Spin exchange dominated by charge fluctuations of the Wigner lattice in the chain cuprate Na${}_5{\text{Cu}}_3{\text{O}}_6$

Na${}_5$Cu${}_3$O${}_6$, a new member of one-dimensional charge-ordered chain cuprates, was synthesized via the azide/nitrate route by reacting NaN${}_3$, NaNO${}_3$, and CuO. According to single-crystal x-ray analysis, one-dimensional ${}_{\mathrm{\infty }}^{\hspace{0.5em}1}$CuO${}_2$${}^n$${}^{-}$ chains built up from planar, edge-sharing CuO${}_4$ squares are a dominant feature of the crystal structure. From the analysis of the Cu-O bond lengths, we find that the system forms a Wigner lattice. The commensurate charge order allows the explicit assignment of the valence states of either $+2$ or $+3$ to each copper atom, resulting in a repetition according to ${\mathbf{Cu}}^{2+}$-Cu${}^{3+}$-${\mathbf{Cu}}^{2+}$-${\mathbf{Cu}}^{2+}$-Cu${}^{3+}$-${\mathbf{Cu}}^{2+}$. Following the theoretical analysis of the previously synthesized compounds Na${}_3$Cu${}_2$O${}_4$ and Na${}_8$Cu${}_5$O${}_{10,}$ the magnetic susceptibility was expected to show a large dimer gap. Surprisingly, this is not the case. To resolve this puzzle, we show that the magnetic couplings in this compound are strongly affected by excitations across the Wigner charge gap. By including these contributions, which are distinct from conventional superexchange in Mott-insulators, we obtain a quantitatively satisfying theoretical description of the magnetic susceptibility data.

Figure 1

Crystal structure with unit cell (green sticks) of Na${}_5$Cu${}_3$O${}_6$: (a) showing the periodicities of Na and CuO${}_2$ units with Na ions forming a honeycomb pattern (emphasized by white sticks) with the channels occupied by cuprate ribbons; (b) CuO${}_2$ chains in Na${}_5$Cu${}_3$O${}_6$: showing the periodicities of Cu${}^{3+}$(red squares) and Cu${}^{2+}$ (blue squares) within each ribbon. The copper and oxygen atoms are numbered corresponding to Table .

Figure 2

Wigner lattice melting of Na${}_5$Cu${}_3$O${}_6$ as detected by (DSC) at $T$ $=555$ (540) K on heating (cooling), respectively.

Figure 3

Scattered x-ray intensity for polycrystalline sample of Na${}_5$Cu${}_3$O${}_6$ at $T$ $=$ 298 K as a function of diffraction angle 2θ (λ $=$ 1.54059 Å), showing the observed pattern (diamonds), the best Rietveld-fit profile (—) based on single crystal data, reflection markers (vertical bars), and difference plot Δ $=$ ${\mathrm{I}}_{\mathrm{obs}}$-I${}_{\mathrm{calc}}$ ($+$) (shifted by a constant amount). Note that the higher angle part is enlarged by a factor of 4 starting at 2θ $=$ 46 °.

Figure 4

(a) Specific heat ($C_{\mathrm{p}}$/T) at zero field as a function of $T$ of polycrystalline sample of Na${}_5$Cu${}_3$O${}_6$. The $C_{\mathrm{p}}$/$T^2$ plot as a function of $T$ around $T_{\mathrm{N}}$ [panel (a)], and $d$/dT (${\chi }_{\mathrm{mol}}$ × $T$) [Fisher's heat capacity, panel (b)] emphasize the pristinely perfect sample devoid of magnetic defects.

Figure 5

Measured susceptibility (symbols) compared to a Curie law fit (dashed line) for 150 K < $T$ < 680 K with $C$ $=$ 0.40 K emu/mol and θ $=$ −40.5 K and to an independent dimer model (solid line) with exchange constant $J_2=$ 145 K and ${\chi }_0=$ −0.9 × ${10}^{-4}$ emu/mol.

Figure 6

(a) Chain with Cu${}^{2+}$ (filled circles, spin 1/2) and Cu${}^{3+}$ ions (open circles, no spins) and magnetic exchange couplings $J_1$, $J_2$, and $J_3$. (b) Simplified magnetic structure showing only magnetic Cu${}^{2+}$ ions and couplings $J_1$ and $J_2$. In this approximation the model is an alternating ferro-antiferromagnetic Heisenberg chain. (c) Also taking $J_3$ into account, the magnetic model is equivalent to a Heisenberg ladder.

Figure 7

(a) Schematic view of the density of states N(E) of a Mott-Hubbard insulator: Excitations from the ground state at $E_0$ to double-occupied configurations in the upper Hubbard band with energy $U$ lead to conventional antiferromagnetic superexchange interaction ∼$4t^2$/$U$. (b) The charge excitations $D_{\mathrm{n}}$ of a Wigner crystal are in general small compared to $U$. Higher-order processes of such charge excitations can contribute to superexchange with antiferro- or ferromagnetic interactions and thereby compete with the conventional superexchange. (c) When the Wigner lattice melts, e.g., at high temperature, charge gaps disappear and the system changes into a correlated metal.

Figure 8

Sequence of electron transitions, 1-2-3, with Cu${}^{2+}$ ions denoted by filled circles and Cu${}^{3+}$ ions by open circles. Two distinct exchange processes are shown: (a) Virtual excitation across the Wigner gap giving a contribution ∼$t_1$${}^2$$t_2$/D${}^2$ to $J_2$. (b) Superexchange process yielding a contribution $J_2$ ∼ $t_1$${}^2$$t_2$/(UD). Exchange process (b) is a virtual excitation across the Mott-Hubbard gap U, while exchange process (a) only involves the Wigner gap D.

Figure 9

Exchange constants, Eq. (5), as a function of the energy scale for charge fluctuations D for U $=$ 3.5 eV, $t_1=$ −0.08 eV, $t_2=$ −0.12 eV, and $t_3=$ 0.05 eV. Shown are the total exchange constants (solid lines), the superexchange contributions involving U (dashed lines), and the terms involving virtual excitations across the Wigner gap (dot-dashed lines).

Figure 10

Experimental data for the susceptibility (symbols) compared to the susceptibility for a Heisenberg model with $J_1=$ −43 K, $J_2=0$ K, $J_3=69$ K, and ${\chi }_0$ $=$ –1.6 × ${10}^{-4}$ emu/mol as calculated numerically using the TMRG algorithm.