Transport in close-packed solids with stacking defects

Abstract

Lithium and sodium are the only solids that are known to lose crystalline order upon cooling. The seemingly-disordered low-temperature phase shows signatures of various close-packed structures. The lack of order has been attributed to a hidden gauge symmetry that arises when electrons from one layer can hop to a neighbouring layer but not further. It makes all close-packed structures nearly degenerate and leads to ``structural frustration''. In this article, we examine whether this symmetry is reflected in transport signatures. Taking advantage of in-plane translational periodicity, we map the bulk Bloch Hamiltonian to an effective one-dimensional chain, with stacking disorder mapping to random phases of the hopping amplitudes. We derive an explicit analytic form for the Green's function of electrons and use it to calculate conductance of a bulk crystal. When hopping in the effective one-dimensional chain is restricted to nearest neighbours, conductance is completely insensitive to phase disorder, which indicates that all close-packed structures exhibit the same conductance. We show that the leading correction that can differentiate between close-packed structures arises from hopping to the next-nearest-neighbour layer, equivalent to second-neighbour hopping in the chain model. This process appears when a pair of next-neighbour layers are aligned in a certain way, e.g., at an hcp-like stacking fault within an fcc background. With this hopping included, conductance becomes sensitive to the precise arrangement of layers. When multiple stacking faults are present, the conductance decreases with increasing system size, as expected from Anderson localization. Our results are applicable to pressurized lithium and sodium, where conductance measurements can identify and characterize stacking faults.

Figures (9)

• Figure 1: 1D tight-binding chain with open boundary conditions. Hopping amplitudes (with phases) between neighbouring atoms are indicated above the arrows.
• Figure 2: Atomic environment of a representative atom (center; shown in blue) within one layer of any close-packed structure. The 1nn atoms in the layer above are located at the sites marked '$1+$' if the corresponding chirality variable in the Hägg code is '$+$', or at the sites marked '$1-$' if the corresponding chirality variable in the Hägg code is '$-$'. In the same way, 2nn sites are '$2+$' or '$2-$', depending on the Hägg code entry.
• Figure 3: a) Local atomic environment in the Barlow sequence $(\cdots BAC\cdots)$. The layers are offset along the $\hat{\bm{c}}$ axis for visual clarity. The reference atom is marked "0" and is shown in blue. The sets of all 1nn and 2nn atoms are marked "1" and "2", shown in cyan and magenta respectively. b) Atomic environment in the sequence $(\cdots ABA\cdots)$. Because the corresponding Hägg code, $(\cdots+-\cdots)$, now contains the substring $(+-)$, the "$0$" atom has a 3nn in the second layer above (marked "3"). c) Atomic environment in the sequence $(\cdots ABC\cdots)$. The "0" atom now has no 3nn [up to the distance cutoff considered, see the discussion below Eq. (\ref{['eq:close_packed_chi_definition']})], since the Hägg code entries do not change sign.
• Figure 4: 3D rendering of the stacking fault described by Eqs. (\ref{['eq:stacking_fault_barlow_sequence']}) and (\ref{['eq:stacking_fault_hagg_code']}). The left and right leads consist of repeating $(CAB)$ and $(BAC)$ units, respectively, while the device is taken as the three-layer sequence $(CAC)$. Note that the only 3nn hopping process occurs between the $C$-layers in the device, i.e., for every atom in the left $C$-layer in the device, there is a corresponding atom in the right $C$-layer at a displacement of $\sqrt{8/3}~\bm{\hat{z}}$.
• Figure 5: Lead-device-lead arrangement for transmission through a stacking fault. The device and lead sites are represented by open and closed circles, respectively. First-neighbour hopping with phases $\phi_m$ exist between all sites in the system, while a single second-neighbour hopping process acts between the endpoints of the device.