Theory of momentum resolved tunneling into a short quantum wire

Motivated by recent tunneling experiments in the parallel wire geometry, we calculate results for momentum resolved tunneling into a short one-dimensional wire, containing a small number of electrons. We derive some general theorems about the momentum dependence, and we carry out exact calculations for up to $N=4$ electrons in the final state, for a system with screened Coulomb interactions that models the situation of the experiments. We also investigate the limit of large $N$ using a Luttinger-liquid type analysis. We consider the low-density regime, where the system is close to the Wigner crystal limit, and where the energy scale for spin excitations can be much lower than for charge excitations, and we consider temperatures intermediate between the relevant spin energies and charge excitations, as well as temperatures below both energy scales.

Figure 1

Schematic geometry of electron tunneling between two parallel quantum wires. Electrons are assumed to tunnel between an infinitely long lower wire and a short upper wire of length $L$. The wires are separated by a center-to-center distance $d$. The upper (lower) wire has a width $W_u$ $\left(W_l\right)$ and an average electron density ${\overline{n}}_u$ $\left({\overline{n}}_l\right)$. A magnetic field $\vec{B}$ is applied perpendicular to the plane of the wires to allow a field-dependent momentum boost $\hslash q_B=eBd$ in the tunneling. The energy of the tunneling electrons can be changed by adjusting the voltage applied between the two wires. In the experiments of Ref. 7, $d=30\mathrm{nm}$, $W_u=20\mathrm{nm}$, $W_l=30\mathrm{nm}$, and $L=2–10\mu \mathrm{m}$. Electron densities $\overline{n}$ can range from $=0$ to $100\mu {\mathrm{m}}^{-1}$. For much of this paper we will assume ${\left({\overline{n}}_l{a}_B\right)}^{-1}⪡1$ and ${\left({\overline{n}}_u{a}_B\right)}^{-1}⪢1$, where $a_B$ is the Bohr radius, so that the upper wire is close to the Wigner crystal regime and the lower wire can be treated as noninteracting.

Figure 2

(Color online) Momentum resolution of one-particle box states. Shown is the modulus squared of Eq. (15) for various values of $\tilde{N}$. Many of the features in the noninteracting case remain when interactions are included and the Fourier transform of a many-body wave function is taken.

Figure 3

Effective interaction potential $V_{\mathrm{eff}}$, as a function of the separation $z$ between two electrons in the upper wire, taking into account the finite width $W_u$ of the upper wire and screening due to a parallel lower wire of width $W_l$ at a distance $d$. See Fig. 1. Dashed and dotted lines show the bare Coulomb interaction $z^{-1}$, and the unscreened interaction softened at short distances, ${(z^2+W_u^2)}^{-1∕2}$. Parameters are $d=1$, $W_u=0.1$, and $W_l=0.15$. Region $0<z<0.5$ is expanded in the inset. In the main plot $1∕z$ and ${(z^2+W_u^2)}^{-1∕2}$ are indistinguishable in the range $0.5<z<4$. In the inset the interaction potential $V_{\mathrm{eff}}\left(z\right)$ and ${(z^2+W_u^2)}^{-1∕2}$ are indistinguishable in the range $0<z<0.5$.

Figure 4

Top: Ground-state density of a four-electron system with spin. Scaled densities are measured in units of $1∕L$. In the label of the individual plot, “density” means the average physical density $N∕L$ of the system, in units of $\mu {\mathrm{m}}^{-1}$. Bottom: Ground-state density-density correlation of a four-electron system with spin. The density-density correlation function is defined as $1∕(1-x){\int }_0^{1-x}\rho \left(x^{\prime }\right)\rho (x^{\prime }+x)d{x}^{\prime }$. The prefactor before the integral is to take into account the finite length of the system.

Figure 5

Top: Ground-state density of a four-electron system without spin. Bottom: Ground-state density-density correlation of a four-electron system without spin. See Fig. 4 for a definition of the density-density correlation function.

Figure 6

Top: Squared matrix elements of tunneling from a two- to three-electron system with spin vs scaled wave vectors $kL∕\pi$. Here, the $L=0.1\mu \mathrm{m}$ curve is indistinguishable from noninteracting limit. Bottom: Same as top, only tunneling from three to four electrons.

Figure 7

Top: Squared matrix elements for tunneling from a two- to three-electron spinless system vs scaled wave vectors $kL∕\pi$. Bottom: Same as top, only tunneling from three to four electrons.

Figure 8

Comparison of the gap $\Delta$ between the ground state and the first excited state energy in a system of two electrons in a box, obtained by direct calculation vs by first-order perturbation theory. The unit is ${\Delta }_0$, the noninteracting energy gap between the ground and the first excited state.

Figure 9

Estimate of the spin exchange energy per electron from a two- and four-electron system using the interaction in Eq. (28). Both of the axes are in physical units: the $x$ axis is the physical density, in units of one electron per $\mu \mathrm{m}$; the $y$ axis is the physical exchange energy $J$ in units of $\mu \mathrm{eV}$. Typical experimental temperatures are in the range $250\mathrm{mK}–2\mathrm{K}\approx 25–200\mu \mathrm{eV}$.

Figure 10

Momentum dependence of tunneling conductance between one-electron and two-electron states for a wire of length $L=0.4\mu \mathrm{m}$. Dashed curve shows line shape ${\mid M_s\left(k\right)\mid }^2$ for tunneling into the singlet ground state of the two particle system, applicable at $T=0$ when the Zeeman energy $E_Z$ is smaller than the exchange gap $\Delta$. Short-dashed curve shows ${\mid M_t\left(k\right)\mid }^2$, for tunneling into the triplet ground state, applicable at $T=0$ when $E_Z>\Delta$. Solid curve is a weighted average, applicable if $T$ is large compared to both $E_Z$ and $\Delta$ but small compared to the energy of the lowest charge excitation.

Figure 11

Momentum dependence of ground-state tunneling for $N=10$ electrons and various interaction strengths, obtained by calculating ${\mid M\left(k\right)\mid }^2$ from the effective wave function (39). Each curve has been divided by a normalization constant $N_g$, chosen so that the plotted curves have equal areas. The norm $N_g$ decreases rapidly with increasing interaction strength (decreasing $g$), but the width of the peak broadens only slightly. Note that for $g=1$ the results are equivalent to the $\tilde{N}=5$ case in Fig. 2.

Figure 12

Spectral function $A(k,\overline{u})$, which determines the momentum dependence of tunneling in the free-spin regime. The quantity $\overline{u}$ is the root-mean-square electron displacement, due to quantum fluctuations, from the sites of a classical Wigner crystal, and $a$ is the lattice spacing. When $\overline{u}\gtrsim a$, the momentum distribution is single lobed and peaked about zero momentum. In the opposite limit, when $\overline{u}⪡a$, the momentum distribution exhibits a doubled-lobed structure with peaks near $k=\pm {\tilde{k}}_F=\pm \pi ∕a$. The peaks have width slightly less than ${\tilde{k}}_F$, in agreement with Eq. (60) obtained from the Green’s function (51).

Figure 13

Polarization dependence of the spectral function given by Eqs. (67, 68). Here, $x=E_Z∕k_{B}T$ with $E_Z$ the Zeeman energy and $T$ the temperature. We have used $\overline{u}=0.6a$ for each curve. (See Fig. 12 for the dependence on $\overline{u}$ at $E_Z=0$.) As the polarization increases, the peaks at ${\tilde{k}}_F=\pi ∕a$ become narrower and higher as described in the text and the weight between the peaks decreases.