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Title: Wannier-Stark states of graphene in strong electric field

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Journal Article: Publisher's Accepted Manuscript
Journal Name:
Physical Review B
Additional Journal Information:
Journal Volume: 90; Journal Issue: 8; Journal ID: ISSN 1098-0121
American Physical Society
Country of Publication:
United States

Citation Formats

Kelardeh, Hamed Koochaki, Apalkov, Vadym, and Stockman, Mark I. Wannier-Stark states of graphene in strong electric field. United States: N. p., 2014. Web. doi:10.1103/PhysRevB.90.085313.
Kelardeh, Hamed Koochaki, Apalkov, Vadym, & Stockman, Mark I. Wannier-Stark states of graphene in strong electric field. United States. doi:10.1103/PhysRevB.90.085313.
Kelardeh, Hamed Koochaki, Apalkov, Vadym, and Stockman, Mark I. Thu . "Wannier-Stark states of graphene in strong electric field". United States. doi:10.1103/PhysRevB.90.085313.
title = {Wannier-Stark states of graphene in strong electric field},
author = {Kelardeh, Hamed Koochaki and Apalkov, Vadym and Stockman, Mark I.},
abstractNote = {},
doi = {10.1103/PhysRevB.90.085313},
journal = {Physical Review B},
number = 8,
volume = 90,
place = {United States},
year = {Thu Aug 28 00:00:00 EDT 2014},
month = {Thu Aug 28 00:00:00 EDT 2014}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record at 10.1103/PhysRevB.90.085313

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Cited by: 14works
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  • With the application of a spatially constant electric field, the degeneracy of electronic energy levels of geometrically equivalent sites of a crystal is generally lifted. As a result, the electric field causes the electronic eigenstates of a one-dimensional periodic chain to become localized. In particular, they are Wannier-Stark states. With sufficiently large electric-field strengths these states become sufficiently well localized that it becomes appropriate to consider electronic transport to occur via a succession of phonon-assisted hops between the localized Wannier-Stark states. In this paper, we present calculations of the drift velocity arising from acoustic- and optical-phonon-assisted hopping motion between Wannier-Starkmore » states. When the intersite electronic transfer energy is sufficiently small so that the Wannier-Stark states are essentially each confined to a single atomic site, the transport reduces to that of a small polaron. In this regime, while the drift velocity initially rises with increasing electric field strength, the drift velocity ultimately falls with increasing electric-field strength at extremely large electric fields. More generally, for common values of the electronic bandwidth and electric field strength, the Wannier-Stark states span many sites. At sufficiently large electric fields, the energy separation between Wannier-Stark states exceeds the energy uncertainty associated with the carrier's interaction with phonons. Then, it is appropriate to treat the electronic transport in terms of phonon-assisted hopping between Wannier-Stark states. The resulting high-field drift velocity falls with increasing field strength in a series of steps. Thus, we find a structured negative differential mobility at large electric fields.« less
  • In the one-particle single band approximation, which is the basis of the original Wannier result, commonly referred to as the Wannier-Stark ladder (WSL), we have extended the concept by predicting the existence of noncanonical WSLs which are a set of evenly spaced levels (in the middle of the tilted band) with noncanonical level spacing equal to the Plank constant times (1{minus}2m{sup {prime}}/m){sup {minus}1} times Bloch oscillation frequency. To observe a particular WSL, the certain voltage must be applied. The latter is related to the numbers m=3,4,{hor_ellipsis} and m{sup {prime}}=1,2,{hor_ellipsis}{lt}m/2. We also show that, if the electrostatic energy due to appliedmore » voltage is larger than the zero-field band width, the quantization of surface localized states smoothly changes from the Airy type (at the spectrum edges) to the Wannier-Stark type with a pronounced energy interval in between, where the level spacing doubles that of canonical WSL. Analytical results are derived within the exactly solvable model of finite tilted tight-binding band. Their experimental implications and further-to-go directions are addressed to dielectric crystalline layers and superlattices, whose thickness (length) admits the direct tunneling.« less
  • The local density of states [rho]([ital x],[ital E]) is calculated for a Bloch electron in an electric field. Depending on the system size, we can see one or more sequences of Wannier-Stark ladders in [rho]([ital x],[ital E]), with Lorentz-type level widths and apparent spatial localization of the states. Our model is a chain of [delta]-function potential barriers plus a steplike electric potential, with open boundary conditions at both ends of the system. Using a wave-tunneling picture, we find that the level widths shrink to zero as an inverse power of the system size as the system size approaches infinity, confirmingmore » an earlier result. The level width will not approach zero if the [delta]-function barriers are replaced by the Kronig-Penney potential or smoother ones, as is commonly believed.« less
  • Electron and hole emission from states of a ten-layer system of tunneling-coupled vertically correlated InAs/GaAs quantum dots (QDs) is studied experimentally by capacitance-voltage measurements and deep-level transient spectroscopy. The thickness of GaAs interlayers separating sheets of InAs QDs was {approx}3 nm, as determined from transmission electron microscope images. It is found that the periodic multimo-dal DLTS spectrum of this structure exhibits a pronounced linear shift as the reverse-bias voltage U{sub r} applied to the structure is varied. The observed behavior is a manifestation of the Wannier-Stark effect in the InAs/GaAs superlattice, where the presence of an external electric field leadsmore » to the suppression of coupling between the wave functions of electron states forming the miniband and to the appearance of a series of discrete levels called Wannier-Stark ladder states.« less
  • Raman lasers are used as a spectroscopic probe of the state of atoms confined in a shallow one-dimensional (1D) vertical lattice. For sufficiently long laser pulses, resolved transitions in the bottom band of the lattice between Wannier Stark states corresponding to neighboring wells are observed. Couplings between such states are measured as a function of the lattice laser intensity and compared to theoretical predictions, from which the lattice depth can be extracted. Limits to the linewidth of these transitions are investigated. Transitions to higher bands can also be induced, as well as between transverse states for tilted Raman beams. Allmore » these features allow for a precise characterization of the trapping potential and for an efficient control of the atomic external degrees of freedom.« less