Fragment-orbital-dependent spin fluctuations in the single-component molecular conductor $\left[\mathrm{Ni}{\left(\mathrm{dmdt}\right)}_2\right]$

Motivated by recent nuclear magnetic resonance experiments, we calculated the spin susceptibility, Knight shift, and spin-lattice relaxation rate ($1/T_{1}T$) of the single-component molecular conductor $\left[\mathrm{Ni}{\left(\mathrm{dmdt}\right)}_2\right]$ using the random phase approximation in a multi-orbital Hubbard model describing the Dirac nodal line electronic system in this compound. This Hubbard model is composed of three fragment orbitals and on-site repulsive interactions obtained using ab initio many-body perturbation theory calculations. We found fragment-orbital-dependent spin fluctuations with the momentum $\mathbf{q}=\mathbf{0}$ and an incommensurate value of the wave number $\mathbf{q}=\mathbf{Q}$ at which a diagonal element of the spin susceptibility is maximum. The $\mathbf{q}=\mathbf{0}$ and $\mathbf{Q}$ responses become dominant at low and high temperatures, respectively, with the Fermi-pocket energy scale as the boundary. We show that $1/T_{1}T$ decreases with decreasing temperature, but starts to increase at low temperature owing to the $\mathbf{q}=\mathbf{0}$ spin fluctuations, while the Knight shift keeps monotonically decreasing. These properties are due to the intramolecular antiferromagnetic fluctuations caused by the characteristic wave functions of this Dirac nodal line system, which is described by an $n$-band ($n\ge 3$) model. We show that the fragment orbitals play important roles in the magnetic properties of $\left[\mathrm{Ni}{\left(\mathrm{dmdt}\right)}_2\right]$.

Figure 1

(a) Crystal structure in the $b$–$c$ plane of [$\mathrm{M}{\left(\mathrm{dmdt}\right)}_2$]. This material consists of $\mathrm{M}{\left(\mathrm{dmdt}\right)}_2$ molecules. (b) Crystal structure along the $b$-axis. The red, blue, brown, and cyan balls represent Ni, S, C, and H atoms, respectively. The black frames represent the unit cell.

Figure 2

Schematic of a $\mathrm{Ni}{\left(\mathrm{dmdt}\right)}_2$ molecule and the fragment orbitals A, B, and C. The red, blue, brown, and cyan balls represent Ni, S, C, and H atoms, respectively. The red dots in orbitals A, B, and C indicate the location of the Ni atom as a guide to the eye. (a) Side view of the molecule. The black solid line represents the $\mathrm{Ni}{\left(\mathrm{dmdt}\right)}_2$ molecule. (b) Vertical-axis view of the molecule. The red dashed circles indicate C atoms, which are replaced with ${}^{13}\mathrm{C}$ in the ${}^{13}\mathrm{C}$-NMR experiment [60]. These figures are plotted by VESTA. Different colors represent different signs of the wave functions. The cut-off of the normalized wave functions is 0.01.

Figure 3

Hopping networks between fragment orbitals in tight-binding model of [Ni(dmdt)2]. (a) Schematic illustration of the 2D network of the transfer integrals (shown by double-headed arrows) in the crystalline $bc$-plane. The dashed square stands for the unit cell. (b) Schematic illustration of the 3D hopping network including the transfer integrals along the $a$-direction. The black chain lines and vertical bold lines in (b) are guides to the eyes: The first lines are parallel to the $a$-direction, while the second lines connect the molecules in the same $bc\text{−}\mathrm{plane}$.

Figure 4

(a) Feynman diagram of ${\widehat{\chi }}^{zz,s}(\mathbf{q},i{\omega }_l)$. (b) Feynman diagram of ${\widehat{\chi }}^{\pm ,s}(\mathbf{q},i{\omega }_l)$. The solid lines represent the noninteracting Green's functions, which describe the quasiparticles. The dashed lines represent the interactions. The open circles are the vertexes connecting the noninteracting Green's functions and the interaction. The black dots represent the spin operators.

Figure 5

(a) Energy dispersion of [$\mathrm{Ni}{\left(\mathrm{dmdt}\right)}_2$] in the $k_b$–$k_c$ plane, where $k_a=-\pi /2$. Dirac cones exist between each pair of bands. Those between bands 1 and 2 draw the Dirac nodal line. (b) Fermi surface in the first Brillouin zone. The electron and hole pockets are drawn in magenta and green, respectively. The inset shows the Dirac nodal line in the first Brillouin zone. (c) Schematic of the relationship among the Dirac nodal line, Fermi surface, and wave number $k_a$. The red curved line shows the Dirac nodal line. The dotted transverse line shows $E_F$.

Figure 6

The local density of state $D_{\alpha }\left(\omega \right)$ for the fragment orbital $\alpha =\text{A}$, B, and C in the unit cell of [$\mathrm{Ni}{\left(\mathrm{dmdt}\right)}_2$]. The red dotted and blue dashed lines show $D_A\left(\omega \right)$[=$D_C\left(\omega \right)$] and $D_B\left(\omega \right)$, respectively. The black solid line shows the total density of states $D_{\mathrm{tot}}\left(\omega \right)=D_A\left(\omega \right)+D_B\left(\omega \right)+D_C\left(\omega \right)$. The integral value of $D_{\mathrm{tot}}\left(\omega \right)$ over $\omega$ is equal to 6 due to the three orbitals and the spins. Its unit is the number of electrons.

Figure 7

Wave function of orbital B in band 2, $|d_{B,2,\sigma }{\left(\mathbf{k}\right)|}^2$, in the $k_b$–$k_c$ plane, where $k_a=-\pi /2$. The up arrows indicate the corresponding positions of the Dirac points formed between the Bands 1 and 2 in Fig. 5, while the down arrows indicate those between the Bands 2 and 3. The color bar represents the magnitude of $|d_{B,2,\sigma }{\left(\mathbf{k}\right)|}^2$.

Figure 8

The momentum dependencies of the diagonal elements of the spin susceptibility in the absence of $U$. (a) ${\chi }_{AA}^0(\mathbf{q},0)$ in the $q_b$–$q_c$ plane, where $q_a=0$. (b) ${\chi }_{BB}^0(\mathbf{q},0)$ in the $q_b$–$q_c$ plane, where $q_a=0.2\pi$. (c) $\mathrm{Im}\left[{\chi }_{AA}^0(\mathbf{q},{\omega }_0)\right]$ in the $q_b$–$q_c$ plane, where $q_a=0$. (d) $\mathrm{Im}\left[{\chi }_{BB}^0(\mathbf{q},{\omega }_0)\right]$ in the $q_b$–$q_c$ plane, where $q_a=0.2\pi$. The temperature $T=0.003$ eV.

Figure 9

(a) Spectral weight ${\rho }_A(\mathbf{k},0)$. It is not zero on the Fermi surface. (b) Spectral weight ${\rho }_B(\mathbf{k},0)$. A part of it is zero because of a ZR. In the both figures, $k_a=\pi , -0.65\pi <k_b<-0.50\pi$, and $0.20\pi <k_c<0.30\pi$. Color bars represent the magnitude of the spectral weight ${\rho }_{\alpha }(\pi ,k_b,k_c,0)$. The yellow region indicates that the spectral weight at $E_F$ is high.

Figure 10

Nesting vector $\mathbf{Q}$ in the momentum space. $\mathbf{Q}$ connects the regions where the spectral weight of orbital B is high (in the right figure). Color bars represent the magnitude of the spectral weight ${\rho }_B(\pi ,k_b,k_c,0)$. The electron and hole pockets are drawn in magenta and green, respectively (in the left figure).

Figure 11

Temperature dependence of ${\chi }_{\alpha \beta }^0(\mathbf{0},0)$. The red dashed, blue solid, green dotted, and purple chain lines represent ${\chi }_{AA}^0(\mathbf{0},0), {\chi }_{BB}^0(\mathbf{0},0), {\chi }_{AB}^0(\mathbf{0},0)$, and ${\chi }_{AC}^0(\mathbf{0},0)$, respectively. The inset shows the band-dependent spin susceptibilities ${\chi }_{AB,12}^0(\mathbf{0},0)$ and ${\chi }_{AC,12}^0(\mathbf{0},0)$ by the green dotted and purple chain lines, respectively.

Figure 12

Temperature dependence of $K_{\alpha }$ in the absence of $U$. The red dotted and blue dashed lines show $K_A$ and $K_B$, respectively. The black solid line shows the total Knight shift $K_{\text{total}}=K_A+K_B+K_C$. The dotted longitudinal line indicates $T=T^{*}\sim 0.01$ eV. The green thin line which is proportional to $T$ is drawn for the eye guide.

Figure 13

Temperature dependence of ${(1/T_{1}T)}_{\alpha }$ in the absence of $U$. The red solid and blue dotted lines show ${(1/T_{1}T)}_A$ and ${(1/T_{1}T)}_B$, respectively. The dashed longitudinal line indicates $T=T^{*}\sim 0.01$ eV. The green thin line which is proportional to $T^2$ is drawn for the eye guide.

Figure 14

Temperature dependence of the Stoner factors ${\xi }_s\left(\mathbf{0}\right)$ and ${\xi }_s\left(\mathbf{Q}\right)$. The red solid and blue dotted lines show ${\xi }_s\left(\mathbf{0}\right)$ and ${\xi }_s\left(\mathbf{Q}\right)$, respectively.

Figure 15

First- and second-order perturbation terms of ${\chi }_{AA}^s(\mathbf{0},0)$ shown by the brown solid and magenta dotted lines, respectively. The horizontal axis represents the temperature.

Figure 16

(a) Temperature dependence of ${\chi }_{AA}^s(\mathbf{0},0), {\chi }_{BB}^s(\mathbf{0},0), {\chi }_{AB}^s(\mathbf{0},0)$, and ${\chi }_{AC}^s(\mathbf{0},0)$ shown by the red dashed, blue solid, green dotted, and purple chain lines, respectively. The inset shows an enlarged view of the region around ${\chi }_{BB}^s(\mathbf{0},0)$ and ${\chi }_{AB}^s(\mathbf{0},0)$. (b) Temperature dependence of $\mathrm{Re}\left[{\chi }_{AA}^s(\mathbf{Q},0)\right], \mathrm{Re}\left[{\chi }_{BB}^s(\mathbf{Q},0)\right], \mathrm{Re}\left[{\chi }_{AB}^s(\mathbf{Q},0)\right]$, and $\mathrm{Re}\left[{\chi }_{AC}^s(\mathbf{Q},0)\right]$. The combination of matrix elements and lines is the same as in (a).

Figure 17

Schematic illustrations of spin polarization. (a) Intramolecular antiferromagnetic spin fluctuations linked to the $\mathbf{q}=\mathbf{0}$ response shown in Fig. 16. (b) Spin correlations within a molecule given by the $\mathbf{q}=\mathbf{Q}$ response in Fig. 16.

Figure 18

The momentum dependencies of the diagonal elements of the spin susceptibility in the presence of $U$ (a) ${\chi }_{AA}^s(\mathbf{q},0)$ in the $q_b$–$q_c$ plane, where $q_a=0$. (b) ${\chi }_{BB}^s(\mathbf{q},0)$ in the $q_b$–$q_c$ plane, where $q_a=0.2\pi$. (c) $\mathrm{Im}\left[{\chi }_{AA}^s(\mathbf{q},{\omega }_0)\right]$ in the $q_b$–$q_c$ plane, where $q_a=0$. (d) $\mathrm{Im}\left[{\chi }_{BB}^s(\mathbf{q},{\omega }_0)\right]$ in the $q_b$–$q_c$ plane, where $q_a=0.2\pi$. The temperature $T=0.003$ eV.

Figure 19

Temperature dependence of the Knight shift $K_{\alpha }$ for $U=0.802$ and $\lambda =0.95$. The red dashed and blue dotted lines show $K_A$ and $K_B$, respectively. The black solid line shows the total Knight shift $K_{\mathrm{tot}}=K_A+K_B+K_C$.

Figure 20

Temperature dependence of ${(1/T_{1}T)}_{\alpha }$ for $U=0.802$ and $\lambda =0.95$. The red solid and blue dotted lines show ${(1/T_{1}T)}_A$ and ${(1/T_{1}T)}_B$, respectively.

Figure 21

Correspondence table between the fragment orbitals, shape of the Fermi surface, momenta at which the noninteracting spin susceptibilities have peaks, Stoner factors, and $1/T_{1}T$.