Abrikosov's 1972 theory of the gapless state of bismuth-type semimetals derived the low-energy spectrum near the annihilation of the direct gap at the L point of the Brillouin zone - quadratic along the electron-pocket elongation axis and linear in the perpendicular directions at the gapless point, a dispersion now called a three-dimensional semi-Dirac point - and showed that the tuning parameter can be varied continuously by hydrostatic pressure or by alloying with antimony. The analysis explicitly left open four issues: the microscopic dependence of the spectral parameters on pressure and antimony concentration; the effect of omitted higher-order terms at ultra-low energies; discriminating near-threshold infrared experiments on ultra-pure crystals under controlled pressure and composition; and a controlled many-body treatment of the marginal mixed dispersion beyond the leading order-alpha screening estimate. What remains open is the material-specific completion: a validated microscopic map from pressure and composition to the effective L-point spectrum (still acknowledged as uncertain as of 2010), and a controlled account of electron-electron interactions at the gapless point of the real anisotropic multi-band material, sharpened by the 2010 discovery that bismuth's pressure-driven semimetal-semiconductor transition is inconsistent with purely band-like behavior.

Background

Bismuth is the canonical low-density semimetal: a semimetal is a solid whose conduction and valence bands overlap slightly, so that small electron and hole pockets coexist at zero temperature, in contrast to the finite band gap of a semiconductor. In bismuth the electrons occupy three tiny ellipsoidal pockets at the L points (edge centers of the rhombohedral Brillouin zone) and the holes sit at the T point (zone corner); both the band overlap (of order 40 meV) and the direct gap at L (of order 15 meV) are exceptionally small, so modest hydrostatic pressure or alloying with antimony, forming Bi1−x_{1-x}Sbx_x, can continuously drive the L-point gap through zero. A continuous change of Fermi-surface connectivity or pocket occupancy as a band edge crosses the chemical potential is called a Lifshitz (electronic topological) transition. Abrikosov (1972) analyzed this 'gapless state' starting from a multi-band Hamiltonian for the L-point vicinity and derived the effective low-energy spectrum

ε=pz22m1±(γ−γ0+pz22m2)2+(vxpx)2+(vypy)2+λ2\varepsilon = \frac{p_z^2}{2m_1} \pm \sqrt{\left(\gamma - \gamma_0 + \frac{p_z^2}{2m_2}\right)^2 + (v_x p_x)^2 + (v_y p_y)^2 + \lambda^2}

where γ−γ0\gamma-\gamma_0 measures the distance from gap annihilation (γ0\gamma_0 being its critical value, tunable by pressure or antimony concentration), m1m_1 and m2m_2 are effective masses along the elongation axis zz of the electron ellipsoid, vxv_x and vyv_y are velocities in the perpendicular plane, and λ\lambda is a residual gap generated by higher-order terms, with λ∼(γ0−γ)3/2\lambda \sim (\gamma_0-\gamma)^{3/2}. Exactly at γ=γ0\gamma=\gamma_0 the two branches are linear in px, pyp_x,\,p_y and quadratic in pzp_z: the 'marginal' mixed dispersion, known today as a three-dimensional semi-Dirac point, with vanishing density of states at the annihilation point. In the same 1972 issue, Brandt, Ponomarev and Chudinov realized the pressure-induced gapless state in Bi1−x_{1-x}Sbx_x experimentally, and Brandt and Chudinov reported a magnetic-field-induced excitonic insulator phase (an interaction-driven paired state of electrons and holes) when the band overlap between the L electrons and T holes falls below about 1 meV - an early indication that interactions become decisive precisely in the near-gapless regime. Abrikosov estimated Coulomb interaction effects within a simplified isotropic model of the marginal spectrum and found them parametrically small, of order α∼e2/(ε0v)≪1\alpha \sim e^2/(\varepsilon_0 v) \ll 1, with a non-analytic screening correction. The paper's own analysis left four issues open: the microscopic dependence of the spectral parameters on pressure and antimony concentration; the role of the omitted higher-order strain and spin-orbit terms (relevant only below roughly 0.1 K); dedicated near-threshold infrared absorption experiments under controlled pressure and composition to discriminate the alternative spectral sign assignments (the predicted optical threshold ω0=0.0675\omega_0 = 0.0675 eV versus 0.07760.0776 eV for the alternative sign, against the measured 0.0686 eV); and a controlled many-body computation to confirm the absence of stronger instabilities at the marginal mixed dispersion.

Problem Statement

The open problem is the completion of the microscopic theory of the L-point gapless state of bismuth-type semimetals - elemental bismuth and its alloys Bi1−x_{1-x}Sbx_x - under the two experimental control variables, hydrostatic pressure PP and antimony concentration xx. It has two jointly required components. (i) Microscopic spectrum map: derive, from the electronic structure of the real material, the quantitative map (P,x)↦(γ−γ0, m1, m2, vx, vy, λ, higher-order coefficients)(P,x) \mapsto (\gamma-\gamma_0,\, m_1,\, m_2,\, v_x,\, v_y,\, \lambda,\, \text{higher-order coefficients}) - the gap-annihilation offset, effective masses and velocities, the residual gap λ∼(γ0−γ)3/2\lambda \sim (\gamma_0-\gamma)^{3/2}, and the symmetry-allowed higher-order terms - over the range spanning the semimetallic, gapless, and semiconducting regimes, to an accuracy directly comparable with the available experimental data on these materials (for example optical absorption thresholds). (ii) Interactions at the gapless point: give a controlled account of electron-electron interactions for the anisotropic multi-band spectrum at the gap-annihilation point of the real material, beyond the leading-order estimate α∼e2/(ε0v)\alpha \sim e^2/(\varepsilon_0 v) of Abrikosov (Ref. 1) in a simplified isotropic model, covering long-range Coulomb screening, spectral renormalization, and symmetry-breaking channels including excitonic pairing of the remaining electron and hole pockets. The two components serve one objective - a quantitative, experimentally validated microscopic description of the gapless state of bismuth-type semimetals - and are coupled: the interaction analysis takes the microscopic spectrum as input, while the interpretation of the near-threshold optical properties of the real material requires the interaction analysis. The ultra-low-temperature consequences of the omitted strain and spin-orbit terms, and the optical sign discrimination, lie outside the scope of this task, while the symmetry-allowed higher-order terms of the effective spectrum remain required in component (i). The map must have its parameters obtained from the electronic structure or independent input data rather than fitted to the near-threshold optical data used to test it, and must reproduce the near-threshold optical absorption threshold value of 0.0686 eV given in Ref. 1, to within its stated experimental uncertainty, with the agreement to be established for the electronic-structure-derived spectrum itself, independently of the component (ii) interaction corrections. The interaction corrections for the anisotropic multi-band spectrum must reach a controlled order beyond the leading estimate α∼e2/(ε0v)\alpha \sim e^2/(\varepsilon_0 v), with a quantified uncertainty for the long-range Coulomb screening, the spectral renormalization, and each symmetry-breaking (excitonic) channel, and the analysis must determine whether any many-body instability survives at the gapless point of the real material.

References

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