The behaviour of electrons and holes in a crystal lattice is a fundamental quantum phenomenon, accounting for a rich variety of material properties. Boosted by the remarkable electronic and physical properties of two-dimensional materials such as graphene and topological insulators, transition metal dichalcogenides have recently received renewed attention. In this context, the anomalous bulk properties of semimetallic WTe2 have attracted considerable interest. Here we report angle- and spin-resolved photoemission spectroscopy of WTe2 single crystals, through which we disentangle the role of W and Te atoms in the formation of the band structure and identify the interplay of charge, spin and orbital degrees of freedom. Supported by first-principles calculations and high-resolution surface topography, we reveal the existence of a layer-dependent behaviour. The balance of electron and hole states is found only when considering at least three Te-W-Te layers, showing that the behaviour of WTe2 is not strictly two dimensional.

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Layer-dependent quantum cooperation of electron and hole states in the anomalous semimetal WTe2.

It is shown that in semimetals of the bismuth type under certain conditions the direct energy gap in the vicinity of theL point of the Brillouin zine can be closed. The energy spectrum is found for values of the parameters near the gap annihilation point (γ=γt0):

$$\varepsilon = p_z^2 /2m_1 \pm \sqrt {\left[ {\gamma - \gamma _0 + ({\text{p}}_{\text{z}}^{\text{2}} /2m_2 )} \right]^2 + (v_x p_x )^2 + (v_y p_y )^2 + \lambda ^2 } $$

where the parameter γ−γ0 can be altered by variation of the external pressure or of the alloy concentration;m1>m2; thez axis corresponds to the direction of “elongation” of electron “ellipsoid”; and γ∼(γ0−γ)3/2 for γ0−γ>0. It is shown that it is permissible to assume γ−γ0≈0 for pure bismuth. The qualitative and quantitative characteristics of the spectrum, including the optical properties, are well explained in this case. The role of the interaction of electrons in the gapless state is analyzed (the energy maximum at theT point is below the band contact level in the vicinity of theL point). The carrier mobility in the gapless and semiconducting states is found as a function of the temperature, the parameter γ−γ0, and the number of carriers injected by alloying.

Gapless state of bismuth-type semimetals

Astrophysics and the many body problem: edited by K. W. Ford. Brandeis Lectures in Theoretical Physics, Vol. 2. 438 pages, 6 × 9 in. New York, W. A. Benjamin Inc., 1963. Price, $9.00 (cloth); $5.95 (paper)

Pressure-induced metal-semiconductor transitions in bismuth-antimony alloys in a strong magnetic field (up to 70 kOe) at helium temperatures have been investigated. It is found that for values of the “overlap-gap” |G|≲1 meV the alloy forms an excitonic insulator (EI) in magnetic fields above a certain “threshold” (30–40 kOe). It is inferred that the EI energy gapΔ increases with the magnetic field. The maximum gap observed in fields of ∼70 kOe turns out to beΔ00∼7.5 K. An analysis of the results shows that transitions to the EI phase are observed from both the semimetal and the semiconducting states. The critical transition temperatureTc is related to the EI gapΔ by the expressionTc⋍0.7Δ. Arguments are advanced in support of the fact that the formation of the EI phase involves the pairing of electrons at theL point with holes at theT point.

Investigation of the excitonic insulator phase in bismuth-antimony alloys

A previous measurement of the anomalous temperature dependence of the giant quantum attenuation of longitudinal sound waves in bismuth under strong magnetic fields was extended down to 1.06 K, and further evidence for the occurrence of the excitonic phase transition was obtained. The plot of sound attenuation versus temperature in the anomalous case showed a roundish convex curve near a maximum located at a point a little below T c , a temperature parameter to be identified with the phase transition point. This temperature dependence of the attenuation is well explained in terms of the critical fluctuation and a modified form of Maki-Nakanishi's formula of the attenuation of the excitonic state in the absence of field. The transition temperature, the critical exponent and other parameters to specify the anomalous attenuation curve near T c are evaluated for several cases of a parameter related to the energy difference between the hole and electron Landau levels in the normal state.

Further Study" of the Occurrence of the Excitonic Phase Transition in Bismuth under Strong Magnetic Fields by Ultrasonic Attenuation

Accurate and detailed measurements of the temperature dependence of the longitudinal magnetoresistance of single-crystal Bi-Sb alloys have been made, with static magnetic fields in the range 0–100 kG oriented parallel to the trigonal axis. Alloy concentrations were in the range 8–12 at.% Sb, and temperatures in the range 1–35 K. At very high fields the resistance increases with increasing temperature in a metallic manner with “ideal” and “residual” components, in contrast to the semiconductor behavior observed at zero field or low fields. For the high-field semimetal regime the electrical resistance behaves in a simple manner similar to a metal in zero field, in contrast to the complicated magnetoresistance phenomena for metals in low fields. This behavior can be understood in terms of a simple quasi-one-dimensional extreme-quantum-limit regime. The magnetic-field-induced semiconductor-semimetal transition is associated with an energy gap and changes of the energy-band structure which are of order 1 meV. Thermal activation energies for electrical conduction manifest this gap only at temperatures below approximately 20 K. Activation energies an order of magnitude larger which have been measured at considerably higher temperatures are apparently the direct gap at theL-point in the Brillouin zone and are not directly connected with the semiconductor-semimetal transition. Our results indicate that the zero-field indirectL-T energy gap increases from zero somewhere near 7–8 at. % Sb to values only as large as approximately 1.5 meV at 12 at. % Sb. At the magnetic-field induced transition there occurs evidence of an intermediate “excitonic insulator” phase, a resistance minimum below 10 K reminiscent of the Kondo alloy behavior. This anomalous regime is a property of the semiconductor-to-semimetal transition and cannot be associated with the well-known temperature and magnetic-field “freeze-out” of charge carriers in extrinsic semiconductors, or with magnetic ordering of the Kondo type.

Magnetic-field-induced semiconductor-semimetal transition in Bi-Sb alloys

The electrical resistivity of semimetals in the extreme quantum limit is strongly affected by the presence of anomalous charge—density waves at low temperatures. Anomalous Green's function and perturbation theories of the charge—density wave regime are related and extended to calculation of the imaginary part of the electron self-energy. The electrical resistivity due to electron—hole scattering is calculated. The electron—hole scattering reaches a peak value near a scaling temperatureT

 c
 with a lnT decrease at higher temperatures. When combined with electron—impurity and electron—phonon scattering contributions, the total resistivity behaves in a manner which has been observed experimentally in Bi-Sb alloys.

Electrical resistivity of semimetals in the extreme quantum limit

A weak-coupling theory of the itinerant-electron antiferromagnet forT>T

 c
 is described, Vertex and self-energy parts are treated self-consistently using Ward-Pitaevskii identities The weak-coupling criterion is that the scalar three-vertex must be suitably small This condition is not satisfied right down toT

 c
 ; however, it is satisfied in a region nearT

 c
 where the corrections to the bare electron-electron interaction are not small Expressions are derived for the specific heat, temperature derivative of the electrical resistivity, etc, which are all proportional to the weakly singular form[T

 c
 /(T−T

 c
 )]
 n
 , withh«1 An analogous theory applies for superconductors In this case the weak-coupling exponenth is approximately equal to(T

 c
 /E

 F
 )2 This quantity is typically 10−8–10−7 for superconductors, and in the weak-coupling temperature regime the lambda curve singularity would not be observable experimentally However, for an itinerant-electron antiferromagnet such as chromium, the quasicubic shape of the electron and hole Fermi surfaces causes enhancement ofh to a value of order 10−1 For chromium the theory is consistent with experimental results and with a recent theory by Dzyaloshinskii and Kats (using a different method)

Phase transition of an itinerant-electron antiferromagnet

Theory of the phase transition of an itinerant electron antiferromagnet is similar to superconductivity theory, including singular contributions to the specific heat and other properties at the transition temperature. The effects of a small concentration of normal impurities, which are pair breaking for the antiferromagnet, are calculated for temperatures nearT c . The principal effect of normal homovalent impurities is temperature smearing of the phase transition due to inhomogeneous electron-impurity scattering. This temperature smearing removes the singularities, leaving peaks with finite amplitudes which depend on the impurity concentration in a simple manner. Recent experimental results for chromium are discussed.

E. W. Fenton

Papers

Magnetic-field-induced semiconductor-semimetal transition in Bi-Sb alloys

Electrical resistivity of semimetals in the extreme quantum limit

Phase transition of an itinerant-electron antiferromagnet

Effects of impurities on the phase transition of an itinerant electron antiferromagnet