Low-momentum strong-coupling electron and hole spectra of gapless semiconductors with strongly unequal masses ($m_h\gg m_e$)

In gapless (zero-gap) semiconductors such as HgTe and gray tin, conduction and valence bands touch quadratically at the Brillouin-zone center and the hole effective mass $m_h$ greatly exceeds the electron mass $m_e$. Abrikosov (1975) computed the interaction-corrected spectra in the window $k_0\ll k\ll(m_h/m_e)k_0$, where $k_0=m_ee^2/\kappa$ is the momentum at which the Coulomb energy equals the electron kinetic energy: both bands acquire an identical linear term $\frac{3\pi e^2}{32\kappa}k$ and a logarithmic term $-\frac{2}{\pi}\omega_0\ln(k/k_0)$, making the hole branch effectively linear. In the opposite, strong-coupling regime $k\ll k_0$, however, the ordering assumption $\delta\varepsilon_e\gg\delta\varepsilon_h$ on the interaction-induced corrections $\delta\varepsilon_{e,h}$ to the full single-particle energies $\varepsilon_{e,h}$ (written in the source with the same symbols, $\varepsilon_e\gg\varepsilon_h$) becomes self-contradictory, signaling the failure of the random phase approximation, and the self-consistent small-$k$ solution attempted in the paper was internally inconsistent and could not be matched to the high-$k$ branch. The open problem is the determination of the true $k\to0$ electron and hole spectra $\varepsilon_e(k)$, $\varepsilon_h(k)$ and their connection to the perturbative regime; in post-2013 literature the question has been recast as the infrared fate of the three-dimensional quadratic band touching (Luttinger) semimetal with electron-hole mass asymmetry under long-range Coulomb interactions.

Background

Gapless (zero-gap) semiconductors are inverted-band materials — canonical examples are HgTe, gray tin (α-Sn), and the Hg$_{1-x}$Cd$_x$Te alloy — in which the conduction and valence bands meet at the Brillouin-zone center $\Gamma$ with quadratic dispersion and zero gap. Near $\Gamma$ the four relevant states of total angular momentum $j=3/2$ are described by the Luttinger Hamiltonian, the standard effective $k\cdot p$ Hamiltonian for these four bands; in an isotropic two-band reduction each band carries an effective mass, $m_e$ for the (light, conduction-like) electrons and $m_h$ for the (heavy, valence-like) holes, with $m_h\gg m_e$ in the materials of interest.

Because there is no gap and, at zero doping, no Fermi surface, the long-range Coulomb interaction $e^2/\kappa k$ (with $\kappa$ the static dielectric constant and $e$ the electron charge) is only weakly screened. Two scales control the problem: the momentum $k_0=m_ee^2/\kappa$ at which the Coulomb energy equals the electron kinetic energy $k^2/2m_e$, and the corresponding energy $\omega_0=m_ee^4/2\kappa^2$. For $k\gg k_0$ the interaction is a perturbation; for $k\ll k_0$ it dominates and standard weak-coupling tools fail. The random phase approximation (RPA) — computing screening from the noninteracting polarization bubble, valid when the interaction only weakly perturbs the carriers — is the standard weak-coupling tool. Abrikosov and Beneslavskii had earlier shown, for the symmetric case $m_e=m_h$, that the $k\ll k_0$ regime is a genuine strong-coupling regime with interaction-dominated scaling behavior and no ordinary quasiparticles; the modern literature calls the corresponding scale-invariant state the Abrikosov–Beneslavskii, or Luttinger-Abrikosov-Beneslavskii (LAB), fixed point.

Abrikosov's 1975 paper treated the realistic unequal-mass case $m_h\gg m_e$. For the broad window $k_0\ll k\ll(m_h/m_e)k_0$ — shifted upward by $m_h/m_e$ for holes and experimentally accessible — second-order perturbation theory and the RPA give identical corrections to the two bands,

$$\varepsilon_{e,h}(k)=\frac{k^2}{2m_{e,h}}+\frac{3\pi e^2}{32\kappa}k-\frac{2}{\pi}\omega_0\ln\frac{k}{k_0},$$

so the hole branch is dominated by the linear term, $\varepsilon_h\approx v_hk$ with $v_h=3\pi e^2/32\kappa$. At temperatures $T\gg\omega_0$ the spectra in the essential momentum regions remain unchanged, and the thermally excited carrier densities follow. The problem arises in the opposite limit $k\ll k_0$: the analysis rests on the ordering assumption $\delta\varepsilon_e\gg\delta\varepsilon_h$ on the interaction-induced corrections $\delta\varepsilon_{e,h}$ to the full single-particle energies $\varepsilon_{e,h}$ (the source writes this assumption with the same symbols as the full energies, $\varepsilon_e\gg\varepsilon_h$), which the same analysis shows to become inconsistent there, indicating that the RPA (and the perturbative expansion built on it) fails. The attempted self-consistent small-$k$ hole self-energy, $\Sigma_h\sim\frac{k^2}{50m_e}\ln(k_0/k)$, came with an internal inconsistency (the ansatz parameter $\eta=1-0.08\xi$, with $\xi=\ln(k_0/k)$, violates the assumed $\eta\approx1$ once $\xi\gtrsim1$) and could not be matched onto the high-$k$ spectrum, so it was flagged as possibly nonphysical, with the transition to the strong-coupling region for $m_e\ll m_h$ described as posing considerable mathematical difficulties.

The modern literature has renamed and generalized this problem: the same band structure is called a three-dimensional quadratic band touching (QBT), or Luttinger semimetal (materials: HgTe, α-Sn, pyrochlore iridates such as Pr$_2$Ir$_2$O$_7$, half-Heuslers such as YPtBi), and $N$ counts the number of QBT points (equivalently, the number of independent four-component fermion flavors; HgTe and α-Sn have a single touching point at $\Gamma$, so the physical value is $N=1$). The Coulomb interaction is a relevant perturbation below the upper critical dimension $d=4$, and the theory is studied by renormalization-group (RG) methods in $4-\epsilon$ and $2+\epsilon$ expansions, in large-$N$, and by functional RG. Within these approaches the strong-coupling regime is the LAB non-Fermi liquid with anomalous scaling; the electron-hole mass asymmetry is the perturbation $k^2/2m'$ added to the symmetric Luttinger Hamiltonian, parametrized by the heavy-to-light band-mass ratio $r_m=m/m'=m_h/m_e\gg1$, where $m=m_h$ and $m'=m_e$ are the heavy-hole and light-electron band masses. This is the language in which all later work on Abrikosov's question is phrased.

Problem Statement

Determine the true small-momentum ($k\ll k_0$, with $k_0=m_ee^2/\kappa$) electron and hole single-particle spectra $\varepsilon_e(k)$ and $\varepsilon_h(k)$ — equivalently, the infrared limit of the electron and hole two-point functions — of the isotropic gapless semiconductor with strongly unequal masses $m_h\gg m_e$ and long-range Coulomb interaction $e^2/\kappa k$, in the regime where the ordering assumption $\delta\varepsilon_e\gg\delta\varepsilon_h$ on the interaction-induced corrections $\delta\varepsilon_{e,h}$ becomes inconsistent and the random phase approximation fails. The answer must give the actual low-$k$ spectra and their connection to the established perturbative regime $k_0\ll k\ll(m_h/m_e)k_0$, in which

$$\varepsilon_{e,h}(k)=\frac{k^2}{2m_{e,h}}+\frac{3\pi e^2}{32\kappa}k-\frac{2}{\pi}\omega_0\ln\frac{k}{k_0},$$

with $\omega_0=m_ee^4/2\kappa^2$. The answer must determine what becomes of the electron-hole mass asymmetry at the lowest energies — equivalently, whether the mass-asymmetric three-dimensional quadratic band touching (Luttinger) semimetal at $N=1$, $d=3$ is a scale-invariant Luttinger-Abrikosov-Beneslavskii non-Fermi liquid or gives way to a symmetry-broken (nematic or excitonic) ground state.

References

1. A. A. Abrikosov, "On the energy spectrum of gapless semiconductors with considerably differing masses of electrons and holes", Journal of Low Temperature Physics 18, 185–199 (1975). DOI 10.1007/BF00118152, https://doi.org/10.1007/BF00118152
2. A. A. Abrikosov and S. D. Beneslavskii, "Some properties of gapless semiconductors of the second kind", Journal of Low Temperature Physics 5, 141–154 (1971). DOI 10.1007/BF00629569, https://doi.org/10.1007/BF00629569
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