Electronic structure and properties of a few-layer black phosphorus Mikhail Katsnelson

Transcription

1 Electronic structure and properties of a few-layer black phosphorus Mikhail Katsnelson Main collaborators: Sasha Rudenko Shengjun Yuan Rafa Roldan Milton Pereira Sergey Brener

2 Motivation Plenty of 2D materials starting from graphene Semimetals (graphene), semiconductors, metals, superconductors, broad-gap insulators... For historical and basic review:

3 Ling et al, PNAS 2015 What is BP? Bridgman (1914) Black P stable allotrope of P at ambient conditions Layered compound, 4 atoms per unit cell, bond lengths within layer and nm, between layers 0.53 nm Tunable gap, depending on number of layers

4 Other allopropes of phosphorus White phosphorus is most know to a general public (e.g., phosphorescence )

5 Computations of electronic structure Gap problem: conventional density functional (LDA or GGA) usually strongly underestimates gaps in semiconductors more advanced methods should be used like GW

6 Computations of electronic structure II GW self-consistency loop

7 Computations of electronic structure III Bulk BP And self-consistency makes the gap accurate, 0.35 ev!

8 Computations of electronic structure IV A very strong dependence of the gap on number of layers!

9 Computations of electronic structure V Cohesive energy: QMC calculations VdW interactions: weak and can hardly be described by conventional density functional How can it be? How weak cohesive energy is consistent with a huge dependence of interlayer hopping to the gap?!

10 Computations of electronic structure VI GW is qualitatively important also for single layer

11 Mapping on tight binding model Near band edges p z orbitals play the most important role; let us map on them only

12 Mapping on tight binding model II Single layer BP

13 Mapping on tight binding model III

14 Mapping on tight binding model IV Minimal model for single-layer BP (phosphorene)

15 Mapping on tight binding model V Minimal model for single-layer BP

16 Mapping on tight binding model VI Optimal TB model for multilayer BP vs GW

17 Mapping on tight binding model VII The main difference with graphene: a very large and positive second-neighbour hopping in plane; interlayer hopping is roughly of the same order of magnitude

18 Effect of interlayer bias: anisotropic Dirac cones Linear crossing in x direction and parabolic in y direction Bias leads to gap opening and formation of anisotropic conical points

19 Experiment: K deposite ARPES

20 Calculations: K deposite Indeed, very similar to just perpedicular electric field effect

21 Optics Chebyshev polynom expansion, etc. up to 10 9 sites Successfully applied to graphene

22 Optics II

23 Large scale TB simulations for disordered BP Point defects: missing atoms Puddles: Gaussian electrostatic potential

24 Large scale TB simulations for disordered BP II Anisitropy of optical adsorption

25 Low-energy Hamiltonian and Landau level quantization Effective Hamiltonian (single layer) Parameters are expressed in terms of hopping integrals In magnetic fields:

26 Low-energy Hamiltonian and Landau level quantization II At B < 30 T the spectrum of low-lying levels are almost linear Cyclotron frequency For holes

27 Quantum Hall effect in biased BP S. Yuan, MIK, R. Roldan, arxiv: Insulator-semimetal transition with formation of anisotropic Dirac cones

28 Quantum Hall effect in biased BP II Kubo-Bastin formula

29 Quantum Hall effect in biased BP III Transition to half-integer quantum Hall effects like in graphene: plateaus at ±1 instead of 0

30 Plasmons: role of defects Kubo formula, TB propagation method Static screening X zigzag direction Y armchair direction

31 Plasmons Single Layer II

32 Plasmons Single Layer III

33 Plasmons: Bilayer Acoustic and optical plasmon branches, with square-root and linear dispersion, respectively

34 Plasmons: Bilayer II

35 Plasmons: Biased Bilayer Insulator-semimetal transition (as discussed above)

36 Plasmons: Biased Bilayer II Plasmon velocity is sensitive to bias (may tune plasmons be electric field!) Plasmon have larger spectral weight in semimetal phase Strong anisotropy of plasmon spectrum, essential difference betwen zigzag and armchair directions

37 Intrinsic mobility in single-layer BP In graphene, two-phonon processes involving flexural phonons are more important that single-phonon processes involving in-plane phonons (except very low temperatures)

38 Intrinsic mobility in single-layer BP II A. N. Rudensko, S. Brener, MIK, arxiv: What about black phosphorus? Kubo-Nakano-Mori formula (corresponds to variational solution of Boltzmann equation) Phonon energies at q = 2k F are supposed to be much smaller than k B T (the most relevant case) Parameters of interactions with in-plane and out-of-plane phonons are calculated from first principles

39 Intrinsic mobility in single-layer BP III Single-phonon processes Elastic energy Deformartion potential of interactions with acoustic phonons

40 Intrinsic mobility in single-layer BP IV Double-phonon processes Elastic energy Deformartion potential of interactions with acoustic flexural phonons For simplicity: Logarithmic cutoff of small q is requred (like in graphene)

41 Intrinsic mobility in single-layer BP V Ab initio calculations Extracting deformation potentials from band shifts

42 Intrinsic mobility in single-layer BP VI Energy dependence of effective masses are important (non-parabolic bands)

43 Intrinsic mobility in single-layer BP VII Conclusions: (1) Contrary to graphene, single-phonon processes are more (2) Intrinsic limit of for holes (electrons) contrary to 10,000 for graphene; (3) Anisotropy of mobility is much strongre for holes than for electrons (2),(3) seem to be in qualitative agreement with experiments on a few-layer phosphorus

44 Resume (1) Completely different physics from both graphene and transition-metal dichalcogenides (2) Strong difference of elecrtonic structure from monolayer to bulk despite small cohesive energy (3) Insulator-semimetal transition under bias/potassium doping with anisotropic Dirac cones in bilayer black phosphorus (4) Tunable plasmonics?! (5) Mechanisms of intrinsic mobility quite different from graphene, much stronger interaction with acoustic phonons, strong limits on mobility MANY THANKS FOR YOUR ATTENTION