Orbital magnetism of graphene nanostructures: Bulk and confinement effects

We consider the orbital magnetic properties of noninteracting charge carriers in graphene-based nanostructures in the low-energy regime. The magnetic response of such systems results both from bulk contributions and from confinement effects that can be particularly strong in ballistic quantum dots. First we provide a comprehensive study of the magnetic susceptibility $\chi$ of bulk graphene in a magnetic field for the different regimes arising from the relative magnitudes of the energy scales involved, i.e., temperature, Landau-level spacing, and chemical potential. We show that for finite temperature or chemical potential, $\chi$ is not divergent although the diamagnetic contribution ${\chi }_0$ from the filled valance band exhibits the well-known $-B^{-1/2}$ dependence. We further derive oscillatory modulations of $\chi$, corresponding to de Haas–van Alphen oscillations of conventional two-dimensional electron gases. These oscillations can be large in graphene, thereby compensating the diamagnetic contribution ${\chi }_0$ and yielding a net paramagnetic susceptibility for certain energy and magnetic field regimes. Second, we predict and analyze corresponding strong, confinement-induced susceptibility oscillations in graphene-based quantum dots with amplitudes distinctly exceeding the corresponding bulk susceptibility. Within a semiclassical approach we derive generic expressions for orbital magnetism of graphene quantum dots with regular classical dynamics. Graphene-specific features can be traced back to pseudospin interference along the underlying periodic orbits. We demonstrate the quality of the semiclassical approximation by comparison with quantum-mechanical results for two exemplary mesoscopic systems, a graphene disk with infinite mass-type edges, and a rectangular graphene structure with armchair and zigzag edges, using numerical tight-binding calculations in the latter case.

Figure 1

Rough schematic overview of the orbital magnetic behavior of graphene in a perpendicular field for the energy regimes studied in Sec. 3d. Here $\mu$ is the chemical potential, $k_{B}T$ is the thermal energy and ${\Delta }_{LL}\propto \sqrt{B}$ represents the Landau-level spacing. Blue (red) regions refer to parameter regimes where graphene shows diamagnetism (paramagnetism). The color intensity roughly indicates the strength of the magnetic response. The magnetic susceptibility $\chi$ is diamagnetic in almost all areas except the de Haas–van Alphen regime, $\mu >{\Delta }_{LL}>k_{B}T$, with susceptibility oscillations of $\chi$ being linear in ${\mu }^2$ (and $1/B)$.

Figure 2

Energy spectrum of triangular quantum dots with armchair edge (upper panel) and zigzag edge (middle panel) as a function of the normalized magnetic flux through each system. The dashed orange lines refer to the lowest Landau energies $E_n$. In the case of zigzag geometry the zigzag edge states clearly appear close to $E\approx 0$ and contribute to the zeroth Landau level, as the enlarged section of the blue shaded region in the lower right panel shows. The lower left panel represents the green shaded region in the energy spectrum of the armchair triangular quantum dot, where no edge states occur. Insets: Sketch of the geometries; the actual systems considered are much larger: $\mathcal{L}\simeq 100a$.

Figure 3

(a) Comparison of the average level spacing (in units of hopping energy $t)$ of the lowest 600 electron states of the two triangular quantum dots, see Fig. 2, with the Dirac model for bulk graphene (red solid line) calculated from Eq. (12) as a function of the magnetic flux. The blue dotted (dashed-dotted) line represents the average level spacing of the zigzag system with (without) considering edge states. Due to the edge states $\Delta E$ is smaller than in the armchair system (green dashed line). (b) Mean energy of the lowest 600 electronic energies as a function of the magnetic flux. The full (red), dotted (blue), and dashed (green) lines correspond to the bulk, zigzag, and armchair system, respectively.

Figure 4

Susceptibility contribution ${\chi }_T$ for bulk graphene calculated from Eqs. (44) and (45) at $1/{\beta }^{\left(\mathrm{bulk}\right)}\approx 3\times {10}^{-3}t$. (a) ${\chi }_T$ shown as a function of ${\mu }^2$ for fixed $\varphi =15{\varphi }_0$. (b) ${\chi }_T$ plotted as a function of the inverse magnetic flux for fixed $\mu =0.3t$.

Figure 5

Oscillatory susceptibility contribution ${\chi }_T$ for triangular graphene cavities as a function of the chemical potential at $\varphi =15{\varphi }_0$. (a) ${\chi }_T$ for armchair (green dashed) and zigzag (blue dotted) confinement with side length $\mathcal{L}\approx 100a$ at $1/{\beta }^{(ac,zz)}\approx {10}^{-3}t$. The red arrows indicate the peak positions in the case of bulk graphene where only Landau levels exist. (b) Comparison of ${\chi }_T$ of the finite systems (dashed and dotted) at a slightly higher thermal energy $1/{\beta }^{(ac,zz)}=5\times {10}^{-3}t$ with the corresponding bulk result (solid) at $1/{\beta }^{\left(\mathrm{bulk}\right)}\approx 3\times {10}^{-3}t$.

Figure 6

Susceptibility contribution ${\chi }_T$ as a function of $\varphi$ (for same triangular quantum dots as in Fig. 5) for $\mu =0.3t,1/{\beta }^{\left(\mathrm{bulk}\right)}\approx 3\times {10}^{-3}t$ and $1/{\beta }^{(ac,zz)}=5\times {10}^{-3}t$. The maxima coincide well with the bulk case for $\varphi \gtrsim 10{\varphi }_0$, i.e., the flux range where the bulk theory is applicable to the spectra of the finite quantum dots (see Fig. 2).

Figure 7

Flux dependence of the temperature-dependent susceptibility contribution ${\chi }_T$ of bulk graphene compared with the numerically calculated contribution of triangular nanostructures with (a) armchair and zigzag (b) edges of side length $\mathcal{L}\approx 100a$. The chemical potential is chosen at $\mu =0$ and the thermal energies are $1/{\beta }^{(ac,zz)}\approx {10}^{-2}t$ and $1/{\beta }^{\left(\mathrm{bulk}\right)}\approx 1.5\times {10}^{-1}t$ in panel (a) and $1/{\beta }^{\left(\mathrm{bulk}\right)}\approx 1.7\times {10}^{-1}t$ in panel (b), such that Eq. (18) holds true. The scaling factors ${\gamma }^{\left(ac\right)}\approx 2.5\times {10}^{-2}$ and ${\gamma }^{\left(zz\right)}\approx 3.8\times {10}^{-2}$ are obtained by fitting.

Figure 8

Same as Fig. 7 for smaller thermal energies $1/{\beta }^{(ac,zz)}\approx {10}^{-3}t$ and $1/{\beta }^{\left(\mathrm{bulk}\right)}\approx 1.44\times {10}^{-1}t$ in panel (a) and $1/{\beta }^{\left(\mathrm{bulk}\right)}\approx 1.42\times {10}^{-1}t$ in panel (b), such that Eq. (18) holds true. The scaling factors ${\gamma }^{\left(ac\right)}\approx 3.9\times {10}^{-3}$ and ${\gamma }^{\left(zz\right)}\approx 7.8\times {10}^{-3}$ are obtained by fitting.

Figure 9

Comparison of the temperature-dependent susceptibility contribution ${\chi }_T$ for bulk graphene with that of a triangular armchair (a) and zigzag (b) graphene flake as a function of the chemical potential at a magnetic flux of $\varphi =5{\varphi }_0$ and $1/{\beta }^{(ac,zz)}=5t$. The corresponding values used in the analytic expression for the bulk are $1/{\beta }^{\left(\mathrm{bulk}\right)}\approx 2.1t$ in (a) and $1/{\beta }^{\left(\mathrm{bulk}\right)}\approx 2.9t$ in (b). The fitted scaling factors are ${\gamma }^{\left(ac\right)}=0.44,{\gamma }^{\left(zz\right)}=1.7$.

Figure 10

Comparison of the numerical data of the orbital magnetic susceptibility contribution ${\chi }_T$ of a triangular armchair (a) and zigzag (b) graphene flake at $\varphi =5{\varphi }_0$ with the analytic result for bulk graphene in the limit $\mu \to 0$. In both cases the correspondence is convincing for $t/\beta >1$ as for lower thermal energies this approximation loses validity. The scaling factors attain ${\gamma }^{\left(ac\right)}\approx 5.8$ and ${\gamma }^{\left(zz\right)}\approx 8.6$ which is in agreement with the condition $|{\overline{E}}^{\left(\mathrm{bulk}\right)}-{\overline{E}}^{\left(ac\right)}|<|{\overline{E}}^{\left(\mathrm{bulk}\right)}-{\overline{E}}^{\left(zz\right)}|$. The insets show both the full orbital magnetic susceptibility $\chi$ in a double logarithmic plot and confirm the scaling behavior $\chi \propto -\beta$ at the Dirac point.

Figure 11

Classical periodic orbits in the circular billiard. Panel (a) shows representatives of one fundamental orbit family. Panels (b) and (c): Pairs of counterpropagating orbits in the presence of a perpendicular magnetic field. For weak magnetic fields, panel (b), the bending of the classical orbits can be neglected and the enclosed areas (green shaded) are approximately equal. Panel (c) shows the same orbit pair for stronger magnetic field.

Figure 12

Trajectories representing families of classical periodic orbits in the disk billiard. $w$ denotes the winding number, whereas $v$ labels the total number of boundary reflections. Enclosed areas are marked in green.

Figure 13

(a) Length spectrum of ${\chi }^{\mathrm{osc}}$ calculated quantum mechanically from the eigenenergies of a graphene disk, Eq. (78), at $\varphi =0,1/\beta \approx {10}^{-3}t$ and using $R\approx 200a$. Peak positions correspond to orbit families $(w,v)$. Due to pseudospin interference, orbits with an odd number of reflections do not contribute. Their lengths are marked by green arrows. (b) Comparison of the semiclassical prediction (84) (red dashed line) for the orbital magnetic susceptibility with the quantum-mechanical result (solid blue) at $B=0$ and $1/\beta \approx {10}^{-3}t$. The (green) horizontal line represents the typical value, Eq. (88), of the magnetic susceptibility. All susceptibilities are normalized by $X={\chi }_0[{\varphi }_0/\left(2\mathcal{A}\right)]$ and $\sqrt{k_{F}R}$.

Figure 14

Example of a typical rectangular graphene quantum dot with ${\mathcal{L}}_{ac}=11/\sqrt{3}a$ in $x$ and ${\mathcal{L}}_{zz}=9a$ in $y$ direction.

Figure 15

Three representative members of the orbit families $(1,1)$ and $(1,2)$. The enclosed area varies depending on the position of the reflection point $x_0\in [0,{\mathcal{L}}_{zz}/n]$ at the lower boundary. In the case of the $(1,2)$ family the net directed enclosed area is zero due to opposite propagation direction along both trajectory parts.

Figure 16

Panels (a) and (b): Length spectra calculated from the Fourier transform of the tight-binding results for ${\chi }^{\mathrm{osc}}$; panels (b) and (d): Comparison of ${\chi }^{\mathrm{osc}}$ in semiclassical approximation with corresponding numerical tight-binding results. Results shown in (a) and (b) are obtained for a rectangular graphene quantum dot with $K{\mathcal{L}}_{zz}=0\mathrm{mod}2\pi$ and side lengths of ${\mathcal{L}}_{zz}=201a$ and ${\mathcal{L}}_{ac}\approx 201.207a$, respectively. Panels (c) and (d) display results for a cavity with side lengths ${\mathcal{L}}_{zz}=202a$ and ${\mathcal{L}}_{ac}\approx 202.073a$ such that $K{\mathcal{L}}_{zz}=\pi /3\mathrm{mod}2\pi$. Green arrows in panels (a) and (c) mark positions of orbits that do not contribute to the susceptibility due to their vanishing directed enclosed area ${\mathcal{A}}_{M,N}$. ${\chi }^{\mathrm{osc}}$ is normalized by $X/\sqrt{k_E{\mathcal{L}}_{ac}}$ in all cases, with $X$ as defined in Eq. (85).