Wigner crystal phases in confined carbon nanotubes

We present a detailed theoretical analysis of the Wigner crystal states in confined semiconducting carbon nanotubes. We show by detailed semimicroscopic calculations that the effective exchange interaction has an SU(4) symmetry, and can reach values as large as $J\sim 100\mathrm{K}$ in weakly screened, small diameter nanotubes, close to the Wigner crystal—electron liquid crossover. This large value of the exchange coupling in the crossover region also follows from robust scaling arguments. Modeling the nanotube carefully and analyzing the magnetic structure of the inhomogeneous electron crystal, we recover the experimentally observed “phase boundaries” of Deshpande and Bockrath [V. V. Deshpande and M. Bockrath, Nat. Phys. 4, 314 (2008)]. Spin-orbit coupling only slightly modifies these boundaries, but breaks the spin symmetry down to $\mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right)$, and in Wigner molecules it gives rise to interesting excitation spectra, reflecting the underlying SU(4) as well as the residual $\mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right)$ symmetries.

Figure 1

(a) “Phase diagram” for an unscreened nanotube with $\epsilon =2$ and of a radius $R=1.6\mathrm{nm}$, as a function of particle number $N$, and external magnetic field $B$. Red and black curves denote the phase boundaries based upon our theoretical calculations with spin-orbit coupling, ${\mathrm{\Delta }}_{\text{SO}}=0.186\mathrm{meV}$ (${\mathrm{\Delta }}_{\text{SO}}\approx 2.1\mathrm{K}$). The fully polarized, orbitally polarized, and unpolarized states are displayed in different colors. Green and red symbols indicate the boundaries for the experimental phase diagram of Ref. [25]. (b) Overall magnetization change at charge transitions as a function of $N$ for $B=4\mathrm{T}$. The vertical dashed lines indicate the jumps corresponding to the phase boundaries. For $N>26$ the Wigner crystal starts to “melt”. The melted region is indicated by the red shaded areas.

Figure 2

Sketch of the effective potential, $V\left(z\right)$ (blue), and the single-particle density (red), as obtained from the solution of the two-body problem, Eq. (8). $d$ is the distance between two neighboring electrons.

Figure 3

Contour plot of the background potential $U_{\text{tot}}$ in Eq. (10) in terms of relative and center of mass coordinates. Tunneling between the two minima ($z_{12}=\pm d,Z=0$) gives rise to the exchange splitting $J$. The black arrow indicates the semiclassical tunneling path used here.

Figure 4

The effective SU(4) spin-isospin exchange coupling $J$ as a function of $n_{e}R$ for various nanotube radii with $\epsilon =2$. The vertical dashed line shows the limit of the Wigner crystal regime.

Figure 5

Energy levels of an $N=2$ Wigner molecule, as obtained from exact diagonalization of the effective Hamiltonian. On the left-hand side, the ${\mathrm{\Delta }}_{\mathrm{SO}}=0$ spectrum is shown, with states classified by SU(4) representations, indicated by the Young-tableaux. For ${\mathrm{\Delta }}_{\mathrm{SO}}\ne 0$, states are classified in terms of the residual $\mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right)$ symmetry.

Figure 6

Energy levels of a $N=3$ Wigner molecule. Only the first two SU(4) multiplets and their spin-orbit coupling induced splittings are shown.

Figure 7

Site dependent exchange couplings for a nanotube with $\epsilon =2$ and $R=1.6\mathrm{nm}$ when $N=26$ electrons are confined inside. The inset shows the dimensionless parameter $r_s$ as a function of the site index. Notice that the stability condition $r_s⪆r_s^{*}\simeq 3.3$ is fulfilled, and the nanotube is in the Wigner crystal regime, though for larger values of $N$ the Wigner crystal is expected to melt at the center of the nanotube.

Figure 8

Retailored Brillouin zone (BZ) of graphene (black rectangle). The red lines denote states allowed for a zigzag carbon nanotube, $K$ and $K^{\prime }$ mark the two Dirac points, $a$ is the lattice constant. Thick solid red lines indicate the segments $L_{\pm Q}$.

Figure 9

The Hartree potential $V_i^{\left(1\right)}\left(z\right)$ confining the $i\mathrm{th}$ electron, defined in Eq. (B6).

Figure 10

The exchange interaction as function of the dimensionless ratio ${\tilde{r}}_s$ for a Wigner molecule. The radius of the nanotube was fixed to $R=2\mathrm{nm}$ and the confinement energy is $\hslash {\omega }_0=7.8\mathrm{meV}$. The solid lines are obtained using the pure Coulomb interaction (B4) while for the dashed lines the effective potential in Eq. (B2) with $\overline{\alpha }=0$ has been used.

Figure 11

Dimensionless charging energy as function of the particle number $N$.

Figure 12

The “length” of the Wigner crystal $L_N$, as a function of the number of charges $N$, for various potential depths $\alpha$ and dielectric constants. The length of the nanotube is $L=500\mathrm{nm}$. (Inset) $L_N/l$ for an infinite nanotube with parabolic confinement. $L_N/l$ is a universal function of $N$.

Figure 13

The dimensionless parameter $r_s$ as a function of the number of charges $N$ in the middle of a $500\mathrm{nm}$ long nanotube for $\alpha =0.015\mathrm{meV}/{\mathrm{nm}}^2$ and $\epsilon =2$. The Wigner crystal starts to melt at the center for $N=26$, once $r_s$ decreases below the crossover value $r_s^{*}\approx 3.3$.