid line and for $x=0.025(\zeta =1.6)$ by the dashed line. The result of the calculation with $\omega =2\pi \times 81.4\text{MHz}$ for $x=0.035(\zeta =1.3)$ coincides with the dashed line. The contribution to ${}^{17}(1∕T_1)$ from the wave vectors in the vicinity of $(\pi ∕a,\pi ∕a)$ for $x=0.035$ with $\zeta =1$ is shown by the upper dotted line and by the lower dotted line with $\zeta =1.3$.

Figure 14

Temperature and doping dependence of the plane copper nuclear spin-lattice relaxation rate ${}^{63}(1∕T_1)=2W$ (solid lines) fitted to the experimental data for ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$ with $x=0$, $x=0.012$, $x=0.024$, and $x=0.03$ from Ref. 45 and with $x=0.04$ from Ref. 58. The values of the fitting parameter $\zeta$ are shown in the Table . The dotted line shows the contribution to ${}^{63}(1∕T_1)$ from spin diffusion for $x=0$.

Figure 15

Recalculated plane copper nuclear spin-lattice relaxation rate ${}^{63}(1∕T_1)=2W$ with fixed $\zeta =1$ is shown by solid lines from up until down with increased doping. The dotted line shows the contribution to ${}^{63}(1∕T_1)$ from spin diffusion for $x=0.04$ with $\zeta =1$. The notation is the same as in Fig. 14.