Orbital magnetism in ensembles of gold nanoparticles

The last two decades have witnessed various experiments reporting very unusual magnetic properties of ensembles of gold nanoparticles surrounded by organic ligands, including ferromagnetic, paramagnetic, or (large) diamagnetic responses. These behaviors are at odds with the small diamagnetic response of macroscopic gold samples. Here we theoretically investigate the possibility that the observed unusual magnetism in capped gold nanoparticles is of orbital origin. Employing semiclassical techniques, we calculate the orbital component to the zero-field susceptibility of individual as well as ensembles of metallic nanoparticles. While the result for the orbital response of individual nanoparticles can exceed by orders of magnitude the bulk Landau susceptibility in absolute value, and can be either diamagnetic or paramagnetic depending on nanoparticle size, we show that the magnetic susceptibility of a noninteracting ensemble of nanoparticles with a smooth size distribution is always paramagnetic at low magnetic fields. In particular, we predict that the zero-field susceptibility follows a Curie-type law for small nanoparticle sizes and/or low temperatures. The calculated field-dependent magnetization of an ensemble of diluted nanoparticles is shown to be in good agreement with existing experiments that yielded a large paramagnetic response. The width of the size distribution of the nanoparticles is identified as a key element for the quantitative determination of the orbital response.

Figure 1

Example of families of classical periodic orbits on an equatorial plane of the sphere labeled by the topological indexes $(\nu ,\eta )$, with $\nu$ the winding number and $\eta$ the number of bounces.

Figure 2

Grand-canonical zero-field susceptibility ${\chi }^{\left(1\right)}$, in units of the absolute value of the Landau susceptibility ${\chi }_{\mathrm{L}}$, as a function of the radius $a$ (scaled with the Fermi wave vector $k_{\mathrm{F}}$). Blue line: semiclassical result from Eq. (19). Black line: quantum-mechanical result from Eq. (A5). In the figure, room temperature ($T/T_{\mathrm{F}}=5\times {10}^{-3}$) is chosen and $\chi =0$ is indicated by the dashed gray line.

Figure 3

Same quantity from Eq. (19) as in Fig. 2, in a restricted $a$ interval for $T/T_{\mathrm{F}}=5\times {10}^{-3}$ (blue line) and $T/T_{\mathrm{F}}=5\times {10}^{-4}$ (violet line). The quantum-mechanical result (A5) (black line: $T/T_{\mathrm{F}}=5\times {10}^{-3}$; orange line: $T/T_{\mathrm{F}}=5\times {10}^{-4}$) and approximate semiclassical result (23) for $T/T_{\mathrm{F}}=5\times {10}^{-4}$ (green line) are shown for comparison purposes.

Figure 4

Grand-canonical magnetic moment ${\mathcal{M}}^{\left(1\right)}$ in units of the Bohr magneton ${\mu }_{\mathrm{B}}$ for three different nanoparticle sizes as a function of the cyclotron frequency ${\omega }_{\mathrm{c}}\propto H$ (in units of $E_{\mathrm{F}}/\hslash$). Blue lines: semiclassical result from Eq. (18). Black lines: perturbative quantum result from Eq. (A4). In the figure, $T/T_{\mathrm{F}}=5\times {10}^{-3}$.

Figure 5

Zero-field susceptibility of a nanoparticle ensemble with average radius $\overline{a}$ for various size dispersions $\delta a$ at $T/T_{\mathrm{F}}=5\times {10}^{-3}$, from Eq. (29). The diagonal contribution to the averaged susceptibility ${\chi }_{\mathrm{ens}}^{\mathrm{d}}\left(\overline{a}\right)$ [Eq. (30)] is shown for comparison as a black solid line.

Figure 6

Ensemble-averaged magnetic moment per particle from Eq. (28) as a function of magnetic field with mean radius $k_{\mathrm{F}}\overline{a}=20$ (black lines), 60 (red lines), 100 (blue lines), and for size dispersions $k_{\mathrm{F}}\delta a=5$ (dashed lines) and 10 (dotted lines). The diagonal contribution ${\mathcal{M}}_{\mathrm{ens}}^{\mathrm{d}}\left(\overline{a}\right)$ [cf. Eq. (34)] is shown as solid lines. In the figure, $T/T_{\mathrm{F}}=5\times {10}^{-3}$.

Figure 7

Log-log plot of the diagonal contribution ${\chi }_{\mathrm{ens}}^{\mathrm{d}}\left(\overline{a}\right)$ to the ensemble-averaged zero-field susceptibility (in units of $|{\chi }_{\mathrm{L}}|$) scaled with $k_{\mathrm{F}}\overline{a}$ as a function of $k_{\mathrm{F}}\overline{a}\frac{T}{T_{\mathrm{F}}}$. Circles: numerical evaluation of Eq. (30). Solid line: Curie-type law (31), with $C$ as given in Eq. (32). Dashed line: exponential fit (33), with $c_1\simeq 22$ and $c_2\simeq 12$. Inset: Linear-scale plot of the same data, but for a different range of $k_{\mathrm{F}}\overline{a}\frac{T}{T_{\mathrm{F}}}$.

Figure 8

Diagonal contribution to the ensemble-averaged magnetic moment from Eq. (34) scaled with ${\left(k_{\mathrm{F}}\overline{a}\right)}^2$ as a function of the reduced magnetic flux $\overline{\varphi }/{\varphi }_0$ for $k_{\mathrm{F}}\overline{a}\frac{T}{T_{\mathrm{F}}}={10}^{-3}$ (red solid line), ${10}^{-2}$ (black dashed line), and ${10}^{-1}$ (blue dotted line). Inset: Same as the main figure for a different range of $\overline{\varphi }/{\varphi }_0$.

Figure 9

Blue lines: grand-canonical (a) free energy $\mathrm{\Delta }{F}^{\left(1\right)}$ at $H=0$ (scaled with the Fermi energy $E_{\mathrm{F}}$) from Eq. (15) at a temperature $T/T_{\mathrm{F}}=5\times {10}^{-3}$ and (b) corresponding zero-field susceptibility ${\chi }^{\left(1\right)}$ from Eq. (19) (cf. blue line in Fig. 2) as a function of $k_{\mathrm{F}}a$. Red lines: canonical contribution $\mathrm{\Delta }{F}^{\left(2\right)}$ from Eq. (17) [panel (a)] and zero-field susceptibility ${\chi }^{\left(2\right)}$ from Eq. (21) [panel (b)].

Figure 10

Low-temperature $T/T_{\mathrm{F}}=5\times {10}^{-4}$ results for the contributions to (a) the free energy and (b) the susceptibility. As in Fig. 9, blue lines represent the grand-canonical contributions from Eqs. (15) and (19), and red lines depict the canonical contributions of Eqs. (17) and (21).

Figure 11

Dots and triangles: measured magnetization $M_{\mathrm{ens}}$ of an ensemble of Au nanoparticles functionalized with various ligands (in electromagnetic units per gram of gold nanoparticles in the sample) as a function of applied field $H$ (in œrsted), with mean diameter $2\overline{a}=2.5\mathrm{nm}$, size dispersion $2\delta a=0.4\mathrm{nm}$, and at a temperature $T=1.8\mathrm{K}$. The data are taken from the experiments of Refs. [17, 18, 19]. The corresponding ligands are: polyvinyl pyrolidone (PVP) [red dots (Refs. [17, 18]) and red triangles (Ref. [19])], polyacrylonitrile (PAN) [green triangles (Ref. [19])], and polyallyl amine hydrochloride (PAAHC) [blue triangles (Ref. [19])]. Solid line: nondiagonal magnetization from Eq. (28) with $2\delta a=0.4\mathrm{nm}$. Dashed line: diagonal approximation (34), corresponding to $2\delta a\to \infty$.

Figure 12

Black lines: exact spectrum $E_{\mathrm{ex}}$ of the mean-field Hamiltonian (4) (scaled by $E_0={\hslash }^2/2m{a}^2$) of a sphere as a function of the magnetic flux $\varphi =\pi a^{2}B$ in units of the flux quantum. Red lines: perturbative spectrum $E_{nl{m}_z}$ from Eq. (A1), showing the perturbative regime ($0<\varphi /{\varphi }_0\lesssim 5$, solid red lines) and the region where perturbation theory starts to depart from the exact result ($5\lesssim \varphi /{\varphi }_0\lesssim 10$, dotted red lines).

Figure 13

Topological indexes $(\nu ,\eta )$ contributing to the double sums in Eqs. (B1) and (C1) (red dots). The critical value ${\eta }_{\mathrm{c}}=\alpha L_T/2a$ separates the two summation regions with different values of ${\nu }_{\max }\left(\eta \right)$ given by the explicit formulas (solid lines). The dotted line depicts the limiting value of ${\nu }_{\max }$ when $\eta \to \infty$.